Generalized Musical Intervals and Transformations

Generalized Musical Intervals

and Transformations

This page intentionally left blank

Generalized Musical Intervals

and Transformations

David Lewin

OXFORDUNIVERSITY PRESS

2007

OXFORDUNIVERSITY PRESS

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Originally published 1987 by Yale University PressPublished by Oxford University Press, Inc.

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Library of Congress Cataloging-in-Publication Data

Lewin, David, 1933-2003.Generalized musical intervals and transformations /

David Lewin.p. cm.

Originally published: New Haven: Yale University Press, c1987.Includes bibliographical references and index.

ISBN 978-0-19-531713-81. Music intervals and scales. 2. Music theory. 3. Title.

ML3809.L39 2007781.2'37—dc22 2006051121

1 3 5 7 9 8 6 4 2

Printed in the United States of Americaon acid-free paper

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For June and Alex


Contents

Foreword by Edward Gollin ix

Preface xiii

Acknowledgments xxvii

Introduction xxix

Mathematical Preliminaries 1

Generalized Interval Systems (1):Preliminary Examples and Definition 16

Generalized Interval Systems (2): Formal Features 31

Generalized Interval Systems (3):A Non-Commutative GIS; Some Timbral GIS models 60

Generalized Set Theory (1): Interval Functions; Canonical Groups andCanonical Equivalence; Embedding Functions 88

Generalized Set Theory (2): The Injection Function 123

Transformation Graphs and Networks (1):Intervals and Transpositions 157

Transformation Graphs and Networks (2):Non-Intervallic Transformations 175

Transformation Graphs and Networks (3): Formalities 193

Transformation Graphs and Networks (4): Some Further Analyses 220

Appendix A: Melodic and Harmonic GIS Structures;Some Notes on the History of Tonal Theory 245

Appendix B: Non-Commutative Octatonic GIS Structures;More on Simply Transitive Groups 251

Index 255

1.

2.

3.

4.

5.

6.7.

8.

9.10.11.

12.


Foreword to the Oxford Edition

Edward Gollin

It has been nearly twenty years since the initial publication of David Lewin's Gen-eralized Musical Intervals and Transformations (GMIT), and the work has agedwell. This is due in part to the foundational nature of the book's subject matter.The work, a methodical examination of the concept of a musical interval, exploreshow the familiar notion of interval as "a distance extended between pitches in aCartesian space" is merely one specific case of a more general idea, one that canembrace different kinds of musical objects (durations, meters, Klangs, timbres,and so on), different (i.e. non-Euclidean) geometries, and different orientationalperspectives (interval as action or gesture rather than as simply measurement ofdistance between things). Along the way, the work recasts set theory, the conceptsof transposition and inversion, and notions of musical time in this generalizedimage. But the work has maintained its relevance and importance as well becauseof the brilliance and musicality of its author. David had a gift for finding musicallysignificant examples for his sometimes abstract concepts, and a gifted musicalimagination that delighted in finding new ways to hear and understand familiarmusical passages. While GMIT does not offer the extended musical analyses ofhis later books, Musical Form and Transformation or Studies in Music with Text,the work is nonetheless rich with smaller analytical gems.

To be sure, transformational theory has evolved in the years since GMIT firstappeared—the analytical use of Klumpenhouwer networks, the development ofneo-Riemannian theory, and the resurgence of spatial methodologies and metaphorsin analysis all postdate David's seminal study. But each of these subsequent de-velopments can find its basis in the framework David sets forth in GMIT: Klum-penhouwer networks apply the Generalized Interval System (GIS) concept recur-sively to create networks of networks; neo-Riemannian theory, which emerged fromexplorations begun in chapter 8 of GMIT, takes families of contextual transforma- ix


tions to be the formal intervals between the familiar set of harmonic triads or sev-enth chords; spatial methodologies simply extend the idea of transformational net-works to create graphs that embrace all members of a family of objects (pitches,pitch sets, rhythmic durations, and so on) related by certain contextually significantintervals.1

One notable new feature of this edition is an author's addendum (the preface),drawn from a previously unpublished typescript titled "Updating GMIT," whichpresents, in a sometimes synoptic form, concepts or musical examples David hadplanned for a future edition of GMIT. The document was likely written in the sum-mer of 1987 and was used as the handout for a talk given at the Eastman Schoolof Music in the fall of that same year. It should not be surprising to those whoknew David's incredible industry and the speed with which he could read and sug-gest revisions to others' work that David would have been drafting plans for a newedition of GMIT so soon after its publication—for David, it was often difficult tostop thinking about a project, or tinkering with its ideas, once begun, and the docu-ment clearly represents David's residual energy following the writing of GMIT. Theexamples explored in the addendum are diverse, although certain themes recur.For one, David seems to have been particularly concerned with examples that in-volve non-commutative groups of operations, no doubt because such groups oftendefy our accustomed and familiar intuitions about the way intervals work. For an-other, David seems to have been interested in finding examples that do not simplyinvolve individual pitch classes (transformations of melodies, of Lagen in triplecounterpoint, of ordered hexachords), again because these are less familiar, andoften reveal less intuitive aspects of interval.

Although the document is perfectly intelligible, some sections of "UpdatingGMIT" deserve additional comment.

1. The error in figure 8.2 (g minor instead of g# minor) that prompted David'scommentary in section I has been corrected in this edition. The first section ofDavid's notes was expanded to become his article "Some Notes on AnalyzingWagner: The Ring and Parsifal" (19th-century Music 16.1, 1992, reprinted in DavidLewin, Studies in Music with Text [Oxford University Press, 2006]).

2. David developed and expanded section IV into a pair of unpublished exer-cises for his math and music course at Harvard University. Exercise 5 (2 pages) di-rects the student to discover the elements of the Q-X group acting on the aug-mented triads of sc (014589) and then find transformations of the "rapture of the

1. David has written articles on each of these topics subsequent to the publication of GMIT.Klumpenhouwer networks are the topic of two articles: "Klumpenhouwer Networks and Some Iso-graphies that Involve Them," Music Theory Spectrum 12.1 (1990): 83-120, and "A Tutorial onKlumpenhouwer Networks, Using the Chorale in Schoenberg's op. 11, no. 2," Journal of Music The-ory 38.1 (1994): 79-101. David's most significant post-GMIT contribution to neo-Riemannian theoryis the article "Cohn Functions," Journal of Music Theory 40.2 (1996): 181-216. Two of David's con-tributions to graphical methods of analysis are "The D-major Fugue Subject from WTCII: Spatial Sat-uration?" Music Theory Online 4.4 (1998), and "Notes on the Opening of the F# Minor Fugue fromWTC I," Journal of Music Theory 42.2 (1998): 235-239.x


strife" figure under Q4, Q8, and X5 in Schoenberg's Ode to Napoleon (as inDavid's example 2 from the addendum). An optional part of that exercise encour-ages students to explore transformations of characteristic tetrachords in Schoen-berg's Ode using the members of the same Q-X group. Exercise 8 (3 pages) ex-plores the simple transitivity of the Q-X group and has the student find the(interval-preserving) elements of the commuting group, {T0, T4, T8, I,, I5, I9).David's article "Generalized Interval Systems for Babbitt's Lists, and for Schoen-berg's String Trio" (Music Theory Spectrum 17.1 [1995]: 81-118), in particular"Part 5: Background on Non-Commutative GISs," explores the relationship be-tween non-commutative GISs and their commuting groups.

3. David similarly developed and extended section V into an exercise for hismath and music course (exercise 9,4 pages). The Daniel Harrison article to whichDavid refers was published as "Some Group Properties of Triple Counterpoint andTheir Influence on Compositions by J. S. Bach" (Journal of Music Theory 32.1[1988]: 23-49). David inserted a manuscript page into the "Updating GMIT" type-script that presents a TPERM and VPERM analysis of Bach's c-minor fugue fromthe Well-Tempered Clavier, Book I. The manuscript notes that the diagram is mod-eled after Schenker's "Table of Voices" from "Das Organische der Fuge" in DasMeisterwerk in derMusik, Band II, p. 59, and further observes that the Lagen sym-bol "'A' can mean 'Subject,' 'B' can mean 'Countersubject' and 'C' can mean 'anythird part of roughly characteristic rhythm'" (emphasis Lewin's), suggesting thatthe methodology is not bound to works in strict triple counterpoint. David's dia-gram, however, has not been incorporated into the author's addendum of this vol-ume because David wrote no accompanying text for it—creating new text wouldhave adversely disrupted David's prose in the rest of the section. David, however,did use the c-minor fugue analysis as part of exercise 9 in his math and musiccourse, which I present below for interested readers to explore if they wish (ter-minology has been adapted to conform to the text of "Updating GMIT"):

PART I OF EXERCISE 9: (a) Complete the partially-filled diagrambelow, which pertains to the c-minor fugue in Book I:

Meas. Stufe Lage <hi-mid-lo> TPERM interval VPERM interval

111

152026.5

imV

ii

<B-C-A><A-C-B><B-A-C><A-B-C><C-B-A>

(b) Discuss features of the construction which you find revealed bythe double intervallic analysis. For instance, does the use of 3-cyclesbring out any aspect of the structure? Do the TPERM and VPERM xi


analyses coincide as they did [in the A-major Prelude]? What aspectsof the piece are bound together by repetition of TPERM intervals?By repetition of VPERM intervals?

4. Section VI considers the GIS structure of a family of 12-tone-row transfor-mations that David first explored in his article "On Certain Techniques of Re-Ordering in Serial Music" (Journal of Music Theory 10.2 [1966]: 276-287).David refers in the section to "an excellent work, as yet unpublished" by AndrewMead. That work was published in two parts as "Some Implications of the Pitch-Class/Order-Number Isomorphism Inherent in the Twelve-Tone System: Part One"(Perspectives of New Music 26.2 [1988]: 96-163) and, more pertinent to Lewin'saddendum, "Some Implications of the Pitch-Class/Order-Number IsomorphismInherent in the Twelve-Tone System Part Two: The Mallalieu Complex: Its Ex-tensions and Related Rows" (Perspectives of New Music 27.1[1989]: 180-233).

David, of course, never created a second edition ofGMIT, an undertaking that,he wrote, would have involved "[fixing] a lot of errata & corrigenda; some majorrewrites here and there; a reasonable amount of bibliographic updating."2 This edi-tion ofGMIT, while retaining the text of the original, does incorporate the correc-tions indicated by David's errata list. Moreover, while it does not attempt to identifyor alter passages that David felt needed rewriting, the articles cited in this fore-word give a picture of David's evolving ideas about transformational theory. Andwhile David may have wanted a new edition of GMIT, rather than a second print-ing, he was also eager to make GMIT available to students and scholars. In theserespects, this Oxford edition fulfills David's wishes—that his ideas be available toall who seek them, so that they may grow, evolve and multiply.

2. 1995 e-mail correspondence, recipient unknown.

xii

Preface

I. The following figures redo those of figure 8.2 on p. 179. Music examples laand b present scores of the relevant passages.

L = LEITTONWECHSEL; +- = MAJOR-MINOR;S = "BECOMES SUBDOMINANTOF".

xiiiEXAMPLE la

Preface

EXAMPLE Ib

The analysis is better than that in the book. It brings out a clear isography betweenthe passages. Figure 8.2a in the book is not a well-formed "graph" by the later defi-nition. (SUBM is not = LT SUED on major as well as minor Klangs: (C,+)SUBM = (e,-) but (C,+)LT SUED = (e,-)SUBD = (b,-).) The symbol "(G,-)"on figure 8.2a is a misprint for (Gt,-).1 The discussion of section 8.1.2, pages179-180, still applies: a group that contains L, S, and +— operations on Klangswill not be simply transitive in equal temperament. (For instance, (C,+)SSSS=(E,+), but (C,+)L +- also = (E,+).)

1. See item 1 in the foreword, p. x.xiv

b) Modulating section of Valhalla, Rheingold II,5ff.

Preface

Later in the Ring, Wagner develops the relationship of Valhalla and Tarnhelmthemes very ambitiously. Figures c) through f) below analyze a transformationthat occurs at the climax of Walkure 11,2: Woton, coming to realize the full impli-cations of Valhallagate, ironically gives his blessing to Hagen ("So nimm meinenSegen, Niblungen Sohn!"). Music examples Ic through le are coordinated with thefigures.

EXAMPLE Ic-e

Figure Ic) shows the Valhalla Kopf put into At major and 4/4 meter, with theoriginal harmonization. Figure d) is the +- transform of c). Figure e) transformsd) so that the subdominant inflection of c)—d) is applied not to the tonic but to theLeittonwechsel of the tonic; also the inflected Klangs change mode as they go, via+ — . Music example le is essentially the upper part of the accompaniment forWotan's pronouncement (there is more beneath!). The Tarnhelm network infectsthe diatonic aspect of Valhalla here. Figure f) brings that out by rewriting e) in aformat that suggests a). In the Waltraute scene of Gotterdammerung, the idea getseven more overloaded ... rather like the picture of Dorian Grey. JCV

II. An interesting transformation network is used by Lora L. Gingerich, ". . .Melodic Motivic Analysis in ... Charles Ives," MTS 8 (1986), 75-93. The net-work appears as her example 24, page 90.

III. Let s be the twelve-tone row of Schonberg's Fourth Quartet. Let S be the fam-ily comprising the 48 forms of s. Let TTO be the group of forty-eight twelve-toneoperations. TTO is simply transitive on S (given forms s and t, there exists aunique member OP of TTO such that OP(s) = t.) It follows that we can develop aCIS structure for S in such wise that the members of TTO are exactly the formaltransposition operations for the GIS (GMIT1A. 1, pp. 157-58). The standard prac-tice, in which forms of the row are labeled by their TTO-intervals from a "tonic"referential row-form—as "RI3," "17," etc.—instances the LABELing practice dis-cussed in chapter 3 of GMIT.

If s is any one of the 48 forms, then there exists a unique inverted form of s (inthis case) which shares the same three tetrachordal segments with s. Define atransformation TETRA on S: given a sample s, TETRA transforms s into this in-verted tetrachordal associate. For instance:

TETRA (Ob78 312a 6549) = 780b 4659 123a;TETRA (5012 6a9b 4378) = 6ba9 5120 7843.

The transformation TETRA is a formal interval-preserving operation of theGIS under discussion (GMIT 3.4.6, p. 48). Similar operations for this particularrow, like TRI and HEXA, are also interval-preserving operations. In his disserta-tion on Moses undAron (Yale, 1983), Michael Cherlin argues that transformationsof this sort, engaging the forms of the Moses row, are highly constructive featuresof Schonberg's compositional method in the opera.

IV. Appendix B in GMIT outlines two possible non-commutative GIS structuresfor the octatonic set. It develops two simply transitive groups of operations on thatset; either may be taken as the group of formal transpositions for a GIS; the otherthen becomes the group of formal interval-preserving operations.

A similar situation obtains for set-class 6-20. Taking S as [modeled by] the sixnumbers 0,1,4,5,8, and 9 mod 12, two simply transitive groups of operations maybe defined on S as follows. The group Gl comprises the operations R0= identity,R4 = pc transposition by 4, R8 = pc transposition by 8, Jl = pc inversion withindex number 1, J5 = pc inversion with index number 5, and J9 = pc inversionwith index number 9. (In the GIS determined by this simply transitive group, allthe six operations are formal "transpositions" for that GIS.)

The group G2 comprises the six operations RO, Q4, Q8, XI, X5, and X9, de-fined as follows:xvi

Preface

Preface

RO = identity operation.Q4 takes pcs 0,4, and 8 to pcs 4, 8, and 0 resp.; takes pcs 1,5, and 9

topes 9,1, and 5 resp.Q8 takes pcs 0,4, and 8 to pcs 8,0, and 4 resp.; takes pcs 1,5, and 9

to pcs 5, 9, and 1 resp.The Qs are "queer" operations, as opposed to the "rotations" R.XI exchanges each pc of 6-20 with that pc which lies ic 7 away.

Thus XI maps 0 to 1,1 to 0,4 to 5, 5 to 4, 8 to 9, and 9 to 8.X5 exchanges each pc with the pc that lies ic 5 away.X9 exchanges each pc with the pc that lies ic 3 away.

Both the groups Gl and G2 are simply transitive on S. Either group may betaken as the group of formal transpositions for a formal GIS involving S; the othergroup thereupon becomes the group of interval-preserving transformations.

The pertinence of G2 is manifest in Schonberg's Ode to Napoleon. Music ex-ample 2 shows some prominent thematic motives of the piece, all interrelated byoperations of G2. Example 2a projects a six-note series that is mapped into ex-ample 2b by Q8. 2b' retrogrades 2b; 2c shows the series of 2b' in action. Example2d is the Q4-transform of series 2a; 2e shows series 2d in action. Example 2f is the

EXAMPLE 2 xvii

Preface

X5-transform of series 2a; 2g is a combinatorial inversion of series 2f, and 2hshows series 2g in action. The motives of 2a, 2c, 2e, and 2h appear frequently inthe work, at a variety of pitch levels, retrograded (= inverted), etc.; the various six-note series generate characteristic tetrachordal segments that are ubiquitous mo-tivic germs in the music. (These tetrachords are all G2-forms of one another).

V. Daniel Harrison, in a recent study of triple counterpoint, has made interestinganalytic use of GIS structures.2 I adapt his procedures to my terminology here.

Let us suppose three tunes, A, B, and C, that work in triple counterpoint. Letus suppose three voices, 1, 2, and 3, in which the tunes can appear. We can con-sider the six various possible dispositions of the three tunes in the three voices; let uscall each such disposition a "Lage." We can model each Lage by a three-elementseries: thus the series <B-C-A> models "tune B in voice 1, tune C in voice 2,and tune A in voice 3."

Let LAGEN be the family of the six possible Lagen. Given two members ofLAGEN, there are two "natural" ways to conceptualize a transformation takingthe first Lage into the second. For instance, suppose s and t are the Lagen <B-C-A> and <C-A-B> respectively. We can imagine the tunes as being permuted,to get from s to t: tune B (in voice 1) becomes tune C; tune C (in voice 2) becomestune A; and tune A (in voice 3) becomes tune B. Thus, in getting from s to t, wepermute tune B to tune C, tune C to tune A, and tune A to tune B. We can sym-bolize this permutation of tunes by the symbol (ABC): A becomes B, B becomesC, and C becomes A. But there is also another "natural" way of conceptualizinggetting from s to t: we can imagine the voices as being permuted. Thus, in passingfrom s = <B-C-A> to t = <C-A-B>, we can note that tune B, in voice 1 fors, goes into voice 3 for t; tune C, in voice 2 for 5, goes into voice 1 for t; tune A,in voice 3 for 5, goes into voice 2 for t. In sum, voice 1 of s becomes voice 3 of t;voice 3 of s becomes voice 2 of t; and voice 2 of s becomes voice 1 of t. We cansymbolize this permutation of voices by the symbol (132): 1 becomes 3, 3 be-comes 2, and 2 becomes 1.

There are six possible permutations on the symbols {A,B,C}; the six permu-tations can be used to label six transformations on LAGEN; those six transforma-tions form a group of operations on Lagen which we shall call TPERMS, for"tune-permutations." There are six possible permutations on the symbols {1,2,3};those six permutations can be used to label six transformations on LAGEN; andthose six transformations form a group of operations on Lagen which we shall callVPERMS, for "voice-permutations."

Both the groups TPERMS and VPERMS are simply transitive on LAGEN.Either group can be taken as the group of formal transpositions for a GIS whose

xviii 2. See item 3 in the foreword, p. xi.

Preface

family is LAGEN; the other group thereupon becomes the group of formal interval-preserving operations for the GIS. This situation is as in the last paragraph of ap-pendix B, GMIT.

Harrison analyzes the D-major 3-part invention, observing most of the fol-lowing structure. "A" is the lead-off theme in the rh; "B" is the counterpoint thatruns along in sixteenths; "C" is the counterpoint which steps down in leisurelysuspensions.


3.5

6

10

V

I

vi

<C-A-B>

<B-C-A>

<A-B-C>

(ACB)

(ACB)

(123)

(123)

(BIG MIDDLE SECTION)

19

21.5

23.5

IV

I

I

<C-B-A>

<B-A-C>

<A-C-B>

(ACB)

(ACB)

(132)

(132)

The columns headed "TPERM interval" and "VPERM interval" are read asfollows: from Lage <C-A-B> (m.3.5) to Lage <B-C-A> (m.6) the formal in-terval of transposition in the TPERM GIS is (ACB), while the formal interval oftransposition in the VPERM GIS is (123). From Lage <C-B-A> (m.19) to Lage<B-A-C> (m.21.5) the formal interval of transposition in the TPERM GIS is(ACB), while the formal interval of transposition in the VPERM GIS is (132).

Harrison points out that all six Lagen appear. He notes that the articulationinto the two subfamilies of 3 Lagen each, before and after the middle of the piece,is "natural." He points out that in the first half of the piece, the tunes "sweep down"through the voices, while in the second half of the piece, the tunes "sweep up"through the voices. (He does not use the VPERM GIS to discuss this, but expressesit by investigating specific properties of the group TPERMS.) He makes a numberof other cogent observations about the TPERM structure of the piece. Amongthose, he notes that the second half of the piece is TPERM-isographic to the firsthalf, even though the tunes "sweep down" the voices in the first half and "sweepup" the voices in the second half. In GMIT terminology, this can be expressed bynoting that in the VPERM GIS, the second half of the piece is ann'-isographic tothe first half: (132) is the inverse of (123) in VPERMS.

Harrison analyzes other works, including the f-minor invention. Here is myanalysis of Lagen in the A-major Prelude from Book I: xix

Preface


1

4

8.5

12

16.5

19

I

V

I

vi

I

I

<A-B-C>

<B-C-A>

<C-A-B>

<A-B-C>

<B-C-A>

<A-C-B>

(ABC)

(ABC)

(ABC)

(ABC)

(AB)

(132)

(132)

(132)

(132)

(13)

This analysis is useful to contrast to the D-major invention. Here only four Lagenare used. The idea seems to be that the final tonic Lage has a special function here:it breaks the otherwise incessant chain of (ABC) or (132) intervals. Harrison'sanalysis of the f-minor invention provides still a different idea, for laying out vari-ous Lagen.

The whole enterprise smells of Marpurg; perhaps the way in which he formu-lated "Rameau's" (i.e. his) theories of chord inversion might bear similar updat-ing, perhaps even in a somewhat isomorphic vein.

xx

VI. Let us consider the family SPECIAL of 12-tone rows whose order-rotationbeginning on order-number 4 is the same as their T4-transpose. An example of aSPECIAL row is Ob56439a8712: starting the row at order-number 4 and proceed-ing therefrom, we derive 439a8712 [and around the end to] Ob56; this order-rotationis the same as T4 of the original row.

To fix a notation, we consider each SPECIAL row as a function s mapping theorder number ord [mod 12] into the pc number s(ord) [mod 12]. The SPECIALrow of the above paragraph is thus conceived as a function s: s(0) = 0, s(/) = b,s(2) = 5, . . . ,s(a) = l,s(fc) = 2.

Using this notation, we can write out the algebraic property that characterizesSPECIAL rows:

SPECIAL PROPERTY: for all ord, s(ord + 4 = s(ord) + 4.[all addition mod 12]

Of interest to us here is the fact that the family of SPECIAL rows admits asimply transitive group-of-operations G in a natural way. Therefore, according tothe discussion of GMIT, the family of SPECIAL rows has a natural GIS structure,a structure in which the operations of G play the role of formal transpositions.

What follows is a semi-formal development of the group G, and a semi-formalindication that G is simply transitive on SPECIAL.

Preface

To begin with we consider certain operations ADD{j,m}, where j is somemultiple of 4 mod 12 and m is some multiple of 3 mod 12; i.e. j = 0, 4, or 8, andm = 0,3,6, or 9. The operation ADD{j,m}, when applied to the SPECIAL row s,adds the pc interval j to the mm, the (m+4)th, and the (m+S)th notes of s. For ex-ample, let us take the SPECIAL row of the first paragraph above and apply the op-eration A{8,3} to it, adding the pc interval 8 to its 3rd, 7th, and b\h notes:

o r d numbers: 0 1 2 3 4 5 6 7 8 9 a bpc numbers of s: O b 5 6 4 3 9 a 8 7 1 2

We add 8 to 3rd, 7th, and Mi, +8 +8 +8obtaining pc numbers of ADD{8,3}(s): O b 5 2 4 3 9 6 8 7 1 a

In the example, we note that ADD {8,3}(s) is still a row. That is because s isSPECIAL: since s(ord + 4) = s(ord) + 4, it follows that the pc numbers s(3), s(7),and s(&)—that is the 3rd, 7th, and 6th notes of s—form an augmented triad. In theexample above the augmented triad comprises the pc numbers 6, a, and 2. Whenwe add the interval j = 8 to each of these pc numbers, we simply permute themembers of that augmented triad among themselves, without disturbing the otherpcs in the other order-positions of the row.

Thus, in the above example, order-positions 3, 7, bcontain pcs 6, a, 2

of the row s; when 8 is added to each ofthose pc numbers, the same order-positions

then contain pcs 2, 6, aof the row ADD{ 8,3}(s), while the other pcs of s "carry ondown" to ADD{8,3}(s), unchanged in their order-positions.

This observation can be made rigorous and general, to show that each opera-tion ADD{j,m}, when applied to any SPECIAL row s, yields a row. Furthermore,it can be proved what is intuitively obvious: the new row ADD{j,m}(s) will itselfbe SPECIAL.

The following formulas are easily verified, for j and k any multiples of 4 mod12, and for m and n any multiples of 3 mod 12:

FORMULA 1: ADD{j,m} ADD{k,m} = ADD{j+k,m}FORMULA 2: ADD{j,m} ADD{k,n} = ADD{k,n}ADD{j,m}

ADD{0,m} is the identity operation, for each m: it leaves [the pcs of] any sampleSPECIAL row unchanged. It follows, via formulas 1 and 2, that the collection ofall operations that can be written in form

ADD{jO,0} ADD{j3,3} ADD{j6,<5} ADD{j9,9}

is a group of operations. We will call this group "ADDINGS." The group is com-mutative. It has 3-times-3-times-3-times-3 members, ie 81 members. xxi

Preface

Now we shall develop another group of operations on SPECIAL rows, a groupwe shall call "PERM." A PERM operation X{p} is defined by any permutation pthat acts upon the four symbols 0,3, (5, and 9. Here the permutation p is to be con-sidered as any 1-to-l function that maps the family of four symbols onto itself. ThePERM operation X{p} is determined by the

PERM DEFINITION: X{p}(s)(m +/) = s(p(m) + j )where m symbolizes a multiple of 3 mod 12

and; symbolizes a multiple of 4 mod 12.

Fix m = 0 in the formula of the definition, and let; run through the values 0, 4,and & The formula tells us that the Oth, 4th, and 8th notes of the X{p}(s) will berespectively the p(0)th, (p(0)+4)th, and (p(0)+5)th notes of s. Similarly [for m = 3]the 3rd, 7th, and £th notes of X{p}(s) will be respectively the p(3)rd, (p(3)+4)th,and (p(3)+S)th notes of s. And so forth [for m = 6 and m = 9].

For an example, fix p to be the permutation p(0) = 3, p(3) = 0, p(6) = 6, p(9)= 9. Then, according to the work we have just gone through, X{p}(s) will have inits Oth, 4th, and 5th order-positions the 3rd, 7th, and bth notes of s respectively,while X{p}(s) will have in its 3rd, 7th, and bth order-positions the Oth, 4th, and5th notes of s respectively; otherwise X{p}(s) will maintain the [other] notes of sin their respective order positions. The diagram below shows this X{p} applied tothe specimen special row used before.

o r d numbers: 0 1 2 3 4 5 6 7 8 9 a bpc numbers of s: 0 4 8

b 5 3 9 7 16 a 2

pc numbers of X{p}(s): 6 a 2b 5 3 9 7 1

0 4 8

For any SPECIAL row s, and any permutation p, X{p}(s) is a SPECIAL row.If p and q are permutations, then we have

FORMULA 3: X{p} X{q) = X{qp}.

The PERM operations on SPECIAL rows form a group (anti)-isomorphic to thegroup of permutations on the four symbols 0,3,6,9. PERM therefore has 4! = 24members. The group is not commutative.

The following formula can be proved:

FORMULA 4: ADD{j,m} X{p) = X{p} ADD{j,p(m)}].

In general, therefore, members of PERM do not commute with members ofADDINGS. However, formula 4 tells us that the collection of all operations whichcan be expressed as some-ADDING-following-some-PERM is a closed family of

xxii

Preface

EXAMPLE 3 xxiii

Preface

operations: following one such by another such will generate a third such. It fol-lows that this collection of operations is a group of operations. It is our desiredgroup G. A specimen member of G can be written in

CANONICAL FORM: ADD{jO,0} ADD{j3,3} ADD{j6,<5}ADD{J9,P} X{p}.

The chromatic scale is one SPECIAL row. It is straightforward, if tedious, toshow that given any SPECIAL row s, there is a unique member of our group Gwhich transforms s into the chromatic scale. (Set kO = s(0), k3 = s(3), k6 = s(6),k9 = s(9); since s is SPECIAL, each of the k's must lie within a different aug-mented triad; apply four appropriate ADDs to obtain a new s' in which the set ofk'-values is 0,3,6, and 9; permute the row s' into the chromatic scale. Etc. etc.) Itfollows that the group G is simply transitive on SPECIAL rows: given any twoSPECIAL rows s and t, there is a unique member of G, in the canonical formabove, which transformes s to t. (Transform s into the chromatic scale; then trans-form the chromatic scale into t.)

Thus the family of SPECIAL rows has a natural GIS structure, as discussedabove. The group G has cardinality 81-times-24 = 1944; that then is also the num-ber of SPECIAL rows.

SPECIAL rows become more interesting when one notes their relation to"semi-Mallalieu" rows. Andrew Mead, in excellent work as yet unpublished, hasinvestigated semi-Mallalieu rows exhaustively; some interesting insight can beshed on his work by placing it in a GIS setting.3 Pertaining to our SPECIAL rowsare those semi-Mallalieu rows whose every-third-note transform is identical withtheir T4-transposition. Every-ninth-note of such a row will then be its T8-trans-position. (Every-ninth-note = retrograde-of-every-fourth-note.)

Such a row, for example, is a premise of my piano piece Just a Minute, Roger[PNM 16.2 (SS 1978), 143-45]: Ob45732681a9. Right at the opening of the piece,one hears quite clearly that every-third-note of this row is its T4-transpose: inmeas. 1-4 (music example 3a), the total texture is governed by the row, while theright hand picks out every-third-note, thereby projecting T4-of-the-row. Later on,in meas. 30-35 (example 3b), the total texture is governed by the T4-form of therow, while the right hand picks out every-fourth-note-of-T4, thereby projectingthe retrograde of every-ninth-note-of-T4 = the retrograde of T8(T4) = the retro-grade of the original row.

We shall focus in on these rows for the nonce, calling them SEMI-MALLALIEU.The point is, that the family of SEMI-MALLALIEU rows and the family of SPE-CIAL rows, along with their characteristic properties, are mathematically equivalentin structure under a transformation that makes each row in one family correspond

xxiv 3. See item 4 in the foreword, p. xii.

Preface

uniquely to a row in the other. I discuss that transformation in ITO 2.7 (October1976), 8.

The upshot of this is that the SEMI-MALLALIEU rows also form a naturalGIS, whose formal transposition operations are the members of a natural simplytransitive group G' that corresponds to the group G we have just explored.

Mead's procedures enable one to find the equivalent of our G or G' for anyoperation OP on order numbers that can be written as four permutation-cycles oforder 3 on the order numbers. OP generalizes "rotation-by-4" in the case of SPE-CIAL rows, or "every-third-note" in the case of SEMI-MALLALIEU rows.

VII. The injection function, discussed in chapter 6 of GMIT, can be generalizedeven farther. Let S and S' be any two families of objects—we allow here for the pos-sibility that S' may be a different family from S. Let X be a (finite sub)set of S; letY' be a (finite sub)set of S'; let f be any function from S into S'. Then the injectionnumber into Y' for f, denoted INJ(X,Y')(f), is the number of elements s in X suchthat f(s) is a member of Y'. We can also develop a twin concept, not developed inGMIT: the surjection number of X into Y' for f, denoted SURJ (X,Y')(f), is the num-ber of elements s' in Y' such that s' = f(s) for some member s of X. If f is not one-to-one or onto, SURJ(X,Y')(f) may be a very different number from INJ(X,Y')(f).

EXAMPLE:Take S to be the family of numbers

{0,1,2,4,6,8,10,14,16,18,19,20,22,23,24,26,27,28}.

Take S' to be the family of pitches{A3,CH,D4,E4,F4,G4,A4,BI4,D5}.

Take the function f to map S into S' according to the following table:

s= 0 1 2 4 6 8 10 14 16 18 19 20 22 23 24 26 27 28f(s)= D4 E4 F4 E4 D4 A4 D5 A4 BW G4 E4 A4 F4 D4 G4 E4 C#4 A3

The function f is onto but not one-to-one, f models certain aspects of the themefrom Bach's d-minor Concerto.

Take X to be the set of all numbers in S divisible by 4; thusX = {0,4,8,16,20,24,28}.

Take Y' to be the triad {D4,F4,A4} within S'. XXV

Preface

We have f(0) = D4, f(8) = A4, and f(20) = A4. Otherwise f(s) is not a member ofthe set Y' when s is a member of the set X.

The three members 0,8, and 20 of X are mapped by f into members of Y'. HenceINJ(X,Y')(f) = 3.

The two members D4 and A4 of Y' are attacked, during this passage, at time-pointsthat are multiples of 4. Hence SURJ(X,Y')(f) = 2.

xxvi

A cknowledgmen ts

The time and leisure I needed to do this work were afforded by a SeniorFaculty Fellowship from Yale University and by a Guggenheim Fellowship.The Guggenheim Foundation also provided a subvention toward publication.

The excerpt from the score of Elliott Carter's String Quartet No, 1 isused by permission of Associated Music Publishers, New York. Materialfrom Arnold Schoenberg's songs, Opus 15, from his Six Little Piano Pieces,Opus 19, and from Pierrot Lunaire, Opus 21, is used by permission of BelmontMusic Publishers, Los Angeles, California 90049. Arnold Schoenberg'sPhantasiefor Violin with Piano Accompaniment, Opus 47, is copyright 1952 byHenmar Press Inc. It has been used by permission of C. F. Peters Corporation.The excerpt from the score of Anton Webern's Four Pieces for Violin andPiano, Opus 7, and material from the third movement of his Variations forPiano, Opus 27, are used with these permissions: "Anton Webern—FourPieces for Violin and Piano, Op. 7. Copyright 1922 by Universal Edition. Copy-right renewed 1950. All Rights Reserved. Used by permission of EuropeanAmerican Music Distributors Corporation, sole U.S. agent for UniversalEdition" and "Anton Webern—Variations for Piano, Op. 27. Copyright 1937by Universal Edition. Copyright Renewed 1965. All Rights Reserved. Usedby permission of European American Music Distributors Corporation, soleU.S. agent for Universal Edition." Analytic sketches for works by Bartok andProkofieff appear with the following permissions: "Syncopation # 133 fromMikrokosomos (volumes 1-6) Bela Bartok © copyright 1940 by Hawkes &Son (London) Ltd.; Renewed 1967. Reprinted by permission of Boosey andHawkes, Inc." and "Prokofieff: Melody # 1 from Melodies Op. 35. Reprintedby permission of Boosey & Hawkes, Inc. Copyright Owner." xxvii


Introduction

The following overview of the book will provide a good point of departure.Chapter 1 is purely mathematical; it presents terminology and notation thatwill be needed later, along with a few important theorems. I am not happy tobegin a book about music with a mathematical essay. On the other hand, I dofeel that it is helpful for the reader to have this material collated and isolatedfrom the rest of the book. Chapter 1 can be used for quick reference where itstands, and the material obtrudes only minimally into musical discussionslater on. Readers who find themselves put off or fatigued in the middle of thischapter are urged to move on into the rest of the book; they can return tochapter 1 later, when later applications of the material make the referenceback seem natural or desirable.

Chapter 2 takes as its point of departure the general situation portrayedschematically by figure 0.1.

FIGURE 0.1

The figure shows two points s and t in a symbolic musical space. Thearrow marked i symbolizes a characteristic directed measurement, distance, ormotion from s to t. We intuit such situations in many musical spaces, and weare used to calling i "the interval from s to t" when the symbolic points are xxix

Introduction

pitches or pitch classes. Chapter 2 begins by running through twelve examplesof musical spaces for which we have the intuition of figure 0.1. Six involvepitches or pitch classes in melodic or harmonic relations; six involve aspects ofmeasured rhythm. The general intuition at hand is then made formal by amathematical model which I call a Generalized Interval System, GIS forshort. A few basic formal properties of the model are explored. Then thetwelve examples are reviewed to see how each (with one exception) instancesthe generalized structure.

Chapter 3 concerns itself with further formal properties of the GISmodel. In that model, the points of the space may be labeled by their intervalsfrom one referential point; this has advantages and disadvantages. New GISstructures may be constructed from old in various ways. A passage fromWebern is examined in connection with a combined pitch-and-rhythm GISconstructed in one such way. Generalized analogs of transposition and inver-sion operations are explored. So are "interval-preserving operations"; thesecoincide with transpositions in some GIS models but not in others, specificallynot in GISs that are "non-commutative."

The bulk of chapter 4 explores one non-commutative GIS of musicalinterest. The elements of the system are formal time-spans. Extended dis-cussion of a passage from Carter's First Quartet demonstrates the pertinenceof this GIS to exploring music in which there are functional measured rela-tions among time spans, but no one overriding time span that acts as a unit tomeasure all others. After that, chapter 4 presents two examples of timbralGISs, and ends with a methodological note on the relations of music theory,perception, and the intuitions of a listener. Some motivic work by Chopin isconsidered in this connection.

Chapter 5 begins a study of generalized set theory, that is, the interrela-tionships among finite sets of objects in musical spaces. The first constructionstudied is the Interval Function between sets X and Y; this function assignsto each interval i in a GIS the number of ways i can be spanned between amember of X and a member of Y. Then the Embedding Number of X in Yis studied; this is the number of distinct forms of X that are subsets of Y. Tostudy that number, we have to establish what we mean by a "form" of the setX, a notion that involves stipulating a Canonical Group of operations. Boththe Interval Function and the Embedding Number generalize Forte's IntervalVector. Passages from Webern, Chopin, and Brahms illustrate applicationsof the constructs.

Chapter 6 continues the study of set theory, generalizing the work ofchapter 5 even farther. The basic construction is now the Injection Function:Given a space S, finite subsets X and Y of S, and a transformation f mapping Sinto itself, INJ(X, Y) (f) counts how many members of X are mapped by f intomembers of Y. This number is meaningful even when S does not have a GISstructure, and even when the transformation f js not so well behaved as arexxx

Introduction

transpositions, inversions, and the like. Passages from Schoenberg and fromBabbitt are studied by way of illustration.

Instead of starting with a GIS and deriving certain characteristic trans-formations therefrom, it is possible to start with a family of characteristictransformations on a musical space and derive a GIS structure therefrom.That is, instead of regarding the i-arrow on figure 0.1 as a measurement ofextension between points s and t observed passively "out there" in a Cartesianres extensa, one can regard the situation actively, like a singer, player, orcomposer, thinking: "I am at s; what characteristic transformation do Iperform in order to arrive at t?" Chapter 7 explores this conceptual inter-relation between interval-as-extension and transposition-as-characteristic-motion-through-space. After developing the mathematics that shows a logicalequivalence between GIS structures and certain structures of transformationson spaces, the work proceeds by example. Passages from Schoenberg, Wag-ner, Brahms, and Beethoven indicate how suggestive it can be to considernetworks of "intervals" and networks of "transpositions" (modulations, andso forth) as various aspects of the same basic phenomenon.

The morphology of such networks can be carried over to that of networksinvolving other sorts of transformations. Chapter 8 studies networks involv-ing transformations of Klangs in the sense of Riemann, networks involvingserial transformations of various sorts, and networks involving inversionaltransformations. The Beethoven example from chapter 7 is reconsidered, andthere are further examples from Wagner, Webern, and Bach.

Chapter 9 develops the formalities of transformation networks in arigorous way. The structure of a network allows us to assign a formal "input"function to some things and a formal "output" function to other things; thesefunctions seem of considerable musical interest in some cases. The networkshave intrinsic rhythmic properties which can also be studied formally. Net-work structure can accommodate hierarchic levels in a quasi-Schenkeriansetting, as an example shows.

Chapter 10 applies the network concept in a variety of ways to passagesfrom Mozart, Bartok, Prokofieff, and Debussy.

Note on Musical TerminologyAll references to specific pitches in this book will be made according to thenotation suggested by the Acoustical Society of America: The pitch class issymbolized by an upper-case letter and its specific octave placement by anumber following the letter. An octave number refers to pitches from a givenC through the B a major seventh above it. Cello C is C2, viola C is C3, middleC is C4, and so on. Any B# gets the same octave number as the B just below it;thus B#3 is enharmonically C4. Likewise, any Cb gets the same octave numberas the C just above it; thus Q?4 is enharmonically B3. xxxi


1 Mathematical Preliminaries

A mathematician would begin by saying, "Let S be a set." Unfortunately,music theory today has expropriated the word "set" to denote special music-theoretical things in a few special contexts. So I shall avoid the word here.Instead I shall speak of a "family" or a "collection" of objects or members.When I do so, I mean just what mathematicians mean by a "set." For presentpurposes, it will be safe to leave the sense of that concept to the reader'sintuition.

1.1 DEFINITION: Let S and S' be families of objects. The Cartesian productS x S' is the family of all ordered pairs (s, s') such that s is a member of S and s'is a member of S'.

1.2.1 DEFINITION: A function or mapping from S into S' is a subfamily f ofS x S' which has this property:

Given any s in S, there is exactly one pair (s, s') within the family f whichhas the given s as the first entry of the pair.

We say that s', in this situation, is the value of the function f for theargument s; we shall write f (s) = s'.

If we think of fas a table, listing members of S (arguments) in a column onthe left and corresponding members of S' (values) in a column on the right,then the defining property for functionhood stipulates that each member of Sappear once and only once in the left-hand column. (Some members of S' mayappear more than once in the right-hand column. Some members of S' maynot appear at all in the right-hand column.) 1

1.2.2 Mathematical Preliminaries

1.2.2 DEFINITION: Given families S and S', we shall say that the functions fand g from S into S' are the same, writing f = g, if f and g are the same subsetsof S x S', that is if they produce the same table.

This special definition of functional equality is worth stressing. We shallsoon see why.

1.2.3 DEFINITION: Let f be a function from S into S', and let f' be a functionfrom S' into S". Then the composition function f'f is defined from S into S" asfollows: Given an argument s in S, the value (f'f)(s) is f'(f(s)).

1.2.4 Let me draw special attention to the orthographic convention wherebyf' appears to the left of fin the notation for the composition function f'f. Thatconvention follows logically from another orthographic convention, the con-vention of writing the function name to the left of the argument in theexpression "f (s)." The reader is no doubt used to this convention. One canread "f (s)" as "the resulting value, when function f is applied to argument s."Then "f'f(s)" is "the result when f is applied to the result of applying f to s."These conventions will be called left (functional) orthography.

Right functional orthography is preferred by some mathematicians forall contexts and by most mathematicians for some contexts. In right or-thography, one writes "sf" or "(s)f" for "the operand s, transformed by thefunction f." This value is what was written "f(s)" in left orthography. Thecomposition function which we called "f'f" in left orthography is called "ff "in right orthography, so as to be consistent: "(s)ff'" in right orthography is"s-transformed-by-f, all transformed by f'." This is what was notated "f f (s)"in left orthography.

In the following work we shall use left orthography almost exclusively.We shall use right orthography only once, when its intuitive pertinence seemsoverwhelming. At that point in the text, the reader will be reminded of thisdiscussion. Right orthography would abstractly be more suitable for oureventual purposes, but the reader's presumed familiarity with left or-thography seemed decisive to me in making my choice.

1.2.5 Suppose that f t and f2 are functions from S to S'; suppose that f{ and f'2are functions from S' to S"; suppose that f" is a function from S to S". We canconsider the truth or falsity of functional equations like f^ = f", fif t = f2f2,and so on. Our discussion of "functional equality" in 1.2.2 tells us how tounderstand these equations, in evaluating their truth or falsity. The firstequation above asserts, "for any sample s, the result of applying f[ to ^(s) isthe same as the result of applying f" to the given s." The second equationabove asserts, "for any sample s, applying f{ to f^ (s) yields the same result asapplying f2 to f2(s)."2

Mathematical Preliminaries 1.3.1

For an example, let us take S, S', and S" all to be the family of positiveintegers. Let f^s) = s + 3, f{(s) = 2s, f2(s) = 2s, and f^s) = s + 6. The fourspecified functions satisfy the functional equation f'1f1 = f^. That is, givenany integer s, if we compute fjfi(s), multiplying by two the result of adding 3to s, we obtain the same net result as we do when we compute f^Cs), adding 6to the result of multiplying s by two.

For another example, let us take S, S', and S" all to be the family of thetwelve pitch-classes. Let f(s) = s transposed by 2, f(s) = s inverted withrespect to the pitch class C, and f"(s) = s inverted with respect to the pitchclass B. The three specified functions satisfy the functional equation f'f = f".That is, given any pitch class s, if we compute f T(s), inverting about C theresult of transposing s by 2, we obtain the same net result as we do when wecompute f"(s), inverting the given s about B.

1.2.6.1 DEFINITION: The function f from S into S' is onto S' if every member ofS' is the value of some argument. (Every member of S' appears at least once inthe right-hand column of the function table.)

1.2.6.2 DEFINITION: The function f from S into S' is 1-to-l if no two distinctarguments share the same value. (No member of S' appears more than once inthe right-hand column of the function table.)

1.2.6.3 DEFINITION: Let f be a 1-to-l function from S onto S'. Then f"1, theinverse function off, is defined as the family of pairs (s', s) within S' x S suchthat (s, s') is a member of f.

1.2.6.4 THEOREMS: Given the situation as in 1.2.6.3 above, then f ~* is indeeda function in the sense of 1.2.1. f-1 is in fact a 1-to-l function from S' onto S.The inverse function off"1 is, of course, f.

The theorems are stated without proof.

1.2.6.5 THEOREM: Let f and f' be functions from S into S' and from S' into Srespectively. Suppose that the functions satisfy the two conditions (A) and (B)following. (A): for every s in S, f'f(s) = s. (B): for every s' in S', ff'(s') = s'.Then f and f' are both 1 -to-1; they are respectively onto S' and onto S; and theyare inverse functions, each of the other.

The theorem is given without proof.

1.3.1 DEFINITION: A function from a family S into S itself will be called atransformation on S. If the function is 1-to-l and onto, it will be called anoperation on S. 3


1.3.2 DEFINITION: Given a family S, a collection F of transformations on S iscalled closed if, given any members f and g of F, the composition fg is amember of F. A closed collection of transformations on S will also be called asemigroup of transformations on S.

1.3.3.1 DEFINITION: The identity operation on a family S is that operation 1on S which assigns the value 1 (s) = s to any argument s.

1.3.3.2 THEOREM: For any transformation f on S, the functional equationsIf = f and fl = fare true (in the sense of 1.2.5 above).

1.3.3.3 THEOREM: A transformation f on S is an operation (i.e., 1-to-l andonto) if and only if there exists a transformation f on S satisfying thefunctional equations f T = 1; ff' = 1. If this be the case then f' is the inverseoperation of f.

The theorem follows from the various matters studied over section 1.2.6.

1.3.4 DEFINITION: By a group of operations on S we shall mean a family (i.e.collection) G of transformations on S which satisfies conditions (A) and (B)following. (A): G is a closed family, a semigroup of transformations in thesense of 1.3.2. (B): Given any member f of G, there exists a member f of Gsatisfying f'f = ff = 1.

Condition (B) guarantees that the members of G are indeed operations,via 1.3.3.3. (B) also guarantees that G contains the inverse operation for eachof its member operations. (A) and (B) together imply that G contains theidentity operation 1, provided that G contains any members. Whether we callG a "collection" or a "family" is immaterial; for us the terms are synonymouswith each other as they also are with the terms "ensemble" and "set-in-the-mathematical-sense."

1.3.5 The work of section 1.3 so far has explored certain algebraic behaviorcharacteristic of transformations on S. The transformations compose onewith another, f with g to form the transformation fg. There is an identitytransformation 1, which composes left or right with any f to yield f itself:If = fl = f. Certain transformations, the operations, have inverses; if f is suchthen f"1 is characterized by the algebraic relations f-1f = ff"1 = 1.

These algebraic features of the situation are abstracted and generalizedby the study of "abstract" semigroups and groups, a study we shall shortlycommence. Before we do so, we should note one more aspect of transforma-tion algebra which the abstract study will generalize. This is the associativity4


of transformational composition. That is, the composition of transformationsobeys the Associative Law f(gh) = (fg)h: Given any sample s, the result ofapplying f to the (gh)-transform of s is the same as the result of applying (fg) tothe h-transform of the given s.

1.4 Now we begin the abstract study. We fix a family (i.e. collection) X ofabstract objects x, y, z,..., and develop abstract algebraic systems that modelthe behavior of transformational algebra. First we must specify how theobjects of X are to "compose" one with another.

1.4.1 DEFINITION: A binary composition on X is a function BIN that mapsX x X into X. We write BIN(x, y) for the value of BIN on the pair (x, y).

1.4.2 DEFINITION: A binary composition on X is associative if BIN(x,BIN(y, z)) = BIN(BIN(x, y), z) for all x, y, and z.

A familiar non-associative binary composition on the natural numbersis exponentiation: BIN(x, y) = x-to-the-y-power. For example BIN(3,BIN (2,3)) = 3-to-the-(2-cubed)-power, or 3-to-the-eighth-power, whileBIN(BIN(3,2), 3) = (3-squared)-to-the-third-power, or 9-cubed. Nine-cubedis 3-to-the-sixth, not 3-to-the-eighth.

1.4.3 DEFINITION: A semigroup is an ordered pair (X, BIN) comprising afamily X and an associative binary composition BIN on X.

It is traditional to write the binary composition for a semigroup usingmultiplicative notation when there is no reason to use some specific othernotation. Thus we shall generally write "xy" to signify BIN(x, y) in a semi-group, failing some reason to write "x + y" or "x * y" and the like. TheAssociative Law for BIN then reads "x(yz) = (xy)z." This notational conven-tion simplifies the look of the page. It is important, though, not to carry overinto our general study intuitions about numerical multiplication which maynot be valid within a specific semigroup at hand.

It is also important to remember that in order to define a particularsemigroup, we must specify not only the family X of elements but also thecomposition BIN under which the elements combine. Despite this, it is cus-tomary to refer (improperly) to "the semigroup X" when the binary compo-sition is clearly understood in a given context.

1.5.1 DEFINITION: A left identity for a semigroup is an element 1 such that forevery x, Ix = x. A right identity is defined dually: For every x, xr = x. Anidentity is an element e which is both a left identity and a right identity. 5


1.5.2 THEOREM: If a semigroup has both a left identity 1 and a right identity r,then 1 and r must be equal. Hence there can be at most one identity for asemigroup. If a semigroup has one, we can therefore speak of "the" identityelement.

Proof: lr must equal r since 1 is a left identity. Ir must also equal 1 since r isa right identity.

There are, incidentally, semigroups that have an infinite number of leftidentities. (By the theorem above, a semigroup that has more than one leftidentity cannot have any right identities.) There are, in fact, both finite andinfinite semigroups in which every element is a left identity. To illustrate this,take any family X and define on X the composition BIN(x, y) = y for all x andall y; (X, BIN) is such a semigroup.

1.6.1 DEFINITION: Given a semigroup with identity e; given an element x, aleft inverse for x is an element 1 satisfying Ix = e. A right inverse for x is anelement r satisfying xr = e. An inverse for x is an x' which is both a left inverseand a right inverse.

1.6.2 THEOREM: If an element x of a semigroup with identity has both a leftinverse 1 and a right inverse r, then 1 = r. Hence x can have at most one inverse.If x has one, we can therefore call it "the" inverse of x.

Proof: 1 = le = l(xr) = (lx)r = er = r.

1.6.3 In multiplicative notation for a semigroup with identity, the inverse ofan element x that has one is denoted x"1.

1.7 DEFINITION: A group is a semigroup with identity in which every elementhas an, inverse.

The abstract definitions of "semigroup" and "group" (1.4.3; 1.7) areconsistent with the earlier use of those terms in connection with families oftransformations (1.3.2; 1.3.4).

1.8.1 DEFINITION: Given a binary composition BIN on a family X, elements xand y commute if BIN(y, x) = BIN(x, y), that is, if yx = xy in multiplicativenotation. The composition BIN is commutative if every pair of elementscommutes. A semigroup or group is commutative if its binary composition iscommutative.

The group of transposition and inversion operations on the twelve pitch-classes is non-commutative. To illustrate this, let T2 be the operation of6


transposing-by-2; let I, J, and K be the respective operations of inverting-about-C, inverting-about-B, and inverting-about-Cft. Then, as we observedearlier, IT2 = J (1.2.5). On the other hand, T2I = K. Thus the operations T2

and I do not commute. (Remember that we are using left orthography."IT2 = J" means: "Given any sample pitch-class s, if you invert-about-C the2-transpose of s, you will obtain the inversion-about-B of the given s.""T2I = K" means: "Given any sample pitch-class s, if you transpose-by-2the inversion-about-C of s, you will obtain the inversion-about-C# of thegiven s.")

1.8.2 DEFINITION: Given a binary composition BIN on a family X, anelement c of X is central if c commutes with every x in X. The family of allcentral c is the center of the system (X, BIN).

1.9 In this section we shall develop the conceptual structure and terminologyfor equivalence relations on a family S (not necessarily a semigroup). Weshall see in particular how the notion of an equivalence relation is inti-mately connected with the idea of mapping S onto another family S' by somefunction f.

1.9.1 DEFINITION: Given a family S, an equivalence relation on S is a sub-family EQUIV of S x S that satisfies conditions (A), (B), and (C) following.(A): For every s in S, (s, s) is in EQUIV. (B): If (s, t) is in EQUIV, then so is(t, s). (C): If (r, s) and (s, t) are in EQUIV, then so is (r, t).

The three conditions are called the "reflexive," "symmetric," and "tran-sitive" properties of the relation. The conditions express formally some of ourintuitions about things that are "equivalent." (A) matches our intuition thatany object s should be equivalent to itself. (B) matches our intuition that if s isequivalent to t, then t should be equivalent to s. (C) matches our intuition thatif r is equivalent to s and s is equivalent to t, then r should be equivalent to t.

1.9.2 THEOREM: Let f be a function from S onto S'. Define a relation EQUIVon S by putting the pair (s, t) in the relation if and only if f (s) = f (t). ThenEQUIV is an equivalence relation.

Proof: (A) f (s) = f (s), so (s, s) is in the defined relation. (B) If (s, t) is in thedefined relation, f (s) = f (t). Then f (t) = f (s), so that (t, s) is in the definedrelation. (C) If f(r) = f(s) and f(s) = f(t) then f(r) = f(t).

We shall see soon that every equivalence relation on S can be regarded asbeing generated in precisely the above fashion, for some suitable choice of S'andf. 7


1.9.3 THEOREM: Let EQUIV be an equivalence relation on a family S. Foreach s in S let E(s) be the subfamily of S comprising exactly those members of Swhich are in the EQUIV relation to s, i.e. those t such that (s, t) is in theEQUIV relation. Then, giveu any s and any t in S, either (A) or (B) below willbe true.

(A): s and t are equivalent; E(s) and E(t) are the same collection.(B): s and t are not equivalent; E(s) and E(t) are disjoint (have no

common member).Proof: Suppose first that s and t are equivalent. Then, by the symmetric andtransitive laws, r is equivalent to s if and only if r is equivalent to t. In otherwords, r is a member of E(s) if and only if r is a member of E(t). Thus E(s) andE(t) are the same collection. (A) of the theorem obtains.

Now suppose that s and t are not equivalent. Then there can be no r whichis both a member of E(s) and a member of E(t). For if there were such an r,then r would be equivalent to both s and t; by the symmetric and transi-tive laws, we could infer that s was equivalent to t, which we have supposedis not the case. Thus E(s) and E(t) are disjoint. (B) of the theorem obtains,q.e.d.

1.9.4 By virtue of Theorem 1.9.3, an equivalence relation partitions S intothe set-theoretic union of mutually disjoint subfamilies E^,.. .,En,... Thesesubfamilies are called the equivalence classes of the relation. For each s in S,there is precisely one equivalence class En to which s belongs, s is a member ofthe class En if and only if En = E(s), where E(s) is the family defined in 1.9.3,the family of objects equivalent to s.

Indeed, it would be possible to define any equivalence relation by par-titioning S into mutually disjoint subfamilies S l f ..., Sn, ... One could thendefine the pair (s, t) to be in a relation REL if both s and t lie in the samesubfamily of the partition. One could show that REL is an equivalencerelation, and that the members S j , . . . , Sn , . . . of the given partition are exactlythe equivalence classes for that equivalence relation.

1.9.5 DEFINITION: Given an equivalence relation EQUIV on a family S, thefamily of equivalence classes is called the quotient family of S modulo EQUIV.We shall denote it symbolically by S/EQUIV.

The function E of 1.9.3 maps S onto S/EQUIV, mapping each argument sto the value E(s), the member of the quotient family that contains s. Thefunction E is called the natural map of S onto S/EQUIV.

We may now regard every equivalence relation as potentially generatedin the manner of 1.9.2. Given EQUIV on S, take S' = S/EQUIV and takef = E, the natural map of S onto S'. s and t are then equivalent under the givenrelation if and only if f (s) = f (t).8


1.9.6.1 EXAMPLE: Let S be the family of all pitches under twelve-tone equaltemperament. Define EQUIV by putting (s, t) in EQUIV if s and t have thesame letter name, give or take enharmonic equivalence. The quotient familyS/EQUIV comprises the twelve pitch classes. The natural map E takes eachpitch s into its pitch class E(s).

1.9.6.2 EXAMPLE: Let S be the family of all beats in a certain waltz. Define afunction f from S into the numbers 1,2, and 3: f (s) = 1 if s is the first beat of itsmeasure; f(s) = 2 if s is the second beat of its measure; f(s) = 3 if s is the thirdbeat of its measure. A dancing master might construct this function by calling"one-two-three," over and over again as the beats go by. The function finduces an equivalence relation on S by the method of 1.9.2: s and t areEQUIValent if they share the same f-value. The three equivalence classes canbe called the "beat classes" of the relation; they comprise the first beats, thesecond beats, and the third beats of the waltz.

1.9.6.3 EXAMPLE: Let S be the family of all collections of pitch classes. Put thepair (s, t) into the relation SAMETYPE if the collection t is a transposed orinverted form of the collection s. (Transposition-by-zero is considered aformal transposition here.) SAMETYPE is an equivalence relation. Theequivalence classes are Forte's set-types.1 The class 3-11, in Forte's nomen-clature, contains the twenty-four major and minor triads. The class 3-12contains the four augmented triads. And so on.

1.9.7 OPTIONAL: This section of the work is for those who are curious toexplore the material in a bit more depth.

Given a function f from S onto S', define EQUIV as in 1.9.2; i.e. put (s, t)into the EQUIV relation if and only if f(s) = f(t).

Given any member s' of S'; since f is onto, there is some s in S satisfyingf(s) = s'. The family of all s satisfying f(s) = s' is an equivalence class En; En

contains just those arguments for f having the given s' as their f-value. Wewrite En = ARGS(s').

ARCS is a function from S' into S/EQUIV. ARGS maps S' ontoS/EQUIV: Given any equivalence class En, let s be a member of En and lets' = f(s); then ARGS(s') = En; the given En is an ARGS-value. The functionARGS is also 1-to-l: If s' and t' are distinct members of S', then the equiva-lence class ARGS(s'), comprising those s such that f(s) = s', is obviouslydifferent from the equivalence class ARGS(t'), comprising those t such thatf (t) = t'.

By the method of its construction above, the function ARGS satisfies

1. Allen Forte, The Structure of Atonal Music (New Haven and London: Yale UniversityPress, 1973). 9

1.10 Mathematical Preliminaries

formula (A) below.

(A): ARGS(f (s)) = E(s) for all s in S.

We have observed that the function ARGS is 1-to-l and onto. Hence ithas an inverse function, which we shall call f/EQUIV. ARGS maps S' 1-to-lonto S/EQUIV as in formula (A) above. f/EQUIV maps S/EQUIV 1-to-lonto S'. Applying f/EQUIV to both sides of formula (A), we obtain formula(B).

(B): f (s) = (f/EQUIV) (E(s)) for all s in S.

f/EQUIV is called the induced map on S/EQUIV. While f may map manyof its arguments onto the single value f (s) in S', f/EQUIV maps exactly one ofits arguments onto that value. Via formulas (A) and (B), the mutually inversefunctions ARGS and f/EQUIV set up a 1-to-l correspondence between themembers f (s) of the image family S' and the members E(s) of the quotientfamily S/EQUIV, the family of equivalence classes.

1.10 When we shift our attention from an arbitrary family S to a semigroup(X, BIN), certain sorts of equivalence relations on X are of special interestbecause of the ways they interact with the algebraic structure of the semi-group. We shall study here some special equivalence relations called con-gruences. They interrelate with special sorts of functions on semigroups,functions called homomorphisms. Homomorphisms map semigroups oneinto another in a special way that engages algebraic structure.

1.10.1 DEFINITION: An equivalence relation on a semigroup is a congruence ifit has this property: Given xt equivalent to yx and x2 equivalent to y2, then\1x2 is equivalent to yâ-

1.10.2 THEOREM: Given a congruence on a semigroup, let C^ and C2 be anycongruence classes (equivalence classes for the congruence). Then there is aunique congruence class C3 such that whenever XA and x2 are members of Ct

and C2 respectively, the composition x tx2 is a member of C3.Proof: Take any specimen y: in Q and any specimen y2 in C2. Let C3 be

the congruence class containing y^. C3 is the class whose existence thetheorem asserts. To see this, suppose that x t and x2 are any members of Cx andC2 respectively. Since X j is congruent to y: and x2 is congruent to y2, x tx2 willbe equivalent to y^ (1.10.1). Hence XjX 2 will lie within the same congruenceclass as y xy 2 . That is, xxx2 will lie within the constructed C3. q.e.d.

1.10.3 THEOREM: Let CONG be a congruence on the semigroup (X, BIN).Then the quotient family X/CONG (i.e. the family of congruence classes)becomes a semigroup itself under the binary composition BIN/CONG de-fined as follows. Given congruence classes Cj and C2 (members of X/CONG),w

Mathematical Preliminaries J.JO.4.2

the composition (BIN/CONG) (C1,C2) is the congruence class C3 ofTheorem 1.10.2, that is the unique congruence class which containsBIN(x1,x2) whenever x1 belongs to Ct and x2 belongs to C2.

Sketch of proof: The heart of the theorem is that the binary compositionBIN/CONG is well defined. Given Cj and C2, the value of C3 does not dependat all on the specimen \v and x2 we may select to represent Ct and C2. C3

depends only upon the classes C1 and C2 themselves.Having noted that, it is not hard to prove that BIN/CONG is associative.

1.10.4.1 EXAMPLE: Let (X, BIN) be the group of all integers, positive, nega-tive, or zero, under addition. Define a relation CONG: the pair of integers(x, y) is in this relation if the difference y — x is an integral multiple of 12.CONG is reflexive: (x, x) is in the relation since x — x = 0 = 0-times-12 is anintegral multiple of 12. CONG is symmetric: If (x,y) is in the relation, thenthere is some integer n such that y — x = n-times-12; then there is some integerm such that x — y = m-times-12 (take m = — n); then (y, x) is in the relation.CONG is transitive: If y — x = m-times-12 and z — y = n-times-12, thenz — x = (z — y) + (y — x) = (n + m)-times-12.

So CONG is an equivalence relation. It is in fact a congruence, for itsatisfies the requirement of 1.10.1: If yt — x t is a multiple of 12 and y2 — x2 isa multiple of 12, then (yt + y2) — (xj + x2) is a multiple of 12.

We write C(x) for the congruence class containing x. Since C(x) = C(x-plus-or-minus-any-multiple-of-12), every congruence class is one of the twelveclasses C(0), C(l), ..., C(ll). The quotient semigroup, then, contains justthose twelve members. For each i between 0 and 11 inclusive, the class C(i)contains exactly those integers that can be written as i-plus-some-multiple-of-12. Composition of the twelve congruence classes within the quotient semi-group follows the rule of addition modulo 12. That is, C(i) + C(j) = C(i + j)if i + j is less than 12; otherwise C(i) + C(j) = C(i + j - 12). ThusC(5) + C(8) = C(l). According to 1.10.2, we can read this as stating cor-rectly: "Any number divisible by 12 with a remainder of 5, added to anynumber divisible by 12 with a remainder of 8, produces some number divisibleby 12 with a remainder of 1." The equation "C(5) + C(8) = C(l)" in theabove context is customarily abbreviated: "5 + 8 = 1 (mod 12)."

The quotient semigroup is called "the integers modulo 12." It is in fact agroup. We shall see later that the quotient semigroup of any group must itselfbe a group. If we replace the modulus 12 in the above construction by anarbitrary integer N greater than 1, we obtain "the integers modulo N" as aquotient group.

1.10.4.2 EXAMPLE: Let (X, BIN) be the group of all rational numbers thatcan be expressed as x = 2a3b5c, where a, b, and c are integers (positive,negative, or zero); BIN is multiplication. We can consider these numbers tomodel all possible ratios of pitches in just intonation. 11

1.11 Mathematical Preliminaries

Define a relation CONG: The pair (x, y) is in this relation if the number yis some power of 2 (positive, negative, or zero) times the number x. In ourintervallic model, this will be the case when the intervals x and y differ by somenumber of octaves.

For example, any one of the numbers 12/5, 6/5, 3/5, and 3/10 is in thisrelation to itself or to any other one. The four numbers model the fourintervals of a minor tenth up, a minor third up, a major sixth down, and amajor thirteenth down.

As an exercise, using the procedure of 1.10.4.1 as a guide, the reader mayverify that CONG is a congruence. (Remember to verify first that it is anequivalence relation!) The quotient group models all pitch-class intervals injust intonation. That is, each congruence class consists of one interval, give ortake any number of octaves.

Mathematically, C(x) = C(2x) = C(4x) = • • • = C(x/2) = C(x/4) = . . From this, it can be proved: Given any x, there is a unique member x' of C(x)which lies between 1 (inclusive) and 2 (exclusive). In this way, the members x'of X that lie between 1 and 2 provide a plausible system of labels for thecongruence classes C(x'). (The various pitch-intervals between the unison andthe rising octave can be used to label the various intervals-modulo-the-octave.)

It can also be proved: Given any x, there is a unique member x" of C(x)which can be expressed as x" = 3b5c. So the numbers x" that have factors of 3and 5 only in their rational expressions provide another plausible system oflabels for the congruence classes C(x"). (x" = 3b5c labels the pitch-class inter-val of "b dominants and c mediants, modulo the octave.")

1.11 When we studied an equivalence relation on a family S, we made anumber of observations about the natural map E, the function that maps eachelement s of S into the equivalence class E(s) of which s is a member. When thefamily S is a semigroup X and the equivalence relation is a congruence, weshall replace the name "E" of this natural map by the name "C": C maps eachelement x of the semigroup X into the congruence class C(x) of which x is amember. We have already used this nomenclature in examples 1.10.4.1 and1.10.4.2 above.

Everything that we observed earlier about the natural map E (1.9.3,1.9.4,1.9.5,1.9.7) is true for the natural map C, which is only a special notation for Ein the particular event that S is a semigroup and the equivalence relation is acongruence. Beyond that, C has special properties that engage the algebraicstructure of the semigroup X and the quotient semigroup X/CONG. Specifi-cally, the natural map C of X onto X/CONG satisfies law (A) below.

(A): C(x!)C(x2) = C(XiX2) for all xx and x2.

Indeed we defined the binary composition "C(x1)C(x2)" in X/CONG72


precisely so as to satisfy this law. That was the work of 1.10.2 and 1.10.3.Mathematicians express property (A) above by saying, "C is a homomorph-ism of X onto X/CONG." The crucial term "homomorphism" is defined in1.11.1 below.

1.11.1 DEFINITION: A function f from a semigroup (X, BIN) into a semigroup(X', BIN') is a homomorphism if it satisfies the law:

BIN'(f(Xl),f(x2)) = f(BIN(Xl,x2))

for all Xi and all x2 in X. One can express this law colloquially by saying, "Thecombination of the images is the image of the combination." Using multi-plicative notation for both semigroups, the law looks simpler:

f(Xl)f(x2) = f(Xlx2).

Certain sorts of homomorphisms are of special interest.

1.11.2 DEFINITION: A homomorphism is an isomorphism (into) if it is 1 -to-1If f is an isomorphism of (X, BIN) onto (X', BIN'), we say the two semigroupsare isomorphic (via f). In that case the inverse map f-1 is an isomorphism of(X', BIN') onto (X, BIN).

1.11.3 OPTIONAL: Let f be a homomorphism of a semigroup (X, BIN) onto asemigroup (X', BIN'). We have already seen (in 1.9.2) that an equivalencerelation is defined if we select as equivalent just those pairs (x, y) satisfyingf (x) = f (y). We can show that the relation in this case is in fact a congruenceCONG.

From earlier work (1.9.7) we know that the mapping ARGS of X' intoX/CONG is 1-to-l and onto. (ARGS(x') is the congruence class comprisingexactly those x such that f (x) = x'.) In this case, f being a homomorphism, wecan show that ARGS is a homomorphism of the semigroup (X', BIN') into thequotient semigroup (X, BIN)/CONG.

Here is a sketch for the proof of that. We want to show that for all \l andfor allx2,ARGS(f(x1))ARGS(f(x2)) = ARGS(f(x1)f(x2)). In this equation,the symbolic product of the two ARGS-values on the left means the binarycomposition of those values in the quotient semigroup; the symbolic productf(Xi)f(x2) within the equation means the binary composition of thosetwo f-values in the semigroup (X',BIN'). Now f(x t)f(x2) = f(x1x2), sincef is a homomorphism, and ARGS (f (any thing)) = C(that thing), as per1.9.7(A). Hence the equation we have to show reduces to the equationC(x1)C(x2) = C(\1x2). And the latter equation is indeed true, since CONG isa congruence (1.11 (A)).

Since ARGS is 1-to-l, onto, and a homomorphism, it is an isomorphismof the two semigroups (X', BIN') and (X, BIN)/CONG. Colloquially speak- 13


ing, we can say that the image semigroup is isomorphic with the quotientsemigroup in this context. f/CONG, the inverse map of ARGS, the inducedmap of the quotient semigroup onto the image semigroup, is therefore also anisomorphism.

This is very significant. It means that any homomorphic image (X', BIN')of a semigroup (X, BIN) "is essentially" some quotient semigroup of(X, BIN), and the generic homomorphism f of (X, BIN) onto that image "isessentially" the natural map of (X, BIN) onto that quotient. The words "isessentially" here must be interpreted with some care. They express the idea ofidentification up to within isomorphism of the image semigroups. With thatunderstanding, we can say that it suffices to study the possible congruencerelations on (X, BIN), in order to know all possible homomorphisms whichcan map (X, BIN) onto other semigroups, and all possible other semigroupswhich are homomorphic images of (X, BIN).

1.11.4 One more term should be introduced here. An anti-homomorphism ofone semigroup into another is a function f satisfying f(x!)f(x2) = f^x^.

Given a semigroup (X, BIN) we can define another binary compositionANTIBIN on the family X: ANTIBIN (x l 5x2) = BIN(x2,x1). ANTIBINis associative, so (X, ANTIBIN) is a semigroup. (X, ANTIBIN) is anti-isomorphic to (X, BIN) under the map f (x) = x. In the obvious sense, everyanti-homomorphism of (X, BIN) is a homomorphism of (X, ANTIBIN) andvice-versa. Thus we will not normally have to concern ourselves with anti-homomorphisms. We will only have to do so when we have to deal with bothhomomorphisms and anti-homomorphisms of the same semigroup at thesame time.

Such a situation will in fact arise later on. We shall be studying a certaingroup whose elements are i, j, k ...; we shall also be studying various familiesof transformations on a certain family of objects. One such family will becalled "transpositions"; for each i there will be a corresponding transposition-operation TJ. Another such family will be called "interval-preserving oper-ations"; for each i there will be a corresponding interval-preserving operationPJ. The P-operations will combine according to the rule PjPj = P^; the T-operations will combine according to the rule T;Tj = T^. The map of i to P;

will be an isomorphism, while the map of i to T, will be an anti-isomorphism ofthe same group.

In such a situation we must perforce deal with the concept of anti-homomorphism. We could change BIN to ANTIBIN in the index group i, j,k . . . so as to make the mapping of i to Tj an isomorphism, but then the map-ping of i to PJ would become an anti-isomorphism. Using right orthographyfor the operations T; and P; would have the same effect.

1.12.1 THEOREM: Let f be a homomorphism of the semigroup (X, BIN) ontothe semigroup (X', BIN'). If e is an identity for (X, BIN) then f (e) is an identity14

Mathematical Preliminaries 1.13

for (X', BIN'). In that case, if x has an inverse x * in (X, BIN) then f(x-1) is aninverse for f(x) in (X', BIN').

Proof: f(e)f(x) = f(ex) = f(x); f(x)f(e) = f(xe) = f(x). Thus f(e) is anidentity for the family of all f(x). Since f is onto, every member of X' can bewritten as the value f (x) of some argument x. Hence f (e) is an identity for all ofX'. If x has an inverse then f(x-1)f(x) = f(x-1x) = f(e) = the identity in X';likewise f(x)f(x-1) is the identity in X'. So f(x-1) is the inverse in X' for f(x).Thatis, f(x~1) = (f(x)r1.

1.12.2 THEOREM: A homomorphic image of a group is a group.

The theorem follows at once from 1.12.1.

1.12.3 THEOREM: Any quotient semigroup of a group is a group.

The theorem follows from 1.12.2, since the natural map of the givengroup onto its quotient semigroup is a homomorphism (1.11 (A)).

The quotient construction is one common way to derive new semigroupsor groups from old. Another way is to form "direct products" as sketchedbelow.

1.13 Let SGPi = (Xt, BINO and SGP2 = (X2, BIN2) be semigroups. Thedirect product of SGPj and SGP2 is a semigroup SGP3 = (X3,BIN3)constructed as follows. X3 is the Cartesian product Xt x X2. Given(xlsx2) and (y1}y2) in X3, BIN3((xl5x2), (yl5y2)) is defined as theelement (BIN^x^yj), BIN2(x2,y2)) of X3. In multiplicative notation,(Xi,x2)(y1 ,y2) is defined = (x1y1,x2y2).

BIN3 as defined is associative, so that SGP3 is indeed a semigroup. Tosymbolize that SGP3 is the direct product of SCPj and SGP2 we writeSGP3 = SGP1®SGP2.

If ej and e2 are identities for SGPj and SGP2, then e3 = (else2) is anidentity for SGP3. If xl in Xt and x2 in X2 have inverses in their respectivesemigroups, then (x^SxJ1) is an inverse for the element (Xj,x2) of SGP3. Itfollows: If SGPj and SGP2 are both groups, then so is their direct productSGP3.

75

2 Generalized Interval Systems(1): Preliminary Examplesand Definition

In conceptualizing a particular musical space, it often happens that we con-ceptualize along with it, as one of its characteristic textural features, a familyof directed measurements, distances, or motions of some sort. Contemplatingelements s and t of such a musical space, we are characteristically aware of theparticular directed measurement, distance, or motion that proceeds "from s tot." Figure 0.1 on page xi earlier symbolized this awareness, using an arrowmarked i extending or moving from a point marked s to a point marked t tohelp us render visible our intuition. When s and t are pitches or pitch classes weare comfortable with the word "interval" as a term to use in connection withthe i-arrow. That is why I have decided to keep using the word "interval"when generalizing our intuitions about the i-arrow to musical spaces whoseobjects s, t, and the like are not necessarily pitches or pitch classes.

It will be helpful to explore some specific musical spaces informally in thisconnection, before proceeding more formally later on. "int(s, t)" will provi-sionally denote our intuition of a directed measurement or motion behavinglike an "interval from s to t." Later on we shall attach a more formalsignificance to the expression "int(s, t)."

2.1.1 EXAMPLE: The musical space is a diatonic gamut of pitches arranged inscalar order. Given pitches s and t, int(s, t) is the number of scale steps onemust move in an upwards-oriented sense to get from s to t. Thus int(C4, C4)= 0, int(C4,D4) = 1, int(C4,E4) = 2, and int(C4,C5) = 7. Int(C4,A3) =— 2, since moving " — 2 steps up" amounts to moving 2 steps down.

Using these measurements, if we take 2 steps up (e.g. from C4 to E4) andthen take 2 more steps up (in this case, from E4 to G4), we have taken 4 stepsup in all (in this case, from C4 to G4). Symbolically, int(C4, E4) = 2,16

Generalized Interval Systems (1) 2.1.5

int(E4, G4) = 2, int(C4, G4) = 4, and 2 + 2 = 4. The intervallic mea-surements of the model thus interact effectively with ordinary arithmetic. Thisobviates a defect in the traditional measurements which tell us, for example,that a "3rd" and another "3rd" compose to form a "5th." (3 + 3 = 5 ???)

2.1.2 EXAMPLE: The musical space is a gamut of chromatic pitches undertwelve-tone equal temperament. Given pitches s and t, int(s, t) is the numberof semitones one must move in an upwards-oriented sense to get from s to t,not counting s itself. Thus int(C4, D4) = 2, int(C4, G4) = 7, int(C4, C5) =12, int(C4, F3) = -7, and int(C4, F2) = -19.

2.1.3 EXAMPLE: The musical space comprises the twelve pitch-classes underequal temperament. If we arrange the pitch classes around the face of a clockfollowing the order of a chromatic scale, then int(s, t) is the number of hoursthat we traverse in proceeding clockwise from s to t. For instance, if s is at 8o'clock and t is at 1 o'clock, int(s, t) = 5. Note that the number int(s, t) doesnot depend on which pitch class is positioned at 12 o'clock. In any case,int(E, E) = 0, int(E, F) = 1, and int(F, E) = 11.

2.1.4 EXAMPLE: The musical space comprises seven pitch-classes, corre-sponding to the seven mode degrees of system 2.1.1. If we wrap the scalearound the face of a seven-hour clock, then int(s, t) is the number of hoursthat we traverse on that clock, in proceeding clockwise from s to t. Thusint(D, D) = 0, int(D, E) = 1, and int(D, C) = 6.

We could produce analogs for the linear spaces of examples 2.1.1 and2.1.2, using other sorts of scales. And, for systems in which octave equivalenceis functional, we could derive analogs for the modular spaces of examples2.1.3 and 2.1.4. For example, we could investigate octatonic-scale space in themanner of 2.1.1 and 2.1.2; we could derive there from a modular space of eightpitch-classes, wrapping the octatonic scale around an eight-hour clock andmeasuring intervals modulo 8.

2.1.5 EXAMPLE: This musical space, harmonic rather than melodic, com-prises pitches available from a given pitch using just intonation. If we writeFQ(s) to denote the fundamental frequency of the pitch s, then int(s, t) is thequotient FQ(t)/FQ(s). That quotient will be some number of the form 2a3b5c,where a, b, and c are integers, positive, negative, or zero.

It is not immediately clear what intuitions of "distance" or "motion"we are measuring by these intervals. Personally, I am convinced that ourintuitions are highly conditioned by cultural factors. In particular, I do notthink that the acoustics of harmonically vibrating bodies provide in them-selves an adequate basis for grounding those intuitions. For instance, when we 17

2.7.5 Generalized Interval Systems (1)

write int(C4, F#4) = 45/32 (= 2~5 32 5), I do not believe that we are intuiting acommon partial frequency F#9 for both C4 and F#4, a partial which isintuited forthwith in some harmonic space as both the 32nd partial of F#4 andthe 45th partial of C4. Nor do I believe that we intuit a path in harmonic spacewhich corresponds directly to a compound series of individual multiplicationsand divisions by 2,3, and 5. That is, if we take the 5th partial of the 3rd partialof the 3rd partial of C4, and then find the frequency of which that is the 2ndpartial, and then find the frequency of which that is the 2nd partial, continuingon in this way and so arriving eventually at F#4,1 do not believe that the wayof getting from C4 to F#4 which we have intellectually reconstructed inharmonic space is in any sense an intuition of distance or gesture beingmeasured by the composite ratio 45/32 = 5 times 3 times 3 divided by 2divided by 2 and so on.

In order to describe an actual harmonic intuition, I would rather proceedas follows. When we hear or imagine the succession C4-F#4 in its own contextand try to intuit a harmonic sensation, we intuit a tonic followed by the leadingtone of its dominant. And we intuit the secondary leading tone harmonicallyas the third of a harmony whose root is the dominant of the dominant. Con-structing a fundamental bass representative for that root, i.e. some D belowthe F$4, we will locate that D in register as D3, to keep it completely beneaththe "soprano line" C4-F#4. For the same reason, in constructing a funda-mental bass for the note G4 that we imagine following F#4 in the soprano, afundamental bass representing the implicit role of dominant harmony in thecontext, we will locate the bass G as G3. In this way we intuit the enlargedharmonic context of figure 2.1 (a) from the given stimulus C4-F#4.

FIGURE 2.1

The arrows on figure 2.1(b) show the path in harmonic space which Ibelieve we actually intuit in this enlarged context. Starting at C4 as a localtonic, the first arrow takes us to G3, a fundamental bass for the dominantharmony where we hear the implicit enlarged context closing. The secondarrow shows G3 inflected by its own dominant immediately preceding; thearrow points to the fundamental bass D3 for that event. The third arrowpoints to F#3, the major third of the harmony over D3. The fourth arrowpoints to F#4, the octave above F#3. Collating the entire path, we can retraceit and express it in prose: F#4 lies an octave above the major third of that18

Generalized In terval Systems (1) 2.1.5

dominant which lies a fourth below that dominant which lies a fourth belowC4.

Now we can finally see in what sense the number 45/32 is a valid measure-ment for some intuition of a characteristic way from C4 to F#4 in harmonicspace. We do have clear intuitions for a number of basic harmonic moves; wecan measure those moves, and we have intuited (not just constructed) a chainof them. Our belief in the validity of mathematics carries us the rest of the way.Specifically, we intuit clearly the relation "t lies an octave above s," and weaccept empirically the measurement FQ(t) = 2FQ(s) as a valid reflection ofthat intuition. We also intuit clearly the relation "t is the major third of thes harmony," and we accept the measurement FQ(t) = (5/4)FQ(s) as validin connection with that intuition. Finally, we intuit clearly the relation "tis that dominant which lies a fourth below s," and we accept the measure-ment FQ(t) = (3/4)FQ(s) as valid in that connection. Applying those basicmeasurements to the arrows of figure 2.1(b), we get FQ(F#4) = 2FQ(F#3),FQ(F#3) = (5/4)FQ(D3), FQ(D3) = (3/4)FQ(G3), and FQ(G3) =(3/4)FQ(C4). Applying mathematics to this chain of measurements,we infer that the equation FQ(F#4) = 2FQ(F#3) = 2(5/4)FQ(D3) =2(5/4)(3/4)FQ(G3) = 2(5/4)(3/4)(3/4)FQ(C4) is valid as measuring anintuited chain of intuitions, that is, not simply as an empirical fact. Observethat the number int(C4, F#4) = FQ(F#4)/FQ(C4) = 45/32 arises here notas the product of 2~5 and 32 and 5, which is its most "natural" mathematicalfactorization. Rather, 45/32 arises as the product of the four factors 2, (5/4),(3/4), and (3/4), reflecting its "natural" way of measuring an intuited chain ofintuitions in the given situation.

I should stress again not only the sophistication and complexity of thissystem (compared, for example, to the melodic system of example 2.1.1) butalso its heavy reliance on cultural conditioning. A brief review of just how thenoteheads got onto figure 2.1 will emphasize the point: Cultural conditioningis obviously important in our construction of the extended mental/aural con-text, given only the acoustical stimulus C4-F#4 and the intent to think/hear"harmonically." That intent, in turn, is itself a cultural phenomenon. Imaginea culture whose members, when they hear the notes of figure 2.1, are able tooverride the melodic relation of F#4 to the G4 which follows, as a primarystructural determinant for a system of music theory that addresses suchpassages! The amount of time it takes the reader to discover that the lastsentence is ironic indicates the greater or lesser extent to which we are all stillwithin the grip of that culture.

Then too, our interest in the harmonic system under study depends to aconsiderable extent on a cultural predilection for harmonically resonatinginstruments (with one degree of freedom) producing sustained (steady-state)sounds. In fact, much of the traditional theory pertinent to the system wasdeveloped under the supposition that the domain of investigation could be 19

2.1.6 Generalized In terval Systems (1)

adequately represented by aspects of one or more stretched strings. Despite itsproblems, that representation had the great advantage of enabling theorists toconnect their intuitions of musical intervals with the measurement of visible,tangible material distances along the strings. This Cartesian modelling ofharmonic intervals as res extensae, in a space outside the minds and bodies ofthe musicians, enabled harmonic theorists of earlier times to avoid having toconfront such complex and problematic gestural intuitions as those of figure2.1.

2.1.6 EXAMPLE: The musical space comprises the pitch classes generated bythe space of example 2.1.5 above. Given pitch classes s and t, int(s, t) is theordered pair of integers (b, c) such that t lies b dominants and c mediants froms. Here, if p and q are pitches belonging to the classes s and t respectively, thenthere is some integral power of 2, say 2a, such that the interval from p to q inthe system of 2.1.5 above is 2a3b5°.

We have int(C, G) = (1,0), int(G, D) = (1,0), int(D, F#) = (0,1), int(C,F#) = (2,l), int(C,F) = (-l,0), int(C,Ab) = (0,-l), and int(C,Db) =(—1, —1). (Since we are in just intonation, there are many distinct pitchclasses that have any given letter name C, G, D, F#, F, A(7, D|?, and so on. I amsupposing above that we are considering the simplest possible harmonicrelationship in each case. Later discussion will clarify the exact issuesinvolved.)

This system modularizes the system of 2.1.5 by reducing out all octaverelationships among both pitches and intervals. Its utility is clear in connec-tion with figure 2.1. The most salient aspect of our harmonic intuition therewas that the pitch class F# represented the mediant of the dominant of thatdominant of the pitch class C. Our having to manipulate precise registers inthe figure, particularly registers for a fundamental bass, made the working outof that intuition more complicated and problematical than necessary in thiscase. Now that we have available the modular harmonic system of the presentexample, we could get from C4 to F#4 more simply by a new chain ofintuitions. Contemplating the succession of pitches C4-F#4 as before, wearrive as before at the intuition that F#4 is a leading tone to a dominant of C,and therefore functions harmonically as a mediant for a dominant of thedominant. We have now intuited int(C, F#) = (2,1) in modular harmonicspace. We know that the pitch interval between two pitches whose pitchclasses are in a tonic/dominant relation is validly measured by some-power-of-2 times 3/2. We also know that the pitch interval between two pitches whosepitch classes are in a root/mediant relation is validly measured by some-power-of-2 times 5/4. Hence we infer from our intuition of the pitch-clasinterval int(C, F#) = (2,1) the validity of measuring the pitch intervalint(C4, F#4) as = some-power-of-2 times (3/2)2 times (5/4). We know that arising interval between two pitches lying closer than an octave is validly20


measured by a number between 1 and 2. So we infer that int(C4, F|4) is validlymeasured by the number 2a times (3/2)2 times (5/4), where the integer a makesthe product greater than 1 and less than 2.

We can visualize modular harmonic space as a two-dimensional map orgame board in the format of figure 2.2.

On this map, if int(s, t) = (b, c) then the pitch class t lies b places to theright of s and c places above s. (—b places to the right is b places to the left.)The subscripts help us keep track of the mediant dimension: If s has subscriptm and t has subscript n then int(s, t) will be of form (something, n - m).The subscripts and the visual format more generally clarify the functional distinc-tions in this space between pitch classes with the same letter-name but differ-ent subscripts; they are at different places on the map. Since we are assumingjust intonation, the frequencies involved will differ by factors involving thesyntonic comma and the pitch classes C_,, C0, C,, ... will be distinct acousti-cally. But even in equal temperament, the visual format of figure 2.2 portraysa conceptually infinite harmonic space, whose distinct places correspond tosubscripted pitch-classes. (Subscripted pitch-classes can be expressed formallyas ordered pairs; for example A\>3 can be expressed as the pair (A|>, 3).)

We can conceptualize the intervals of modular harmonic space as charac-teristic "moves" on the game board of figure 2.2. For example, in going fromCQ to Ffj, we move 2 squares to the right and 1 square up. The number-pair(2, 1), which is int(C0, F|j), can thereby be conceptualized as a label for thisparticular form of the "knight's move" on the game board, i.e. the knight'smove east by northeast. The same move (interval) takes us from Al to D#2, orfrom DL, to G0. 27

FIGURE 2.2


We could reduce the system of 2.1.6 farther if we considered pitch classesto be equivalent when they shared the same letter name, differing only bysubscript. Then C_ l 5 C0, Cj, C2 , . . . Cn , . . . would all mean the same thing; sowould £_!, E0, E l5 E 2 , . . . , En. In this case, moving one square north on thegame board of figure 2.2 would be functionally equivalent to moving foursquares east. The north/south dimension of the board would functionallydisappear, and we could reduce our map to a one-dimensional east/westsuccession of dominant-related pitch classes... E|?0, 6(7 0, F0, C0, G0, D0, A0,E0, ... Because of the equivalence relation that led to this series, we may aswell consider the reduced pitch classes to represent pitches in quarter-commamean-tone temperament: Four new "fifths" (that were steps east on figure 2.2)are pitch-class equivalent to a "major third" (that was a step north on figure2.2). There is no reason to keep the zero subscripts in the series of mean-tonepitch classes, so we can just write... E[?, 8(7, F, C, G, D, A, E,... The intervalsof this reduced system are integers measuring steps "east" on that chain; sincewe have lost the north/south dimension of our earlier figure, we may as wellsay "to the right" rather than "east."

We could reduce the mean-tone system even farther by declaring theenharmonic equivalence of Gl? with F#, of Dfr with C#, and so on. The infiniteseries of mean-tone pitch classes thereby gets wrapped around the face of aclock, and we find ourselves back at the system of 2.1.3, only now measuringintervals-modulo-the-octave by (equally tempered) fifths rather than bysemitones.1

We have explored six examples, and suggested some further examples, ofmusical spaces in connection with which we traditionally use the word "inter-val" to denote a directed measurement, distance, or motion. All six of thesemusical spaces, melodic or harmonic, had pitches or pitch classes for theirelements. Now we transfer our attention to some musical spaces whoseelements are measured rhythmic entities of various sorts. We presuppose acontext that makes us sensitive to time in segments that can be measured bysome temporal unit, whether this unit is some local pulse within a piece orsome conceptual span, like the minute that underlies metronome markings.

2.2.1 EXAMPLE: The musical space is a succession of time points pulsing atregular temporal distances one time unit apart. Given time points s and t,

1. Maps like figure 2.2 have been especially common in German theories of tonality sincethe eighteenth century, generally in connection with key relationships rather than root relation-ships (though some theories do not dwell on such a distinction). The closest precedent I can findfor the actual configuration of figure 2.2 itself appears in Hugo Riemann, Grosse Kompositions-lehre, vol. 1, Der homophone Satz (Melodielehre und Harmonielehre) (Berlin and Stuttgart: W.Spemann, 1902). Riemann's map is on page 479. He illustrates intervals as moves on the board, onpage 480.22


int(s, t) is the number of temporal units by which t is later than s. (—x unitslater is x units earlier.)

2.2.2 EXAMPLE: The musical space is the preceding one, wrapped around theface of an N-hour clock. We can imagine this as modeling the imposition of anN-unit meter on the earlier space, so that barlines appear regularly every Npulses. The present space has N members, which we shall call "beat classes,"labeling them by numbers from O through N — 1. Beat-class 0 comprises allthe pulses of 2.2.1 that occur at some bar-line; beat-class 1 comprises all thepulses of 2.2.1 that^ occur one unit after some barline;...; beat-class (N — 1)comprises all the pulses of 2.2.1 that occur one unit before some barline. If sand t are beat classes, int(s, t) is the number of hours clockwise that t lies froms on the N-hour clock. Thus, in twelve-eighths meter (N = 12) the intervalfrom beat-class 10 to beat-class 5 is 7.

We discussed the notion of beat classes earlier (1.9.6.2), as exemplifyingthe concept of equivalence classes. We observed there that a dancing masteroften calls out beat classes over and over as the pulses go by, using numbers 1through N rather than 0 through N — 1, e.g. "ONE-two-three, ONE-two-three,..." Conductors and conducting students will also be familiar with thenotion kinetically. For them, beat classes are associated with definite spatialpositions of the hand, positions which are numbered on pedagogical dia-grams. Intervals of 1 between beat classes correspond to minimal unbrokenhand gestures for the conductor, gestures that proceed from each beat class tothe next along smooth arrows on the diagrams, tracing a characteristicgestural path through this modular space over and over again. This is the pathalong which we ride "from s to t," making int(s, t) gestural articulations alongthe way.

Milton Babbitt has worked with a system of 12 beat classes that behavesformally exactly like the traditional 12-tone system for pitch classes.2

2.2.3 EXAMPLE: The musical space is a family of durations, each durationmeasuring a temporal span in time units. And int(s, t) is the quotient of the tand s measurements, t/s. If s spans 4 time units and t spans 3 time units, thenint(s, t) = 3/4. t is "3/4 the length of" s.

We may, if we wish, identify each duration with the beat for a certaintempo. The numerical quotients of our durations then measure the inversequotients of the corresponding tempi in tempo-space.3

2. He describes the system in "Twelve-Tone Rhythmic Structure and the ElectronicMedium," Perspectives of New Music vol. 1, no. 1 (Fall 1962), 49-79.

3. Influential compositions whose rhythmic textures involve such proportions includeElliott Carter's String Quartet no. 1 (1950-51), Conlon Nancarrow's Studies for Player Piano(1951-), and Gyorgy Ligeti's Poeme symphonique for 100 metronomes (1962). 23

2.2.4 Generalized Interval Systems (I)

2.2A EXAMPLE: We reduce the system of 2.2.3 by a durational modulus Mgreater than 1. Two durations are conceived as equivalent if one is someintegral power of M times the other. This leads us to a modular musical spacewhose elements are duration-classes (i.e. equivalence classes of durationsunder the defined equivalence relation).

The intervals of 2.2.3 are reduced in the same manner: Two numericalquotients or proportions are conceived as equivalent if one is some power ofM times the other. The ratio-classes can be used as formal intervals in thereduced system. Mathematically, the reduction from 2.2.3 to 2.2.4 is exactlythe same as the reduction from a system of pitches and pitch-ratios, to asystem of pitch classes and ratio-classes modulo powers of M = 2, that is pitchclasses and intervals-modulo-the-octave.

To illustrate, let us take M = 2 for a rhythmic modulus: Two durations,or two numerical proportions, are conceived as equivalent if one is twice theother, or four times the other, or eight times the other, or half the other, orone-quarter of the other, and so on. One equivalence class of durations is thenthe family r = (...,5/32,5/16,5/8,5/4,5/2,5,10,20,...). Another equiva-lence class is the family s = (..., 1/96,1/48,1/24,1/12,1/6,1/3,2/3,4/3,8/3,...). Yet another equivalence class is the family t = (..., 7/80,7/40,7/20,7/10,7/5,14/5,28/5,...). With reference to these particular classes r, s, and t,int(r, s) is the ratio-class that contains the number 16/15. The members of thisratio-class are exactly the numbers of form (some-power-of-2)-times-(16/15).If p is any member of class r and q is any member of class s, then the ratio q/pis (some-power-of-2)-times-(16/15). In similar wise, int(s, t) is the ratio-classthat contains 21/20; the members of this class are exactly the numbers of form(some-power-of-2)-times-(21 /20).

If we allow irrational durations (or tempi with respect to the time-unit),we can consider the equivalence class u = (..., n/4, n/2, n, 2n, 4rc,...).Int(s, u) is the ratio-class that contains the number 3?r/8; the members of thisclass are exactly the numbers of form (some-power-of-2)-times-(37c/8).4

2.2.5 EXAMPLE: The musical space is a family of durations. Int(s, t) is thedifference (NB not the quotient) of time units between s and t: Int(s, t) =(t — s) units. So if r, s, and t are respectively 3, 4, and 8 units long, thenint(r, s) = (4 — 3)units = 1 unit, int(s, t) = (8 — 4)units = 4 units, andint(t,r) = (3 — 8)units = — 5 units. In the earlier system of quotients (2.2.3),the corresponding intervals would have been 4/3, 2, and 3/8.

4. Karlheinz Stockhausen argues the plausibility of system 2.2.4, with M = 2, in "... howtime passes ...," trans. C. Cardew, Die Reihe (English version) vol. 3, pp. 10-40. He also arguesthere the interrelatedness of his rhythmic system with traditional pitch systems. A clear view ofhow Stockhausen uses these ideas for the tempi of Gruppen (1955-57) is provided by JonathanHarvey, The Music of Stockhausen (Berkeley and Los Angeles: University of California Press,1975), 55-76.24


The additive system now under study measures intervals in units of time;the earlier system of quotients measured intervals as pure numbers (ratios).The difference between the systems here becomes striking if we set the timeunit as "one sixteenth note." Then the durations r = 3, s = 4, and t = 8 can besymbolized respectively by a dotted eighth, a quarter note, and a half note.The additive intervals int(r, s), int(s, t), and int(t, r), computed above as 1 unit4 units, and — 5 units, can be expressed as "plus a sixteenth," "plus a quarter,"and "minus a-quarter-tied-to-a-sixteenth." The corresponding multiplicativeintervals are simply the numbers 4/3, 2, and 3/8, numbers that express ratiosinvolving the durations.

2.2.6 EXAMPLE: To simplify matters, we restrict our attention to the dura-tions of 2.2.5 that are exactly the positive integral multiples of some basicsmall duration, which we take as the temporal unit. We wrap these durationsaround an M-hour clock, accordingly reducing the system to a modularsystem. The modular space comprises M duration-classes: Two durationsbelong to the same duration-class if their lengths differ by some integralmultiple of M. The interval between duration classes s-units-mod-M and t-units-mod-M is (t — s)-units-mod-M. t — s is the number of hours clockwisefrom s to t on the M-hour clock. The duration t is int(s, t) units longer than s,give or take any number of M-unit "measures".

For example take M = 16; take s = 8 and t = 4 units mod 16. Thenint(s,t) = 4 — 8= — 4 = 12 units mod 16. If we represent the unit as a six-teenth note, then the M-unit "measure" lasts a whole note. The duration-classs = 8 is represented by a half note, give or take any number of whole notes tiedon. The duration-class t = 4 is represented by a quarter note, give or take thesame. The interval int(s, t) = 12 is represented by "plus a dotted half," give ortake the same. Our arithmetic mod 16 above reflects this observation: Aquarter note, tied to an extra whole note for free, is a dotted half longer than ahalf note.

We have now explored six rhythmic spaces as well as six tonal ones. Tothe extent we intuit these spaces, we intuit "intervals" in connection withthem. Later on we shall explore yet other spaces, including some morerhythmic ones and some timbral ones. At this point, though, it will be helpfulto stop and develop some formal generalities.

All of the examples in this chapter so far have certain structural featuresin common. Foremost among these is our intuition in each case of a group,explicitly or implicitly defined, within which the intervals lie. If i and j areintervals (characteristic measurements, distances, motions, or the like) weintuit being able to compose them in some characteristic way (e.g. by addition,addition mod 12, multiplication, multiplication mod powers of 2, concatena- 25


tion of moves on a game board, and so on). And we intuit the composition ijof the intervals i and j to be itself an interval of the system (characteristicmeasurement, distance, motion, and the like). Indeed, we intuit that for anyelements r, s, and t of the musical space, the interval-from-r-to-s composeswith the interval-from-s-to-t to yield the interval-from-r-to-t. Symbolically:int(r, s)int(s, t) = int(r, t).

We intuit the composition of intervals to be associative: i(jk) = (ij)k. Weintuit an identity interval e, that composes with any interval j to yield j: ej =je = j. Indeed, we intuit that each object s of the space lies the identity inter-val from itself: int(s, s) = e.

We intuit that each interval has an "inverse interval" in the sense ofmeasurement, distance or motion: i""1 measures, extends, or moves in thereverse sense from i. We intuit that this intuitive inverse is also a group inverse:'i-1i = ii"1 = e. Indeed, we intuit that if i is the interval from s to t, then i"1 willbe the interval from t to s. Symbolically: int(t, s) = int(s, t)"1.

We can collate all these intuitions to construct a formal generalizedsystem. As we shall see, all our examples so far with one exception suggestspecific instances of the generalized system.

2.3.1 DEFINITION: A Generalized Interval System (GIS) is an ordered triple(S, IVLS, int), where S, the space of the GIS, is a family of elements, IVLS, thegroup of intervals for the GIS, is a mathematical group, and int is a functionmapping S x S into IVLS, all subject to the two conditions (A) and (B)following.

(A): For all r, s, and t in S, int(r, s)int(s, t) = int(r, t).(B): For every s in S and every i in IVLS, there is a unique t in S whichlies the interval i from s, that is a unique t which satisfies the equationint(s, t) = i.Condition (B) of the definition is a new idea. We shall discuss it shortly.

Condition (A) has already been discussed. But what about the other equationsinvolving the function int, equations we also discussed above? Should we notalso stipulate these other equations in defining a GIS? It turns out that we donot have to, because they are logically implied by the group structure andCondition (A). We demonstrate that in the form of a theorem.

2.3.2 THEOREM: In any GIS, int(s, s) = e and int(t, s) = int(s, t)"1 for every sand t in S.

Proof: int(s, s)int(s, s) = int(s, s), via Condition (A). Multiply both sides ofthat equation by int(s, s)"1; we obtain int(s, s) = e as asserted.

int(s, t)int(t, s) = int(s, s) via Condition (A). We have just provedthat int(s, s) = e; hence int(s, t)int(t, s) = e. Multiply both sides of thatequation on the left by int(s, t)"1; we obtain int(t, s) = int(s, t)"1 as asserted.26


Now let us turn our attention to Condition (B) of the definition. Essen-tially, it guarantees that the space S is large enough to contain all the elementswe could conceive of in theory. The idea is: If we can conceive of an element sand if we can conceive of a characteristic measurement, distance, or motion i,then we can conceive of an element t which lies the interval i from s. In certainspecific cases, application of this idea may require enlarging practical familiesof musical elements, to become larger formal spaces that are theoreticallyconceivable while musically impractical. For instance, we shall need to con-ceive supersonic and subsonic "pitches" in order to accommodate the idea ofbeing able to go up or down one scale degree from any note, in connection withexample 2.1.1. Figure 2.2 affords another good example: Obviously no finitemusical context can explore the entire extent of this map, which accommo-dates the idea of being able to conceive the dominant, mediant, subdominant,and submediant of any pitch class.

This is the methodological point: We must conceive the formal space of aGIS as a space of theoretical potentialities, rather than as a compendium ofmusical practicalities. In a specific compositional or theoretical context, thespace S of a GIS might be perfectly accessible in practice. Such is the case, forexample, with the "twelve-tone" GIS pertaining to example 2.1.3: Every oneof its twelve pitch-classes is easily referenced by any pertinent music. On theother hand, in other compositional or theoretical contexts, the space S of aGIS might be pertinent as an entirety only to the extent it is suggested orimplied by the actually stated musical material, plus the characteristic re-lationships actually employed. In just this way a painting or statue mightsuggest or imply the entire extension of Euclidean two-or-three-dimensionalspace, or some other geometrical space. (I am thinking in particular of theparabolic space in some of Van Gogh's late work.)

Let us consider figure 2.2 yet again in this connection. In order to conceivethe extension of the entire map, we need only three things: one (tonic) place onthe map, the characteristic idea of a "just dominant" relation involving pitchclasses, and the characteristic idea of an independent "just mediant" relation.In other words, we need only the pitch classes and the intervals we can inferfrom one tonic triad (!) in order to generate the entire conceptual group ofintervals, and thereby to infer the conceptual extension of the entire map, as aterrain within which a particular composition or theory may occupy someparticular region.

Another feature of Condition (B) also requires discussion. Given s and i,the condition demands not just some t that satisfies int(s, t) = i, but a uniquesuch t. We might consider weakening the condition, replacing the words "aunique," where they appear in 2.3.1(B), with the word "some." Let us call theweakened condition "(weak B)". Could we gain even greater generality byusing (weak B) instead of (B)? Not really. Under condition (weak B), the spaceS would be partitioned into equivalence classes: s and s' would be equivalent ifand only if int(s, s') = e. Given s' equivalent to s and t' equivalent to t, it would 27

2.4 Generalized Interval Systems (1)

be true that int(s', t') = int(s, t). We could thus think of the intervals as beingfrom one equivalence class to another. We could replace S by the quotientfamily S/EQUIV, the family of equivalence classes, and obtain a GIS thereby.(That is, Condition (B) would apply to the function int on equivalenceclasses.) It is hard to see what we could possibly want to do with S that wecould not do as well or better with the reduced space S/EQUIV of equivalenceclasses.

2.4 At this point, let us briefly review all our examples, noting the variousGIS structures which they suggest. 2.1.1 suggests a GIS in which S is theindicated gamut, extended indefinitely both up and down. In this GIS, IVLS isthe group of integers under addition and int(s, t) is the number of steps upfrom s to t. — i steps up is i steps down. The reader may verify that Conditions(A) and (B) of Definition 2.3.1 are satisfied. As we observed informally duringthe earlier discussion of this example, the non-traditional numbering of thescalar intervals is necessary so that the algebra of Condition (A) can obtain.

The GIS suggested by example 2.1.2 consists of the indicated space S, achromatic scale extended indefinitely up and down, the group IVLS = in-tegers under addition, and the function int(s, t) = number of semitones upfrom s to t, not counting s. Again, — i up is i down. The reader may verify thatConditions (A) and (B) of 2.3.1 are satisfied.

The GIS for example 2.1.3 consists of the space S = the twelve pitchclasses as indicated, the group IVLS = the integers under addition modulo 12,and the function int(s, t) = number of hours clockwise from s to t on a 12-hourclock. The GIS for example 2.1.4 consists of the space S = the seven modedegrees as indicated, the group IVLS = the integers under addition modulo 7,and the function int(s, t) = number of hours clockwise from s to t on a 7-hourclock. The GIS for example 2.1.5 has for its space S the extended family of all"pitches" conceptually available from a given pitch using just intonation. Thegroup IVLS here is the multiplicative group comprising all rational numbersthat can be expressed in the form 2a3b5c, where a, b, and c are integers. Thefunction int here is the quotient int(s, t) = FQ(t)/FQ(s).

To conceive the GIS for example 2.1.6, we take as our space S the "gameboard" of figure 2.2. IVLS here is the group of ordered pairs of integers (b,c)under componentwise addition: (b^c^ + (b2,c2) = (bt + b2,C! + c2).(IVLS is thus the direct product of the-integers-under-addition with itself.)The function int is as discussed: int(s, t) = (b, c), where t lies b squares east andc squares north of s on the game board.

The GIS for example 2.2.1 has as its space S the indicated succession oftime points, conceptually extending indefinitely both backwards and for-wards in time. IVLS here is the integers under addition, and int(s, t) is thenumber of time units by which t is later than s, — i later meaning i earlier. Forexample 2.2.2, the GIS consists of S = the N beat-classes, IVLS = the integers28

Generalized Interval Systems (1) 2.4

under addition modulo N, and int(s, t) = the number of hours clockwise thatt lies from s on an N-hour clock.

The GIS for example 2.2.3 depends upon just what proportions amongdurations we wish to allow. IVLS in any case will be some multiplicative groupof positive numbers. If we allow only the basic proportions of 2 and 3, thenIVLS will comprise those numbers of form 2a3b, where a and b are integers; Swill then comprise exactly such durations as are in those proportions to anygiven duration. S is conceptually extended to allow indefinitely short andindefinitely long "durations." If we allow basic rhythmic proportions of 5 and7, as well as 2 and 3, then IVLS will comprise all numbers of form 2a3b5c7d,and S will contain the corresponding extra members. Or we might take, asbasic rhythmic proportions, the square root of 2 (2(1/2)) and the square root of3 (3(1/2)); then IVLS will contain exactly those numbers of form 2(a/2)3(b/2),where a and b are integers, and S will contain the corresponding durations inproportion to any one given duration. And so on. In each case, the functionint(s, t) is given by the quotient t/s = (t units)/(s units).

When we reduce the space of 2.2.3 to the space of 2.2.4 by the modulus M,we also reduce the group of intervals for 2.2.3 to a quotient group, the newgroup of intervals for 2.2.4. Specifically, intervals i' and i for 2.2.3 are declaredcongruent if one is the other multiplied by some integral power of M: i' = iMa.This relation is indeed a congruence in the group-theoretic sense (1.10.1): Itis an equivalence relation, and if j' is congruent to j (f =jMb), then i'j'is congruent to ij (i'j' = (ij)M(a+b)). The congruence generates a quotientgroup, imposing a group structure on the family of congruence-classes. Thosecongruence-classes are the "ratio-classes" of example 2.2.4. The function intfor 2.2.4 can be derived mathematically from the function int for 2.2.3, alongwith the declared equivalence and congruence relations on the space andgroup of 2.2.3. We shall explore this derivation formally in chapter 3, wherewe shall study the general construction of a "quotient GIS" from a given GISand a given congruence on its group of intervals.

Example 2.2.5, exceptionally, does not lead at once to a GIS structure. Ifwe try to find a pertinent GIS, we will take IVLS to be some additive group ofnumbers i, j, ..., for as durations s and t vary in the family S of durations, tmay be ±i, ±j, . . . units longer or shorter than s. And int(s, t) = (t — s)unitsBut S, IVLS, and int here cannot satisfy Condition (B) of Definition 2.3.1. Forinstance, try s = 3 units and i = — 8 units; then there is no duration t in Ssatisfying int(s, t) = i. If there were such a t, then (t — s)units = (t — 3)unitswould have to equal i units = — 8 units, t — 3 would equal — 8, and t would be— 5 units in duration. But S does not contain "negative durations," and failingsome convention not yet specified, it is not clear what intuition we couldpossibly be modeling, when we stipulate a duration t that lasts not only lessthan no time at all, but also measurably less than no time at all. The modelingproblem here is different in kind from those involved in other spatial exten- 29

2.4 Generalized In terval Systems (1)

sions we have made conceptually. E.g., while we cannot hear a "pitch" of .001Hertz fundamental, or of five trillion Hz fundamental, we can conceive suchpitches. That is, we are at ease with the notion of periodic vibration at thoserates, and we can imagine that other creatures might be sensibly aware ofthem. In the same sense, we can conceive indefinitely short and indefinitelylong "durations," and we can conceive "time points" that lie indefinitely farback in the past or ahead in the future. But we can not conceive, in such a sense,a duration lasting precisely 5 units less than no time at all, which is therebyprecisely 2 units longer than a duration lasting precisely 7 units less than notime at all. So example 2.2.5 does not lead to a GIS.

Example 2.2.6 can be regarded as one means of salvaging example 2.2.5 inthis connection, by providing a convention that attaches meaning to theconcept of a negative duration-class. E.g. we can think of duration-class" — 5" as that class containing all durations lasting just 5 units less than somemultiple of the modulus duration. " — 5" thus means the same as "M — 5," orthe same as " — 5, modulo M." Example 2.2.6 has a GIS structure. The space Scomprises the M duration-classes. The group IVLS is the additive group ofintegers modulo M. int(s, t) is the number of hours t lies clockwise from s whenthe duration-classes are wrapped around an M-hour clock. The reader canverify that (S, IVLS, int) is indeed a GIS. In particular, Condition (B) of thedefinition, which failed for example 2.2.5, obtains for example 2.2.6.

30

Generalized Interval Systems

There is a familiar convention whereby the twelve pitch classes in equaltemperament are labeled by their intervals from a referential pitch class C. C,C#, D, ..., Bb, B are thereby labeled 0, 1, 2, ..., 10, 11 (mod 12). Thisconvention can be generalized. That is, in any GIS we can always use theintervals of the group IVLS to label the members of the space S by theirrespective intervals from an assumed referential object in S. We can make thenotion formal by introducing some new terminology.

3.1.1 DEFINITION: Given a GIS (S, IVLS, int) and a fixed referential member"ref" of S, the function LABEL, mapping S into IVLS, is defined by theequation

LABEL(s) = int(ref, s).

3.1.2 THEOREM: Whatever the element ref, the function LABEL maps S1-to-l onto IVLS, and it satisfies the formula

int(s,t) = LABEL(s)~1LABEL(t).

Proof: Given the element ref and any interval i, there is one and only one sin S satisfying int(ref,s) = i, by Condition (B) of Definition 2.3.1 for a GIS.Since int(ref, s) is LABEL(s), we have observed the following: Given any i,there is some s satisfying LABEL (s) = i; furthermore, there is only one such s.Thus the function LABEL is onto; furthermore it is 1-to-l.

Now we prove the formula of the theorem. By definition 3.1.1,LABEL(s)"1LABEL(t) = int(ref,s)~1int(ref,t). In that equation we cansubstitute int(ref, s)"1 = int(s, ref), via 2.3.2. The equation then states:LABEL(s)~1LABEL(t) = int(s, ref)int(ref, t). And the expression on the

3(2): Formal Features

31


right side of that equation equals int(s, t) as desired, via Condition (A) ofDefinition 2.3.1. q.e.d.

The LABEL function can be very useful, particularly for computationsinvolving members of S. On the other hand, its use in some contexts can beproblematic, both conceptually and computationally. Conceptually, theremay not always be adequate musical reasons for assigning a special referentialstatus to ref. Why, for example, should I assign this status to the pitch class C apriori? Perhaps, in a certain context, I hear no referential pitch class at all, butrather notes related only to each other, and not to any one given note. Orperhaps I hear E as referential in this piece; then why should I call E "4" in a C-LABELing system, rather than "0" in an E-LABELing system? But theviolins and string basses have actually decided how E will sound by tuningtheir Es a certain interval from A; then would it not be methodologically mostaccurate to use an A-LABELing system? And so on. Some of the conceptualissues will be familiar from similar issues in fixed-do and movable-do systemsof solfege. Conceptual difficulties aside, computations themselves can bemuddled, when we use a LABELing system, by the algebraic influence ofirrelevant intervals, intervals arising from irrelevant relations of ref to theobjects s, t , . . . , whose interrelations we actually wish to compute.1

Now we shall explore some ways in which we can formally construct newGIS structures from old. We have observed one such construction usedinformally again and again during our survey of examples 2.1 and 2.2 earlier;that is the construction of a "quotient GIS" from a given GIS and a stipulatedcongruence relation on IVLS. Whenever we spoke informally about "modu-larizing" one system to obtain another, we invoked such a notion. Soon weshall make the notion formal. By way of preparation before doing so, we shallexamine a bit more carefully just how it applies in one specific and familiarcase.

We consider for this purpose how the "chromatic scale" GIS of 2.1.2gives rise to the "twelve-tone" GIS of 2.1.3. Let us call the two systemsrespectively GISj = (SÎVLSîntO and GIS2 = (S2,IVLS2,int2). St is achromatic scale extended conceptually up and down indefinitely. IVLSi isthe integers under addition. int: (s, t) is the number of semitones up from s to t.When "modularizing" GISX to obtain GIS2, we use a certain congruence onthe group IVLSi. Intervals i and i' are congruent if i' = i plus or minus someintegral multiple of 12. This relation is indeed a congruence in the group-theoretic sense of 1.10.1. That is, it is an equivalence relation; furthermore,i' + j' is congruent to i + j whenever i' is congruent to i and j' is congruent to

1. I discuss these matters further in "A Label-Free Development for 12-Pitch-ClassSystems," Journal of Music Theory vol. 21, no. 1 (Spring 1977), 29-48.32


j. As we saw in chapter 1, the congruence gives rise to a quotient groupIVLSJCONG. The elements of this group are the congruence classes,and the binary combination of these elements in the quotient group is welldefined (!) by the formula (class-containing-i) 4- (class-containing-j) = (class-containing-(i + j)). In this case the quotient group is the integers modulo 12.And that is IVLS2, the group of intervals for GIS2.

In modularizing GlSi to GIS2 we also invoked a certain equivalencerelation on S^ Pitches s and s' were declared equivalent if they differed bysome integral number of octaves. The condition for equivalence of s and s' canbe expressed by using the congruence relation on IVLSj (and that is impor-tant): s and s' are equivalent as defined if and only if int^s, s') is divisible by12, which is the case if and only if intl (s, s') is congruent to the identity intervaloflVLSj.

The equivalence relation reduces the family S! of pitches into the familyS2 of pitch classes. S2 = S1/EQUIV: The 12 pitch classes are precisely the 12equivalence classes of pitches, under the constructed equivalence relation.

Finally, in modularizing GlSt to GIS2, we implicitly invoked a signifi-cant interrelation between the congruence on IVLS^ and the equivalence relationon Sv: If pitches s and s' are equivalent (different by some number of octaves),and if pitches t and t' are also equivalent, then the intervals int^s.t) andint^s'jt') are congruent (different as integers by some multiple of 12). Wecan put this another way, significant for our purposes: The congruence class towhich intj(s, t) belongs depends only on the equivalence class to which sbelongs and the equivalence class to which t belongs, not on any specific s' andt' chosen to represent those equivalence classes.

This feature of the situation enables us to see how the function int2 worksfor our present example: Given pitch classes (equivalence classes) p andq, int2(p, q) in IVLS2 is the congruence class (interval mod 12) to whichint1(s, t) belongs, whenever s and t are any pitches belonging to the pitchclasses (equivalence classes) p and q respectively. For example, let p and qbe the pitch classes C and F. If s and t are middle C and Queen-of-the-NightF, then int^s.t) = 29 (semitones). If s' and t' are high C and contra F,then intjCs'.t') = -55 (semitones). The integers 29 = 24 + 5 and -55 =— 60 + 5 belong to the same congruence class mod 12, "congruence-class 5."If s" and t" are any other pitches belonging to p and q (i.e. named C and F),then intj (s", t") also belongs to congruence-class 5. That is what gives rigorousmeaning to our saying "int2(C, F) = 5." Otherwise, our clock-face model forGIS2 would remain only arbitrarily or vaguely related to GISj.

Indeed, we could use the relationships under present discussion to defineint2'. int2(p,q) is that unique congruence class which contains any and allvalues of inti(s, t), s being any member of p and t being any member of q.

Everything noted about OK^ and GIS2 in the example just studied can begeneralized so as to define a "quotient GIS," given any GIS and any con- 33


gruence relation on its group of intervals. The work of section 3.2 followingwill be devoted to this generalization.

3.2.1 THEOREM AND DEFINITION: Let (S, IVLS, int) be a CIS; let CONG be acongruence on the group IVLS. Then an equivalence relation EQUIV isinduced on S by declaring s and s' to be equivalent whenever int(s, s') iscongruent to the identity e in IVLS. EQUIV will be called the inducedequivalence on S.

Proof: int(s, s) = e, so EQUIV as defined is reflexive. If int(s, s') iscongruent to e, then int(s', s) = int(s, s')"1 is congruent to e; thus EQUIV asdefined is symmetric.2 Finally, if int(s, s') and int(s', s") are both congruentto e, then int(s, s") = int(s, s')int(s', s") is congruent to e-times-e = e. ThusEQUIV is transitive.

3.2.2 LEMMA: Let (S, IVLS, int) be a GIS; let CONG be a congruence onIVLS. Then the following is true. Suppose s and s' are equivalent in S under theequivalence induced by CONG; suppose t and t' are also equivalent under thatequivalence relation; then int(s, t) and int(s', t') are congruent members ofIVLS.

Proof: int(s,t) = int(s,s')int(s',t')int(t',t). Now both int(s,s') andint(t', t), by supposition, are congruent to e. Hence int(s, t) is congruent toe • int(s', t') • e. That is, int(s, t) is congruent to int(s', t'). q.e.d.

3.2.3 THEOREM: Let (SÎVLSînti) be a GIS. Let CONG be any con-gruence on IVLSj; let EQUIV be the induced equivalence relation on S^Let S2 be the quotient space Sj/EQUIV, the family of equivalence classeswithin §! under the induced equivalence. Let IVLS2 be the quotient groupIVLS ! /CONG, whose members are the congruence classes within IVLS^

Then a function int2 from S2 x S2 into IVLS2 is well defined by thefollowing method: Given equivalence classes p and q (members of S2), thevalue int2(p, q) is that congruence class (member of IVLS2) to which int1(s, t)belongs, whenever s and t are members of p and q respectively.

Furthermore, (S2,IVLS2,int2) is itself a GIS.Proof (optional): Lemma 3.2.2 assures us that int2 is well defined by the

indicated procedure: Given p and q, if s and s' are any members of p, and if tand t' are any members of q, then int^s', t') belongs to the same congruenceclass as int^s, t). That congruence class thus depends only on the equivalenceclasses p and q, and not on the particular s-and-t, or s'-and-t', which we choose

2. The reader may prove as an exercise the necessary lemma: If x is congruent to e in agroup, then x"1 is also congruent to e. After that, prove this: If x is congruent to y, then x"1 is con-gruent to y-1. (Hint: x"1 y is congruent to x"1 x; apply the preceding lemma. Alternatively, but lesselegantly, one can bludgeon out all the desired results as corollaries of 1.11, 1.12.1, and relatedresults already established.)34


to represent p and q. And it is that congruence class which is thus well defined,asint2(p,q).

We now prove that (S2, IVLS2,int2) is a GIS. To do so, we must showthat Conditions (A) and (B) of Definition 2.3.1 are satisfied. We show firstthat (A) is satisfied. Given equivalence classes o, p, and q in S2, we want toshow that int2(o, p)int2(p, q) = int2(o, q). Let r, s, and t be elements of Sj thatare members of classes o, p, and q respectively. The congruence class contain-ing inti(r, s) combines in the quotient group with the congruence class con-taining int^s, t), to yield the congruence class containing int^r, s)int1(s, t)(1.11). And, since (SÎVLSîntJ is a GIS, int^r, 8)111̂ (8,1)1 = int^M).Hence the congruence class containing int^r, s) combines in the quotientgroup with the congruence class containing int^s, t), to yield the congruenceclass containing intj(r, t). Or, by the definition of int2 in the theorem,int2(o,p)int2(p,q) = int2(o,q), as desired. So the system (S2,IVLS2,int2)satisfies Condition (A) of 2.3.1.

It remains to show that the system also satisfies Condition (B) of thatdefinition. Given any equivalence class p (member of S2) and any congruenceclass J (member of IVLS2), we must show that there is a unique equivalenceclass q that satisfies the equation int2(p, q) = J. Let s be a member of the givenp; let j be a member of the given J. Since (S:, IVLSi, intl) is a GIS, we can finda (unique) t in Sj which satisfies the equation intt(s, t) = j. Let q be theequivalence class containing this t. q is that class (member of S2) for which weare searching. First of all, q satisfies the equation int2(p, q) = J. That is sobecause int^s, t) = j, s belongs to p, t belongs to q, and j belongs to J; int2 wasdefined precisely so as to make this happen. Furthermore, q is a uniquesolution for the equation int2(p, q) = J. To see this, let us suppose thatint2(p,q') = J; we shall show that q' must equal q. Let t' be a member of q';then by the nature of int2, in^ (s, t') lies in the congruence-class J. So intj (s, t')is congruent to int^s, t). Thence, applying the second lemma of footnote 2,we infer that int^s, t')"1 is congruent to int^s, t)"1. Or: int^t', s) is congruentto intt(t,s). Then int^t', s^nt^s,t) is congruent to int^t,s)int1(s,t). Or:inti(t', t) is congruent to int^t, t) = e. But then t' is equivalent to t under theinduced equivalence relation: The interval between them is congruent to e.Hence t' and t lie in the same equivalence class; that is, q' = q as asserted,q.e.d.

3.2.4 DEFINITION: Given the situation as in 3.2.3, the GIS(S2,IVLS2,int2)will be called the quotient GIS of (SÎVLSîntJ modulo CONG. We writeGIS2 = GIS1/CONG.

Thus, to review yet once more the specific example discussed just preced-ing 3.2.1, let (Sl5 IVLS15 intx) be the GIS such that St is an infinite chromaticscale, IVLSj is the additive group of integers, and intx(s, t) is the number of 35


semitones up from s to t. Let CONG be the relation on IVLSj that makes icongruent to i' when the intervals differ by any integral multiple of 12 semi-tones. Then the quotient GIS(S2, IVLS2, intj), constructed by the method ofsection 3.2, has these components: S2 is the family of twelve pitch-classes,IVLS2 is the integers modulo 12, and int^p, q) is the reduction modulo 12of the integer intj(s, t), where s and t are any pitches belonging to the pitchclasses p and q respectively.

Here is another specific example. S, is the family of all pitches that can beconceptually generated from a given pitch using just intonation. IVLSj is thegroup under multiplication of all rational numbers that can be written inthe form 2a3b5c, where a, b, and c are integers. int,(s, t) is the fundamentalfrequency of t divided by the fundamental frequency of s. (Sp FVLSj, intj)is a GIS. It was studied earlier in example 2.1.5. When we informally "modula-rized" that GIS to obtain the "modular harmonic space" of example 2.1.6,we were actually constructing a quotient GIS, using a certain congruence rela-tion on IVLSj. Specifically, we declared intervals i and i' within IVLSj to becongruent if i' was some-power-of-2 times i. As we noted earlier (in example1.10.4.2), this relation is indeed a congruence on IVLSj: It is an equivalencerelation; further, if I' is some-power-of-2 times I and j' is some-power-of-2times j, then i'j' is some-power-of-2 times ij.

Using this congruence, we can note how the formal constructions of sec-tion 3.2 go through for the GIS at hand, producing the GIS for example 2.1.6as a quotient GIS. By Definition 3.2.1, the induced equivalence on Sj makess equivalent to s' if and only if intj(s,s') is some-power-of-2. Thus s and s'enjoy the induced equivalence relation here if and only if the pitches lie somenumber of octaves apart. The equivalence classes (members of S2) are thus pre-cisely the pitch classes determined by the pitches of S,. And, given pitch classesp and q, int^p^) in the quotient group is well defined as the congruence classto which intj(s, t) belongs, whenever s and t are members of p and q. If theintervals, i, i', i", . .. all belong to this congruence class, then we can writei = 2a3b5c, i' = 2a'3b5c, i" = 2a"3b5c, . .. and so on. So we may identify thecongruence class with the number-pair (b, c). When s and t are members of pand q, then t, give or take some number of octaves, lies b twelfths and c major-seventeenths from s. Taking some octaves, we can say that t, give or take somenumber of octaves, lies b fifths and c major-thirds from s. Returning our at-tention to the pitch classes p and q, of which s and t are members, we cansay that q lies b dominants and c mediants from p. That is, on the game boardof figure 2.2, q lies b steps east and c steps north from p. The quotient GIShere is thus the GIS associated with that game board in example 2.1.6.

As a further specific example, the reader can work out how the formalconstructions of section 3.2 apply to the "diatonic scale" GIS associated withexample 2.1.1, together with the congruence which collects octave-related in-tervals (those here which differ by multiples of 7 scale steps), to give rise to a36


quotient GIS which is in fact the GIS of example 2.1.4, a GIS of seven scale-degrees. In similar fashion the GIS associated with the time-point space ofexample 2.2.1, together with the congruence which collects time intervalsdiffering by multiples of "N beats," gives rise to a quotient GIS which is in factthe GIS of example 2.2.2, the GIS associated with the space of N beat-classes.Likewise, the GIS of example 2.2.3, together with a suitable congruence, givesrise to the GIS of example 2.2.4 as a quotient GIS; the discussion of example2.2.4 may be reviewed in this connection.

The reader may also review further the discussion of example 2.1.6, to seehow the GIS of just pitch-classes, corresponding to the game board of figure2.2, can give rise to a quotient GIS whose space is an infinite one-dimensionalchain of dominant-related pitch classes in quarter-comma mean-tone tem-perament. The congruence at issue declares two "moves" (b, c) and (b', c') onthe game board of figure 2.2 to be congruent if there is some integer N suchthat b' = b + 4N while c' = c — N, that is, if b' steps east is the same as b stepseast plus 4N steps east, while c' steps north is the same as c steps north andthen N steps south. ( — N east, north, etc. means N west, south, etc.) If we takeb = 0, c = 0, and N = 1 in the above arithmetic, then b' = 4 and c' = — 1.Thus going four squares east and one square south on the game board is amove congruent to "staying still." The reader may verify that the definedrelation is a congruence, and that the induced equivalence relation on thegame board renders equivalent just those pitch classes that share the sameletter-name (differing, if at all, only by subscript).

Each of the specific examples above instances, in one setting or another,the general and abstract relation of a given GIS, modulo a given congruence,to a quotient GIS. We can see thereby how ubiquitous the quotient construc-tion really is, as a method of generating new GIS structures from old. Anotheruseful method is the construction of a "direct-product GIS." Before proceed-ing to an abstract discussion of that idea, let us study some examples of it.

3.3.1 EXAMPLE: Let GISj be the GIS of example 2.1.3: Sj is the twelveequally-tempered pitch classes; IVLSi is the integers modulo 12; int^p, q) isthe number of semitones up, mod 12, from any pitch within class p to any pitchwithin class q. Let GIS2 be the GIS of example 2.2.1: S2 is an indefinite seriesof equally spaced time-points; IVLS2 is the integers under addition; int2(s, t) isthe number of beats from time-point s to time-point t.

We construct a new GIS, GIS3, as follows. S3 is the Cartesian productof Sx and S2, that is, S3 = Si x S2. S3 is thus the family of ordered pairs(p, s), where p is a pitch class in Sj and s is a time point in S2. IVLS3 is thedirect-product group of the groups IVLS! and IVLS2; that is, IVLS3 =IVLSj (g) IVLS2. A sample member of IVLS3 thus has the form (ij, i2), wherei j is an integer mod 12 (member of IVLSj) and i2 is an integer (member ofIVLS2). The members of IVLS3 combine under the law of 1.13 for direct- 37


product groups: (ilt\2) + (JiJ2) = Oi + J i , i 2 + J2>; here the sum h + Ji iscomputed mod 12 in IVLSl9 while the sum i2 + J2 is the ordinary integersum in IVLS2. The function int3 works as follows. Given two members(p, s) and (q, t) of S3, each a pitch-class-cum-time-point pair, the intervalint3((p, s),(q, t)) is taken as the pair (int1(p,q),int2(s,t)) within the groupIVLS1(g)IVLS2 = IVLS3.

(S3, IVLS3, int3) as constructed is in fact a GIS. The reader may take thison faith or verify it as an exercise. Later on we shall call this GIS the directproduct GIS of GlSi and GIS2, writing GIS3 = GISj ® GIS2. Right now, letus explore what GIS3 models.

One sample member of S3 is the pair (C#, 35), which models a reference topitch class C# at time 35. Another member of S3 is the pair (F,46), whichmodels a reference to pitch class F at time 46. The interval between the twocited objects is (int^Cft, F),int2(35,46)) = (4,11). The compound interval(4,11) models the spanning of pitch-class interval 4 between events happeningat time-points 11 beats apart.

FIGURE 3.1

38

Figure 3.1 will be used to suggest the relevance of GIS3 for musicalanalysis and theory. It transcribes the opening of the third movement fromWebern's Piano Variations op. 27. The brackets on the figure display someGIS3-intervals of interest, using the written quarter note as the temporal unit.Temporal intervals between time-points are calculated between attacks of thenotes. Thus the bracket from E|? to D on figure 3.1 is labeled by the GIS3-interval (11,5): A pitch-class interval of 11 is spanned between the attack ofthe Eb and the attack of the D, 5 beats later. Similarly, the bracket on thefigure between D and C# is labeled by the GIS3-interval (11, 5): A pitch-classinterval of 11 is again spanned between two time-points 5 beats apart, herebetween the attack of the D and the attack of the C#. The GIS3-interval (11,1)labels the brackets extending from B to B[? attacks, from C# to C attacks, andfrom A to G# attacks; in each case, the pitch-class interval 11 is spannedbetween events happening 1 beat apart.

The recurrences of GIS3-intervals on figure 3.1 are of analytic interest,


for not many GIS3-intervals occur more than once in this passage. The re-current GIS3-interval (11, 1) is associated with a quiet slurred figure in theleft hand which we shall call "the accompaniment figure"; this figure occursin the music at B-B|> and again A-G#. However the musical presentation ofC#-C, also associated with the GIS3 interval (11, 1), does not project the ac-companiment figure; rather, it is loud and staccato. The recurrence of theGIS3 interval (11, 1) on figure 3.1 imbues the pitch-class interval 11 with aspecial function: As the passage unfolds in time, pitch-class interval 11 be-comes bound up with defining the beat. That is, pitch-class interval 11 re-curs significantly in connection with beat-interval 1, forming the GIS3-inter-val (11, 1). And no other pitch-class interval recurs in conjunction withbeat-interval 1.

This special beat-defining function for pitch-class interval 11 gives specialmeaning in turn to the recurrence of GIS3-interval (11, 5) on figure 3.1, aGIS3-interval which also involves pitch-class interval 11. Via the recurrentGIS3-interval (11, 5), the temporal interval of 5-beats-later becomes associ-ated with the pitch-class interval 11, a pitch-class interval of special mensu-ral status. "5 beats later" thereby acquires a special mensural status itself,linking it with "1 beat later" via the pitch-class interval 11 that is shared bythe recurrent GIS3-intervals (11, 5) and (11, 1). This special mensural statusfor "5 beats later" is not the only reason many analysts hear the music "in *meter," but it does endow the possibility of such a hearing with a specialmeaning and thematic richness.3 I myself believe that pertinent statements in-volving GIS3-intervals provide a more exact and less problematic account ofthe mensural structures at issue here, then does the notion of "^ meter."

Figure 3.1 also shows the recurrence of the GIS3-interval (3, 2), spanningB-attack to D-attack and also E-attack to G-attack. The recurrent (3, 2) seemsto have a cadential function in the music, to my ear, "2 beats later," the "2"of the GIS3-interval (3, 2), engages the notated meter of the music, which "5beats later" does not. In these connections, the pitch-class interval 3, as it re-curs within the GIS3-interval (3, 2), has its own special function, to my eara cadential function. Hearing a cadence on the D and another on the G isconsistent with the maximal amount of time we wait, after the attack of the Dand again after the attack of the G, before hearing the next attack in each case.It is curious that the next attack after this maximal wait comes precisely "5beats later" in each case. Indeed if we start measuring spans of 5 beats starting

3. A \ "cross-meter" is heard as one of several contending temporal patterns by Edward TCone, "Analysis Today," Problems of Modem Music, ed. Paul Henry Lang (New York: W. W.Norton, 1962), p. 44. Elsewhere, we can read that the first three notes we hear form a rhythmicgroup which "is clearly *; the motion of the two quarter notes into the next downbeat defines themeter precisely." This is the hearing of James Rives Jones, "Some Aspects of Rhythm and Meterin Webem's Opus 27," Perspectives of New Music vol. 7, no. 1 (Fall-Winter 1968), p. 103. Later,still on page 103, he refers to "the \ meter" as "already ... established" by the cited criterion. 39

3.3.1 Generalized Interval System (2)

at the attack of the opening El., then the D and the G on figure 3.1, with therests that follow them, each fill one such span. I do not know what to makeof this. The idea of quintuple "perfections" seems a better metaphor for myhearing this aspect of the music than does the idea of quintuple "meter."

Figure 3.1 also shows the recurrent GIS3-interval (2, 7), interlocking themensural function of "7 beats later" with the pitch-class interval 2. The first(2, 7) recurrence links the two notes of the B-B[> accompaniment figure tothe corresponding two notes of the C|-C figure, 7 beats later. The secondrecurrence (third occurrence) of (2, 7) links the attack of the cadential G tothe beginning of the accompaniment figure A-G| 7 beats later. In sum, therecurrent GIS3-interval (2, 7) links aspects of the recurrent accompanimentfigure with other events of the music.

40

Figure 3.2 summarizes the discussion of figure 3.1 so far, in the form ofa table. It shows how the recurrence of GIS3-intervals gives special meaningsto the pitch-class intervals 11, 3, and 2, as those interrelate with the temporal

FIGURE 3.2


intervals 1,5,2, and 7, all in connection with various compositional features ofthe music.

Figure 3.3 applies to the opening of the passage a theoretical constructionsuggested by the temporal aspect of GIS3. The left-hand column of the figurelists the first six members of S3 in their order of appearance during the passage.First comes pitch-class Eb at time-point 0, instancing element (Eb,0) of S3.Next comes pitch-class B at time-point 3, instancing element (B, 3) of S3. Atthis time (i.e. just after time-point 3), we become aware of a 2-element S3-set,that is, the set ((Eb, 0), (B, 3)). The elements of this set form one GIS3-intervalwith a positive time-component, that is, the interval (8,3) from (Eb,0) to(B, 3). The GIS3-interval (8,3) is listed in the second row of the second columnin figure 3.3; that interval is the sole constituent within the interval vector ofthe 2-element S3-set.

Now the pitch class 8(7 occurs at time-point 4, providing the newS3 element (Bb,4) for the third row of column 1 on figure 3.3. At thistime (i.e. just after time-point 4), we become aware of a 3-element S3-set,((Eb,0),(B,3),(Bb,4)). Besides the GIS3-interval of (8,3) already listed inrow 2, the elements of the 3-member set produce new GIS3-intervals of (7,4)(= int3((Eb,0),(Bb,4))) and (11, !)(= int3((B,3),(Bb,4))). These new con-stituents for the interval vector of the expanded 3-element S3-set are listed inthe third row of column 2 on figure 3.3; the old interval vector expands toinclude the occurrences of the two new intervals.

When pitch class D enters at time-point 5, the 3-element S3-set expandsto a 4-element S3-set ((Eb,0),(B,3),(Bb,4),(D,5)), and the interval vectorof the 3-element set expands to adjoin new occurrences of the GIS3-intervals(11,5)(= int3((Eb,0),(D,5))), (3,2)(= int3((B,3),(D,5))), and (4,1)(= int3((Bb,4),(D, 5))). The new intervals are listed in the fourth row ofcolumn 2, on figure 3.3. At this time (i.e. just after time-point 5), we become 41

83-elements GIS3-interval vectors of S3 sets

FIGURE3.3


aware for the first time that some pitch-class intervals are predominating overothers, and that some temporal intervals are predominating over others, as wenote the various intervals going by. In earlier work I have suggested that ourbecoming aware of such predominances is associated with our marking such atime-point as an "ictus." 4 The present analysis supports that theoretical idea,since time-point 5 is both a notated barline, indeed the first notated barline,and also audible to some extent as marking the attack of an intuitively"strong" quarter.

These considerations lie behind my bracketing the first four entries ofcolumn 1 on figure 3.3, as belonging together in a special way. According tothe theory just sketched, it is only when the 4-element S3-set has been com-pletely exposed, i.e. it is at and only at a moment just after time-point 5, as welisten along, that we first become sensitive to any regular mensural structuringin the passage. Just after time-point 5 we become aware that temporal interval1 is predominating over other temporal intervals; we can then (and only then)hear temporal interval 1 as a beat with which to measure other temporalintervals. In that sense the GIS3-structure of figures 3.1-3.3 really "begins"for a listener at (and only at) time-point 5, the first written barline and the firstperceived ictus; any GIS3-structure preceding time-point 5, according to thistheory, is reconstructed^ a listener at (or following) time-point 5. Not only isthe beat established at time-point 5, the pitch-class interval 11 is also es-tablished at the same time, as a predominating pitch-class interval. Pitch-classinterval 11 is thus bound psychologically to the establishment of mensuralstructure in the piece, as part of one and the same Gestaltist experience that alistener will be having just after time-point 5.

Figure 3.4 symbolically collates the ideas discussed just above. The figurealso suggests how the temporal interval of "5 beats later" has already acquireda special significance at the moment the listener hears time-point 5. "5 beatslater," namely, spans the temporal distance from the opening attack to thefirst ictus. Figure 3.4 shows how that temporal distance is already clearlyassociated with the structural function of the recurrent pitch-class interval 11,

4. David Lewin, "Some Investigations into Foreground Rhythmic and Metric Pattern-ing," Music Theory, Special Topics, ed. Richmond Browne (New York: Academic Press, 1981),101-37.

FIGURE 3.4

42


the pitch-class interval between the opening Eb and the D of the first ictus, 5beats later. These considerations enable us to analyze with greater precisionhow the idea of "being in 4" might arise, and how it would become associatedwith the compositionally thematic E(?-D gesture.5

Let us now inspect the fifth row of figure 3.3, investigating how ourimpressions develop when C# is attacked at time-point 10, introducing newmanifestations of the GIS3-intervals (10,10), (2,7), (3,6), and (11,5). We nowhear a second pitch-class interval of 3, but simultaneously we also hear a thirdpitch-class interval of 11. The latter interval, by recurring yet again, continuesto predominate over other pitch-class intervals. Indeed, its predominationitself recurs. Furthermore, we now (i.e. just after time-point 10) hear for thefirst time the recurrence of a GIS3-interval (not just of a temporal interval orpitch-class interval). The recurring GIS3-interval is (11,5), recently discussedin connection with the possibility of asserting a "thematic | meter" listening at time-point 5. The sensations prompting such a possible assertionare thereby intensified at time-point 10. Time-point 10 is experienced as an"ictus" in the formal sense of the theory mentioned earlier. Dynamic andregistral accents at time-point 10, the loud low C# attack, support the possiblehearing of a "strong beat" there, should one want to assert "| meter" beyondpurely mensural considerations. We can then expand figure 3.4 to figure 3.5,which portrays a provisional impression one might have while listening justafter time-point 10.

Now let us turn to the sixth row of figure 3.3, investigating our im-pressions when C is attacked at time-point 11, introducing new manifestationsof the GIS3-intervals (9,11), (1,8), (2,7), (10,6), and (11,1). (11,1) here is arecurring GIS3-interval; it confirms our already developed sensations aboutthe beat-defining and other mensural functions of the pitch-class interval 11.The GIS3-interval (2,7) also recurs at time-point 11. This builds up another

5. My analysis of listener psychology just after time-point 5 partly elaborates, partly quali-fies, and partly takes issue with the thought-provoking approach to this passage by ChristopherHasty, in his important methodological and analytic study, "Rhythm in Post-Tonal Music: Pre-liminary Questions of Duration and Motion," Journal of Music Theory vol. 25, no. 2 (Fall 1981),183-216.

FIGURE 3.5

43


mensural matrix that tries to expropriate (11,1) for its own purposes, trying toput an ictus at time-point 11, rather than time-point 10. Figure 3.6 sketchesthis notion.

The ictus on C in figure 3.6 corresponds to a written barline; this was notthe case with the conflicting ictus on C# in figure 3.5. That C# was 5 beats afterthe D-ictus of figure 3.4; the C of figure 3.6, which picks up the HauptstimmeD in register, is 6 beats after the D-ictus. The mensural conflict of "5 unitsafter" (D-ictus to C#-ictus) and "6 units after" (D-ictus to C-ictus) is highlythematic in op. 27 as a whole.6

The mensural reading of figure 3.6 tries to associate the C#-C event in themusic with the "accompaniment figure," the figure that projected the B-B[?event. In contrast, the mensural reading of figure 3.5 tried to associate theD-C# descent with the thematic E[?-D descent. Figure 3.5, of course, "did notknow about" the forte and staccato C natural coming up right after the forteand staccato C#. We have already discussed how the recurrent GIS3-interval(2,7) interacts with the accompaniment figure more generally.

This completes the exegesis of figure 3.3. The theoretical notion of an"unfolding interval vector," made abstractly available by the temporal aspectof GIS3, was analytically useful for examining our impressions of figure 3.1 asthose developed note-by-note, and for discussing to a significant extent ourimpressions of the music beyond that. GIS3 was particularly useful because itenabled us to consider pitch-class structure and mensural rhythmic structurein conjunction with each other, rather than as independent features of thepassage. That is the essence of GIS3 in its capacity as what we shall soon callthe direct product of GISt and GIS2.

6. A conflict between mensural distances of 5 and 6 units figures in the relation of the rhyth-mic ostinato to the written meter at the opening of the first movement. A reprise of this rhythmicconflict occurs at the opening of the coda in the last movement (mm. 56-62). The written metersare functional, in ways too complex to indicate here. Special accents attach to the loud and densetrichord-pairs of the middle movement. The lower chords of the trichord-pairs attack at the bar-lines of measures 4, 9, 4 bis (6 measures after 9), 9 bis, 15, 20, 15 bis (6 measures after 20), and20 bis. The resulting pattern projects alternating spans of 5 and 6 written measures. Indeed thismay well be the strongest mensural function for the written measure as a temporal span in themovement.

FIGURE 3.6

44


3.3.2 EXAMPLE: Let GISj be the GIS involving time-points, that just figuredas "GIS2" in the preceding example. For the present example, let GIS2 bea GIS of durations as in 2.2.3 earlier: S2 is a certain family of "durations"x, y,. . . related by certain stipulated proportions; IVLS2 is the multiplicativegroup of such proportions; int2(x, y) is the quotient y/x.

We construct a new GIS, GIS3, as follows. S3 is the Cartesian productSx x S2, that is, the family of pairs (s,x), where s is a time-point and x is aduration. We can conceive (s, x) as modeling an event that begins at time-points and extends for a duration of x (units) thereafter. IVLS3 is the direct-productgroup IVLSj (8) IVLS 2. Each member of IVLS3 is a pair (il5i2), where ilisa member of IVLSj (representing a number of beats between time-points)and i2 is a member of IVLS2 (representing a quotient of durations). (i lsi2)and (Ji,j2) combine in the direct-product group IVLS3 according to therule (i1 , i2)(Ji»J2) = fli + Ji^zJa)- int3((s»x)>(t,y)) is defined as (int^s,!),int2(x,y)), that is, loosely speaking, as (t — s,y/x). To put it another way, ifint3((s, x), (t, y)) = Oi, i2), then event (t, y) begins i t units of time after event(s, x) and extends for i2 times the extent of event (s, x).

The reader can take it on faith that GIS3 as constructed above is indeed aGIS. It will not be necessary to produce an analytic example, I think, in orderto convince the reader that this GIS is a useful theoretical tool. It combinestwo aspects of our mensural rhythmic intuition, as they impinge upon usconjointly, into one compound structure. We are still and again presuming afixed unit of time here, by which we measure durations and distances-between-time-points.

In both example 3.3.1 and example 3.3.2 we combined a given OK^ with agiven GIS2 in a certain manner, obtaining a new GIS that modeled theconjoint action of the two given GIS structures. We can make our proce-dure formal and general, as a means of combining any given GlSi and anygiven GIS2 into a "direct-product GIS." The following definition gives theprocedure.

3.3.3 DEFINITION: Given G^ = (Sl5IVLSîntJ and GIS2 = (S2, IVLS2,int2), the direct product of GISa and GIS2, denoted GIS1 <g) GIS2, is thatGIS3 = (S3, IVLS3, int3) which is constructed as follows.

S3 is Sj x S2, the Cartesian product of Sx and S2. That is, the elements ofS3 are pairs (sl5 s2), where Si and s2 are elements of Sj and S2 respectively.

IVLS3 is IVLSÎVLSi, the direct-product group of IVLSj andIVLS2. That is, the members of IVLS3 are pairs Oi,i2), where ij and i2 aremembers of IVLSi and IVLS2; further, the members (i l5i2) and ( j l f j 2 )ofIVLS3 combine (to form a group) under the rule (ii , i2)(j l 5 j2) = (iJi.iJa)-

The function int3, from S3 x S3 into IVLS3, is given by the rule

int3((s1,s2),(t1,t2)) = (int1(s1,t1),int2(s2,t2)). 45


It is straightforward to verify that GIS3, as defined above, is indeed aGIS, i.e. that GIS3 satisfies Conditions (A) and (B) of Definition 2.3.1.

This finishes our investigation into methods of constructing new GISstructures from old in a general abstract setting. Now we shall see how thenotions of "transposing" and "inverting" elements of a space S arise naturallyin any GIS; we shall see how operations of transposition and inversioninterrelate characteristically with intervallic structure, and we shall explorehow the operations combine among themselves. We shall also explore othercharacteristic operations, the "interval-preserving operations"; these may ormay not be the same as the transposition operations, depending upon whetherthe group of intervals is or is not commutative.7

3.4.1 DEFINITION: Given a GIS; given an interval i of IVLS; then transpo-sition by i, denoted Tj, is defined as a transformation on S as follows.

Given a sample element s of S, the i-transpose of s, T;(s), is that uniquemember of S which lies the interval i from s. That is, T;(s) is well defined by theequation

int(s,T,(s)) = i.

This definition conforms to our abstract intuition that the i-transpose of agiven element s should lie the interval i from s. The definition also conforms tothe way in which we already use the word "transposition" in connection withsome GIS structures, namely those involving pitches and pitch classes. T;(s) isindeed well defined by the equation within the definition. Condition (B) of2.3.1 assures us that given i and s, there exists a unique t such that int(s, t) = i;it is precisely this t which we are now calling "the i-transpose of s," Tj(s).

3.4.2 THEOREM: Each Ts is an operation; that is, it is 1-to-l and onto as atransformation on S. The transposition operations form a group of operationson S. That group is anti-isomorphic to the group of intervals. Specifically, letus consider the map f, defined from IVLS onto the family of transpositions bythe formula f(i) = T^ then f is an anti-isomorphism. That is, f is 1-to-l as wellas onto; and TjTj = T^.

Pr00/(optional): We shall prove the assertions of the theorem in an orderdifferent from that in which the theorem states them. First we show that f is ananti-homomorphism, i.e. that TjTj = T^.

Given intervals i and j, given a sample s in S, then we write int(s, Tj(Tj(s)))as the group product of the two intervals int(s, Tj(s)) and int(Tj(s), T;(Tj(s))),using Condition (A) of 2.3.1. Now int(s, Tj(s)) = j, by the defining equation ofDefinition 3.4.1. And by the same equation, int(Tj(s), Tj(Tj(s))) = i. Hence

7. All the groups in our specific examples of GIS structure so far have been commutative.Later on, in chapter 4, we shall study a non-commutative GIS of musical interest.46


int(s, Ti(Tj(s))) is expressed as the group product of the two intervals j and i.That is, int(s,Ti(Tj(s))) = ji. Thus Tj(Tj(s)) lies the interval ji from s. So it isequal to TJJ(S). We have shown: For any sample s, TjTj(s) = TJJ(S). So thetransformation TjTj has the same effect on S as does the transformation T^;the functional equation TjTj = T^ is true, as claimed.

So the map f is an anti-homomorphism. We show now that it is an anti-isomorphism. We have only to show that f is a 1-to-l map. Supposing that thefunctional equation Ts = Tj is true, we wish to prove that i and j must be thesame interval. Fix any s. Since Ts = Tj by assumption, Tj(s) = Tj(s). Then, byDefinition 3.4.1, i = int(s,Ti(s)) = int(s,Tj(s)) = j, which is what we had toshow.

It remains only to prove that each Tj is an operation, and that the familyof transpositions is a group of operations. To prove all this, it suffices to showthat the family of transpositions satisfies Conditions (A) and (B) of 1.3.4earlier, namely (A) that the family is closed and (B) that for each Tj there is aTJ satisfying TjTj = TjTj = 1. Condition (A) is evident from the formulaTjTj = Tjji The composition of two transpositions is a transposition. In orderto prove Condition (B), we shall prove a lemma: Te is the identity operation 1 onS. That is true since, given s in S, Te(s) is the member of S which lies the identityinterval from s; hence Te(s) = s. Or Te(s) = l(s); that being the case for anysample s, Te = 1 as asserted by the lemma. Now we can prove Condition (B).Given any interval i, take j = i"1. Then ji = ij = e. Tj is the transformationdemanded by Condition (B): TjTj = Tjs = Te = 1; likewise TjTj = 1. q.e.d.

3.4.3 THEOREM: Fix some referential member ref of S; then LABEL(Ti(s)) =LABEL(s) • i.

Proof: LABEL(Ti(s)) = int(ref, T,(s)) by definition of LABEL in 3.1.1,= int(ref, s)int(s, Ti(s)) by 2.3.1 (A),= LABEL(s)int(s, Tj(s)) by 3.1.1,= LABEL(s)-i by 3.4.1.

Theorem 3.4.3 tells us that no matter what ref is chosen for LABELingpurposes, the label for the i-transpose of s is the label of s, right-multiplied byi in IVLS. The natural question arises: What happens when we left-multiplyref-LABELs by i? We shall explore that right now.

3.4.4 DEFINITION: Fix some referential member ref of S. Given any intervali, the transformation Pj (more exactly P[ef)is defined on S by the formula

LABEL(Pj(s)) = i • LABEL(s),

which is to say the relation

int(ref, Pi(s)) = i • int(ref, s). 47

3,4.5 Generalized Interval Systems (2)

Given ref, i, and s, Condition (B) of 2.3.1 tells us that a unique memberof S, call it t, satisfies the relation int(ref, t) = i • int(ref, s). Definition 3.4.4takes that unique t, given ref, i, and s, and calls it Pj(s). Note that the specificvalue of Pj(s) depends on ref as well as on i and s. In contrast, the value ofT.(s) was well defined in 3.4.1 independent of any choice of ref.

3.4.5 THEOREM: The transformations P. form a group of operations isomor-phic to IVLS under the map f(i) = P}. In particular, the formula PjP. = Pr

is valid.Proof (optional): LABEL(Pj(p.(s))) = i • LABEL(P.(s)) = i • (j • LABEL(s))

= (ij) • LABEL(s) = LABEL'S)). The elements Pj(Pj(s)) and Py(s) thushave the same LABEL (lie the same interval from ref). So they are the same:PjPj(s) = Pjj(s). This being the case for any sample s, the functional equa-tion PjP. = P.. is true.

The map f of the Theorem is thereby a homomorphism of IVLS onto thefamily of transformations P.. To prove that f is an isomorphism, it remainsonly to show that f is 1-to-l. Suppose P. = P.; we are to infer that i = j.Fix any s; then i • LABEL(s) = LABEL(pj(s)) = LABEL(Pj(s)) by supposi-tion; and that = j • LABEL(s). In sum, i • LABEL(s) = j • LABEL(s). Hence,multiplying that equation through on the right by the inverse of LABEL(s),i = j as desired.

We can now use arguments just like those we used in the proof of 3.4.2,to show that the family of P. is a group of operations.

The operations P. have a special property. They are what we shall call the"interval-preserving" transformations. 3.4.6 and 3.4.7 develop the formalities.

3.4.6 DEFINITION: Given a GIS (S, IVLS, int), a transformation X on S willbe called "interval-preserving" if X has this property: For each s and each t,

int(X(s), X(t)) = int(s, t).

3.4.7 THEOREM: No matter what ref is chosen, the interval-preserving trans-formations on S are precisely the P..

Proof (optional): We show first that P. preserves intervals. We can writeinters), P,(t))

=(i • LABELS(s))-!(i • LABEL(t)) by 3.4.4,=(LABEL(s)~1 • i-])(i • LABEL(t))=LABEL(s)-1LABEL(t)=int(s, t) by 3.1.2.

Thus P. preserves intervals. Now suppose X is any interval-preserving48

=LABEL(P i(s))-1LABEL(Pi(t)) by 3.1.2,


transformation on S. We shall show that there exists some interval i such thatX-P,.

The i we want here is LABEL(X(ref)) = int(ref, X(ref)). Given any s, wecan then write LABEL(X(s)) = int(ref, X(s))

= int(ref,X(ref))int(X(ref), X(s)) via 2.3.1 (A),= i • int(X(ref), X(s)) by construction of i here,= i • int(ref, s) because X is interval-preserving,= i • LABEL(s) by the definition of LABEL,= LABEL(Pi(s)) via 3.4.4.

In sum, we have LABEL(X(s)) = LABEL(Pj(s)). Since X(s) and Ps(s)have the same LABEL, we infer that X(s) = Pj(s). Since s here is an arbitrarysample member of S, X = Pi as a transformation on S. Thus our interval-preserving X is in fact this particular P{. q.e.d.

Because of 3.4.7 we can conceive the transformations P5 as somewhat lessdependent on the choice of ref. It is true that a transformation labeled "Pj" byone choice of ref might be labeled "Pj" by another choice of ref. However theinterval-preserving transformations en masse, literally "as a group," remainthe same family of transformations en masse, no matter what ref is chosen.The interval-preserving property does not depend on the choice of ref, andthat property is sufficient to define the family of transformations as a group.

3.4.8 THEOREM: Given an interval i, Conditions (A) through (D) below areall logically equivalent: If any one of them is true, then they are all true.

(A): T; preserves intervals.(B): For some choice of ref, Tj = P}.(C): For any choice of ref, Tj = Pj.(D): i is central in IVLS. That is, i commutes with every j in IVLS (1.8.2).Proof (optional): Suppose (A) is true; we prove that (C) follows.

Fix any ref. Since Tt preserves intervals by assumption, 3.4.7 tells us thatTj = Pj for some j. For any s, LABEL(s) • i = LABEL(T;(s)) by 3.4.3that = LABEL(Pj(s)) since T, = Pjj and that = j • LABEL(s) by 3.4.4. Insum, LABEL(s) • i = j -LABEL(s) for any s. Consider s = ref in particularLABEL(ref) • i = j • LABEL(ref). But LABEL(ref) = int(ref, ref) = e. Henci = j. Tj = Pj as desired.

Clearly the truth of (C) entails the truth of (B). Now suppose (B) is true;we prove that (D) follows. We are supposing Ts = Pj for some ref. Then forevery s, LABEL(Tj(s)) = LABEL(Pj(s)). Then for every s, LABEL(s) • i =i • LABEL(s) (3.4.3 and 3.4.4). It follows that i commutes with every j, whichis Condition (D) as desired. For given any j, find the s which lies the intervalj from ref; then LABEL(s) = j; substituting j for LABEL(s) in the most recentequation involving i, we obtain ji = ij; i commutes with the given j. 49


Now we close the logical chain by showing that the truth of (D) entails thetruth of (A). When we have done this, we shall have shown that (A) implies(C), (C) implies (B), (B) implies (D), and (D) implies (A); hence the truth of anyone entails the truth of all the others.

Supposing (D) is true, then, we show that (A) will be true. Fix any ref.Given any s and any t, we write int(Ti(s), Tj(t))

= LABEL(Ti(s))-1LABEL(Ti(t)) by 3.1.2= (LABEL(s) • i)~1(LABEL(t) • i) by 3.4.3= (r1 • LABEL(s)'1) (LABEL(t) • i)= i"1 • (LABEL(s)-1LABEL(t)) • i= i - 1-mt(s, t)- i by 3.1.2= int(s, t) by the assumption

of Condition (D), that i is central. In sum, assuming Condition (D), thenint(T;(s), T;(t)) = int(s, t) for every s and t. Or, assuming Condition (D), Tjwill be interval-preserving. Or: (D) implies (A) as desired, q.e.d.

3.4.9 COROLLARIES: (A): In a commutative GIS (a GIS whose group ofintervals is commutative), the transposition operations are precisely theinterval-preserving operations.

(B): In a non-commutative GIS, there exist transposi-tions that do not preserve intervals, and there exist interval-preservingoperations that are not transpositions.

3.4.10 THEOREM: Any transposition operation commutes with any interval-preserving operation.

Proof: Fixing any ref, consider T; and Pj. We apply 3.4.3 and 3.4.4 tovarious LABELS. Taking any sample s, we have LABEL(Pj(Ti(s))) = j •LABEL(Tj(s)) = j • (LABEL(s) • i) = (j • LABEL(s)) • i = LABEL(Pj(s)) • i =LABEL(Tj(Pj(s))). Thus PjTj(s) and T,Pj(s) have the same LABEL; theylie the same interval from ref; so PjTj(s) = TjPj(s). This being the case forany sample s, the functional equation PjTj = TjPj is true: Ts commutes withPr q.e.d.

We are now ready to define and study "inversion operations" on anabstract GIS. For each u and each v in S (v may possibly equal u), we shalldefine an operation I*, which we shall call "u/v inversion." Figure 3.7 helpsus visualize an appropriate definition for the operation, conforming to ourspatial intuitions.

The figure shows how we conceive any sample s and its inversion I(s)(short for !„($) here) as balanced about the given u and v in a certain intervallicproportion. I(s) bears to v an intervallic relation which is the inverse of therelation that s bears to u. The inverse proportion is symbolized by the mirror50


relation of the two arrows on figure 3.7. The interval from v to I(s), which is theinverse of int(I (s), v), will then be the same as the interval from s to u. That is, weintuit int(v, I(s)) = int(s, u). We can use this equation to define !„ formally inany GIS.

3.5.1 DEFINITION: Given any u in S and any v in S, the operation !„ of u/vinversion is defined by the equation

int(v, Iu(s)) = int(s, u) for all s.

The operation is well defined by the equation: Given any s, set i =int(s, u) and find the unique t which lies the interval i from v. That t, whichsatisfies the equation int(v, t) = int(s, u), is precisely the value for IJ^s).

We have been referring to the "operation" I* prematurely; so far we haveconstructed a transformation, but we have not verified that the transforma-tion is indeed an operation, i.e. onto and 1-to-l. The reader may verify, as anexercise, that I* as defined is onto and 1-to-l. (Given t, find an s such thatru(s) = t. Prove that if IJ(s') = IJ(s), then s' = s.)

3.5.2 THEOREM: Fix a referential element ref of S. Set i = LABEL(v) andj = LABEL(u). Then

LABELTOs)) - i • LABEL(s)'1 • j.

Proof: To save space, we write "I" for "!„" here.

int(v, I(s)) = int(s, u) (3.5.1). Soint(v, ref )int(ref, I(s)) = int(s, ref )int(ref, u) (2.3.1 (A)).

Thence,

LABEL(v)'1LABEL(I(s)) = LABEL(s)~1LABEL(u) (3.1.1; 2.3.2)

Or: T1 • LABEL(I(s)) = LABEL(s)'1 • jOr: LABEL(I(s)) = i • LABEL(s)"1 • j q.e.d.

FIGURE 3.7

51


The formula of this theorem is very useful despite the dependence of allthe LABELs (including i and j) on a possibly arbitrary ref. We shall use theformula to analyze this question: When are lv

n and I* the same operation on S?In the familiar GIS of twelve chromatic pitch classes, for instance l£ = l£{ =IA' = ID» = IB'anc* so on- 'îat *s' ^^ mversi°n nas the same effect on anysample pitch class as does F|/F# inversion, or A/Df inversion, or DJt/A in-version, or B/C| inversion, and so on. In this special GIS one can intuit thatl£ and I* will be the same operation if and only if the pitch class w is the C/Cinversion of the pitch class x. One might thereby conjecture that I* and I* willbe the same operation, in a general GIS, if and only if the thing w is the u/vinversion of the thing x. This conjecture is valid if the GIS is commutative.When the GIS is not commutative, we must be content with the broader viewprovided by the following theorem.

3.5.3 THEOREM: I* = I* as an operation on S if and only if w = I*(x) andthe interval int(x, u) is central.

Proof (optional): Imagine ref fixed. Let i, j, k, and m be the respectiveLABELS for v, u, w, and x. Then, via 3.5.2, given any s,

LABEL(I^(s)) = i LABEL(s)-1 j, whileLABEL(I*(s)) = k LABEL(s)-1 m.

Hence the condition that I* be the same operation as I* is equivalent to thecondition that

i LABEL(s)'1 j = k LABEL(s)-1 m for all s in S.

Now as s runs through the various members of S, the inverse of its LABELruns through all the various intervals in IVLS. Hence the condition under dis-cussion is equivalent to the condition that

inj = knm for every n in IVLS.

And that condition is equivalent, via group theory, to Condition (A) below.

(A): (k-1i)n = nOnj"1) for every n in IVLS.

In sum, F[ = I* as an operation if and only if Condition (A) above is satisfied.Now we shall prove that Condition (A) is satisfied if and only if w = I*(x)and int(x, u) is central. That will prove the theorem as stated.

Suppose, then, that Condition (A) is true; we shall show that w = IJJ(x)and int(x, u) is central. Condition (A) being true by supposition, it is true forn = e; therefore k~!i = mj"1. Let us call this interval c. Condition (A) thentells us that en = nc for every n in IVLS, so c is central.

c was taken equal to mj"1. It turns out that c also equals j"1!!!. To see that,we write mj"1 = c; thence m = cj; thence, since c is central, m = jc; thencej-1m = c as asserted.52


Let us review: Assuming Condition (A), we have so far shown that k 4 =mj-1 = j^m = c is central interval. Now k~M = LABEL(w)~1LABEL(v) =int(v, w), via 3.1.2. Likewise j-1m = int(x, u). Thus we have shown: int(v, w)= int (x, u) and int(x, u) = c is central. Now the relation int(v, w) = int(x, u)is exactly the relation which tells us that w = IJj(x) (Definition 3.5.1). Thus,assuming Condition (A), we have proved that w = I^(x) and int(x, u) is cen-tral as desired.

Now we prove the converse half of the theorem: Supposing that w =I*(x) and that int(x, u) is central, we prove that Condition (A) is satisfied.

w = I*(x) (by supposition). Soint(v, w) = int(x, u) (3.5.1). Or:

LABEL(w)-1LABEL(v) = LABEL(u)~1LABEL(x) (3.1.2). Or:k~U = j-1m (meaning of i, j, k, m).

Furthermore, we have supposed that j^m, which is LABEL(u)~1LABEL(x),which is int(x, u), is a central interval. Let us call this central interval c.j^m = c; thence m = jc; thence m = cj; thence mj"1 = c. So

k H = mj l = c, a central interval.

c being central, en = nc for every n in IVLS. Substituting k~!i = mj"1

for c, we obtain

(k~]i)n = n(mj-1) for every n in IVLS.

And that is precisely Condition (A), q.e.d.

We used ref and LABEL in our proof of Theorem 3.5.3. Note, however,that the statement of the theorem does not depend on a choice of ref.

3.5.4 CORROLARY: I* = IJJ if and only if int(v,u) is central.Proof: Applying the formula of Definition 3.5.1 to the algebraic truism

int(v, u) = int(v, u), we infer that u = I*(v). Then, by the logic of Theorem3.5.3 just proved, 1^ = IJJ if and only if int(v, u) is central.

The corollary tells us that in a general GIS, v/u inversion may well not bethe same operation as u/v inversion, despite the relations v = I*(u) = I"(u);u = I^(v) = IJJ(v). The operations I* and l^ both transform u to v, and v to u.But there may be some other s, other than u or v, such that IJj(s) is not thesame element as I^(s). Indeed the corollary assures us there will be somesuch s if int(v, u) is not central. These considerations indicate how carefullyand rigorously we must proceed hereabouts; intuitions based on our familiar-ity with a number of commutative GIS structures will not always be reliable.

3.5.5 COROLLARIES: In a commutative GIS, I* always = I|J; generally, I* =I* if and only if w = I»(x). 53


In a non-commutative CIS, there will always be some inversion operationIy which is not the same as 1 .̂ (For there will always be some int(v, u) whichis not central.)

Now we shall see how inversion operations I* combine with transposi-tions Tn and interval-preserving operations P.

3.5.6 THEOREM: For any transposition Tn and any inversion I*,(A): Tnl^ = I* where x = Tn(u).(B): Iv

uTn = I* where w = T-^V).(C): Tn commutes with 1^ if and only if n is central and nn = e.Proof (optional): We shall fix some ref and use LABELs to help our

computations. Let i = LABEL(v); let j = LABEL(u). Then

LABEL(TnIvu(s)) = LABEL(Iv

u(s)) - n (3.4.3)= i-LABELS)-1- j-n (3.5.2)= LABEL(I^(s)),

where x is the member of S whose LABEL is jn (3.5.2). Since LABEL(x) =jn and LABEL(u) = j, x = Tn(u) (3.4.3).

Now Tnl*(s) I*(s) have the same LABELs, via the chain of computationsabove. Hence Tnl*(s) = I*(s). (Both lie the same interval from ref.) Since swas an arbitrary sample member of S, Tnl^ = I* as an operation. This proves(A) of the theorem.

To prove (B), we start with a similar chain of computations.

LABEL(PuTn(s)) = iCLABELCT^s)))-1] (3.5.2)= i(LABEL(s) • n)-1] (3.4.3)= in~1LABEL(s)-1j= LABEL(I-(s)),

where w is the member of S whose LABEL is in -1 (3.5.2). Since LABEL(w)= in"1 and LABEL(v) = i, w is the n"1 transpose of v (3.4.3). Since T"1 =T"1, we can write w = T~!(v). (The map of n to Tn is an anti-isomorphism;n"1 maps to T"1.)

We go on to infer the operational equality of the operations I*Tn and 1 ,̂exactly in the way we inferred an analogus equation when proving (A) ofthe theorem. This proves (B) of the theorem.

Using the formulas of (A) and (B) that we have now established, we seethat Tnl^ = I*Tn if and only if I* = I* x and w being as in (A) and (B) ofthe theorem. By Theorem 3.5.3, this will be the case if and only if w = I*(u)and the interval int(u, x) is central. That will be so, according to 3.5.1, if andonly if int(v, w) = int(u, x) and int(u, x) is central. But int(u, x) = n, sincex = Tn(u), and int(v, w) = n"1, since w = T"1^). So, in sum, Tn willcommute with I* if and only if n = n"1 and n is central. This proves (C).q.e.d.54


(C) of the theorem shows that, given any interval n in a general GIS, eitherTn commutes with every inversion operation or Tn commutes with no inversionoperation. In the familiar GIS of twelve chromatic pitch-classes, T6 commuteswith every inversion operation: If you invert and then transpose by a tritone,the net result is the same as if you transpose by a tritone and then invert (aboutthe same center or axis). In that GIS, no other interval of transposition has thisproperty, save for the trivial interval of zero. (T0 is the identity operation.) Infact, no other Tn in that GIS will commute with any inversion operation. (C) ofthe theorem shows us that this situation is related to the fact that 6 + 6 = Omod12, while n + n does not = 0 mod 12 for any other non-zero interval mod 12.

Theorem 3.5.6 gave us insight into how inversions combine with transpo-sitions. An analogous theorem will give us analogous insight into how inver-sions combine with the interval-preserving operations P.

3.5.7 THEOREM: For any interval-preserving operation P and any inversionIu»

(A): PI* = I? where w = P(v).(B): i;;P = I* where x = P~1(u).(C): P commutes with !„ if and only if P = Tc for some transposition Tc

such that c is central and cc = e.Proof (optional): We fix a referential element ref. Setting n =

int(ref, P(ref)) = LABEL(P(ref)), we write P = Pn in the manner of 3.4.4earlier. We can then manipulate pertinent LABELs to prove (A) and (B)exactly as we proved (A) and (B) for Theorem 3.5.6 above.

To prove (C), we begin in a manner similar to that by which we proved3.5.6 (C). Using (A) and (B), we note that PI* = I^P if and only if I£ = I*,where w and x are as in (A) and (B). Via 3.5.3, this will be the case if and only ifint(v, w) = int(u, x) and int(u, x) is central.

Now LABEL(u)=j and, since x = P~'(u) = Pn-»(u), LABEL(x) =n-1 • LABEL(u) = n-1j (3.4.4). Therefore LABEL(x)~1LABEL(u) =(n'M)"1 j = T1 nj. And int(u, x) is precisely LABEL(x)"1 LABEL(u) (3.1.2). Soint(u, x) = j"1 nj. We have now showed: P = Pn commutes with I* if and only ifint(v, w) = int(u,x) = j-1nj and the interval j-1nj is central.

A little group theory provides the proof for the following lemma: Theelement j-1nj of a group is central if and only if n is central. Thus either n iscentral, in which case j"1 nj is of course simply n; or else n is not central, in whichcase j~*nj is not central. We have now proved: P = Pn and I* commute if andonlyifint(v,w) = int(u,x) = n and n is central. The rest follows from 3.4.8 and3.5.6(C). q.e.d.

We have now seen how inversion operations combine with interval-preserving operations. Earlier we saw how inversions combined with transpo-sitions (3.5.6), and how transpositions commuted with inversion-preservingoperations (3.4.10). Earlier still, we noted that the transpositions formed a 55


group of operations among themselves (3.4.2), combining according to a cer-tain formula; the interval-preserving operations also formed a group amongthemselves (3.4.5) combining according to another formula. It remains to ex-plore how the inversion operations combine with each other, and we proceedto do so.

3.5.8 THEOREM: Fix ref, and let the LABELs of v, u, w, and x be respectivelyi, j, k, and m. Then

T V T W = p— IT~IJA u A x r imx k

Proof: Given any s, then 3.5.2 tells us that

LABEL(I^(s)) = i-LABEL(s)-1-] whileLABEL(I*(s)) = k • LABEL(s)-1 • m. Hence

LABEL(IvuI-(s)) = KLABELa^s)))-1]

= i(k • LABEUXT1 • mr'j= i(m-1LABEL(s)k~1)j= (im-1)LABEL(s)(k-1j)= LABEUPrjT-^s)) (3.4.3; 3.4.4).

In sum, I*I*(s) and P^T^s) have the same LABELs, and are thereforethe same element of S. Since s was an arbitrary sample member of S, thefunctional equation of the theorem is true, q.e.d.

3.5.9 COROLLARY: I|J is the inverse operation to I*.Proof: Given u and v, take x = v and w = u in the formula of Theorem

3.5.8 above. Then m = i and k = j, as those intervals are defined in thattheorem. So im"1 and k-1j are both equal to e; the formula of the theorem tellsus in this special case that !„!" = PeTe, which is of course the identity oper-ation. By the symmetry of the situation (reversing the roles of u and v), PJI*is also the identity operation, q.e.d.

3.5.10 COROLLARIES: Let T and I be any transposition operation and anyinversion operation in a commutative GIS. Then

(A): r1 = I and(B): IT = T-'I.

Proofs: (A) follows at once from 3.5.9 and the first remark of 3.5.5.To prove (B), set J = IT. Via 3.5.6(B) we know that J is an inversion

operation. Then, invoking (A) just proved above, we infer that J = J"1. Itfollows: IT = J = J-1 = (IT)"1 = T'1!'1 = T~]I; the last step is again aconsequence of (A) just proved, q.e.d.

We have now explored how various types of operations on the space ofan abstract GIS compose among themselves and with each other. Standing back56


from the niceties of the specific formulas involved, we can get a useful globalpicture.

3.5.11 THEOREM: Let PETEY be the family of all operations on S that can beexpressed as (functionally equal to) something of form PT, where P is someinterval-preserving operation and T is some transposition. Let PETINV bethe family PETEY plus the family INVS of all inversion operations. Then(A) PETEY is a group of operations and (B) PETINV is also a group ofoperations.

Proof (optional): We already know that PSVS, the family of interval-preserving operations, and TNSPS, the family of transpositions, are eachgroups of operations (3.4.5; 3.4.2). Let PT and PT' be any two members ofPETEY. Set P" = PP' and T" = TT. Since PSVS and TNSPS are groups,P" is interval-preserving and T" is a transposition; then P"T" is a memberof PETEY as PETEY was defined. Furthermore, the composition of thegiven PT with the given FT' is precisely P"T", this member of PETEY. For(PD(FT') = P(TP')T = P(PT)T' (3.4.10) = P"T". So PETEY is a closedfamily of transformations.

To prove PETEY a group, it suffices via 1.3.4 to show that PETEY-as-defined contains the inverse of each of its member operations. Given PT inPETEY, then P"1 and T-1 are respectively members of PSVS and TNSPS, sothat p-1T-1 is a member of PETEY-as-defined. And p^T"1 is the inverse ofthe given PT: PT = TP (3.4.10), so PT(P~1T~1) = TP(P~1T~1) = 1 and(p-iT-i)PT = (P-1T-1)TP = 1. So (A) of the theorem is proved.

We use the criteria of 1.3.4 again to prove (B) of the theorem. We knowthat PETEY, since it is a group, contains the inverse of each of its members; wealso know via 3.5.9 that INVS contains the inverse of each of its members.Hence PETINV, the set-theoretic union of the two families PETEY and INVS,contains the inverse of each of its members. It remains only to prove thatPETINV is a closed family of operations.

Suppose that X and Y are members of PETINV; we have to show that XYis (operationally equal to) a member of PETINV. We can distinguish fourpossible cases, which we take up one at a time below.

Case 1: X and Y are both members of PETEY. Then XY, being a memberof PETEY, is a member of PETINV.

Case 2: X is a member of PETEY and Y is a member of INVS. Say X =PT and Y = I. Now TI is some inversion-operation J (3.5.6(A)). And PJ issome inversion-operation K (3.5.7(A)). Then XY = PTI = PJ = K is a mem-ber of INVS, and therefore a member of PETINV as desired.

Case 3: X is a member of INVS and Y is a member of PETEY. By anargument analogous to that of Case 2, now using 3.5.6(B) and 3.5.7(B), weconclude that XY is a member of INVS, and hence of PETINV as desired.

Case 4: X and Y are both members of INVS. Then XY is a member ofPETEY (3.5.8). So XY is a member of PETINV as desired, q.e.d. 57


We have seen that transpositions are naturally related in a number of waysto interval-preserving operations. We might conjecture that inversions shouldbe naturally related to "interval-reversing" transformations, in the sense of thefollowing definition.

3.6.1 DEFINITION: A transformation Y on the space S of a GIS will be calledinterval-reversing if

int(Y(s),Y(t)) = int(t,s)

for all s and all t in S.

There is something to our conjecture above. Specifically, if the GIS iscommutative, then the inversions are precisely the interval-reversing operationson S. But if the GIS is now-commutative, then there will not be any interval-reversing transformations at all! We shall now prove these facts, starting with alemma.

3.6.2 LEMMA (optional): Let Y be an interval-reversing transformation; let refbe fixed; then there is an interval i such that

LABEL(Y(t)) = i • (LABEL(t))-1

for every t in S.Proof: int(Y(s), Y(t)) = int(t, s) by supposition. So

LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)~1LABEL(t) (3.1.2).

Take s = ref in the above equation, so that LABEL(s) = e. Set i =LABEL(Y(ref)) = LABEL(Y(s)) in the above equation. Then

LABEL(Y(t))-1 • i = LABEL(t). SoLABEL(Y(t))~1 = LABEL(t) • T1 andLABEL(Y(t)) = i • LABEL(t)-1, for any t.

3.6.3 THEOREM (optional): If IVLS is commutative, then the inversion oper-ations reverse intervals, and every interval-reversing transformation is someinversion-operation.

Proof: Fix some ref. The result of Lemma 3.6.2, in conjunction with theformula of Theorem 3.5.2, tells us that any interval-reversing transformationmust be some inversion, specifically some Vtcf. It remains to show that any I*reverses intervals. Setting i = LABEL(v) and j = LABEL(u), we can write

intai(s), W)) = LABELS))-1 LABEL(I^(s)) (3.1.2)= (iLABEL(t)~1j)-1(iLABEL(s)-1j) (3.5.2)= j~1LABEL(t)i-1iLABEL(s)-1j= LABEL(s)~1LABEL(t),58


since IVLS is commutative! And that

= int(t,s) via 3.1.2. q.e.d.

3.6.4 THEOREM (optional): If IVLS is non-commutative, then there exists nointerval-reversing transformation on S.

Proof: We suppose that Y is an interval-reversing transformation andarrive at a contradiction.

For all s and all t,

LABEL(Y(t))~1LABEL(Y(s)) = LABEL(s)-1 LABEL(t),

as in the proof for Lemma 3.6.2. Having the formula of that lemma at ourdisposal now, we can substitute for the LABELs of Y(s) and Y(t) in the aboveequation, using the special interval i of the lemma. We get the new equation

(i(LABEL(t)r1)-1 • iLABEL(s)-1 = LABEL(s)~1LABEL(t). Or:(LABELCOrîLABELts)-1 = LABEL(s)~1LABEL(t). Or:

LABEL(t)LABELCs)-1 = LABEL(s)~1LABEL(t).

But that equation, holding for all s and t, says that IVLS is a commutativegroup. And that contradicts the premise of the theorem, q.e.d.

59

Generalized Interval Systems(3): A Non-Commutative GIS;Some Timbral GIS Models

During the discussion of transpositions, interval-preserving operations, andinversions, the reader may have been puzzled by the care with which non-commutative GIS structures were separated from commutative. After all, wehave so far not encountered any specimen GIS which is non-commutative.Why should we be concerned at all about the non-commutative case? Why notsave ourselves some trouble, and just stipulate in the definition of a GIS thatthe group IVLS should be commutative? The work of the present chapterwill respond to these concerns by exploring a musically significant non-commutative GIS. I have already presented some of the work elsewhere, butit will have a special impact in the present context.1

4.1.1 DEFINITION: By a time span, we will understand an ordered pair (a, x),where a is any real number and x is any positive real number. The pair ofnumbers is understood to model our sense of location and extension about amusical event that "begins at time a" and "extends x units of time" thereafter.The family of all time spans will be denoted TMSPS.

We have already encountered one rhythmic GIS whose objects werecertain time spans; that was example 3.3.2. There, we restricted the values forthe numbers a to integers, and we restricted the values for the numbers x tocertain proportions. Using the same direct-product construction, we couldalso construct a GIS for time spans in which the number a could assume anyrational value, and the number x any positive rational value. Using the direct-

1. David Lewin, "On Formal Intervals between Time-Spans," Music Perception vol. 1,no. 4 (Summer 1984), 414-23.60

4


product construction, we can also construct a GIS for all time spans in themanner of 4.1.2 following.

4.1.2 EXAMPLE: Take S = TMSPS. Take IVLS to be the direct-productgroup of the real-numbers-under-addition by the positive-reals-under-multiplication. Define the function int, from S x S into IVLS, by the formulaint((a,x),(b,y)) = (b-a,y/x).

Then (TMSPS, IVLS, int) is a GIS. It is commutative. The interval(b — a, y/x) measures our presumed sense that time span (b, y) begins b — aunits after time span (a, x) and lasts y/x times as long.

This commutative GIS is useful and relatively simple, but it is notadequate as a model for the way we perceive time spans interacting under allcircumstances. We shall now investigate why that is so. First we shall examinethe family of time spans as a conceptual space independent of any particularcompositional context; then we shall examine how time spans behave invarious specific musical contexts.

To begin, then, let us focus on the time span (a, x) as a conceptual objectin a conceptual space, modeling our sense that something "begins at timepoint a" and "extends for x time-units" thereafter. We can ask, what is thisabsolute conceptual time-unit? In practice, we often proceed as if it were theminute. We do so, that is, when we write metronome marks which reducevarious contextual units, in various pieces or passages of music, to fractions ofa minute. The minute is not commensurate with our sense of a "beat," but wecan use the second for that purpose if we wish, dividing all the metronomenumbers by 60. Neither the minute nor the second, though, is very satisfactoryas a would-be absolute conceptual time unit; both are derived from certainrelative periodic motions of the earth, the sun, and the moon. Scientists todayfind these motions so erratic and irregular that they use other conceptual unitsof time for precise measurements. But even those units, deriving from certainsub-atomic motions, are clearly contextual. And that does not even begin toengage other technical problems involving Relativity and quantum mechanicsin connection with such sub-atomic "fixed" units of time.

In short, if we declare any one time-unit to have absolute conceptualpriority, that is a matter of computational convenience, or of scientific,sociological, or religious convention, rather then manifest musical reality.Abandoning this approach, we can make our absolute time-unit a matter ofnotational structure: We can call it "the brevis" or "the perfection" or "thewhole note" or "the notated beat," for instance. But then we are throwing thewhole problem back onto some notational convention that is highly restrictedsocio-historically, a convention that indeed already presupposes a highlystructured theory of measuring time by some pre-existing absolute unit. Andthat will not help us in our inquiry. 61


We might try to assert that, though we cannot conceive an absolute time-unit clearly, there will be some clear contextual time-unit, which we canidentify and use for theoretical purposes, in any music that we may want toanalyze. Such an assertion is reasonable in connection with a large body ofmusic, and not only European music of the Classic-Romantic period. But theassertion is still not valid as a universal proposition about music, unless one iswilling to restrict the use of the word "music" circularly, i.e. in precisely thisway. Making that restriction would involve at least a broad aestheticcontention. An even then, many critics who might feel no qualms aboutexcluding as "music" say certain improvisations of John Coltrane, or Stock-hausen's Aus den sieben Tagen, would feel less comfortable excluding asmusic certain Tibetan chants, or sections of Elliott Carter's String Quartetno. 1. Figure 4.1 (pp. 64-65) reproduces measures 22-35 of the Carter score.What could one assert as "the" (one clear contextual) time-unit for measures22-32?2

Figure 4.1 shows that our philosophical musings above do not simplycome down to a matter of mensural versus non-mensural perceptions. Mea-sures 22-32 of Carter's piece have a very strong mensural character, despiteour difficulty in pinning down "the" beat. The mensural profile of the passageis especially—one might even say unusually—strong within each of theindividual instrumental parts. For example, the first violin's A in measure 26and G in measure 28 each last precisely half as long as every other note in thefirst violin from measure 22 to measure 30; a player who does not hear thismensural relation will not project the passage effectively. For anotherexample, the B-F|-D-C|-D# of the cello at measure 32 and following isheard not only in syncopation against the otherwise regular half-note beat ofmeasure 33 and following (half-note = MM90), but also as a rubato of theearlier cello melody C-G-E1>-D-E, appearing eleven semitones lower inmeasures 2729. The earlier melody is presented in notes of constant durationwhose beat, every five written eighths, is at MM48, not MM90. A cellist whodoes not hear the rhythmic relation of the transposed melodies will not projectmeasures 32-35 completely effectively.

In sum, the notion of "an" abstract conceptual time-unit, a unit by whichwe measure the number x of the formal time span (a, x), is a notion fraughtwith methodological problems. The number a of the time span (a, x), as well asthe number x, is measured by our conceptual time-unit. For when we say thatwe perceive something that "begins at time-point a," we mean implicitly that it

2. I am grateful to Jonathan W. Bernard for engaging my interest in this passage throughhis lecture, "The Evolution of Elliott Carter's Rhythmic Practice," delivered to the meeting at Yaleof the Society for Music Theory on November 11, 1983. Bernard's observations engaged manyof the rhythmic relations I shall be discussing. The uses to which I shall put those observations,as I expound my CIS-theories, are of course my own responsibility.62


begins just that number of conceptual time-units after some referential time-point, "time-point zero." Having noted this, we see that we must discuss notonly the conceptual time-unit in this connection, but also the conceptual"time-point zero." What is this abstractly privileged moment that contributestoward measuring the number a of the time span (a, x)? Is it the moment of theBig Bang, or of the Biblical Creation? Is it a completely arbitrary moment verylong ago? (And if so, why should we select an arbitrary moment to play auniquely referential role?) Should we select "time-point zero" by a notationalconvention, e.g. as the first vertical line on the score of whatever piece we areanalyzing at the moment? Or should we presume to assert, explicitly orimplicitly, that there must always be some one uniquely privileged moment, inthe score or the performance of any passage we want to discuss, which we canunequivocally label as a contextual zero time-point for the occasion? Thesemethodological expedients involve difficulties similar to those discussed abovein connection with the referential time-unit.

In one way at least, the choice of a zero time-point is less problematicthan the choice of a temporal unit: The former choice does not affect thenumbers attached to intervals in the GIS of 4.1.2, while the latter choice doesaffect those numbers. To see this, first suppose that we move our referentialzero time-point back by m units into the past. Then the percept that was for-merly manifest over the time span labeled (a, x) in the old scheme will nowbe manifest over the time span labled (a + m, x) in the new scheme: Whatused to begin a units after (old) time-point zero will now begin a + m unitsafter (new) time-point zero. Similarly, the time span labeled (b, y) in the oldscheme will correspond to the time span labeled (b + m, y) in the new scheme.In the GIS of 4.1.2, the interval between the old labels is int((a, x), (b, y)) =(b — a, y/x). In the same GIS, the interval between the new labels isint((a + m, x), (b + m, y)) = (b + m - (a + m), y/x) = (b - a, y/x). So,in transforming each old time span (a, x) to the new time span (a + m, x), wehave not transformed the intervals involved: The interval between a pair oftransformed spans is exactly the same as the interval between the correspon-ding pair of spans prior to transformation. That is, int((a + m, x), (b + m, y))= int((a, x), (b, y)).

Now let us suppose we keep the same referential time-point zero butchange the unit of measurement so that what was x old units becomes xu newunits, the factor u corresponding to the change of scale in measurement. Thenpercepts formerly corresponding to the time span (a, x) and (b, y) in the oldscheme will now correspond to the time spans (au, xu) and (bu, yu) in thenew scheme: What used to begin a old units after time zero and extend x oldunits therefrom will now begin au new units after time zero and extend xu newunits therefrom. We can see that this transformation does change the numbersattached to intervals in GIS 4.1.2: int((a, x), (b, y)) = (b - a, y/x), while 63


FIGURE 4.1

64


FIGURE 4.1 (continued)

65


int((au,xu),(bu,yu)) = (bu — au,yu/(xu)) = ((b — a)u,y/x). Intuitively, thesecond percept we are discussing used to begin b — a units after the firstpercept; now it begins (b — a)u units after the first percept, the "unit" havingchanged. Of course the GIS of 4.1.2 knows nothing of "percepts" or "units";it simply knows that int((a, x), (b, y)) = (b — a, y/x) is not the same pair ofnumbers as ((b — a)u, y/x) = int((au, xu), (bu, yu)).

Lest the host of methodological problems under discussion appear in-superable, we should recall that we can finesse them all by restricting ourattention to music in which we can identify and assert a referential time-unitand a referential zero time-point contextually. Then we can use the GIS of4.1.2 without problems. There are plenty of pieces and passages for which wecan sensibly take this tack. On the other hand, there are also pieces andpassages in which we cannot identify such referential entities contextually,music which we would agree nonetheless to consider highly structured men-surally, music within which it seems analytically valid—even necessary—toarticulate time spans engaged in mensural interrelationships. We have alreadybegun to explore the Carter example in this connection; we shall continue thatanalysis soon. Another example is provided by Stockhausen's KlavierstiickXL Stockhausen tells the pianist to look at the sheet of music and begin withany group of notes from among nineteen such groups dispersed over the score,"the first that catches his eye; this he plays, choosing for himself tempo . . . ,dynamic level and type of attack. At the end of the first group he reads thetempo, dynamic and attack indications that follow, and looks at random toany other group, which he then plays in accordance with the latter indica-tions," and so on and on. "When a group is arrived at for the third time, onepossible realization of the piece is completed." 3 Each of the nineteen groupsis notated quite traditionally as regards pitches and internal rhythmic propor-tions. But each group might be played at any of six tempi, ranging from veryslow to very fast. Indeed, even during one performed realization, any groupmight be played two different times at two different tempi. The tempo of eachperformed group (after the first) depends on the instructions which appear atthe end of the group just played, which might itself occur at any of the sixtempi. In this context it makes no musical sense to speak of "the" referentialtime-unit, beyond the interior of each performed group at its performedtempo. And yet, mensural relations among time spans from different groups(especially consecutive groups) are highly audible, and therefore at leastperceptually functional in any given realization.

We have already mentioned, in footnote 3 of chapter 2, other examples ofhighly mensural music without a fixed time-unit. Nancarrow's Studies forPlayer Piano contain many pitch-canons involving elaborately changing

3. The cited text is from the Performing Directions by the composer on the score (UniversalEdition no. 12654 LW, 3d ed., 1967).66


tempo proportions.4 Ligeti's Poeme symphonique is performed by winding up100 clockwork metronomes to varying degrees of tension, setting them at avariety of tempi, and releasing them, allowing them to run down over thecourse of eighteen to twenty minutes.5 The effect includes an ironic poeticcommentary, among other things, upon the very issue of the "contextual time-unit."

To some extent in all this cited literature, and to a great extent in muchof it, any time span has the potential for becoming a local contextual time-unit, setting a local tempo. By "local," I mean here not only over a certaintemporally connected section of the total texture, but possibly also within acertain part, instrument, or instrumental group. For example, there is a morethan clear mensural structure within the part played by any single metronomewithin Ligeti's piece. For some less extreme examples let us return once moreto figure 4.1, the Carter passage discussed before. The viola moves in notes ofconstant duration from its entrance at measure 25 up to the middle of measure35; these local time-units "beat" the tempo of MM 180. The first violin "beats"constant local time-units at MM36 over measures 22-30, except for the A ofmeasure 26 and the G of measure 28 which, as observed earlier, are each halfthe local time-unit in duration. At measure 33 the first violin starts to project anew constant local time-unit, that beats at MM90. The cello beats constantlocal time-units at MM 120 over measures 22-26; then over measures 27-31 itbeats new constant local time-units at MM48. Finally at measure 32 andfollowing, it stops moving in notes of constant duration as it plays a pitch-variation on measures 27ff., where MM48 began. This variation was discussedearlier. The second violin beats its own constant local time-unit at MM96from measure 22 through measure 26. Then over measures 27-30 it runsquickly through a number of local time-units at MM 120, MM 160 (m. 27j),MM96 (m. 28|), MM80 (to be discussed later), and MM60 (m. 30). Finally itsettles into a more stable local time unit in measure 31, beating at MM90.

4. Some of Nancarrow's recent work involves irrational proportions like n. For the readerwho may at first think such an idea is too bizarre to have any musical meaning, I append a briefexercise in conducting the tempo relation of n. Imagine a horizontal line segment at chest heightin front of you and somewhat to your right. (I am supposing that you conduct right-handed.)Move your hand (arm) back and forth along the line at a constant speed, beating a horizontal \allegro vivo. Now imagine the line segment as a radius of a circle whose center is at the leftmostpoint of your beat. As you reach the rightmost point of your beat, start swinging your hand (armaround and around the circumference of that circle counterclockwise, taking care always to keepyour hand moving at the same constant speed. The amount of time it takes you to swing oncearound the circle is n times the duration of your earlier horizontal \ measure.

5. The composer specifies very precisely how this is to be done. He also specifies that thepiece is to be played after an intermission, so that the returning audience finds the metronomesalready underway, with no persons on stage. The metronomes are to be arranged on risers, like achorus; the slow beaters are at the lowest level and the fast beaters at the highest. I am indebted toMartin Bresnick for supplying me with this report, based on a personal communication from thecomposer. 67


Figure 4.2a collates and tabulates these various metronome marks, that reflectthe various tempi beat by the various local time-units in the individualinstruments over the passage.

FIGURE 4.2b

68

Figure 4.2b takes the numbers of figure 4.2a and represents them aspitches. This device will help clarify for musicians the numerical ratios in-volved among those numbers. The number 180, which labels the fastest beaton figure 4.2a, is represented on figure 4.2b by the highest pitch, high C.Slower tempi, in their numerical ratios to MM 180, are represented on figure4.2b by lower pitches in the corresponding frequency ratios to high C. Forinstance, on figure 4.2a the opening tempo of the cello is MM 120, 2/3 of thetempo MM 180 coming up in the viola. On figure 4.2b the tempo MM 120 is


represented by the opening pitch F5 in the cello, a pitch whose fundamental is2/3 the frequency of the high C coming up in the viola, the high C whichrepresents the tempo MM 180.

The euphony of figure 4.2b is striking. It makes very clear the network of"numerical consonances" displayed by the tempo relations of figure 4.2a,which are also the proportional relations of the various local contextual time-units. If we imagine a quartet actually playing figure 4.2b, we can get an evenclearer idea of how this numerical network affects interactions among theperformers. Here are some, and only some, of the relations the players willheed. (1) The 48 of the cello at measure 27 will lie a good numerical octavebelow the opening 96 of the second violin. (2) The 120 of the second violin atmeasure 27 will match the opening 120 of the cello. (3) The 120-to-160 relationin the second violin during measure 27 will match in its ratio the 36-to-48relation between the first violin and the cello thereabouts. (4) The 80-to-60relation in the second violin over measures 29-30 will retrograde, an octavelower, the 120-to-160 relation discussed in (3) above. (5) The 60-to-90 relationin the second violin, measures 30-31, will match an octave lower the earlier120-to-180 relation between the cello at the opening of the passage and theviola entrance. (6) The 90 of the first violin at measure 33 will match the 90 ofthe second violin at measure 31, which in turn will match, an octave lower, thepersistent 180 of the viola. (7) The 36-to-90 profile of the first violin part as awhole will match in transposed retrograde (or inversion) the 120-to-48 profileof the cello part up to measure 33.

Exactly these numerical relations, and others of the same sort, must beprojected to make the rhythmic structure of Carter's passage come to life andcommunicate, not only between the players and their listeners, but also amongthe four players themselves. We shall discuss the above seven performancenotes some more later on.

On figure 4.2b, it is curious how the D[?4 of the cello can be heard as aroot whose major harmony is elaborated by the symbolic pitches of the figureover measures 22-30, along with a major seventh and an added sixth. To besure, it is doubtful that our tonal perceptions of roots, triads, harmonicsevenths, and added sixths can be used to assert analogous functions in therealm of tempo relations. And yet there is a certain suggestiveness in the ideathat the cello part of measure 27 and following has a rhythmically "ground-ing" function somehow analogous to the tonal root function of the symbolicD|?4 on figure 4.2b. This suggestion is useful for the cellist who wants theMM48 tempo at measure 27 and following to feel stable and referential, ratherthan syncopated and intrusive. The suggestion is also useful for understandingwhy just this tempo of MM48 (symbolized by the pitch D|?4 on figure 4.2b) ispermitted to launch into a wide and free rubato at measure 32 and followingin the cello (symbolized by the "cadenza" on figure 4.2b), just at the time the 69


other three instruments are all finally agreeing upon the new referential tempoMM90-or-180 (symbolized by the prominent pitch class C at m. 32ff. on figure4.2b). Over the second-violin part of figure 4.2b as a whole, D|» moves clearlyto C. But the cello is still refusing to change its earlier D\> for C as we leavethe passage, though its D\> has been abandoned. All of these remarks amplifythe idea suggested above, that the cellist should refer the rhythm of the partover measures 32-35 not just to the beats of the upper instruments at MM90but also to the earlier beat of the cello itself, MM48 at measure 27 and fol-lowing. Figure 4.2b shows by analogy how the symbolic pitches of the "ca-denza" in the cello at measures 32-35 are heard not only in relation to theCs of the upper instruments there, but also in relation to the earlier Dl> of thecello itself at measure 27 following.

However interesting it may be to think of MM48 as a numerical "ground"organizing the tempi of measures 22-31, it is still clear that the players ofthe three upper instruments will not treat the cello part of measures 27-31as a succession of referential time-units. That is, they will not adjust theirown beats to conform in proper proportion with the cello beat of measures27-31. Nor will the cello and the two violins treat the constant beat atMM 180 in the viola from measure 25 on as a succession of referential time-units for the entire passage; at least they will not do so until after measure 30.During measures 22-24, in particular, the MM36, MM96, and MM 120 of thetwo violins and the cello will not be referred to a beat of MM 180 in the viola,for the viola has not yet entered there. Once the viola is in, at measure 25 andthereafter, its constant MM 180 will provide a useful check for the ensemblerhythm without necessarily establishing itself as referential, just as the sus-tained high C on figure 4.2b provides a useful check for the ensemble's into-nation without necessarily establishing itself as a root. Of course MM 180 doeseventually become much more referential during measures 33-35, along withits lower octave MM90; just so, on figure 4.2b, the pitch class C comes todominate the tonal texture.

The players may decide on purely notational grounds to use MM 120(changing to MM 180 by measure 33) or MM60 (changing to MM90 by measure 33) as a referential tempo for the entire passage. This makes a certainpractical sense for early rehearsals, but it can hardly be recommended for aneffective performance. We have all heard and seen players fighting their waythrough slow lyric lines, supposedly tranquillo like that of the first violin inmeasures 22-32 or sostenuto e cantabile like that of the cello in measures27-32, all the while jerking their feet up and down spastically in an erraticapproximation of some distantly related notational "beat." These lyric linesare not syncopated, as such a method of production makes them sound toboth player and listener. Rather, each line has its own autonomous local time-unit, with respect to which it should project an essentially "first-species"character.70


Indeed, the search for any one overriding referential time-unit, to governall of measures 22-31, is bound to fail. It must fail because it misconceives thenature of the temporal space at hand. That space comprises a multitude oflocally referential time-units, in various more-or-less consonant numericalrelationships among themselves. Only by recognizing that character of thespace can one hear the music progress over the passage, and not just "be."What I mean will be amplified by the following review of the earlier perfor-mance notes, (1) through (7).

(1) The MM48 of the cello at measure 27 will lie a good (numerical)octave below the opening MM96 of the second violin. That is, at measure 27the cellist should hear the cello line moving at half the rate the second violinhas been moving so far. This would be easy to hear if the cello came in asixteenth-note later. In fact there is no problem within measure 27, getting theattack of the pitch C2 at the right time-point: The player need only continuebeating MM 120 up to that point. But once the new melody has entered, itstempo may sound arbitrary and its character "syncopated" unless the playerhears the melody taking over, albeit out of phase, from the preceding tempo ofthe second violin, projecting its own local referential time-unit.

(2) The MM 120 of the second violin at measure 27 will match theopening MM 120 of the cello. MM 120 is the notated pulse. Still, the secondviolin should not simply be playing that pulse (with spastic foot-tapping or thelike). What makes measure 27 come alive and communicate in this connectionis an exchange of local referential beats, between the two instruments. Thesecond violin takes over the preceding beat of the cello, while the cello—as wenoted in (1) above—is about to take over the preceding local beat of thesecond violin, a rhythmic octave lower (and out of phase). The cellist and thesecond violinist should hear this "voice-exchange with octave transfer," tohear themselves conversing with each other in a quite familiar tradition ofchamber music. Figure 4.2b, showing the exchange of the symbolic pitchclasses D|? and F in the two instruments at measure 27, serves as a guide to thattradition here. Just as the players would match those pitch classes if playingfigure 4.2b, so they should match their exchange of tempi at measure 27 whenplaying the Carter passage.

(3) The MM 120-to-160 relation in the second violin, later in measure 27,will match the MM36-to-48 relation between first violin and cello hereabouts.Assuming that everything else has gone right so far,this will happen automati-cally if the second violin plays the dotted eighth (beating MM 160) as a precise3/4 of the quarter note (beating MM 120). That is not so easy to do as it is tosay, but the ability should be available to a well-trained player of twentieth-century music. Here, the two quarter notes of the second violin in measure 27will function as locally referential timespans for the player.

(4) The MM80-to-60 relation in the second violin, measures 29-30, willretrograde the MM120-to-160 relation of (3) above, one rhythmic octave 71


The figure transcribes the rhythm of the second violin from the middle ofmeasure 26 through measure 30 into a new notation, using MM 160 as a newnotational tempo of reference, that is, the tempo at which the new notatedquarter beats. This transcription brings out clearly how the rhythmic peripate-tics of the second violin are structured by the indicated relationship, that isMM80-to-60 answering MM120-to-160. The transcription "modulates" ourrhythmic hearing exactly as the second through sixth notes of the secondviolin on figure 4.2b would sound "modulated" if we listened to them in B[>minor, rather than D|? major. Just as the Fs of figure 4.2b would soundprimarily as fourths below the adjacent locally referential Bjjs, in our modu-lated pitch-hearing, just so the tempi of MM 120 and MM60 on figure 4.3sound in rhythmic proportions 3 :4 and 4: 3 to the adjacent locally referentialtempi of MM 160 and MM80, that lie alongside them.

Figure 4.3 demonstrates a logical internal structure for the second-violinpassage as a rhythmic entity in itself; this structure will surely not emerge if theplayer adjusts each individual tempo of figure 4.3 only to the beats of the violaat MM 180 hereabouts, or to the notated beat of the score at MM 120. It isin order to bring out the quasi-palindromic structure of figure 4.3 that the"tempo" of MM80 is represented, exceptionally, by only one time span.

(5) The MM60-to-90 relation in the second violin, measures 30-31, willmatch, a rhythmic octave lower, the earlier MM120-to-180 relation betweenthe cello at the opening of the passage and the viola entrance. This needs nofurther discussion; the relation of the relations will emerge without specialattention if the players are otherwise temporally "in tune."

(6) The MM90 of the first violin at measure 33 will match the MM90 ofthe second violin at measure 31. Obviously. Here MM90 is a referential localtime-unit. Likewise the MM90 of the second violin at measure 31 will havematched the MM 180 of the viola so far, using MM 180 as a referential timeunit.

(7) The MM36-to-90 profile of the first-violin part as a whole over thepassage will match in transposed retrograde (or inversion) the MM120-to-48

lower. Here "MM80" is projected only by the E|?-triad-event in measure 29 ofthe score; the tempo is not beat by any recurrent durational unit. Neverthelessa conceptual MM80 is useful to the player in a manner indicated by figure 4.3.

FIGURE 4.3

72


of the cello part as a whole up to measure 33. It is much harder to hear the largerhythmic proportion here than it is to hear the corresponding symbolic pitchproportion on figure 4.2b. Nevertheless, it will aid communication betweencello and first violin, as well as projection between ensemble and audience, ifthe first violinist hears the MM90 entrance at measure 33 speeding up theearlier MM36 of the instrument in exactly the same ratio as the MM48 ofthe cello at measure 27 slowed down that instrument's earlier MM 120. Theproportion can be sensed when the two instruments rehearse the pertinentmusic by themselves. Via this proportion, the MM90 of the first violin atmeasure 33 engages and completes a large mensural structure functioningover measures 22-35; it is not simply a surrender of the first violin passivelyto the beats of the second violin and viola at measure 31 and following. Ofcourse the first violin will use those beats to find its new tempo at measure 33.

To sum up: When performers confront the score of the Carter passageand the numerical network of local tempi or time-units displayed in figure4.2a, they should not concern themselves with the question, "Which one ofthese is the overall unifying referential tempo?" That question, a rhythmicanalog to the sorts of questions asked about pitch structures by Rameau,Riemann, and Hindemith among others, has no definite answer here. Even ifwe try to force an answer by selecting MM48, or MM 180, or MM60-then-90as a "root" tempo on the basis of this or that criterion, we shall still not beengaging thereby the temporal relationships that make this music progress andcommunicate. Those temporal relationships, some of which were discussed inperformance notes (l)-(7) above, involve patterns of local tempo "conso-nances," patterns in which many different tempi can assume locally referentialroles. This attitude toward the numerical network of figure 4.2a, and thesymbolic pitch network of figure 4.2b, is more in the spirit of Zarlino: It asksnot for one overriding referential unity, but rather for a splendid variety ofconsonant ratios among the entities involved, as they underlay and succeedone another, projecting a logical compositional idea.

In this way of hearing the rhythmic space through which the passagemoves, any time span has the potential for becoming locally referential, orbehaving as if it were. For example, let time span r be the span covered by thefirst note of the cello in measure 22 of the score; let s be that time span coveredby the F# of the first violin in measures 25-26; let t be that time span coveredby the A of the first violin in measure 26. We can say if we-wish that t begins16j r-spans after the beginning of r, and lasts If times the duration of r. Butthis way of listening corresponds to the "foot-tapping" performance of thefirst violin's melody. We can also say that t begins 1 s-span after the beginningof s, and lasts 1/2 the duration of s. And that way of listening corresponds to amuch more musical shaping of the melody. Taking s' as the time span coveredby the opening D of the first violin in measures 22-23, we could also say that tbegins 4 s'-spans after the beginning of s', and lasts 1/2 the duration of s'. This 73


way of listening corresponds to an even more musical shaping of the melody.Adopting the above attitude to Carter's rhythmic space, we implicitly

deny the relevance of GIS 4.1.2 as a model. Given abstract time spans s and t,we want to be able to conceive t as beginning a certain number of s-beats afters, rather than a certain number of possibly irrelevant "referential units" afters. If s is the span (a, x) and t is the span (b,y), the old GIS of 4.1.2 assignedint(s, t) = (b — a,y/x): t begins b — a referential units after s, and lasts y/xtimes as long. We want to replace that old notion of time-span interval by anew function: int(s, t) = ((b — a)/x, y/x). The new interval tells us that t begins(b — a)/x x-lengths (s-beats) after s, and lasts y/x times as long. The newinterval uses s itself as a measuring rod, to tell us how much later t begins.

FIGURE 4.4

Figure 4.4 shows how our new "interval" works. On the figure, fournumerical time-spans are indicated: Sj = (a^Xj), i1 = (b^y^, s2 = (a2,x2),and t2 = (b2, y2). We shall see later that it does not matter at all, for our newmodel, what the formal numerical time-point zero is, or what the formalnumerical time-unit is. That is, it does not matter to what percept we attachthe numerical time-span label (0,1). On the figure, we can imagine an "upperinstrument" projecting sx and ^ at a slow tempo, and a "lower instrument"projecting s2 and t2 at a fast tempo. The dotted slurs arching above the upperinstrument mark off x1 -lengths, durations that mark a contextual (potential)Sj-beat. The dotted slurs arching below the lower instrument mark off x2-lengths, durations that mark a contextual (potential) s2-beat. Using our newinterval construct, we write int^,^) = (4,2): t1 begins 4 Sj-beats after s1?

and lasts twice as long. Arithmetically, (b: — a^/Xj = 4 and y!/xx = 2. Using74

Generalized Interval Systems (3) 4.1.3.2

the new interval construct, we also write int(s2,t2) = (4,2): t2 begins 4 s2-beats after s2, and lasts twice as long. Arithmetically, (b2 — a2)/x2 = 4 andy2/x2 = 2.

Note that our new "interval from s2 to t2" is the same as our new"interval from sl to t^': int(s2,t2) = m^Su^) = (4,2). We shall discuss theimplications of this a lot more later on. Note particularly that sx precedes s2

on figure 4.4 in the obvious sense, while t x , the (4,2)-transpose of s^, followst2, the (4,2)-transpose of s2. One sees that our intuitions about formal "trans-position" will not be completely reliable in our new non-commutative GIS.(Our intuitions about interval-preserving operations will be trustworthy.)

We are of course still far from having constructed a formal GIS in whichour new notion of "interval" is to work. It is high time to do so now.

4.1.3.1 LEMMA: Let IVLS be the family of pairs (i, p), where i is a real num-ber and p is a positive real number. Then IVLS forms a group under thecomposition

0,p)(j,q) = (i + pj,pq).In this group, the identity is (0,1) and the inverse of the element (i, p) is theelement ( — i/p, 1/p). The group is non-commutative.

The proof of the Lemma will be left as an exercise for the interestedreader. Do not forget to show that the defined composition is associative:((i,p)(j,q))(k,r) = (i,p)((j,q)(k,r)).

4.1.3.2 THEOREM: Let int be the function that maps TMSPS x TMSPS intothe group IVLS of Lemma 4.1.3.1 according to the formula

int((a,x),(b,y)) = ((b-a)/x,y/x).

Then (TMSPS, IVLS, int) is a GIS.Proof (optional): We must show that Conditions (A) and (B) of Defini-

tion 2.3.1 obtain.(A): Given time spans (a,x), (b,y), and (c, z), we are to show that

int((a, x), (b, y))int((b, y), (c, z)) = int((a, x), (c, z)). We write

int((a, x), (b, y))int((b, y), (c, z))= ((b-a)/x,y/x)((c-b)/y,z/y)

by the formula defining int in the theorem. This= (((b - a)/x) + (y/x)(c - b)/y, (y/x)(z/y))

by the group composition in IVLS. Canceling factors of y in thenumerators and denominators, we see that this

= ((b — a + c — b)/x, z/x) which = ((c — a)/x, z/x).

And that number pair is indeed int((a, x), (c, z)). 75


So there can be at most one time span t in the desired relation to sand (i, p): That is the time span t = (b, y) = (ix + a, xp). And in fact thisparticular t is in the desired relation: int(s, t) = int((a, x), (ix + a, xp)); this =(((ix + a) — a)/x, xp/x) by the formula defining int; that = (ix/x, p), which isindeed (i, p) as desired, q.e.d.

At long last, we have before us a non-commutative GIS of musicalinterest. The GIS has important formal properties, which we shall now study.

4.1.4 THEOREM: GIS 4.1.3 has properties (A) and (B) below.(A): For any real number h, the interval from time span (a + h, x) to time

span (b + h, y) is the same as the interval from (a, x) to (b, y).(B): For any positive real number u, the interval from time span (au, xu)

to time span (bu, yu) is the same as the interval from (a, x) to (b, y).Proof:

76

(A): int((a + M),(b + h,y))= (((b + h) - (a + h))/x, y/x) (4.1.3.2)= ((b - a)/x, y/x) (algebra)= int((a,x),(b,y)) (4.1.3.2).

(B): int((au, xu), (bu, yu))= ((bu - au)/xu, yu/xu) (4.1.3.2)= ((b - a)/x, y/x) (algebra)= int((a,x),(b,y)) (4.1.3.2).

q.e.d.

Some commentary on this theorem is in order. The time spans (a, x),(b, y), and so on still rely numerically on an implied referential time-unit andan implied time point zero: (a, x) begins the number a of referential units afterthe referential zero time-point, and lasts the number x of referential units. Theessence of Properties (A) and (B) in the theorem above is that the numericalfunction int for the GIS under present discussion does not depend at all on thechoice of time point zero, or on the choice of referential time-unit.

To see this, suppose first that we move the referential zero time-pointback h units into the past (= forward ( —h) units into the future). An eventoriginally associated with the time span (a, x) will now be associated with thetime span (a + h, x): The event will begin a + h units later than the new zero

(B): Given the time span s = (a, x) and the interval (i, p), we are to find aunique time span t = (b, y) which lies the interval (i, p) from the time spans = (a, x). If any such b and y exist, they must satisfy the relation

int((a, x), (b, y)) = (i, p), or((b - a)/x, y/x) = (i, p), or(b — a)/x = i and y/x = p, orb = ix + a and y = xp.


time-point. Similarly, another event originally associated with the time span(b, y) will now become associated with the time span (b + h, y). Property (A)of the theorem says that in GIS 4.1.3, the formal interval between the timespans associated with the two events is not affected by this transformation.Even though the time spans themselves, as number-pairs, change from (a, x) to(a + h, x) and so on, the interval between transformed spans is the same as theinterval between the original spans.

Now suppose we change the referential unit of numerical time, so that theold unit is u times the new unit. A duration of x old units is then the same as aduration of xu new units. And the number a of old units after time-point zerois the same as the number au of new units after time-point zero. Hence theevents that were associated with time spans (a, x) and (b, y) in the old systemwill be associated with time spans (au, xu) and (bu, yu) in the new system.Property (B) of the theorem says that in GIS 4.1.3, the formal interval be-tween the time spans associated with the two events is not affected by thistransformation.

Thus, in the GIS of 4.1.3 the function int(s, t) will always deliver one andthe same pair of numbers (i, p), no matter what the referential time-unit andthe referential zero time-point by which we calculate numerical durations anddistances from time-point zero. To put this intuitively: Given event 1 andevent 2 in a piece, we can play the music whenever we want and at any tempowe want, without affecting at all the pair of numbers (i, p) which GIS 4.1.3 willdeliver to us as the formal interval between the numerical time spans as-sociated with the two events for any particular analysis.

The same can not be said for the commutative GIS of 4.1.2, studiedearlier. In that GIS the interval between time spans (a, x) and (b, y) was(b — a, y/x); accordingly, if we replace the referential time-unit so that eventsonce associated with those time spans now become associated with the newspans (au, xu) and (bu, yu), then the interval between the new spans is differ-ent. It is not (b — a, y/x), but rather (bu — au, y/x). We noted this earlier. Inour present terminology, we can say that GIS 4.1.2 does not enjoy Property(B) of Theorem 4.1.4.

In fact, a remarkable theorem is true. Not only does GIS 4.1.3 enjoy thetwo Properties of Theorem 4.1.4, but it is also essentially the only possible GISinvolving time spans as objects that enjoys those two Properties. The meaningof the word "essentially" in the above sentence is made clear by Theorem 4.1.5following.

4.1.5 THEOREM: Let GIS' = (TMSPS, IVLS', int') be any GIS with timespans for its objects that also enjoys Properties (A) and (B) of Theorem 4.1.4.Then the group IVLS of GIS 4.1.3 and the group IVLS' of the given GIS' areisomorphic via a map f such that, for all time spans s and t,

77int'(s,t) = f(int(s,t)).

4.1.6.1 Generalized Interval Systems (3)

Some commentary is in order before we launch into a proof. The idea ofisomorphism between (semi)groups was discussed in 1.11.1 and 1.11.2 earlier.To review here: If G and G' are abstract groups, a function f from G into G' is"an isomorphism of G with G'" if f is 1-to-1, onto, and a homomorphism. f is ahomomorphism if f (mn) = f (m)f (n) for every m and every n in G.

Supposing f an isomorphism of the abstract groups G and G', then thetwo abstract groups will have exactly the same algebraic structure under theidentification of m in G with its image f(m) in G'. So the first thing Theorem4.1.5 says is that IVLS and IVLS' have essentially the same algebraic structure,when we identify the member m of IVLS with its image f (m) in IVLS'. Second,Theorem 4.1.5 says that if we take the member m of IVLS to be int(s, t) inparticular, then the function f whose existence is asserted makes the imagef (m), a member of IVLS', equal precisely to int'(s, t). Thus the function int' is,so to speak, naught but the isomorphic image of the function int under theisomorphism f whose existence the theorem asserts. In this sense, the givenGIS' is "essentially" the same as GIS 4.1.3.

The (optional) proof of Theorem 4.1.5 is lengthy. To help break it intomanageable sections, we shall prove two lemmas. The lemmas appear belowas 4.1.6.1 and 4.1.6.2. After that, we shall go on to the proof of the Theoremproper.

4.1.6.1 LEMMA (optional): Let G and G' be abstract groups. Let f be a func-tion from G into G' such that for all m and all n in G, ̂ m)"1^) = f^n^n).Then f is a homomorphism.

Proof of (optional) Lemma: Set m = n = e in the given formula; we getf (e)"1^) = f(e). It follows that f(e)"1 is the identity in G'; hence f(e) is theidentity in G', e'.

Now set n = e and let m vary in the formula of the Lemma. We getf(m)"1f(e) = fCnT1). Since f(e) = e', we have f(m)-1 = f(m-1) for all m.Then we can rewrite the formula of the Lemma as

f(m~1)f(n) = ftm^n) for all m and all n.

As m runs through the various members of G, m"1 = o runs through thevarious members of G. Substitute o for m"1 in the rewritten formula; we thenobtain the formula

f (o)f (n) = f (on) for all o and all n in G.

And thus f is a homomorphism, as claimed.

4.1.6.2 LEMMA (optional): Within the group IVLS of 4.1.3.1,

(i,pr1(j,q) = ((j-i)/P,q/p)-Proof of (optional) Lemma: We already verified in 4.1.3.1 that (i,p)-1 in78

Generalized Interval Systems (3) 4.1.6.2

this group was the element (- i/p, 1 /p). Then (i, p)1 (j, q) = (- i/p, 1 /p) (j, q) =((-i/p) + (l/p)j,(l/p)q) = ((j - 0/P,q/P) as asserted.

Now we are ready for the (optional) Proof of Theorem 4.1.5. We take thetime span (0,1) as a referential object within the space of GIS' for purposes ofLABELing. That is, we set ref = (0,1). Then the function f for which we arelooking, the isomorphism of IVLS with IVLS', is defined by formula (i) below.

(i) f(i,p) = LABEL'(i,p) = int'((0, l),(i,p)).

On the left of formula (i) the number-pair (i, p) is considered as aninterval, a member of IVLS, while in the middle and on the right of theformula, the same number-pair is considered as a time span, a span beingLABELed in GIS' by its GIS'-interval from the referential object ref = (0,1).The number-pair (i, p), as a pair of numbers, can be interpreted either way.Now we can write f (i, p)"1 f (j, q)

= LABEL'(i, p)'1 LABEL'(j, q), by formula (i). This= int'((i,p),(j,q)) by 3.1.2. This= int'((0, p), (j — i, q)), since GIS' enjoys Property (A) of Theorem

4.1.4 by supposition. And this= int'((0,1), ((j — i)/p, q/p)) since GIS' enjoys Property (B) of

Theorem 4.1.4 by supposition. And that= f ((j — i)/p, q/p) by formula (i) above. And that= f((i,p)~1(j,q)) by Lemma 4.1.6.2.

Putting together the whole string of equalities we have just noted, substitutingm for (i, p) and n for (j, q), we see that we have proved

f(m)-1f(n) = f(m-1n)

for every m and every n in IVLS. By Lemma 4.1.6.1, we conclude that f is ahomomorphism. Since the LABEL' function is 1-to-l from TMSPS ontoIVLS' (3.1.2), the function f is 1-to-l from IVLS onto IVLS'. Thus f is anisomorphism of IVLS with IVLS'.

It remains to prove that f(int(s,t)) = int'(s,t). Set s = (a, x) and t =(b, y). Then int(s, t) = ((b - a)/x, y/x) and

f(int(s, t)) = int'((0,1), ((b - a)/x, y/x)) (formula (i))= hit' ((0, x), (b - a, y)) (since GIS' enjoys Property (B))= int'((a, x), (b, y)) (since GIS' enjoys Property (A))= int'(s, t). q.e.d.

To recapitulate: Theorem 4.1.5 shows that GIS 4.1.3 is essentially theonly possible GIS involving the family TMSPS whose function int is com-pletely independent of the referential time-unit and referential zero time- 79


point. To put it more intuitively, GIS 4.1.3 is the .only such GIS, essentially,that will return one and the same element of IVLS as the interval between thenumerical time spans associated with two musical events in a piece, regardlessof when you play the piece and what tempo you take. GIS 4.1.3 thereby has aprivileged theoretical status, as well as a special plausibility for modelingevents in the Carter passage and other pertinent music.

Since GIS 4.1.3 is non-commutative, it will provide a useful example forillustrating and reviewing the work of sections 3.4 and 3.5 earlier, work thatformulated the abstract theory of transpositions, interval-preserving opera-tions, and inversions.

4.1.7 NOTES: Within GIS 4.1.3, the following formulas and facts are true:(A): Given an interval (i, p) and a time span (a, x), the transposition of the

given time span by the given interval is

T(iip)(a,x) = (a + ix,px).

(B): If we fix (0,1) as ref, a referential time span, then the number-pair(a, x), as a member of IVLS, is the LABEL for the time span (a, x):

LABEL(a,x) = int((0, l),(a,x)) = (a,x).

(C): The (i, p)-transpose of the time span (a, x) is the number-pair givenby the composition in IVLS of the two intervals (a, x) and (i, p). T(i(p)(a, x) =(a,x)(i,p).

(D): Using the number-pair (a, x) in the same way, as both a time spanand an interval, we can show that the interval-preserving operation P(h)U)

transforms the time span (a, x) into the time spanp<h,u)(a, x) = (h, u)(a, x) = (h + ua, ux).

(E): The only central member of IVLS is the identity interval (0,1).(F): No transposition preserves intervals, and no interval-preserving

operation is a transposition, the identity operation T(0>1) = P(O,D excepted.(G): The operation of (c, z)/(d, w) inversion, applied to the time span

(a, x), yields the time span

I!c;?(a,x) = (d + (c - a)w/x,zw/x)= (d,w)(a,x)~1(c,z).

(H): Given time spans s, t, s', and t', then !£'. = I, as an operation onTMSPS if and only if s' = s and t' = t.

(I): There are no interval-reversing operations on TMSPS.Proofs and commentary: (A): Via 3.4.1, the transposition of (a, x) by (i, p)

is that time span (b, y) which lies the interval (i, p) from the time span (a, x), i.e.which satisfies the equation int((a, x),(b,y)) = (i,p). Thus (b, y) satisfies theequation ((b — a)/x, y/x) = (i, p); whence (b — a)/x = i and y/x = p. So b =a + ix and y = px. The transposed time span (b, y) = (a + ix, px) can be80

Generalized Interval Systems (3) 4.L7

described as follows: b lies i x-spans later than a; y lasts p times as long as x.If we turn back to figure 4.4 (p. 74), we will see that the time span t j of thefigure is the (4,2)-transpose of s x : ilbegins at bj, 4 xt-spans afterat; ttlastsa duration of y t, 2 times the duration Xj of s^^. Likewise, t2 on the figure is the(4,2)-transpose of s2. We noted while studying the figure earlier that sx

precedes s2, while tl follows t2. We may say that transposition operations, inthis GIS, do not only fail to preserve intervals, they even fail to preservechronology.

(B) of the Notes is a straightforward computation: LABEL(a,x)= int((0, l),(a,x) = ((a - 0)/1, Ix) = (a,x).

(C) of the Notes applies (B) to the formula of 3.4.3, and (D) applies (B) tothe formula of 3.4.4. The interval-preserving operation P(h u) first blows up orshrinks the sample time span (a, x) by a factor of u, transforming (a, x) to(ua, ux), and then moves the latter time span backward or forward in time by hor (—h) numerical units, transforming (ua, ux) to (h + ua, ux) = P(a, x). Re-member that these interval-preserving operations are not formal "transpo-sitions" in our non-commutative system!

(E) of the Notes is proved as follows. Suppose the interval (i, p) is centralin IVLS, that is (i,p)(j,q) = (j,q)(i,p) for all (j,q). Expanding the binarycomposition on each side of that equation, we infer (i + pj, pq) = (j + qi, qp)for all j and all positive q. Then i + pj = j + qi for all such j and q, whence(p — l)j = (q — l)i for all such j and q. Take j = 1 and q = 1 as one such j andq; then (p — 1)1 = (1 — l)i or p — 1 = 0; we infer that p must be equal to 1.Now we can go back to our general equation, (p — l)j = (q — l)i; substitutingp = 1, we infer that (q — l)i = 0 for all positive q. But then i is obviously zero.So p = 1 and i = 0; our given central interval (i, p) must be the identity interval(0,1).

(F) of the Notes then follows from Theorem 3.4.8.(G) of the Notes can be computed from 3.5.2 together with (B) of the

Notes. Or it can be computed directly from the defining formula of 3.5.1, usingthe known group structure of IVLS here.

(H) follows from 3.5.3, where we proved that l£ = lls if and only if

t' = Ig(s') and the interval int(s', s) is central. Via (E) above, this will happenif and only if t' = Ig(s') and s' = s. Since Is(s) = t, this will happen if and only ifs' = s and t' = t.

(I) of the Notes simply restates 3.6.4 in the present context.

We may use figure 4.4 yet once again to picture the effect of an inversion.On that figure, we noted that int(s2,t2) = int^,^). Hence, via Definition3.5.1, t2 is the tlls2 inversion of s^ that is, Ij^Sj) = t2.

This concludes our study of a non-commutative GIS which is also arhythmic GIS of musical interest. We shall now study some timbral GISstructures. 81


4.2.1 EXAMPLE: Given positive numbers s(l), s(3), and s(5), let the number-triple s = (s(l), s(3), s(5)) denote the class of all harmonic steady-state sounds(i.e. periodic wave-forms) whose first, third, and fifth partials have respectivepower s(l), s(3), and s(5). Let S be the family of all such number-triples s, ass(l), s(3), and s(5) range over all positive values.

Given positive numbers i(l), i(3), and i(5), let us imagine at hand one ormore "devices" (e.g. computer procedures) which have this property: When-ever a harmonic sound is led as input into such a device, the device outputs aharmonic sound whose first, third, and fifth partials have respectively i(l)times, i(3) times, and i(5) times the power of the corresponding input partials.Given i(l), i(3), and i(5), let the number-triple i = (i(l), i(3), i(5)) denote theclass of devices that transform harmonic sounds according to these propor-tions for the first, third, and fifth partials. Let IVLS be the family of all suchnumber-triples i, as i(l), i(3), and i(5) range over all positive values. IVLS is agroup under the combination ij = (i(l)j(l),i(3)j(3),i(5)j(5)).

Given harmonic class s = (s(l), s(3), s(5)) and harmonic class t =(t(l), t(3), t(5)), take int(s, t) to be that member i of IVLS for which i(l) =t(l)/s(l), i(3) = t(3)/s(3), and i(5) = t(5)/s(5).

Then (S, IVLS, int) is a GIS. That is, Conditions (A) and (B) of Definition2.3.1 obtain. The GIS is commutative. When int(s, t) = i, any sound in class twill have i(l), i(3), and i(5) times the power of any sound in class s, at its first,third, and fifth partials respectively. Another way of regarding the statement"int(s, t) = i" is to think: Any sound of class s, when led as input to any deviceof class i, will cause a sound of class t to be output.

The fundamental frequencies of the sounds are irrelevant here; we areconcerned only with certain aspects of their timbral profiles.

If a given sound is led through a device of class i, and if the resultingoutput is then led through a device of class j, the final output will be a sound ofthe same class as that which would have resulted, had the original sound beenled through a device of class ij. Or, more simply, a device of class i con-catenated with a device of class j forms a device of class ij.

We can make many variations on the specific GIS just discussed. Forexample, instead of considering partials #1, #3, and #5, we could insteadconsider partials #1, #2, and #4. Or we could consider partials #1-through- # 5, or # 1-through- # 8, or # 1-through- # 8-except-for- # 7, and soon.

We can use GIS structures of this sort to build more complex GISstructures of interest. For instance, let GISj be the GIS of the sort justdiscussed which considers partials # 1-through-#8 of harmonic sounds. Weshall call an element s = (s(l),..., s(8)) of GISj a "pertinent spectrum." Nowlet us take as GIS2 a familiar GIS involving the space S2 of "time points." Weimagine a referential zero time-point and a referential time-unit fixed, so thatwe can label the elements of S2 by real numbers a. IVLS2 is the additive group82


of real numbers and, given time points a and b, intâ, b) is the number b — aof time units by which b is later than a. (b — a later = a — b earlier.) Let usexplore the direct product GIS3 = GISj (x) GIS2. The elements of S3 =Sj X S2 are pairs (s, a), where s = (s(l), ... , s(8)) is a pertinent spectrumand a is a time point. The pair (s, a) models a class of sounds having pertinentspectral profile s at time a. A finite set of such pairs, say the set DVSP =((Sj, aj), (s2, a-j), ... , (SN, a^), models a class of sounds that have spectrum Sjat time at, spectrum s2 at time a^ ... , and spectrum SN at time a,̂ . We considerDVSP to be an unordered set of S3-elements, since the chronological order ofthe time points an imposes a natural ordering on the member pairs of DVSP,no matter in what order we list those pairs. For convenience, we shall assumethe members of DVSP to be listed above in chronological order, that is withaj < 82 < ••• < aN. Supposing the time points an to be reasonably close, thenDVSP will model a class of sounds with a certain "developing spectrum."Each sound of this class has pertinent spectrum sn at time an.

FIGURE 4.5

Figure 4.5 displays DVSP, with N = 5 in this case, as an array of num-bers. If we imagine the plane of the page as a base, and erect at each entrysm(n) a spike of heights sm(n) jutting up from that page, we shall obtain a sortof sketch for a relief map that shows how the spectrum of the sound developsover time. Supposing the time points aj through a5 to be dense enough so asto catch enough salient features of the sound-class involved (e.g. times whensome partial has a pronounced local maximum or local minimum value), thenwe can consider this sketch to be a good approximation for a continuous re-lief map that characterizes the class of sounds with respect to its developingspectral "signature." This sort of relief-map representation is in common usetoday as a means for studying various classes of harmonic sounds, including 83


familiar instrumental sounds in particular.6 As we have seen, any such reliefmap can be regarded as approximately a finite (unordered) subset DVSP ofS3. The way that DVSP = ((Sj, ax), (s2, a2),..., (SN, aN)) develops through itsown intrinsic chronology can be studied in exactly the same way we earlierstudied the unfolding interval vector for a chronologically developing set inanother direct-product GIS. That was in section 3.3.1, where we applied ourstudy to the analysis of a passage from Webern's Piano Variations.

4.2.2 EXAMPLE: Fix a lower frequency LO and a higher frequency HI; weshall represent a varying frequency between LO and HI by the variable x. By a"rational spectrum" we shall mean a function s of the variable x, taking onreal values, which satisfies Conditions (A) and (B) below. Condition (A): Thefunction s can be written as the quotient of polynomial functions. That is,there exist polynomial functions p and q in x such that s(x) = p(x)/q(x) forevery x between LO and HI. Condition (B): s(x) is strictly positive for every xbetween LO and HI. We shall take as the family S for a GIS the family of allrational spectra s.

The rational spectra form a group under multiplication. For if s(x) =Pi(x)Ah(x) and t(x) = p2(x)/q2(x), then p3(x) = p1(x)p2(x) and q3(x) =q1(x)q2(x) are polynomial functions. Hence (st)(x) = s(x)t(x) can be writtenas a quotient of polynomial functions: (st)(x) = p3(x)/q3(x). Furthermore,(st)(x) is strictly positive since both its factors, s(x) and t(x), are strictly posi-tive. Thus the product of two rational spectra is itself a rational spectrum.The function 1 (x) = 1 is a rational spectrum; it is a multiplicative identity fothe family of rational spectra. If s is a rational spectrum, so is 1/s, where(l/s)(x) = l/s(x); the spectrum 1/s is an inverse for s within the multiplicativesystem of rational spectra. We shall take as the group IVLS for our GIS thefamily of rational spectra again, now considered as a multiplicative group.

Given rational spectra s and t, considered as members of S, we shall takeas int(s, t) for our GIS the rational spectrum t/s, considered as a member ofIVLS. It is straightforward to verify that (S, IVLS, int) is a GIS.

The GIS models a system of "linear filter classes." With each rationalspectrum s we can associate a class of filters. Any filter in this class can be builtup from two simple kinds of filters, "all-zero" and "all-pole" filters. Any filtein class s will transform an input sound to an output sound in such a way thatthe power of frequency x in the output is equal to s(x) times the power of x inthe input. The manipulation of sounds and filters in this way is characteristic

6. Good examples of the practice can be found in "Lexicon of Analyzed Tones," a series ofanalysis and plotting programs by James A. Moorer and John Grey published in Computer MusicJournal. "Part I: A Violin Tone" appeared in vol. 1, no. 2 (April 1977), 39-45. "Part II: Clarinetand Oboe Tones" appeared in vol. 1, no. 3 (June 1977), 12-29. "Part III: The Trumpet" appearedin vol. 2, no. 2 (September 1978), 23-31. There is also a handsome "relief map" on the cover ofthat issue.84

Generalized Interval Systems (3) 4,3

of certain recent work in computer music.7 We can sensibly talk of "transpos-ing" and "inverting" such filter-classes in our GIS; since the GIS is commuta-tive, these operations will behave in an intuitively familiar way, following thelaws developed in sections 3.4 and 3.5 earlier.

We can vary our GIS by varying the region within the frequency x varies.We can change the values of LO and HI; we can even consider disconnectedregions within which x is to vary.

4.3 METHODOLOGY: In both GIS 4.2.1 and GIS 4.2.2, the formal relationsinvolved match our sonic intuitions only to a certain extent. In either GIS, thatis, we may have int(sp tj) = int(s2, tj), while the intuitive proportion betweenSj and tj does not much "sound like" the intuitive proportion between s2 and^ The models suffer here by comparison with the constructs of WayneSlawson, who has developed an elegant model for an "intuitive" timbralspace.8

Yet such considerations should not necessarily lead us to ignore GIS4.2.1 and GIS 4.2.2. To relate "natural" mathematical structure with intuitionis a problem in connection with virtually all theories involving sensory stimuli.For instance, it is mathematically natural to compare the amplitude of two sinwaves by saying that one wave has i times the amplitude of the other; this isespecially natural if both waves are at the same frequency. Yet if s, and tj =i • Sj are sin waves of the same low frequency, while s2 and t2 = i • s2 are sinwaves of the same middle-range frequency, our intuition about the relativeloudness of Sj and t, may differ considerably from our intuition about therelative loudness of s2 and t2. Still, nobody would propose that we should feelfree to ignore quotients of amplitudes in a study of musical sounds, justbecause they do not always conform to our intuitions of loudness, give or takesome simple transformation.

This is the methodological point: It is unfair to demand of a musicaltheory that it always address our sonic intuitions faithfully in all potentiallymusical contexts under all circumstances. It is enough to ask that the theorydo so in a sufficient number of contexts and circumstances. Perhaps, too, it isfair to ask that the theory be potentially able to address our intuitions in anygiven musical situation, provided that the situation develops in a suitablemusical manner.

To support the methodological point, let us explore certain thematic fea-tures from the first movement of Chopin's Bl»-Minor Sonata.

On figure 4.6, (a) symbolizes aspects of the motive from the opening

7. The techniques are explained lucidly by Richard Cann in "An Analysis/Synthesis Tuto-rial," Computer Music Journal. Part 1 is in vol. 3, no. 3 (September 1979), 6-11. Part 2 is invol. 3, no. 4 (December 1979), 9-13. Part 3 is in vol. 4, no. 1 (Spring 1980), 36-42.

8. "The Color of Sound: A Theoretical Study in Musical Timbre," Music Theory Spectrumvol. 3 (1981), 132-41. 85

4.3 Generalized Interval Systems (3)

Grave measures, written in augmented rhythmic notation to fit the subsequenttempo of the Doppio Movimento. (b) symbolizes the motive of the first theme,(c) symbolizes a motive from the end of the bridge, and (d) symbolizes themotive of the second theme. At the end of the exposition there is a big push tocadence, leading to a big dominant of D[? major. Then there is a repeat sign.Chopin's notation is ambiguous; for this discussion I will assume what I hearstrongly in any case, that the repeat goes back to (a), rather than (b). Then thopening note of (a) completes the Dt? cadence; one thereby hears F|? as thesecond note of (a) all the more strongly, since it is a third of the tonicized Db,despite the notated E natural and the subsequent leading of the note as Enatural in the bass line.

Here is the assertion I wish to study: The first melodic dyad of (b),marked y on the figure, belongs to the same interval class as x, the first melodicdyad of (a). This relation, like the relation of our four sin tones earlier, isformally "true" but intuitively problematic. At least, the relation of x and ydyads is hard to hear when we first hear the first theme, the first time throughthe exposition. But, I claim, the asserted relation has the potential for becom-ing audible, and in fact it does become audible, even highly significant, thesecond time through the exposition.

To hear this one should take motive (b), rather than motive (a), as a poinof departure. By the end of the bridge section, motive (b) has been trans-formed into motive (c). Rhythmically, (c) augments the durations of (b) by factor of 2, and then augments its own last two notes by yet another factor of2. Motive (d) introduces the second theme immediately thereafter. Rhythmi-cally, (d) augments (c) by yet another factor of 2, and exchanges the durationsof the first two notes (counting the rest as part of the note that precedes it).And then, going around the repeat, motive (a) augments (d) rhythmically bystill another factor of 2. Thus the chain of motive-forms (b)-(c)-(d)-(a bis) isgenerated by a very consistent, indeed relentless, process of rhythmic expan-sion. After the repeat, when we continue on from (a bis) to (b bis), we aleaping from the end of the chain (b)-(c)-(d)-(a), back to its beginning and

FIGURE 4.6

86

Generalized Interval Systems (3) 4.3

motivic generator. In this larger and later context, we recognize that (b) and(a), the boundary forms of the chain, are rhythmically transformed variants,each of the other. And so we are much more willing to perceive other sorts ofrelationships linking (b) with (a). In particular it is now much easier, I wouldsay proper and important, to hear dyad y as an ironically scurrying transfor-mation of dyad x, with its portentous weight. We are helped in hearing thisrelationship by the dyad marked z on figure 4.6, which we have by now heardagain and again during the second group. We are also helped by the big D|?cadence prepared at the end of the exposition, which helps us hear F|? at therepeat of (a). The first time around we may have had a certain predilection forE natural because of our associations with the opening of Beethoven's Sonataop. I l l , even specifically with measures 4| to 5^ of that piece. But the secondtime around, when we clearly hear F(? (as well), it is much easier for us toassociate the x dyad, as root-and-minor-third of Dj?, with the y dyad, minor-third-and-root of B(?, particularly since the two dyads both begin on a D(?.

To repeat my methodological claim: One should not' ask of a theory,that every formally true statement it can make about musical events be aperception-statement. One can only demand that a preponderance of its truestatements be potentially meaningful in sufficiently developed and extendedperceptual contexts.

87

Generalized Set Theory (1):

Interval Functions; CanonicalGroups and CanonicalEquivalence; EmbeddingFunctions

In this chapter we shall generalize certain aspects of atonal set theory so as toapply in the context of any GIS.1

5.1.1 DEFINITION: Given a GIS(S, IVLS,int), we shall mean by a set in thepresent chapter a finite unordered subfamily of S.

5.1.2 DEFINITION: If f is any mapping of S into itself and X is any set, wedenote by "f (X)" the set of elements f (s) formed as s varies over X.

That is, if X = (Sj, s2 , . . . , SN), then f(X) is the set whose members aref(Si), f(s2), ..., and f (SN).

If f is not 1-to-l then some of these f-values may not be distinct; then f (X)will have a smaller cardinality than X. If f is 1-to-l then the N f-values listedabove will be distinct, and f (X) will have the same cardinality as X.

5.1.3 DEFINITION: Given a GIS, given sets X and Y, then the XjY intervalfunction is a function IFUNC(X,Y) which maps the group IVLS into thefamily of non-negative integers as follows:

For each interval i in IVLS, the value of the function, IFUNC(X, Y) (i),counts the number of distinct pairs (s, t) in S x S such that s is in X, t is in Y,and int(s, t) = i.

That is, IFUNC(X, Y) (i) tells us in how many different ways the intervali can be spanned between (members of) X and (members of) Y.

Usually, the context will make it clear when we are talking about the

1. The agenda, in chapters 5 and 6, parallels and expands upon the presentation of atonalset theory I developed in my article, "Forte's Interval Vector, My Interval Function, andRegener's Common-Note Function," Journal of Music 77ieor>>vol.21,no.2(Fall 1977), 194-237.88

5

Generalized Set Theory (1) 5.13

function int, which maps S x S onto IVLS, and "the interval function" which,given sets X and Y, maps IVLS into the non-negative integers.

IFUNC does not figure heavily in the standard literature of atonal settheory. Let us study some examples to see how it applies to that theory; thereader may thereby see the point of the construction, both in that specificapplication and more generally.

FIGURE 5.1

Figure 5.1 (a) displays the two pitch-class sets Xj = (E, B|?) and Yt =(F, A, C#) as a sort of symbolic melodic antecedent and consequent. Arrowsdrawn from each note of Xt to each note of Y t show which intervals can bespanned, how many ways, between notes of X^ and notes of \l.Here, theeven-numbered intervals cannot be spanned at all, and each odd-numberedinterval can be spanned in exactly one way. Hence IFUNC(Xl5 YJCi) = 0 ifi is even, and = 1 if i is odd.

Figure 5.1(b) displays the two sets Xj (as before) and Y2 = (G, A, B) in asimilar format. We see that IFUNC(X1, Y2) (i) also = 0 if i is even, and = 1 iiisodd.SoIFUNC(X1,Y2) = IFUNC^Y^ as a function on IVLS, eventhough Y2 is not the same set as Yt—indeed, Y2 is not even a form of Yt.

2

Figure 5.1(c) displays the new sets X2 and Y3 in the same format. HereX2 is different from Xx and Y3 is different from either Y! or Y2; yetIFUNC(X2, Y3) is again the same function of i: Its value is 0 if i is even and1 if i is odd.

Figure 5.1 (d) displays yet another pair of sets, X3 = (E[?, F, G, A, B, C#)and Y4 = (E). The X3/Y4 interval function is still and again the same:IFUNC(X3, Y4)(i) is 0 if i is even, 1 if i is odd. The set X3 here is of different

2. I investigated this sort of phenomenon in an early article, "Intervallic Relations betweenTwo Collections of Notes," Journal of Music Theory, vol. 3, no. 2 (November 1959), 298-301. Itsstyle, unfortunately, makes few concessions to a non-mathematical reader. 89

5.1.3 Generalized Set Theory (1)

cardinality from Xj and X2; Y4 is likewise of different cardinality from Y l 5Y2 ,andY3 .

Figure 5.1 (d) is displayed in a contrapuntal rather than a melodic format.In general, one could also distinguish a set X from a set Y in other formats: byinstrument, by mode of attack (staccato/legato), or by register (as opposed to"voice"), to give only a few examples. The melodic antecedent/consequentformat, which seems particularly suggestive, was proposed by Michael Bush-nell during his theoretical studies at Stony Brook in the 1970s. Bushnell alsoinitiated and carried through the basic work on the specific analysis following,involving a passage from Webern's op. 7, no. 3, the third of the Four Pieces forViolin and Piano.

Figure 5.2(a) reproduces the score through the opening of measure 9. Theright hand of the piano over measures 3-8 comprises two melodic phrases,one filling measures 3-4, the other beginning on the B of measure 5 andextending through the F# of measures 7-8. This is all the lyric melody there isin the piece.

Figure 5.2(b) displays as X and Y the pitch-class sets that underlie thetwo melodic phrases. The pitch noteheads representing the pitch classes E, C#,and Eb within Y have been brought down an octave from the music. This ispartly for convenience, but partly too because of an idea which will emergelater. Figure 5.2(b) also displays the sets Z0 and Z3 projected by the violinostinato that accompanies Y. The repeat sign on the figure indicates that wepass from Z3 to Z0, as well as from Z0 to Z3, during the course of this music.

Figure 5.2(c) displays some interrelations of X, Y, and the Z-forms,interrelations that all involve the pitch-class interval 3. Each arrow on thefigure indicates a T3 relation of one kind or another. The bottommost arrowdepicts the T3 relation between Z0 as a whole and Z3 as a whole, a T3-relationof pitches as well as pitch classes. The arrows directly beneath the lower staffof the figure show T3 pitch-relations between the "fourths" of Z0 and those ofZ3, and also T3 pitch-class relations between the "fourths" of Z3 and those ofZ0, when Z0 is restated directly after Z3. Among all these fourths, Eb-Abwithin Z0 and its T3-transform F#-B within Z3 are of special significance. Athe crossed lines between the staves of figure 5.2(c) indicate, the Eb-Ab thatends ordered Z0 summarizes in retrograde the Ab-Eb that bounds the totalspan of ordered melodic phrase X; analogously, the F#-B that ends orderedZ3 summarizes in retrograde the B-F# that bounds the total span of orderedmelodic phrase Y.

This phenomenon suggests that we explore some sort of T3 relationshipbetween ordered X and ordered Y. On the top staff of figure 5.2(c), the Ab-Ebboundaries of X and the analogous B-F# boundaries of Y are marked bybeamed open noteheads. The medial Bb of ordered X is attached to the X-beam with a stem from a solid notehead; a corresponding C# = T3(Bb) withinordered Y is notated analogously. To hear a function for C#-within-Y analo-90

Generalized Set Theory (1) 5.1.3

FIGURE 5.2a

91


FIGURE 5.2b

FIGURE 5.2c

gous to the function of B(?-within-X, the reader may listen to the followingfeatures of the passage. First, the C# and the F# within Y both receive agogicaccents, marked "a.a." on figure 5.2(c). One may query just what this means,in the piano register of figure 5.2(a) at the given tempo. It surely meanssomething, if only something conceptual; the pianist should be thinking like asinging instrument here. Second, the crescendo that begins at the B[? within X,a crescendo reproduced on figure 5.2(c), is analogous in some degree to thecrescendo that begins at the C# within Y, even though the latter crescendodoes not get all the way to the final note of its phrase. Third, the beamed B, C#,and F# of Y, within figure 5.2(c), occur as every-third-note of ordered Y; thereis a serial regularity about their occurrence. Fourth, once the E, C#, and E[> ofY-within-the-music have been brought down an octave to provide the note-heads for Y-within-figure 5.2(c), it is easy to hear the latter structure as a com-pound gesture, counterpointing the rising beamed B-C#-F# (= T3 (orderedX)) against the falling chromatic counter-gesture-(F-E)-(E[?-D)-. That"falling chromatic line" on figure 5.2(c) fills in the chromatic space betweenC# and F#, the medial and final beamed notes of the rising gesture.92


The uppermost arrow on figure 5.2(c), then, indicates a structural T3

relation between set X and set T3(X)-embedded-in-Y. It seems secure to assertthis relation between the sets, for it seems reasonable to assert much strongerT3 relations, e.g. the structural embedding of T3(ordered X) as every thirdnote of ordered Y, or even the idea that phrase Y as a whole is inter alia adiminuted (ornamented/troped) version of T3 (phrase X).

Now we shall inspect some IFUNC values, to see how they interactwith these analytic ideas. Figure 5.3(a) tabulates four interval-functions,IFUNC(Z0,Z3), IFUNC(X, Y), IFUNC(X,Z0), and IFUNC(Z3, Y).

IFUNC (Z0, Z3) tells us that 4 intervals of 3 are spanned between Z0 andZ3; hence Z3 = T3(Z0). Likewise Z3 = T9(Z0) as a pitch-class set, sinceIFUNC(Z0,Z3)(9) = 4. This T9 relation is concealed, not revealed, by theregistration of the pitches involved in the music. Nevertheless the T9 relationhas a certain rhythmic effect, as portrayed in figure 5.3(b). The pair "(9,7>)"on that figure is a direct-product interval; it means "a pitch-class interval of 9is spanned at a distance of 7 sixteenth-notes between attacks." 93

FIGURE 5.3


The less prominent values of 2 taken on by IFUNC(Z0, Z3) (less promi-nent than 4) also have rhythmic implications. Figure 5.3(c) shows howIFUNC(Z0,Z3)(2) = 2 and IFUNC(Z0,Z3)(10) = 2 articulate the rhythmof the violin obbligato. (The sixteenth-note symbols are now omitted from thedirect-product intervals.) Figure 5.3(d) shows how IFUNC(Z0,Z3)(4) = 2and IFUNC(Z0,Z3)(8) = 2 articulate the same obbligato in a differentrhythmic way.

IFUNC helps us to explore these intervallic/rhythmic substructures morecarefully than we otherwise might. (We have plenty of time to do so at thewritten tempo.) The rhythmic effect of IFUNC(Z0, Z3) (3) = 4 is of course tosupport the quintuple periodicity of the ostinato, as in figure 5.3(e).

Now let us return to the IFUNC table of figure 5.3(a) and inspectIFUNC(X,Y). Since X has cardinality 3 and IFUNC(X, Y)(3) = 3, thisinterval function tells us that T3(X) can be embedded within Y. That is, thefunction informs us that such an embedding is "true," and would lead us toinspect its potential musical significance in the passage at hand if we had notalready done so.

The value of 3 is a maximal possible value for IFUNC(X, Y), just as 4 wasa maximal possible value for IFUNC(Z0,Z3). Both functions take on theirmaximal values on the argument i = 3. This sets up a "true" proportionamong the four sets involved: X is to subsequent Y just as Z0 is to subsequentZ3, so far as a certain property is concerned (having a maximum IFUNCvalue at the argument i = 3). Our discussion of figure 5.2(c) has shown thatthis theoretical truth in fact reflects a musically significant relationship.

IFUNC(Z0, Z3) also took on its maximum value at the argument i = 9.But IFUNC(X, Y) does not. Instead, IFUNC(X, Y) has a maximum at i = 8.This tells us that T8(X) as well as T3(X) can be embedded within Y. Is thefact musically significant as well as true? To explore the matter, let us firstfind the notes of T8(X) as they occur within Y. X = (Ab,B|7,E|7), soT8(X) = (E, Ftf, B). Within ordered-Y, the members of T8(X) appear in therotated order B-E-F#. Inspecting the score again, one asks if these threenotes have any special functions that affect the shaping of phrase Y in themusic. I believe they do. Namely, they are the registral and temporal boundarytones for phrase Y. That is, B is at once the first note and the lowest note of themelodic phrase; E is its highest note; F# is its last note. So one can plausiblyassert an overall shape for phrase Y that uses T8(X) as a bounding frame,along the lines of figure 5.4.

We have been thinking of the set (B, E, F#) as a transposed form of set X,because that transpositional relation is what IFUNC has brought to ourattention. As a series, the succession B-E-F# is a rotation of E-F#-B, whichis T8 (ordered X). B-E-F# can also be generated as the retrograde series ofordered-X inverted about C#. IFUNC cannot suggest this relation to us; in94


chapter 6 we shall generalize the interval function to an "injection function"which can. In any case, our noticing that IFUNC(X, Y) (8) = 3 has suggesteda search that has led us to the musical relationship of figure 5.4, a relationshipthat deserves serious aural attention. The relationship engages the register ofthe high E in the music for phrase Y; we would not hear any boundary functionfor E if it were an octave lower as in figure 5.2(c).3

Let us return again to the table of figure 5.3(a). We can notice there aninteresting resemblance between IFUNC(X,Z0) and IFUNC(Z3,Y). Bothfunctions take on their maximum values at the arguments i = 0, i = 5, i = 6,and i = 11. Figures 5.5(a) through (d) show how the four intervals 0, 5,6, and11 respectively govern both a model/expansion relation of X to Z0 and ananalogous model/expansion relation of (unordered) Z3 to Y.

(a) through (d) of the figure show how the two "fourths" of X, transposedvariously by i = 0, 5,6, or 11, map into the two "fourths" of Z0; this projectsthe relation IFUNC(X,Z0)(i) = 2 in each case, (a) through (d) of the figureshow analogously how the four possible subtrichords of Z3, all of which are inForte's set-class 3-5, can each be transposed by one of the key intervals i = 0,5,6, or 11 so as to map into one of the two subtrichords of Y that lie in the set-class 3-5, both boundary trichords in a certain sense; this projects the analo-gous relation IFUNC(Z3, Y)(i) = 3 in each case.

The latter four mappings, of Z3-trichords into Y-trichords, suggest thatY can be articulated, when heard "against" Z3, as suggested by figure 5.5(e).There we hear Y articulated into a beamed temporal "boundary" comprisingthe union of its two 3-5 trichords, plus a bracketed temporal "interior"comprising C#, E\>, and D. The articulation of Y in this fashion is reinforcedby the coincidence of the interior set (C#, Efr, D) with the cadence setaccompanying the melodic Efr that concludes ordered-X in the music. Figure5.5(f) shows the pitches in the music between the attack of the X-terminatingE|? in measure 4 and the comma in measure 5 that separates X from Y in themelody. The set of pitch classes heard in figure 5.5(f) recurs as the interior setof Y, bracketed on figure 5.5(e).

A methodological note is in order. Some readers may feel confused ratherthan enlightened by the variety of ways in which we have shaped and arti-culated Y during our discussion, specifically in figures 5.2(c), 5.4, and 5.5(e).

3. I am indebted to Taylor Greer for suggesting to me (in another connection) that the"boundary set" of an atonal melodic phrase often has special set-theoretic functions.

FIGURE 5.4

95


96

FIGURE 5.5

If Y "really is" an ornamented version of T3(X), then are we not "wrong"to consider Y as a spatio/temporal gesture framed by T8(X), or to considerY as articulated by figure 5.5(e)? On the other hand, if Y "really is" a spatio/temporal gesture framed by T8(X), then are we not "wrong" ... (and so on)?One often hears such notions accompanied by a thought like, "after all, thepianist must decide which way to play it."

Concerning these issues, the first thing to be said is that the last remarkin the above paragraph seriously under-estimates and misapprehends theresources available to a good pianist (or performer in general) even in acontext as constrained as that of the Y-phrase in the music. What one canhear, one can play. Let us suppose now that the possibly unquiet reader doeshear something significant, or at least engaging, about each of the Y-articulations. (Otherwise there would be no disquiet and no problem.) Thedisquietude arises intellectually, from considering Y as something which


"really is," independent of any specified environment. The interpretation of Ymanifest in figure 5.2(c) is not an attempt to get at the "real" structure of Y in acontext-free environment; rather it asserts that Y, when heard following X andin connection with the many intervals of 3 between notes of X and notes of Y, willtend to be articulated as in that figure. Likewise, figure 5.4 does not pretend toassert something that Y "really is," context-free. Rather, it asserts a structur-ing for Y that tends to emerge in the environment, "remembering X andhearing the many intervals of 8 between notes of X and notes of Y." Finally,figure 5.5(e) does not try to assert a "real" form for Y either. Rather, it saysthat Y will tend to articulate in that way when heard in a complicatedenvironment involving Z3, the idea that Y expands Z3 in a manner analogousto the way Z0 expanded X (as in figures 5.5(a)-(d)), and the effect of thecadence harmony displayed in figure 5.5(f), as that event is recalled in itsvarious contexts.

The various IFUNC values of figure 5.3(a) are useful tools, as we haveseen, for exploring the multifaceted aspects of Y in various of its environ-ments. Figures 5.2(c) and 5.4 explore two different ways in which Y-following-X engages IFUNC; these ways will be of practical interest to thepianist shaping the lyric melody of the right hand over measures 3-7. As wenoted before, that is all the lyric melody there is in the piece, and since themelody articulates musically into two phrases, the pianist will naturally wantto explore listening to the various kinds of "logic" adhering to the way inwhich Y follows X. Figures 5.2(c) and 5.4 provide that theoretical "logic" inthis environment, the one figure in connection with sensitivity to the interval 3,the other in connection with sensitivity to the interval 8.

Figure 5.5(e), in contrast, explores Y as it occurs in a different context,that is as it relates to Z3 in a complex theoretical proportion also involving Xand Z0 as in figures 5.5(a)-(d). The corresponding musical environment isnow not the way the antecedent and consequent phrases of the lyric melody areshaped in the right hand of the piano, but rather the way in which the violinostinato comments upon that melody, and is commented upon by it. The newmusical environment we are now considering involves an interrelation, orrather several interrelations, between the instruments. This will be of partic-ular interest to the violinist, trying to keep the ostinato figure fresh and aliverather than mechanical. In this environment the second appearance of Z3 willsound different from the first. Figure 5.6 shows what I mean.

(a) and (b) of the figure show how the first Z3, at measures 6-7, com-ments upon the opening trichord of Y at the intervals of 5 and 11 respectively,(c) and (d) of the figure, in contrast, show how the second Z3, during measures7-8, comments (as well) upon the fresh trichord (B, F, F#) within Y, now thatF# has appeared on the scene. The new commentary is at the intervals of 0 and6. When executing these commentaries, it will help the violinist to listen to Yarticulated as in figure 5.5(e) earlier. 97


We could of course also consider Y simply in its own environment,spanning various intervals within itself. This would lead us to examineIFUNC(Y, Y), which is essentially Forte's "interval vector of Y."

Our study of the Webern passage has made Z0 seem more subordinatethan it sounds in the piece; the study so far has also underplayed the role of the3-note chromatic set (C#, D, E|?), the set which appeared in figure 5.5(e)-(f).A brief discussion of figure 5.7 will attempt to right this balance, withoutgetting fussy about more IFUNC values.

98

The figure transcribes the noteheads of the ensemble up through thecadence at the bar line of measure 5. Over measures 1 through 3 the musicslowly exposes the set W0 = (A, 6(7, A(?). (At the written tempo, this takesover 20 seconds.) The boundary tones for the exposition of W0 are shown onthe figure with open noteheads: A is lowest and first; Ab is (so far) highest andlast. As the piece continues, the rhythm becomes more active and the registralspace expands. By the cadence at measure 5 we hear new boundary tones, alsoshown with open noteheads: E|?5 is a new highest tone; D3 is a new lowest toneand also a new last tone. The four boundary tones on the figure add upprecisely to a large-scale projection of the set Z0 = (A, D, Efc>, Aj?). So Z0 has a

FIGURE 5.6

FIGURE 5.7


very strong structural meaning in its own right, by the time it enters in theforeground of measure 6 to begin the violin ostinato.

The progression of boundary tones on figure 5.7 suggests an inversionalrelationship: The falling bass, from the opening A3 to the closing D3, sand-wiches a rising melody spanned by the local high tones A|?4-then-E|?5. Indeedthis rising melody is the X-phrase itself. The injection function, to be discussedin chapter 6, will allow us to engage such inversional relations within andbetween sets; IFUNC cannot do so.

The set W5 on figure 5.7 is the cadence harmony shown earlier on figure5.5(f), a set projected in another way by the bracketed "interior of Y" onfigure 5.5(e). Figure 5.7 shows how the cadential W5 responds to the openingW0. W5 is of course T5(W0). The bass A of W0 as presented, flanked by itschromatic neighbors B(? and A|? above, progresses over figure 5.7 to the bassD of W5, flanked by its chromatic neighbors Eb and C# above. So the T5

relation of W0 as a whole to W5 as a whole, as that relation moves structurallyover figure 5.7, expands upon the T5 relation of A to D in the bass register.This is not lost upon our ears when the violin ostinato begins precisely withA-D- at measure 6, presenting thereby a highly charged T5 relation in theforeground. All these considerations dispose us to bracket off W5 as theinterior of Y with somewhat more aural attention than our earlier discussioncould make plausible.

Having explored the pertinence of IFUNC in the setting of traditionalatonal set theory, let us now return to further study of the formal X/Y intervalfunction in generalized set theory. We recall Definition 5.1.3: Given a GIS,given sets X and Y, then IFUNC(X, Y) is that function which assigns to each iin IVLS the number of ways in which i can be spanned from X to Y, that is thenumber of pairs (s, t) such that s is in X, t is in Y, and int(s, t) = i. We shall nowstudy how IFUNC is affected as the sets X and Y are manipulated andtransformed in various ways.

The first theorem to be noted shows that when the roles of sets X and Yare exchanged, IFUNC is in a certain sense "inverted."

5.1.4 THEOREM: IFUNC(Y,X)(i) = IFUNC(X,Y)(i-1).Proof: IFUNC(X, Y) (i-1) is the number of pairs (s, t) such that s is in X, t

is in Y, and int(s, t) = i"1. This is the number of pairs (t, s) such that t is in Y, sis in X, and int(t,s) = i. And that number is IFUNC(Y,X)(i).

The next theorem shows that IFUNC is not affected when X and Y areboth transformed by the same interval-preserving operation P.

5.1.5 THEOREM: Let P be any interval-preserving operation. ThenIFUNC(P(X),P(Y)) = IFUNC(X, Y) as a function on IVLS. 99


Proof (optional): Let PAIRS be the family of pairs (s, t) such that s is in X,t is in Y, and int(s, t) = i. Then IFUNC(X, Y)(i) is the cardinality of PAIRS.Let PAIRS' be the family of pairs (s', t') such that s' is in P(X), t' is in P(Y),and int(s', t') = i. Then IFUNC(P(X), p(Y))(i) is the cardinality of PAIRS'. So,to show that IFUNC(X, Y)(i) = IFUNC(P(X), P(Y))(i), it suffices to show thatPAIRS and PAIRS' have the same cardinality. And that will be the case if wecan map PAIRS onto PAIRS' by some 1-to-l function f. We shall constructsuch a function f.

Given (s, t) in PAIRS, define f(s, t) = (P(s), P(t)). f(s, t) is indeed amember of PAIRS', for P(s) is in P(X), P(t) is in P(Y), and int(P(s), P(t)) =int(s, t) = i (since P is interval-preserving), f is a 1-to-l map because P is1-to-l: If (P(s,), P(t,)) = (P(s2), P(g), then P(s,) = P(s2) and P(t,) = P(t,),whence s, = s2 and tj = tj, whence (s,, t, = (s2, tj). It remains only to showthat f is onto PAIRS'. The interval-preserving operations form a group ofoperations on S; hence P"1 exists and is interval-preserving. Given (s', t') inPAIRS', set s = P~V) and t = P"1^'). The reader may verify that (s, t) isin PAIRS, and that the given (s', t') is the image of (s, t) under the map f. q.e.d.

Now we shall see how IFUNC is affected when X or Y or both aretransposed.

5.1.6 THEOREM: For any transposition operation Tn, the formulas (A), (B),and (C) below obtain.

(A): IFUNC(Tn(X), Y)(i) = IFUNC(X, Y)(ni)(B): IFUNC(X, Tn(Y))(i) = IFUNC(X, Y)(in~])(C): IFUNC(Tn(X), Tn(Y))(i) = IFUNC(X, YXnin'1)

Proof of (A) (optional): Let PAIRS be the family of pairs (s, t) such that sis in X, t is in Y, and int(s, t) = ni. Then the cardinality of PAIRS isIFUNC(X, Y)(ni), the right side of Formula (A) above. Let PAIRS' be thefamily of pairs (s', t) such that s' is in Tn(X), t is in Y, and int(s', t) = i. Thenthe cardinality of PAIRS' is MJNC(Tn(X), Y)(i), the left side of Formula (A).To prove the formula, then, it suffices to show that PAIRS and PAIRS' have thesame cardinality. And we can show that by demonstrating a function f whichis 1-to-l from PAIRS onto PAIRS'.

The desired function is f(s, t) = (Tn(s), t). The reader may verify that fmaps PAIRS into PAIRS', that f is 1-to-l, and that f is onto.

Proof of (B) (optional): IFUNC(X, Tn(Y))(i) = IFUNC(Tn(Y), XXi'1),via 5.1.4. This = IFUNC(Y, X)(ni-1, via Formula (A) just proved. Andthat = IFUNC(X, YXiir1), again via 5.1.4.

Proof of (C) (optional): IFUNC(Tn(X), Tn(Y))(i) = IFUNC(X, TB(Y))(ni),via Formula (A) above. And that = IFUNC(X, Y)(nin~!), via Formula (B) justproved, q.e.d.100


In contrast to the results of Theorem 5.1.6, there is not much to be said ingeneral about the effect of applying an inversion operation to X, Y, or both, sofar as IFUNC is concerned. If we restrict our attention to commutative groupsof intervals, though, then we can say something.

5.1.7THEOREM: If I is any inversion operation in a commutative GIS, thenIFUNC(I(X),I(Y)) = IFUNC(Y,X).

Optional sketch for a proof: In a commutative GIS, the inversions areinterval-reversing operations (3.6.3). The present theorem may therefore beproved by the same technique used to prove 5.1.5 above.

General questions involving IFUNC and other operations on S will bebetter pursued using the Injection Function to be developed in chapter 6; thatfunction will generalize IFUNC among other things. The same holds forgeneral questions involving IFUNC and complement relations among sets,where those are relevant. (We have not assumed that S is finite, so thecomplement of a set need not be a set according to Definition 5.1.1.)

IFUNC can be given an interesting interpretation as a probabilitydistribution.

5.1.8 THEOREM: Let X and Y have respective cardinalities M and N. Select amember s of X at random and a member t of Y at random. Then the numberIFUNC(X, Y) (i)/(MN) measures the probability that int(s, t) will be found toequal i.

Proof: There are MN possible pairs that can be pulled in this way, andIFUNC(X, Y) (i) of these pairs will have the desired property, q.e.d.

Theorem 5.1.8 is interesting because we have used IFUNC so far only as aprecision tool; the theorem shows that it can also be used to portray astatistical texture. For instance, suppose a clarinet is told to improvise for atime upon the notes of a pitch set X, while a flute is told to improvise for thesame span of time upon Y. A statistical field of intervals will result from thisimprovisation, and that field can be modeled by IFUNC(X, Y) according tothe rule of Theorem 5.1.8.

Even when we are not applying IFUNC to such "stochastic" compo-sitional settings, it is still sometimes useful to regard it as providing a statisticalbackdrop for intervallic events. For instance, the notion that a certain intervali appears "often" or "only rarely" between X and Y is implicitly dependent onthis backdrop: i appears "only rarely," e.g., compared to how often otherintervals appear.

We shall look at aspects of the opening from Schoenberg's Violin Fan-tasy op. 47, taking this point of view. The violin projects the pitch-class set 101


Y = (Bb, A, C#, B, F, G), while the piano accompanies the violin with the setX = (Eb, E, C, D, AJ7, Gb). X is the complement of Y and also an inversion ofY but these relations will not concern us explicitly for present purposes. Figure5.8(a) displays the values of IFUNC(X, Y).

102

Figure 5.8(a) shows that IFUNC counts "many" odd intervals from Xto Y (accompaniment-to-solo, lower-instrument-to-upper-instrument), and"few" even intervals. The theoretical point at hand is that two appearancesof an interval is not intrinsically "few"—it would not be few e.g. betweentwo trichords. Rather, the scarce intervals on figure 5.8 (a) are "scarce"only against the statistical backdrop of the table as a whole. The equationIFUNC(X, Y) (0) = 0 expresses the fact that X and Y have no common tones.So the scarce interval 0 does not appear at all between the instruments. Thescarce intervals of 4 and 8 each appear in two different ways between pianoand violin, and those ways are of analytic interest.

Figure 5.8(b) displays the opening noteheads of the piece in order ofsuccession. Figure 5.8(c) shows how the two intervals of 4 between X (piano)and Y (violin) appear. One of those 4-intervals appears between Gb, the lastand lowest note, and Bb, the first note and a provisional low note for theviolin. The other 4-interval appears between Eb, the first and highest note ofthe piano, and G, the last and lowest note of the violin. Thus all the notes infigure 5.8(c) are boundary tones of one sort or another for the passage; thefigure shows how the scarce interval 4 binds this spatio/temporal frame for thephrase.

Figure 5.8(d) shows the scarce interval 8 functioning in a similar way. Wehave already discussed the Bb and the Eb as boundaries; the B is a high

FIGURE 5.8

Generalized Set Theory ( I ) 5.1.8

boundary; the low D is a provisional low boundary until the very end of thephrase.

Many abstractly interesting questions can be asked about our generalizedIFUNC. One family of questions takes the following tack: Given some prop-erty that a function from IVLS to the non-negative integers might have, underwhat conditions on X and Y will IFUNC(X, Y) have that property? For in-stance, we may ask under just what conditions on X and Y IFUNC(X, Y)^"1)will equal IFUNC(X,Y)(i) for all i. Via 5.1.4, this is the same as askingunder what conditions on X and Y IFUNC(Y, X) will be the same functionas IFUNC(X,Y). More generally, we can ask under what conditions therewill exist intervals m and n such that IFUNC(X,Y)(mi~1n) will equalIFUNC(X,Y)(i) for all i. Satisfactory answers for these questions are notknown even in connection with the standard GIS for atonal set theory.4

Another family of questions generalizes in one possible direction thetraditional topic of "Z-sets." In Forte's theory, pitch-class sets Xt and X2

which are not transposed or inverted forms of each other are Z-related if andonly if IFUNCCX^XO = IFUNC(X2,X2), as a function on IVLS. In ageneral GIS setting we may ask under what conditions on Xx and X2 thatequation will obtain. We know that the relation will hold if there is an interval-preserving transformation P such that X2 = PCXJ (5.1.5). The relation willnot automatically hold in a non-commutative GIS when X2 is a transposedform of X t: 5.1.6 shows us that if X2 = T^X^, then IFUNC(X2,X2)(i) willequal IFUNCtX^X^nhT1), but not necessarily IFUNCCX^XJCi).

Going even further, we may ask under what conditions among the foursets X l 5 Y!, X2, and Y2 we will have the relation IFUNCtX^Y^ =IFUNC(X2, Y2) (as a function on IVLS). Figure 5.1 earlier provided someexamples of this state of affairs, in the relatively well-behaved GIS of tradi-tional atonal set-theory. This is all a vast open ground for mathematical andmusical inquiry, even in atonal set-theory.

Our questions can be transferred to a more general mathematical setting.Readers who do not have graduate-level mathematical background shouldskip this paragraph. If we use LABEL to identify S with IVLS, we can see thatwe are treating S = IVLS as a locally compact group under the discretetopology; our "sets" are the compact subsets of IVLS, and IFUNC(X, Y) (i) isthe convolution (f * * g) (i), where f and g are the characteristic functions of thesets X and Y respectively. All of our questions may then be generalized toquestions about the interrelations, in a locally compact group, among thecharacteristic functions of compact subsets. E.g: Given compact subsets X l s

4. Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music vol. 13,no. 1 (Fall-Winter 1974), 191-212. Regener poses essentially equivalent questions in that con-nection, starting at "Among many other things," on page 204. 703


X2, Y t, and Y2, with characteristic functions f l s f2, g l5 and g2, under whatconditions will fr* * gj and f^ * g2 be the same function? As with the specialcase of IFUNC, the study is much simplified when the group is commutative.

Other ways of generalizing our questions about IFUNC will come uplater in connection with the Injection Function.

We now make a big articulation and turn our attention to generalizingForte's Interval Vector. To do so, we shall need some further apparatus, inparticular the notions of a "Canonical Group" of operations on S and a"Canonical Equivalence Relation" among sets. It is worth noting thatIFUNC, and the Injection Function later, can be defined and discussedwithout invoking those notions.

Forte considers pitch-class sets X and Y to be canonically equivalent, byour definition coming up, if Y is a transposition or inversion of X. Here thecanonical group comprises the transposition and the inversion operations.(They form a group here because the GIS is commutative.) In other systems ofatonal set theory, X and Y are considered canonically equivalent if and only ifY is a transposition of X; then the canonical group comprises transpositionsonly. In still other systems the canonical group includes not only the transpo-sitions and the inversions but also the circle-of-fifths transformations andpossibly other transformations as well.5

The idea of canonical equivalence allows us to speak about "the formsof" a set X; those are the sets X' that can be derived from X by operationsin the canonical group, or (what is the same thing) the sets X' which arecanonically equivalent to X. The work coming up in section 5.2 generalizesthese ideas.

5.2.1 DEFINITIONS: In certain connections we shall fix a group of operationson S and call it "the canonical group." It will be denoted CANON. Sets X andX' will be called "canonically equivalent" if there exists some canonicaloperation A such that X' = A(X).

The defined relation is indeed an equivalence. It is reflexive: X =IDENT(X). It is symmetric: If X' = A(X) then X = A~l(X'). It is transitive:If X' = A(X) and X" = B(X'), then X" = (BA)(X).

In any specific context, we suppose there is some good reason for select-ing one particular group of operations on the family S as canonical. Generallythe good reason will involve intervallic relationships of one sort or anotherwithin a GIS for which S is the family of objects. But formally, we do notactually need a GIS structure at all. We could carry through our work if we

5. An extended discussion of such matters can be found in Robert D. Morris, "Set Groups,Complementation, and Mappings among Pitch-Class Sets," Journal of Music Theory vol. 26,no. 1 (Spring 1982), 101-44.104


just started with some family S of objects and some group CANON of opera-tions on S, not concerning ourselves with formal intervals at all. When we lookat things so abstractly, we foreshadow the "transformational" approach to betaken later on in this book.

5.2.2DEFINITIONS: We shall write /X/ to denote the canonical equivalence-class containing the set X. /X/ will be called, for short, the "set class of X."

The term "set class" will grate dreadfully on the ears of any mathematicallogician. Still, it is becoming standard usage for atonal theory. In earlierwriting I used the term "chord type." But that term loses its intuitive per-tinence when we are working with generalized sets of all kinds, includingrhythmic sets, timbral sets, sets in direct-product GIS structures, and the like.

It is important to understand that the notion of set class depends not onlyupon the set X at hand but also upon the canonical group CANON selectedfor the occasion. For example, let us fix the standard GIS of atonal set theory;let us select X = (C, E, G). If we choose CANON to be the group of transpo-sition operations, then /X/, the family of transpositions-of-X, comprises themajor triads. But if we choose CANON to be the group of transpositions andinversions, then /X/, the family of transpositions-and-inversions-of-X, com-prises all the harmonic triads, major and minor.

5.2.3 LOCUTIONS: "X' is a form of X" means that X' is canonically equivalentto X. /X/ may be referred to as "the forms of X."

Given a GIS, it can be a tricky business to decide for any particulartheoretical exercise just which operations on S are to be allowed intoCANON.6 We shall generally want to include at least the interval-preservingoperations in the canonical group. For if P is an interval-preserving operationandX' = P(X),thenIFUNC(X',X') = IFUNC(X,X)(5.1.5).ThusX'-in-its-own-context has the same intervallic structure as X-in-its-own-context; this isa reasonable criterion for wanting X' to be considered "equivalent" to X.When the GIS is commutative, the interval-preserving operations will beexactly the transpositions. When the GIS is non-commutative, we may or maynot wish to include the transpositions, as well as the interval-preservingoperations, in the canonical group for a given exercise.

We can now define the Embedding Number, a construct which general-izes Forte's Interval Vector.

5.3.1 DEFINITION: Given sets X and Y, the embedding number of X in Y,EMB(X, Y), is the number of forms of X (i.e. members of /X/) that areincluded in Y.

6. Morris (ibid.) discusses this at length, in atonal theory. 705


The embedding number depends on the notion of set class, which de-pends in turn upon the canonical group at hand; this cannot be overem-phasized. For example, let us work within the standard GIS of atonal theory;let X be some major triad and let Y be some major scale. If CANON consistsof the transposition operations only, then EMB(X, Y) = 3: three major triadsare embedded in the scale. On the other hand, if CANON consists ofboth transpositions and inversions, then EMB(X, Y) = 6: six harmonictriads are embedded in the scale. Strictly speaking, we should writeEMB(CANON,X, Y) to show that the embedding number varies with thecanonical group as well as the sets X and Y. But our notation is alreadycumbersome enough.

If X' is a form of X then /X'/ = /X/; the members of /X'/ are the mem-bers of /X/ and therefore, via 5.3.1, EMB(X', Y) = EMB(X,Y). If Y' =A(Y) is a form of Y and EMB(X, Y) = N, let X l f X 2 , . . . , Xn be the distinctforms of X embedded in Y. Then ApCJ, A(X2), ..., A(Xn) are the distinctforms of X embedded in Y'= A(Y). So EMB(X,Y') also = N; i.e.EMB(X, Y') = EMB(X, Y). It follows: If X' is a form of X and Y' is a form ofY then EMB(X', Y') = EMB(X,Y). We have proved that Definitions 5.3.2following make sense.

5.3.2 DEFINITIONS: EMB(/X/, Y) will mean the value of EMB(X', Y) for anymember X' of/X/. EMB(X, /Y/) will mean the value of EMB(X, Y') for any Yin /Y/. EMB(/X/, /Y/) will mean the value of EMB(X', Y') for any X' in /X/and any Y' in /Y/.

Let us consider the standard atonal GIS, and let us fix CANON as eitherthe transpositions, or the transpositions plus the inversions. The various 2-note sets will gather into exactly six "2-note set-classes," SC^ SC2,..., SC6.(SC4 for instance contains all the 2-note sets whose notes lie an interval of4-or-8 from each other.) Given a set Y, we can ask for the values ofEMB(SCn, Y) as n runs from 1 through 6, i.e. the values of EMB(/X/, Y) as thevariable /X/ runs through the six 2-note set classes. The function giving usthose six values is Forte's Interval Vector of Y. From our point of view here,we could call it the "dyad-type vector of Y." By analogy we could study the"trichord-type vector of Y," that is the function which gives us the values ofEMB(/X/, Y) as /X/ runs through the various 3-note set classes. (There will benineteen such classes if CANON contains transpositions only; there will betwelve if CANON contains both transpositions and inversions.) Leaving theGIS of atonal theory now, we can generalize such vectors in an abstract settingby the following definition.

5.3.3 DEFINITION: By the "M-class vector of Y," we understand the function106

Generalized Set Theory (1) 5.3.5.2

EMB(/X/, Y) as the variable /X/ runs through the various set-classes whosemembers have cardinality M.

In case S is infinite there may be an infinite number of M-member setclasses. But since Y is finite it has only a finite number of subsets, which canbelong to only a finite number of set classes. Hence EMB(/X/, Y) must be zerofor all but a finite number of /X/.

5.3.4 THEOREM : Let the cardinality of Y be N. Let M be a positive integer lessthan N. Pull M members of Y at random. Then the probability that you havepulled a set of class /X/ is given by the number EMB(/X/, Y)/COMB(M, N).Here COMB(M,N) is the number of combinations of M things that can beextracted from a family of N things; e.g. COMB(13,52) is the number ofpossible hands at bridge.

Proof: There are COMB(M,N) different M-member sets \ve might ex-tract from Y, and EMB(JX|, Y) of those sets will be of class [X|.

COMB(M,N) can be calculated to be Nl/(Mi.(N - M)V), where N! isfactorial N, etc. Theorem 5.3.4 shows us that EMB, like IFUNC earlier, canbe regarded as a statistical measure aside from its uses as a precision tool. Thetheorem enables us derive a very strong formula interrelating various M-classvectors in a general setting. That formula will be proved in 5.3.5.2 below; weshall first prove a lemma involving some numerical computation.

5.3.5.1 LEMMA: Given positive integers L, M greater than L, and N greaterthan M, then

COMB(L,N)/(COMB(L,M)COMB(M,N)) = 1/COMB(N - M, N - L).

Proof (optional):

COMB(L, N)/(COMB(L, M)COMB(M, N))= N!/(L!(N - L)!) divided by the product of M!/(L!(M - L)!)

andN!/(M!(N-M)!)= N!/(L!(N - L)!) times L!(M - L)!/M! times M!(N - M)!/N!.

In this product the factorials of L, M, and N can each be cancelled from thenumerators and the denominators of the participating fractional factors. Thisleaves the product

= 1/(N - L)! times (M - L)! times (N - M)!= (N - M)!(M - L)!/(N - L)!. And that is the multiplicative inverse of

COMB(N - M, N - L), as asserted, q.e.d.

5.3.5.2 THEOREM: Let L, M, and N be as in Lemma 5.3.5.1; let ADJUST bethe fraction calculated in that lemma. Let Z be a set of cardinality N and let X 107

5.3.5.2 Generalized Set Theory (1)

be a set of cardinality L. Then

EMB(X, Z) = ADJUST • SUM(EMB(X, /Y/)EMB(/Y/, Z)),

where the SUM is taken over all M-member set-classes /Y/.Proof (optional): Only a finite number of terms in the sum will be non-

zero, so summing "over all . . . /Y/" makes sense. (As /Y/ varies, only a finitenumber of values EMB(/Y/, Z) can be non-zero.)

Let us imagine first pulling an M-member subset from Z, and then pullingan L-member subset from that M-member set. In the first pull, the probabilitythat we have pulled a set of class /Y/ is prob(/Y/) = EMB(/Y/,Z)/COMB(M,N) (5.3.4). And if we have pulled a set of class /Y/, the proba-bility that our second pull will yield a form of X is prob(/X/-from-/Y/) =EMB(/X/, /Y/)/COMB(L, M) (5.3.4). Probability theory tells us how to mani-pulate these numbers so as to calculate the chances of ending up with someform of X pulled from Z. Namely:

prob(/X/-from-Z) = SUM(prob(/Y/)prob(/X/-from-/Y/)),

where the sum is over all possible intermediate pulls /Y/, that is over all the M-member set-classes /Y/. Now in the probability formula above we can sub-stitute, via 5.3.4,

prob(/X/-from-Z) = EMB(X, Z)/COMB(L, N)prob(/Y/) = EMB(/Y/, Z)/COMB(M, N)

and prob(/X/-from-/Y/) = EMB(X,/Y/)/COMB(L,M).

The probability formula then takes the new form

EMB(X, Z) = FACTOR • SUM(EMB(X, /Y/)EMB(/Y/, Z)), where

FACTOR takes care of the COMB-numbers in the denominators of thevarious probability values above. Calculating FACTOR out, we see thatthis fraction specifically equals COMB(L, N)/(COMB(L, M)COMB(M, N)).And that, via the Lemma, is ADJUST, q.e.d.

The formula of Lemma 5.3.5.1 is not necessary to prove the formula ofthe theorem; we could simply define ADJUST to be COMB(L,N)/(COMB(L,M)COMB(M,N)). But the value 1/COMB(N - M, N - Lwill often be much easier to compute. For instance try L = 5, M = 9, andN = 11: The value 1/COMB(N - M, N - L) gives us 1/COMB(2,6) = 1/1very quickly.

To give us an intuitive sense of why Theorem 5.3.5.2 is interesting, it willbe useful to study a simple example from Fortean set-theory in connectionwith a topological model. Let Z be the pitch-class tetrachord (A, B, C, D); letX be the dyad (A, C). Here L = 2 and N = 4; we will set M = 3 and examinejust what Theorem 5.3.5.2 is telling us.

First let us inspect figure 5.9. It represents the tetrachord Z as a tetra-108


hedron in three-dimensional space; the vertices of the tetrahedron are themember pitch-classes A, B, C, and D of the set Z. When we inquire about the2-note subsets of Z, we are inquiring about the boundary edges of thistetrahedron. The figure lays the boundary edges out for inspection below thetetrahedron. Of the six edges, one belongs to set-class 2-1, two belong to set-class 2-2, two belong to set-class 2-3, and one belongs to set-class 2-5. Theembedding numbers at the lower right of the figure express these counts.

Now let us inspect figure 5.10. It first analyzes the tetrachord into its fourtriangular boundary faces, and then analyzes each triangular face into its threeboundary edges. Two of the triangles are in Forte-class 3-2; these triangles arelabelled Y2 and Y2' on the figure. The other two triangles are in Forte-class3-7; these triangles are labelled Y7 and Y7'.

FIGURE 5.9

709


FIGURE 5.10

110

On figure 5.10 the four triangles are lined up beneath the tetrahedron,and the edges of each triangle are stacked up below that triangle. In the fourresulting stacks, each edge of the original Z-tetrahedron appears twice. That isbecause each edge of the tetrahedron belongs to two of the triangles. (EdgeAB, for instance, belongs both to triangle Y2 and to triangle Y7'.) As a result,when we count how many sticks at the bottom of the figure are in Forte-class2-3, we must divide that count by two, to arrive at the number of sticks in thatForte-class we found on figure 9 earlier. For example, figure 5.9 counted twoedges-of-Z lying in Forte-class 2-3, namely AC and BD. EMB(class


2-3, Z) = 2. Figure 5.10 counts as edges-of-faces-of-Z twice as many sticks ofclass 2-3, namely AC-as-edge-of-Y2, BD-as-edge-of-Y2', AC-as-edge-of-Y7,and AC-as-edge-of-Y7'.

What Theorem 5.3.5.2 does in this connection is to ADJUST the countof sticks at the bottom of figure 5.10, dividing it by two to conform with thestick-count at the bottom of figure 5.9. The theorem knows that two is theproper number to divide by here, because 1/2 is the present value ofADJUST = 1/COMB(N - M, N - L) = 1/COMB(1,2) = 1/2. If Z were heptachordal object in six-dimensional space (N = 7) and the Y hyper-faces were pentachordal objects in four-dimensional space (M = 5) andwe were again interested in counting edges (L = 2), then we would have toADJUST our count of sticks analogously by 1/COMB(N - M, N - L) 1/COMB(2,5)= 1/10.

The probabilistic method we used to prove Theorem 5.3.5.2 will help usunderstand figures 5.9 and 5.10 in a somewhat different light. Inspectingfigure 5.10, we see that if we peel a triangular face at random off the tetra-hedron, the probability is 1/2 that the face will be in Forte-class 3-2 and 1/2that the face will be in Forte-class 3-7. If we pull an edge at random off atriangle of Forte-class 3-2, our expectation is 1/3 that the edge will be inForte-class 2-3. And if we pull an edge at random off a triangle of Forte-class3-7, our expectation is 1/3 that the edge will be in Forte-class 2-3. Hence,according to the theory of probability, our total expectation for pullingan edge of Forte-class 2-3 off the tetrahedron by a random yank is((1/3) (1/2) + (1/3) (1/2)) = (1/6 + 1/6) = 1/3. And this agrees (as our worksays it must) with the probability of that event which we infer from figure 5.9:There we see that of the six tetrahedral edges, two are of Forte-class 2-3; so weinfer that our expectation of yanking an edge of that class in a random pull is2/6, which is 1/3.

To pursue farther what Theorem 5.3.5.2 has to do with figures 5.9 and5.10, and with higher-dimensional analogs of those figures, would lead usdeeply into a branch of mathematics called algebraic topology. That pursuitwould be very much worth undertaking, but it would be out of place here.

In discussing how our generalized embedding number applies to theexample of figures 5.9 and 5.10, I have supposed that the reader is alreadyfamiliar with Forte's use of the interval vector in atonal theory. Now let us seehow our generalized theory applies in a very different context, one with whichthe reader is almost certainly unfamiliar. To that end, we shall study someexamples in connection with the non-commutative GIS of time spans whichwe developed in chapter 4.

The first thing we must do is fix the group CANON for our purposes. Weshall take CANON to be the group of all interval-preserving operations here.We shall not allow transpositions, much less inversions, as canonical oper-ations for this study. Our reason will become clear. Ill


As we observed in 4.1.7(D), the generic interval-preserving operationP = P(h>u) transforms the sample time span (a,x) into the time span(h + ua, ux). The commentary on 4.1.7(D) elaborated upon this: "Theinterval-preserving operation P(h u) first blows up or shrinks the sample timespan (a, x) by a factor of u, transforming (a, x) to (ua, ux), and then moves thelatter time span backward or forward in time by h or ( —h) numerical units,transforming (ua, ux) to (h + ua, ux) = P(a, x)." So if X is a set of time spans(s l 5s2 , . . .sn), where sn = (an,xn), then P(X) is the set (s'^s^,.. . SN), whersj, = P(sn) = (h + uan,uxn). We can imagine the set X here as modelingtemporal aspects of a musical "passage" containing N events; then P(X)models analogous aspects of the passage played u times as slowly (1/u times asfast), starting the tempo change from time-point zero, all this played hnumerical time-units later ( — h earlier).

For example, let us take "the quarter note" as a numerical unit and "thebeginning of the piece" as a numerical time-point zero. Imagine a motiveconsisting of an eighth, a dotted eighth, a sixteenth, and a quarter, playedconsecutively starting 10 quarters after the beginning of the piece. We couldmodel some temporal aspects of this motive by the set X = ((10,|), (10^,f),(Hi,i), (Hi, !))• Remember that X is formally an unordered set; we havelisted its members "in order of appearance" only for convenience here. Let ustake h = 1370 and u = 4. Then the transformed set P(h>u)(X) first augmentsthe entire rhythmic setting by a factor of 4, from time-point 0 on; then P(X)plays the augmented motive beginning h = 1370 quarters later, that is begin-ning 1370 quarters after time point 40, that is beginning at time point 1410. (Theaugmented motive obtained as an intermediate stage began not at time-point10, but at time-point u • 10 = 4 • 10 = 40.) So the transformed motive modeledby P(X) consists of a half note, a dotted half, a quarter, and a whole note,played consecutively starting 1410 quarters after the beginning of the piece.Expressing this in numbers, P(X) = ((1410,2), (1412,3), (1415,1), (1416,4)).One can check that each member of P(X) is mathematically related to thecorresponding member of X via the transformation P(a, x) = (h + ua, ux),here = (1370 + 4a, 4x). For instance, the third-listed members of the sets Xand P(X) are related by the formula P(llii) = (h + u - l l i ,u-±) =(1370 + 4- lli,4-i) = (1370 + 45,1) = (1415,1).

There is no need to restrict our attention to sets modeling consecutiveevents, as X and P(X) did in the preceding example. We could for instanceconsider a passage in which a violin plays four consecutive quarters, while aviola plays three triplet halves, while a cello rests for an eighth and then playstwo consecutive quarters followed by a dotted quarter. We could modeltemporal aspects of this passage by a time-span set Y. Supposing that theonset of the passage comes 16 quarters into the piece, we can writeY = ((16,1), (17,1), (18,1), (19,1), (16, f), (17if), (18f,f), (16i 1), (17*. 1),(18-|, l£)). As P varies over the interval-preserving operations, P(Y) models772


the ensemble passage, played at (all) different tempos and at (all) differenttimes. The elements of the unordered set Y above are listed, not "in order ofappearance," but "by parts," as they were described in the text.

Suppose the numerical time-span set Yt models the above passage forstring trio at the precise time the music was first imagined clearly by thecomposer. Suppose the different numerical set Y2 models the passage at theprecise time it was played during the first performance. Suppose the stilldifferent numerical set Y3 models the passage at the precise time my trioplayed it yesterday, taking a considerably faster tempo. Given one fixedreferential time-point zero and one fixed referential time unit, the numbersdenoting the members of the three sets Y l 9 Y2, and Y3 will be very different.Our formalism, though, enables us to say that the three sets are all (approxi-mately) canonically equivalent.

That is one powerful methodological reason for choosing CANON hereto be the group of interval-preserving operations. Another good reason for thechoice is provided by the way in which this group relates dyad structure tointerval structure in the GIS at hand. We shall now explore that topic.

By a "dyad" we understand a set containing two distinct members s and t.By an attack-ordereddyad(AOD) we shall mean a dyad containing say s and t,ordered in the following way: If s begins before t (as a time span), the order is(s, t); if t begins before s, the order is (t, s); if both time spans begin at the sametime, the shorter of the two spans is listed first. Since s and t are distinct timespans, these criteria are sufficient to order the dyad.

Given an AOD D = (s, t), let (i, p) = int(s, t). Then t begins i s-durationsafter s begins, and t lasts p times as long as s. Because of the ordering criteriaon D, the number i must be non-negative, and ifi = 0 then the number p must begreater than 1. Let us call an interval (i,p) of this form a forwards-orientedinterval. We have seen that if D = (s, t) is an AOD, then int(s, t) is forwards-oriented. The converse is also easily seen: If s and t are time spans such thatint(s, t) is forwards-oriented, then D = (s, t) is an AOD.

We can define (j, q) to be a "backwards-oriented interval" in an analo-gous way: j must be non-positive, and if j = 0 the number q must be less than 1.Now in the group IVLS the inverse of the interval (i,p) is ( — i/p, 1/p). Itfollows that the inverse of a forwards-oriented interval is backwards-oriented,and vice-versa. One sees quickly that the members of IVLS can be partitionedinto three categories: the forwards-oriented intervals, the backwards-orientedintervals, and the identity interval (0,1).

Here now is the crucial manner in which our stipulated canonical groupcomes into play: Given AODsDj = (s1,t1)andD2 = (s2,t2), then Dj and D2

are canonically equivalent if and only if int(sl5 t t) = int(s2, t2). It would taketoo long to include here a formal proof of that theorem; such a proof isappended to the end of the chapter as section 5.6. The theorem is by no meansobvious or trivial. Once we have proved it, we can note that the 2-element set 113


classes correspond 1-to-l with the forwards-oriented intervals. If D = (s, t) is anAOD, then the set class /D/ corresponds to the forwards-oriented interval(i, p) = int(s, t): Every member D' = (s', t') of /D/, when attack-ordered, hasint(s', t') = (i, p); furthermore, if s' and t' are any time spans such thatint(s', t') = (i, p), then the AOD D' = (s', t') is a member of the set class /D/.The forwards-oriented intervals thus play exactly the same role here thatForte's "interval classes" play in his atonal theory: They can be used to labelthe distinct set-classes of dyads. They can be so used, that is, z/we take theinterval-preserving operations as CANONical in constructing those set-classes.

As a result of this structure, we can develop a very strong formal analogfor Forte's interval vector in this particular system (NB). Let X be a setcontaining more than two members; let D be a dyad; then EMB(D, X), thenumber of forms of D embedded within X, is equal to the number of ways theforwards-oriented interval (i, p) can be spanned between members of X, where(i, p) is the interval spanning the attack-ordered members of D. In otherwords, EMB(D,X) = IFUNC(X,X)(i,p). In this sense we can speak ofEMB(/D/, X), when /D/ varies over the dyad-classes, as an "interval vector;"(i,p) will vary concomitantly over the forwards-oriented intervals. Let usstudy some actual interval vectors in this system by way of example.

5.4.1 EXAMPLE: Figure 5.11 shows the mensural skeletons for motives (b),(c), and (d) from the Chopin sonata studied earlier (in section 4.3). Therhythmic motives are modeled by sets of time spans, and their interval vectorsare tabulated on figure 5.12.

Forming and reading these interval vectors becomes easy with practice.The forwards-oriented interval (1,1) labels the set-class of AODs D = (s,t)such that t begins right after s (1 s-length after s begins) and extends the sameduration as s (1 times the length of s). Within set (b) we count three instances ofsuch AODs. The AODs are formed by the first-and-second notes of themotive, its third-and-fourth notes, and its fourth-and-fifth notes. Thus thenumber 3 is entered on the table of figure 5.12, in the row of the table headedby the interval (1,1) and in the column of the table headed "vector of (b)." Set(c) includes only two AODs in the set-class (1,1): the first two notes of motive(c), and the last two notes of the motive. Remember: A pair of successivequarter notes in any tempo at any point in the piece (or any other piece any

FIGURE 5.11

114


FIGURE 5.12

time) is canonically equivalent to such a pair of successive eighths or succes-sive halves or successive quintuplet sixteenths. All the AODs just indicatedbelong to the same set-class, the set-class determined by the interval (1, 1)between the first and second members of each AOD. The two AODs of class(1,1) embedded in set (c) are counted on figure 5.12 by the number 2, enteredto the right of the interval (1, 1) and in the column headed "vector of (c)." Set(d) also embeds two AODs of class (1, 1), namely the pair of half notes andthe pair of whole notes at the end of the motive.

Let us now consider the set-class corresponding to the forwards-orientedinterval (3, 2). An AOD D = (s, t) belongs to this class if t begins 3 s-spanslater than s begins, and lasts twice as long as s. The second quarter of (c) andthe first half-note of (c) form such an AOD. So do the last quarter of (c) and thesecond half-note of (c). These two dyads are tabulated by the entry 2 in the(3, 2)-row and the (c)-vector-column of figure 5.12. The entry of 1 in the(11/2, !/2)-row arises from the AOD formed by the first and third notes ofmotive (d).

In section 4.3 we noted a progressive "expansion" from motive (b), throughmotives (c) and (d). The progressive broadening of note values, from eighthsto quarters to halves to whole notes, is obviously crucial. Our model doesnot address this aspect of the progression. But it does address and ana-lyze well another aspect of interest, something we might call the "progres- 115


sive diversification" of the motives in their internal rhythmic structures. Asone sees from the first column of figure 5.12, motive (b) concentrates on onlya few intervals, most of which appear more than once. Motive (c) projects onlytwo intervals that appear more than once (in the second column of the figure);no interval appears thrice (in that column). Motive (c) projects more intervals,and more diverse intervals, than (b). Motive (d) projects only one interval thatappears more than once; that interval appears only twice. Motive (d) thus con-tinues the process of diversification. The insensitivity of our interval vector tochanges in tempo, a defect in some ways, is useful here: It enables us to com-pare motives (b), (c), and (d), each in its own intrinsic context. We touched onthis idea earlier in connection with IFUNC(Y, Y).

Motive (a), the motive of the opening Grave, does not appear on figure5.11 or figure 5.12. If one ignores the anticipation of F(? = E natural in themusic, then motive (a) is canonically equivalent to motive (d).

5.4.2 In connection with figure 3.3 (page 41), we earlier studied an "unrollinginterval vector" for a set in a different GIS, a set pertinent to Webern's PianoVariations. The present GIS, like the earlier one, has an intrinsic chronology,so we can "unroll" its interval vectors too. The abstract method of doing sowill involve a number of technical finesses.

To begin the abstract study let us consider the imaginary string trio wediscussed a short time ago, and let us imagine another of its passages, whichwe can symbolize as in figure 5.13.

We can model certain temporal aspects of this passage, as we did with thelast one, by a set Y of time spans. The violin projects four time spans, (16,1),(17,1), (18,1), and (19,1); let us call these spans vnl, vn2, vn3, and vn4respectively. The viola projects the two time spans (16,|) and (18f,f); let uscall these spans val and va2. The cello projects the two time spans (16j, 2) and(18^, 1 )̂; let us call these spans vcl and vc2. We can list the members of Y "inparts" as vnl, vn2, vn3, vn4, val, va2, vcl, vc2. Of course that is not their"order of appearance" in the music. But what isl

FIGURE 5.13

116


We might try attack-ordering to list the members of Y "in order ofappearance." Then we would list them as vnl, val, vcl, vn2, vn3, vc2, va2,vn4. For many purposes attack-ordering is natural, and we have seen howcogent it is in connection with dyads, intervals, and the canonical group. Butwe shall not want to use attack-ordering in connection with the way weperceive the members of Y "appearing." To see why not, imagine that we stopthe music of figure 5.13 just after the attack of vn2, that is, just after time-point17, and suppose that we ask just which time spans we have perceived upthrough that time. Obviously we have perceived vnl. But we have not yetperceived any other spans. True, we have heard both the other instrumentsattack other spans. But we do not yet know how long those spans are going tobe, so we cannot claim to have perceived them as spans, using them e.g. toform intervals in an unrolling interval vector. For all we know as we listen attime-point 17, the viola may be intending to hold onto its note for 24 quarters.

Now when we unroll the interval vector for Y in connection with figure5.13, we are going to want precisely to "stop the music" of the figure at variousstages, asking at each stage what intervals we have heard so far. As we havejust seen, when we stop the music at time-point 17, we can be sure of havingperceived only one span, namely vnl; hence we cannot say we have perceivedany proper intervals at all so far. The attack-ordering for Y is deceptive in thisconnection. That ordering, beginning vnl, val, vcl, vn2,..., makes it seem asif the spans val and vcl have "already occurred" by the time vn2 occurs,attacking at time-point 17; hence it seems (wrongly) as if we ought to count theforwards-oriented intervals int(vnl, val) and int(vnl,vcl) and int(val,vcl)as "having already occurred" by the time we "get to" vn2. But this inference iswrong. As we have seen, no intervals have yet "occurred" by time-point 17, sofar as our perceptions of spans and their interrelations are concerned.

To reflect the true order of our span-perceptions, we shall want to use adifferent system of ordering, not attack-ordering but release-ordering. Givendistinct spans s and t, s precedes t in the release-ordering if s ends before t ends,or if they end simultaneously and s is longer. If s and t correspond to musicalevents event 1 and event2, then s precedes t in the release-ordering when weperceive the time span during which event 1 has happened before we perceivethe time span during which event2 has happened, or if we perceive both spanssimultaneously and recall that event 1 began first.

Release-ordering for the set Y thus enables us to articulate the music offigure 5.13 into stages that correspond to our evolving perceptions of timespans "having happened" as we listen. The members of Y in the release-ordering are vnl, vn2, vcl, val, vn3, vc2, va2, vn4. Furthermore, when wearticulate the music into such stages, we shall want to demarcate the stages bythe time points at which various spans are released, not at which they areattacked. Thus the finest possible articulation of the set Y = (vnl, vn2, vcl,val, vn3, vc2, va2, vn4) into stages for our purposes can be realized as follows. 777


Stage 1: We have heard Yt = (vnl, vn2) at time-point 18, the release of vn2.Stage 2: We have heard Y2 = (vnl,vn2,vcl) at time-point 18 ,̂ the release-point of vcl. Stage 3: We have heard Y3 = (vnl, vn2,vcl,val) at time-point18f, the release-point of val. Stage 4: We have heard Y4 = all of Y at time-point 20, the simultaneous release for vc2, va2, and vn4.

By calculating how the interval vectors for Y1} Y2, Y3, and Y4 develop,each expanding the counts of the last among the various intervals counted, weshall be able to model how our sense of intervallic structure evolves as we listento the musical passage. We shall be able to use our formal model analytically,just as we used analogous machinery earlier in connection with the Webernpassage and the expanding interval-counts of figure 3.3.

Our work above can now be generalized. Given any set Y of time spans,first list Y as (s:, s 2 , . . . , SN) in the release-ordering. Next identify N or fewer"stages" associated with certain subsets of Y as follows. Stage 1 is articulatedat the release of s2; it is associated with a certain subset Yx of Y. Yt = ( s t , s2)unless s3 releases simultaneously with s2; in that case Yt = (sr, s2, s3) unless s4

also releases simultaneously at that time; in that case ... (etc. etc.). AfterY! = ($!, s2 , . . . , SM) has been found, Stage 2 is articulated by the release pointof sM+1. Stage 2 is associated with a certain subset Y2 of Y. Y2 = Y1 + (sM+1)unless sM+2 releases simultaneously with sM+1 (etc. etc.). And so on. Even-tually one attains the release point of SN and exhausts the set Y. We can regardthe stages as developing in a simple serial rhythm as stage 1, stage 2, stage 3,and so forth. Or we can regard them as developing in a "perceptual rhythm,"the rhythm of the various release-points at which the stages articulate. (This isinteresting but it oversimplifies the psychology of what is going on.) As thestages develop rhythmically, the evolving interval vectors of Y1} Y2, etc. canbe studied. Care must be taken here because the release-ordering of Y does notnecessarily coincide with the attack-ordering. It is possible for sm to precede sn

in the release-ordering, but to follow sn in the attack-ordering. (The differentlistings of Y in connection with figure 5.13 illustrate the possibility.) Shouldthis happen, when we get to the stage that notices (the release of) sn in therelease-ordering, we shall want to tabulate the forwards-oriented intervalint(sn, sm) in our updated interval vector, not the backwards-oriented intervalint(sm,sn).

The reader who likes to fool with computer programming and who has ahome computer with a color monitor will enjoy writing an "unrolling intervalvector" program. The program will take a set Y of time spans, arrange it inrelease-ordering, determine the articulation-points of the various stages, andfind the corresponding subsets Yj, Y2 , . . . , Y. The program will then computethe interval vector for Y^ and display it on the screen as follows. For eachforwards-oriented interval (i, p) that is counted, a colored dot appears at thepoint (i, logp) on a half-plane grid, (i is always non-negative; log p is positive,zero, or negative.) If the interval appears only once in the set, the dot is violet;118

the more times the interval appears, the more the color of the dot movestoward the red end of the spectrum. (The background of the screen is eitherwhite or black.) After the program has computed the interval vector for Yx , itwill update the count of various intervals so as to obtain the interval vector forY2, changing the color of some dots on the screen as pertinent. Then it willupdate the count of various intervals to obtain the interval vector for Y3, andso on. The updating can be done quickly following the method of figure 3.3.(Remember that you may have to adjoin more than one releasing time-spanat any new stage. Also remember to adjust for any new dyads that may berelease-ordered but not attack-ordered.) The rhythmic updating of the screencan follow either the serial rhythm of the stages or their "perceptual rhythm"as discussed above, either in real time or suitably scaled for visual effect.

5.4.3 EXAMPLE: The technique of unrolling can be applied to EMB-relatedfunctions beyond the interval vector.

The "set" Y of figure 5.14, for example, is articulated into four stages.(We could articulate it farther, but we shall not do so here. Since neither thetime unit nor the point zero is specified, Y is not strictly a numerical "set"within TMSPS, but I am assuming the reader will not mind a certain loosenessin discourse at this point.) Figure 5.14 also displays "sets" X l 5 X2, X'l5 andX'2, all of which can be found embedded within Y. X\ is a canonical form of Xjand X'2 is a canonical form of X2.

Figure 5.15 shows how the embedding numbers of the set-classes /XJand /X2/ within Y develop, as Y develops over the four stages. The values riseat Stage 4 because the dotted half releases there and the appearances of X\ andX'2, augmented (canonical) forms of Xt and X2, can now be counted as "embedded" in Y. Figure 5.15 shows us how the set-class /X2/ comes on lateand strong, pulling ahead of (XJ at Stage 3 and then decisively ahead atStage 4. 779


FIGURE 5.14

FIGURE 5.15

No doubt the reader has recognized Y as interpreting the opening ofBrahms's G-Minor Rhapsody. The idea that "/X2/ comes on late and strong"is reinforced by the end of the closing group in the music, where the closingtheme is liquidated rhythmically down to a succession of X2-forms alternatingon the tonic and dominant of D minor.

The rhythmic interpretation of figure 5.14 does not exclude other pos-sible rhythmic readings of this music. E.g. one could read triplet eighths wherethere are triplet rests on figure 5.14; then /Xl/ and /X2/ would come out in atie on figure 5.15. But such other interpretations just as clearly do not excludethe reading of figure 5.14. The reader who consults the score will find waysenough in which the relation of quarter note to accompaniment changes afterStage 2 so as to support the reading of the figure.

5.4.4 NOTE: Through section 5.4 so far, we have focused upon the intervalvector and more generally the EMB function, in connection with ournon-commutative GIS of time spans. The technique of "unrolling in stages"which we applied to this study could also be applied in connection withIFUNC(X, Y), as we unroll either X or Y or both in stages.

5.5 NOTES: Let us return now to the most general abstract setting, that of afamily S and a group CANON of operations on S. Following the suggestionsof my writings elsewhere, we can explore numbers of interest beyondEMB(X, Y).7 We may define COV(X, Y), for example, the coverfng numberof X in Y, as the number of forms of Y that include X. This is not necessarilythe same number as EMB(X, Y), the number of forms of X that are embeddedin Y. E.g. in atonal theory take X = (C, E) and Y = (C, E, G#); thenEMB(X,Y) = 3 but COV(X,Y)=1. If S is finite then COV(X,Y) =EMB(Y,X), where Y and X are the complements of Y and X.

7. "Some New Constructions Involving Abstract Pcsets, and Probabilistic Applications,"Perspectives of New Music vol. 18, nos. 1-2 (Fall-Winter 1979 and Spring-Summer 1980),

720 433-44.



We may also consider SNDW(X,Y,Z), the sandwich number of Ybetween X and Z; this is the number of forms of Y that both include X and areincluded in Z. If we write 0 for the empty set, then SNDW(0,Y,Z) =EMB(Y, Z); if S is finite then SNDW(X, Y, S) = COV(X, Y). If Y' is a formof Y then SNDW(X,Y,Z) = SNDW(X, Y',Z); hence we can writeSNDW(X,/Y/, Z) without ambiguity. But we cannot use /X/ or /Z/ forsandwich arguments in this way; SNDW(X,Y, Z) depends very much onthe specific forms of X and Z being used as arguments. For example let Zbe the C-major scale; let /Y/ be Forte-class 3-4, which we could write as/(B,C,E)/. Let Xj = (C, E). Then, allowing both transpositions and inver-sions as canonical, SNDW(Xl5/Y/,Z) = 2: There are 2 forms of (B,C,E)that can be sandwiched between (C, E) and the scale, namely (B, C, E) and(C, E, F). Now let X2 = (F, A). X2 and Xj belong to the same set-class, butSNDW(X2,/Y/,Z) = 1, not 2: Only 1 form of (B,C,E) can be sandwichedbetween (F, A) and the scale, namely (E, F, A).

Another interesting number is ADJOIN(X, Y, Z). This is the number offorms Y' of Y satisfying both (A) and (B) following. (A): Y' is disjoint from X.(B): There is some form of Z that includes both X and Y'. To illustrate whatthis number is inspecting, let X = (C, E), Y = (D, G), Z = the C-major scale.(C, E) + (D, G) lies within some major scale; so does (C, E) + (F, 6(7); so does(C, E) + (F#, B); so does (C, E) + (A, D). Exactly the 4 fourths (forms of Y)metioned in the preceding sentence have both the desired properties (A) and(B); other fourths (forms of Y) either contain C or contain E or do not add uptogether with (C, E) to lie within any major scale. So ADJOIN((C, E),(D, G), C-major scale) = 4. Inspecting properties (A) and (B), one sees thatwe can write ADJOIN(X,/Y/,/Z/); it follows that we can even writeADJOIN(/X/,/Y/,/Z/).

5.6 APPENDIX (optional): We prove here the crucial theorem stated earlier,on the relation of dyads and intervals in our non-commutative GIS forTMSPS, using the group of interval-preserving operations as CANONical.Here is that theorem stated again: Given attack-ordered dyads Dx = (s^tj)and D2 = (s2,t2), then Dj and D2 are canonically equivalent if and only ifimXs^tj) = int(s2,t2). The proof follows.

Set Si = (a^xj, t t = (b^yO, s2 = (a2,x2), t2 = (b2,y2). Suppose firstthat D! and D2 are canonically equivalent; we shall show that int(s1,t1) =int(s2, t2). Say that P = P(h%u) is the canonical operation mapping Dx onto D2.P maps the members of the first dyad somehow onto the members of thesecond; conceivably the operation might transform st into t2 and s2 into t t.But in fact that cannot happen in this situation: P transforms st into s2 and tx

into t2. To see that, we use the fact that both dyads are attack-ordered. SinceD! is attack-ordered, either ̂ < bt or (at = bx and xt < yx). If a1 < bl thenua1 < ubi and h + uax < h + ub^ hence the P-transform of Sj begins before 727

5.6 Generalized Set Theory (1)

the P-transform of il. Since D2 is also attack-ordered, that means that the P-transform of st must be s2 and the P-transform of tl must be t2. (Otherwise wewould have b2 < a2, contradicting the attack-ordering of D2.) So the caseax < bi leads to the desired result; let us investigate the case (a^ = bx andxi < vi)- In tnat case» h + uai — h + ubi and ux1t < uy^ again we infer that(P^), P(tj)) is the attack-ordering for D2, and so P(S!) = SjandP^) = t2asdesired. Now that we have established the relations P^) = s2 and P(ti) = t2,the rest is easy: int(s2,t2) = int(P(s1),?(!!)) = int^,^) as claimed, becauseP is interval-preserving.

Now we shall prove the converse. Supposing that int^, t j ) = int(s2, t2),we shall prove that Dx and D2 are canonically equivalent. int(s1,t1) =((bi — aj/x^yjxi) and int(s2,t2) = ((b2 — a2)/x2,y2/x2). So what we areassuming can be expressed by equations (A) below.

(A): (bt - aj/xj = (b2 - a2)/x2; y1/xl = y2/x2.

Using some algebra applied to equation (A), we can derive (B); manipulationof (B) produces (C).

(B): x^-x^ =x1b2-x1a2 ; yiX 2 = x ty2 .(C): x^ - x2aj = Xib2 - Xjb^ x^ = y2l

Let g be the number such thatx ta2 — x2aj = g = Xjb 2 — x2b1. Let u bethe number such that x 2 /Xj = u = y2/y1. From the equation xxa2 — x2a t = gwe infer x^ = g + x2ax; thence we infer a2 = (g/xt) -f (x2/x1)a1, or a2 =(g/Xj) + uaj. In similar fashion, we derive the other equations of (D) below.

(D): a2 = (g/xO + ua f; x2 = uxx

b2 = (g/x1) + ub1; y2 = uy1.

Seth = (g/x1).Then(a2,x2) = (h + uaûxj,while(b2,y2) = (h 4- ub1?

uyj. Thus, taking P = P(hiU), we have s2 = P(s^ and ta = P(tj). Since Pis aninterval-preserving operation and D2 = PCDJ, the dyads D! and D2 arecanonically equivalent, q.e.d

722

Generalized Set Theory (2):The Injection Function

This chapter is a pivot in a sort of modulation from the study of GeneralizedIntervals to the study of Generalized Transformations. The Injection Func-tion can be defined, discussed, and applied to musical analysis without invok-ing the notion of interval or canonical operation at all. In that regard theInjection Function looks forward to the work lying beyond chapter 6. At thesame time the Injection Function is strongly suggested by IFUNC and EMB,either of which it can be used to generalize when there is a GIS or aCANONical group at hand. In that regard the Injection Function logicallycontinues the work of chapter 5; indeed it elucidates some problems thatchapter 5 left hanging.

It was hard for me to decide, as I pondered this dichotomy, how best toarrange my exposition of the material for this chapter. At first it seemednatural to emphasize aspects of continuity, first showing how the InjectionFunction grows out of IFUNC and EMB, and then moving gradually tohigher and higher levels of abstraction. But when I drafted an exposition inthat spirit, I did not like the effect. Looking back at IFUNC and EMB gaverise to so much interesting discussion that an endless time seemed to go bybefore I could get to discuss the more radical abstract features of the newconstruction. So I decided upon a different method of exposition.

That is why I shall begin by discussing the Injection Function at a highlevel of abstraction, emphasizing its novelty in contrast to the material ofchapter 5. Then, once the reader has become familiar with the new construct, Ishall begin stitching in threads from here and there in chapter 5, graduallysewing together the seam I have created. At any rate, that is my plan. Thereader who is not happy with it will I hope be able to follow along neverthelesswithout too much discomfort, once aware of it. 123

6


6.1 CONVENTIONS: We shall be concerned with a family S of objects, andwith various transformations f that map S into itself. We do not assume that thetransformations are necessarily operations (1-to-l and onto S). Operationsare capable of entering into groups, e.g. canonical groups, groups of inter-val-preserving operations and/or transpositions in a GIS, and the like. A trans-formation that is not an operation can have no inverse transformation on S,and so cannot belong to any group of operations on S.

As in chapter 5, we shall use the word "set" to denote a finite subset X ofS; more exactly, we shall do so until explicit notice to the contrary. (Towardthe end of chapter 6 we shall indicate how the work can be generalized to dealwith infinite subsets of an infinite S.)

6.2.1 DEFINITION: Given sets X and Y, given a transformation f on S, thenthe injection number of X into Y for f, denoted LNJ(X, Y)(f), is the number ofelements s in X such that f(s) is a member of Y.

INJ(X, Y)(f) answers the question: "If I apply the transformation f to theset X, how many members of X will I thereby map into members of Y?" If fis 1-to-l, then those distinct members of X will map into distinct members ofY, so that INJ(X, Y)(f) will also be the cardinality of f(X) D Y, that is thenumber of elements that the sets f(X) and Y have in common. But when f isnot 1-to-l, this is not necessarily the case. We might e.g. have 5,273,647distinct members of X all mapping into one member of Y; in this case we mighthave INJ(X, Y)(f) = 5,273,647, while f(X) and Y might have only 1 commonmember.

6.2.2 EXAMPLE: Take S to be the twelve chromatic pitch classes. Define f asfollows: f(s) is the pitch class C when s is any "white note"; f(s) is the pitchclass F| when s is any "black note." Fix X = (C, C|, D, El», E).

Take Y = (B, C#, D, E, F, F|). Then INJ(X, Y)(f) = 2. The 2 blacknotes of X map via f into the member F# of Y; the other notes of X, beingwhite, do not map into any member of Y. They do not because the pitch classC, the image of all white notes under f, is not a member of Y.

Now take Y = (F, F#). Then INJ(X, Y)(f) = 2 still and again, for thesame reasons.

Now take Y = X. Then INJ(X, Y)(f) = 3. The 3 white notes of X all mapvia f into the member C of Y = X; the other notes of X, being black, do notmap into any member of this Y. (Ff, the image of all black notes under f, is nota member of this Y.)

6.2.3 EXAMPLE: Take S to be the twelve chromatic pitch classes. We define atransformation called wedging-to-E, which we shall denote as WE. The trans-

124 formation maps E into E and B\> into B\>. Every other pitch class when wedged


advances one hour towards E along the clock of pitch classes, clockwise orcounterclockwise by whichever route is shorter. The effect of WE on the clockof pitch classes is portrayed by figure 6.1.

FIGURE 6.1

WE is not an operation; it is neither 1-to-l nor onto. Nevertheless it is auseful transformation to have at hand for analyzing "Angst und Hoffen," theseventh song from Schoenberg's Book of the Hanging Gardens op. 15. Figure6.2 shows the pitches in the two chords that form the opening "Angst undHoffen" motto, chords X and Y.

FIGURE 6.2

Here INJ(X, Y)(wE) = 2: When the transformation WE is applied to thenotes of X, 2 of those notes map into notes of Y. The D of chord X,specifically, maps into the E|? of chord Y and the B|? of chord X maps into theBb of chord Y. Thus "two-thirds of X" is mapped into Y by the wedge. If onlythe G(? of X wedged into Y then all of X would wedge into Y. That wouldhappen if the F|j of Y were an F natural. So we can consider the F|j of Y (in thiscontext!) as a "wrong note" or a "blue note"; it substitutes for the F natural 725


that "should" be there. Indeed, there is a lot of musical action later on in thesong that involves the idea of "getting F|? (or E) to move to F."

The F(? of Y is not just a blue note, though; it is also the point to which thewedge converges. In that connection it has a tonic character as a potentialpoint of repose or arrival. We shall see later how this potentiality is realized inthe music.

The operation I = l|j£ = I| is also heavily involved in the effect of themusic sketched by figure 6.2. Inversional symmetry about B|? is very stronglyprojected not only by sonic features but also by the visual layout of the chords,symmetrical about the middle line of the staff. In our new terminology, we canwrite INJ(X, X) (I) = 3: f or all of X maps into X under the transformation I.We can also observe that INJ(Y, Y)(I) = 2: f of Y maps into Y under thattransformation. Once more we can point out that if only F-flat were F-natural, and so on. (We can presume that F|?-for-F is the pertinent substitu-tion here, rather than E|?-for-E in the upper register. Replacing E[? by E in themusic here would result in the syntactically implausible chord (F|?, B[7, E).)

Neither IFUNC nor EMB can adequately engage the ideas we have beenconsidering in connection with INJ here. The injection function enables us todiscuss several thematic functions for the F|? of chord Y: That note is at oneand the same time a substitute for F natural, and the convergence-point of thewedge, and one center for the operation I. INJ also enables us to distinguishvery different structural functions for the transformations I and WE. Specifi-cally, I transforms each chord of figure 6.2 into something very like itself"; WE,in contrast, transforms an antecedent chord into something very like a conse-quent chord. We may think of I as an "internal" transformation and WE as a"progressive" transformation in this musical context. We shall pick up thetheoretical implications of that notion later on.

Figure 6.3 helps us hear how the transformational ideas under discussionpersist, develop, and resolve over the last third of the piece. The figurecomprises mainly the notes of the piano over this section, notes which carrythe harmony in a homophonic texture. Figure 6.3(a) labels the chords in-volved. The last three notes of the vocal part are also included, beamed as anarpeggiated chord Z5 that belongs in this progression. The progression leadsback to X and Y, "Angst" and "Hoffen," now as an outcome rather than apoint of departure.

Figure 6.3(b) indicates those notes of (a) which participate in E-wedgingactivity. Two notes of Z1 wedge into Z2 along the beams, two notes of Z2

wedge into Z3, and two notes (pitch classes) of Z3 wedge into Z4. (Bb of Z3

wedges as a pitch class into Bb of Z4; the registral symmetry of the high andlow Bb pitches about the pitch E4 is nice.) Thus INJ(Zn,Zn+1)(wE) isconsistently = 2 for n = 1,2, and 3. All of Zj-as-2-note-pcset is projected intoits successor set Z2 by the wedge; f of Z2 is projected into its successor, and|ofZ3.726


FIGURE 6.3

The F-with-a-question-mark, notated above the upper beam at Z4, indi-cates a "missing" F that breaks an otherwise consistent pattern of wedgingand inversional balance. In this respect the missing F is exactly like the missingF on figure 6.2 earlier. It is like that F in breaking an E-wedge; it is like that F,too, in leaving the Eb of chord Z4 bereft of its I-partner, just as the E|? of chordY on figure 6.2 was bereft of its I-partner. The note-against-note relations offigure 6.3(b) show that INJ(Zn,Zn)(I) = 2 for n = 1 and 2; INJ(Z3,Z3)(I)actually = 3. (Within the set Z3 of pitch classes, D and F# invert into eachother, while B(? inverts into itself.) Hence the abrupt inversional imbalance atZ4 is all the more strongly felt, since INJ(Z4,Z4)(I) = only 1. The metaphorof "imbalance" caused by "something missing" interacts well with the text:The singer is in a state of emotional discombobulation caused by the lover'sabsence.

Continuing along figure 6.3(b), we hear how, when the progression Zt

etc. recurs as the progression Z\ etc., the singer's final sign-off supplies therequired F natural at the right moment, within the arpeggiated set Z5. Therequired Z5, which continues the patterns of wedging and inversion consis-tently, sets the word begehre = require. (The singer claims not to require theconsolation of any friend.) Once the begehrte Z5 has appeared, the wedge can 727


and does converge all the way to the E of Z6, across the pickup chord Z'4 inthe music. The level of progressive wedge-projection and internal I-projec-tion is thus restored: INJ(Z'3,Z5)(w

E) = 2, INJ(Z5,Z6)(wE) = 3; INJ(Z5,Z5)(I) = 2,INJ(Z6,Z6)(I) = 3.

The convergence of the wedge to its focal point E within Z6 coincides withand supports a big structural downbeat. The sonority of Z6 has earlier beenassociated with the word Seufzer, during the text, "meine Worte sich inSeufzer dehnen (my words trail off in sighs (or groans))." Figure 6.3.(b)portrays the sighing and trailing-off ideas very well. It also shows how thebeamed sighing-progressions are framed by the elements (C, A[?) and (E) ofthe Seufzer-chord, which (for this reason and for others too) takes a bigdownbeat when it appears as Z6.

Figure 6.3(b) shows how the wedging commences yet once more after Z6,continuing through X to Y, which is the end of the piece. Within X and Y, thepitches D5 and Et>5 of the music are brought down an octave, to be displayedas D4 and E|?4 noteheads on the figure; this shows clearly how the pitch classesD and E[? contribute to the final wedge. In particular, it brings out stronglyhow the final progression, Z6-Z7-X-Y, recapitulates the initial wedge-structure of the opening progression Z1-Z2-Z3-Z4 on figure 6.3(b). The"blue note" Fb of the final Y, shown on figure 6.3(a), takes on addedsignificance in its "tonic" function, as it prolongs the downbeat E from theSeufzer-Z6 (Dehnung).

Figure 6.3(c) sketches in a format similar to (b) the influence of asubordinate wedge and inversion over this passage. That is wedging-to-F#and the inversion-operation J = Ipj| = l£. The symbols "A|?-G-F#//," atthe beginning at bottom of the figure, show how the inner voice of the chordsattains the goal and center F# of the wedge, getting there "from above" overchords Zi through Z3. Then from Z\ right on, all the way to X, the outervoices almost succeed in converging to F#(G[?), except that F is missing in thelower voice. Again we run into the thematic and structural "missing F!" Themissing F now has a new structural function: It is missing as a semitoneneighbor to Gj? in a wedge converging to G(?, the bass of the Angst-chord X.The earlier missing F was missing (inter alia) as a semitone neighbor to E inwedges converging to E, the bass of the Hoffen-chord Y.

The T10-relation between the bass note of X and the bass note of Y is thusexpanded into a larger T10-relation, a relation involving the respective wedgesand inversions about those notes. The larger relation can be expressed by the

1. I discuss these ideas and others of the same sort elsewhere, exploring more systematicallyrhythm, meter, text setting, registers, doublings, and other features of the music. The reader whowould like to go deeper into the piece itself will be interested by that article, which is less pre-occupied than we must be here by theoretical constructions of various sorts. The article is "AWay Into Schoenberg's Opus 15, Number 7," In Theory Only vol. 6, no. 1 (November 1981),

128 3-24.


transformational equations WE = T^w^T^1; I = T10JT101. Essentially,

these equations "modulate by T10" the wedging and inversional transforma-tions centering on F#, to obtain the analogous transformations centering onE. We shall explore the idea of such "modulation" more generally and moreformally later on. The present remarks are meant to prepare that later explo-ration. The reader can hear in connection with the example at hand that therelation T10(Gb) = E has to do both with the bass line in the chord-successionX-Y, and also with the relation of WF* and J, in figure 6.3(c), to WE and I, infigure 6.3(b). The reader will then be able to summon that musical experienceto mind during the abstract discussion of transformational "modulation"later on (in connection with Formula 6.7.2(c)).

The formalities of INJ, applied to the transformations WE and I on S,have engaged figure 6.3(b) only partially. The visual layout of that figureconveys a good deal more than our formal machinery has so far described. Forinstance, the figure shows by its note-against-note layout that we do notsimply have an Ab in Zt wedging to a G in Z2 and, independently, a C in Zl

wedging to a C# in Z2; rather the "I-partnership" of (Ab, C) within Z^ wedgesas a whole to the "I-partnership" of (G, C#) within Z2. In a similar sense, themissing F of Z4 is not just a missing note; it is a missing I-partner for Eb,without which the wedge cannot converge; as a missing partner it symbolizesthe absent lover. And in a similar sense, the last line of text, "(that) I requirethe consolation of no friend," is symbolized exquisitely by the pitch class E asgoal of the wedge and center of inversion: The pitch class has and needs nopartner; it gets along by itself, perfectly self-centered in its Seufzer. The visuallayout of the figure brings out such ideas; our transformational machinery hasnot as yet adequately engaged them.

But it can be formally developed so as to do so. The operation I partitionsthe family S into distinct "transitivity sets" (Bb), (A,B), (Ab,C), (G,C#),(Gb,D), (F, Eb), and (E). I transforms the members of each transitivity setamong themselves: I(Bb) = Bb; I(B) = A and I(A) = B; and so on. Suchtransitivity sets enable us to engage the notion of "I-partnerships" in ourformal machinery. Many of the chords under consideration embed an entiretransitivity set; some chords even embed two (e.g. Z3 = X = Angst andZ6 = Seufzer). Frequently a transitivity set embedded in one chord is trans-formed as a whole by WE into a transitivity set embedded in the next chord.Such ideas can be developed very abstractly in connection with the generalizedINJ function. Here, it is formally important that the transformations I and WE

commute. It is not remarkable that the visual aspects of figure 6.3 can bedescribed by formal aspects of the INJ machinery when suitably extended.After all, one could hardly conceive the layout of the figure without someprior intuitions of WE and I as transformations; the formal extensions of themachinery, in that connection, amount only to making the relevant intuitionsexplicit. 729


Our investigations so far have focused on harmony and voice leading, aswe explored the structural functions of various transformations in this song,using INJ. Now let us use INJ to explore some melodic functions of varioustransformations therein.

FIGURE 6.4

Figure 6.4 will help us in this endeavor. The figure transcribes the pitchesfrom the opening of the voice part, where they set the first line of text.Schoenberg's spelling projects throughout the phrase a strong visual inver-sional symmetry about the third line of the staff, where BW would appear asa center of I. Ordinal numbers 1, 2, ... , 10 appear under the first, second, ...tenth notes of the figure. The events of the melody are modeled here in a spaceS whose members are pairs (n, p), n being an ordinal number and p a pitchclass. As a serial structure, the "melody" is modeled by an unordered set often such pairs; the elements of this set are the pairs (2, G!>), (1, D), (10, El>),(3, Et), and so on. "(2, G!>)" can be interpreted as saying "The second noteis Gk"

Arrows on the figure indicate transformational relations that will interestus here. Each arrow is labeled by a pair of symbols comprising a number(1, 2, 3, 5, or 6) and a letter (I or w). The number indicates how many ordinalslater the transformed pitch class appears. The letter indicates a pitch-classtransformation, w standing for WE. Thus the arrow labeled "6, w" which issuesfrom the third note of the series, E\>, indicates that the note is transformed intoa note appearing 6 order positions later, via the wedge transformation. Thearrow from A to Cl>, labeled "2, I," indicates that the A is transformed into anote appearing 2 order positions later, via the / transformation. More for-mally, the transformation 6, w maps the element (3, E\>) of the melody into theelement (3 + 6, w(EI>)) = (9, Fl>); the transformation 2, I maps the element(6, A) into the element (6 + 2, I(A)) = (8, Cl»).

The transformations (6, w) and (2, I) are well defined by this formalmethod on the space S of pairs (n, p). (6, w) maps the pair (n, p) into the pair(n + 6, w(p)); (2, I) maps the pair (n, p) into the pair (n + 2, I(p)). The trans-formations are not operations, w itself is not an operation on the twelve pitch130


classes; even beyond that, the ordinal aspect of the mappings prevents thetransformations from mapping S onto itself. E.g. there is no (n, p) in S (with na positive integer) such that (2,I)(n,p) = (1, Ab).

The dotted arrows and question marks on figure 6.4 arise from thepossibilities of considering the F|?s as substitutes for F naturals. If the flatswere naturals, the dotted arrows would be solid and the question marks woulddisappear.

We shall denote by XJJ, the set comprising the mth through nth events ofthe series (i.e. melody-set). Using the subsets XJJ,, we can apply familiar "un-rolling" techniques to the situation, now using our injection numbers.

For instance INJ(X*, X*) (f) = 2, where f is either (2, w) or (1,1), assum-ing we allow the dotted lines. The equation states: The set comprising (1, D),(2, Gb), (3, Eb), and (4, F(b)) contains "2" members that transform into theset under the transformation (2, w), and "2" members that transform into theset under the transformation (1,1), supposing that Fb is read "as if" F natural.This equation engages a significant "unrolling" when compared to theequation INJ(Xi,Xi)(f) = (only)l for the same transformations f.

We can compare these internal transformations of X| with the internaltransformations of Xf, the next 4-element subset of the melody. Xf comprises(5, C), (6, A), (7, Ab), and (8, Cb). INJ(Xf, Xf) (2,1) = 2: With respect to thenew tetrad Xf, the transformation (2,1) plays the same internal role that (1,1)did in connection with the first tetrad X*. And, as the figure shows, (3,w)plays the same role with respect to Xf that (2,w) played with respect to X^.That is so even though INJ(Xf, Xf) (3, w) is only 1, not 2. Our arrow diagramscapture a certain picture of X* on the figure, as it appears bound togetherinternally in a certain way by (1,1) and (2, w) arrows. The same kinds of arrowshapes capture a similar picture of Xf on the figure, as it appears boundtogether in a similar way by (2,1) and (3, w) arrows.

Our model enables us to observe an interesting augmentation of ordinaldistances, from the arrow transforms binding X| to the arrow transformsbinding Xf. That is, within X* I-relations occur 1 note apart and w-relationsoccur 2 notes apart; within Xf these ordinal distances are expanded: I-relations occur 2 notes apart and a w-relation occurs 3 notes apart. This serialaugmentation is particularly interesting because Xf takes only half the time tosing as did X?, in the clock time of the music.

Our discussion of X*, Xf, I, and w is enhanced by the observation that noI-arrows and no w-arrows lead events of the first tetrad to events of the second,on figure 6.4. In our terminology, INJ(Xf, Xf) (f) = 0 when f is either (n, I) or(n, w), for any n. This observation specifically enhances our sense that themelody articulates into X* + Xf + Xg°, when heard in the context ofw and Irelations. The italicized phrase is meant to recall our earlier discussion inconnection with the various contexts of a melodic phrase within the Webernviolin piece. Our sense of X* + Xf + X^0 in this context is further enhanced 737


by the (6, w) arrow on figure 6.4, extending from inside X* over X| to Xg°.INJ(X}, X*°)(6, w) = 1, after INJ(Xf, X|)(n, w) had been zero for all n.

The ordinal distance over which w functions continues to grow: The firstw-arrow(s) had ordinal span 2 within Xf; the next w-arrow had ordinal span 3within X|; now a w-arrow has ordinal span 6, between X* and Xg°. In thiscontext the F flat of (9, F|?) in the melody is "correct"; it is in fact the goal ofthe wedge. One notes the care with which the high F[? is distinguished by thecomposer from the low F[?.

When we hear the penultimate F|? as "correct" and ignore the last dottedarrow on figure 6.4, we get a sense of "ordinal expansion" over the phrase asregards not just w spans but also I spans. First I projects at ordinal distance 1and w at distance 2; next I projects at distance 2 and w at distance 3; finally Iand w both project at distance 5, and w projects at distance 6.

FIGURE 6.5

The injection function, like IFUNC earlier, enables us to discover andexplore relationships our ears might not otherwise pick up quickly. Figure6.5 (a), for instance, elaborates on the observation that there are 4 transfor-mations f of form (n,T6) such that INJpC|,Xg)(f) is non-zero, allowing Ffrto represent F natural. So there are four T6 arrows on the figure, projecting alof X* progessively into all of Xf at a variety of ordinal distances. This con-trasts sharply with the absence of any I or w arrows on figure 6.4 that led fromanywhere within X* to anywhere within Xf. So far as the two tetrads in thevocal melody are concerned, we may put it that I and w are "internal"transformations, while T6 is "progressive." The structure of figure 6.5(a) ishard to pick up by ear alone because its predominant ordinal rhythm of"3 later" conflicts both with the motivic rhythm of the music, and with the2-later rhythm established in figure 6.4 within X*.752


The structure of figure 6.5(b) is also easy to pick up by inspectingINJ^X^X*), hard to pick up at first by ear alone. It is reasonable to paysome attention to the hexad X* here because that set spans the transfer of thelow F[? to the high Ffj in the melody, the first repeated pitch-class of the series.Inspecting the internal structure of the hexad with our machinery, one notesthat there are relatively many transformations f of form (n, T8)-for-some-nsuch that INJ(X4,X4)(f) is high or positive. The arrows on the figure showhow this works out analytically. For transformations f involving other T,, notas many analogous arrows would appear. The strong ordinal rhythm of thearrows on figure 6.5(b) is supported by the contour of the pitches—C, Ab, andboth the F|?s are all turning points—and also by the rhythm and meter of themusic. In this context, the melodic rise over the octave F[? takes place throughan ornamented arpeggiation of an augmented triad F|?, C, A^, Ffr. Onerecognizes the triadic set of pitch classes from earlier discussion. It is Z6, theSeufzer triad into which the E-wedge will converge at the big downbeat nearthe end of the piece.

And so on. One could combine the pitch classes or the pitches of themelody into formal pairs not only with the ordinal numbers 1 through 10, butalso with the durations of the written notes, or with the time points at whichthe notes are attacked, or with the time spans of the written notes in our non-commutative GIS, and so on. One would get interesting results in each case. Ihave used ordinal numbers here because they furnish some new kinds of ideasabout non-mensural rhythm, and because they guarantee that none of ourtransformations on the space of elements (n, p) can be operations. Even if OPis an operation on pitch classes, like I, the transformation (5, OP) cannot mapour pair-space onto itself: There is no (n, p) such that (n + 5, OP(p)) = (2, q).So using the ordinal-number model gives INJ another opportunity to showhow smoothly it handles transformations that are not operations.

So far as the "new kinds of ideas about non-mensural rhythm" areconcerned, we can take note that our pair-space is one useful way to modelserial melody. Later we shall explore other interesting models for representingseries of pitches or pitch classes (or anything else).

As I mentioned earlier, the reader who is interested can find a more ampleanalysis of "Angst und Hoffen" for its own sake elsewhere in my writings. Theinterested reader might also wish to consult my analytic remarks elsewhere on"Die Kreuze," Number 14 from Pierrot Lunaire.2 Ideas of wedging andinversion are also engaged there. The pertinent wedge transformations arewedging-to-(C/C#) and wedging-to-(F#/G). wc/c# maps F#-to-F-to-E-to-Eb-to-D-to-C#-to-C# and G-to-A|?-to-A-to-B|Ho-B-to-C-to-C. WF*/G trans-forms the pitch classes in analogous wise with respect to the focal goal-dyad

2. "Inversional Balance as an Organizing Force in Schoenberg's Music and Thought,"Perspectives of New Music vol. 6, no. 2 (Spring-Summer 1968), 1-21. The discussion of "DieKreuze" is on pages 4-8. 133


F#/G. Figure 6.6 sketches a sense of how these transformations pertain to theopening of "Die Kreuze."

FIGURE 6.6

134

No pitch class p satisfies wc/c*(p) = F#. Likewise no p satisfies wF*/G(p)= D[?. This feature of the transformations is actually projected by the music,as the figure shows: The first F# and the last Db sit out their respective wedge-games. Iptf = Ic* naturally functions prominently in connection with the twowedges. l£ also figures in the music; it structures the second chord of figure 6.6as an "internal" transformation.

Let us stand back for a moment and think about the analytic uses towhich we have put INJ so far. Nowhere in the discussion of the Schoenbergpieces have we used the word "interval" or even invoked the concept, exceptso far as it is implicit when we label certain operations as T10, T8, and so on.Nowhere, therefore, have we needed to use the fact that the family of pitchclasses is a GIS. Nor did we need to suppose that our melodic space ofelements (n, p) was a GIS, which in fact it was not. We have nowhere needed tosuppose that the transformations we were inspecting were 1-to-l or onto;many in fact were not. From all this we get some idea of how generally the INJconstruct can be applied in how great a variety of situations. We shall increaseour sense of that variety now by studying another application of INJ to asituation not directly involving a GIS for the space S of elements.

6.2.4 EXAMPLE: Our space of elements for this study will be the family PROTof protocol pairs. A protocol pair is an ordered pair (p, q) of distinct (NB)chromatic pitch classes.3 There are thus 132 = 12 times 11 protocol pairs. Atwelve-tone row can be regarded as a certain set within PROT: The pair (p, q)

3. More generally, we could consider protocol pairs of distinct objects from any finitefamily, and the mechanics of our discussion coming up would obtain, so far as the theory wasconcerned.


is in the set if and only if p precedes q in the row. Note that while the rowimposes a certain ordering on the twelve pitch classes, the set under consider-ation is an w«ordered subset of PROT, i.e. an unordered collection of pitch-class pairs.

Besides rows, we can consider other subsets of PROT that are consistentwith our intuitions of "ordering pitch classes." To be consistent in this way, aset X must satisfy two conditions. First, we cannot intuit both p-preceding-qand q-preceding-p. Second, if we intuit p-preceding-q and q-preceding-r, thenwe intuit p-preceding-r. These two conditions translate into the two formalproperties following, PO1 and PO2, which X must satisfy as a collection ofpairs.

(PO1): There is no (p, q) in PROT such that X contains both (p, q) and(q,p).(PO2): If (p, q) and (q, r) are members of X, then so is (p, r).Mathematically, a subset X of PROT that satisfies (PO1) and (PO2) is

called a (strict) partial ordering of the pitch classes. The special partial order-ings that correspond to rows are the "linear" or the "simple" orderings L;these subsets of PROT satisfy in addition the condition (SIMP) below.

(SIMP): For any (p, q) in PROT, either (p, q)or (q, p) belongs to L.

The set-theoretic condition matches our intuition that either p will pre-cede q in the row, or q will precede p.

Representing twelve-tone rows as linear orderings is attractive in manyways. For one thing it makes all rows conceptually equal. That is, it does notassign explicit or implicit priority to one row (e.g. the chromatic-scale row),from which other rows are explicitly or implicitly derived. The model assumesno a priori ordering of the pitch classes; any row orders them as well as anyother row. This is very much in the spirit of the classical twelve-tone method.Other attractive features of the model will become apparent presently.

In connection with the melody from Schoenberg's "Angst und Hoffen" alittle while ago, we brought attention to the way in which series of pitches,pitch classes, and the like could be represented by pairs (n, p) consisting ofordinal numbers n and objects p. Now we have a different way of representingsuch series, provided they are non-repeating (NB). Our new representationallows us to apply set theory to a linear ordering L, together with its varioustransforms and other partial orderings X of interest as subsets of PROT. Theold model represents the row of Schoenberg's Fourth Quartet by a family ofpairs (1, D), (4, B|?), (3, A), (2, C#),... and so on: The first note of the row is D,the fourth note is 6)7, the third note is A, the second note is C#, and so on. Thenew model represents the same row by the family of pairs (A, B(?), (D, B[>),(C#, A), (D, C#),... and so on: A precedes B^, D precedes Bj?, C# precedes A,D precedes C#, and so on. 135


No matter which formal model we use, it will still be convenient to use thenotation D-C#-A-B[?-... for quick perusal.

Partial orderings that are not rows can model many structures of interestin twelve-tone theory. The partial orderings X: and X2 on figure 6.7 exemplifyonly two such types of structure from among many.

FIGURE 6.7

Xj models the small linear motive E-A-Bb. As a subset of PROT, X!contains the three protocol pairs (E, A), (E, 8(7), and (A, 6(7). The reader maycheck formally that this 3-element set satisfies conditions (PO1) and (PO2). Xt

models our being sure of the three cited precedence relations, and unsure of orindifferent to any other precedence relations. Ll is what I take to be "the row"of Schoenberg's Moses undAron; as a subset of PROT, Lj contains the pairs(A,Bb), (A,E),..., (A,C); (Bb,E), (Bb,D),.. . , (Bb,C), (E,D),...,(E,C);...;(B,C).

X2 models an aggregate governing the soprano, alto, tenor, and bassvoices of the four-part texture at the beginning of Variation 3 in Babbitt'sSemi-Simple Variations* X2 contains the twelve pairs (B, D), (B, Eb), (D, Eb);(G, Bb), (G, F), (Bb, F); and so on. L2 models what I take as the row of thepiece, that is, the succession of pitch classes formed by the first twelve notes inthe soprano voice.

We shall be examining various numbers INJ(L1,X1)(f) and INJ(L2,X2)(f) in connection with a few analytic observations on the two pieces.The transformations f on PROT which will attract our interest are of types(l)-(4) following. (1) Transpositions of pairs, T;(p, q) = (Tj(p), Tj(q)); (2) inversions of pairs, I(p,q) = (I(p),I(q)); (3) retrogression of pairs, R(p,q) =(q, p); (4) combinations of types (1) through (3). These transformations areall well defined on PROT; they do not map any protocol pair into some pair(q,q) whose members are not distinct. The transformations are in fact oper-

4. Christopher Wintle provides a very useful analytic study of the piece in "MiltonBabbitt's Semi-Simple Variations" Perspectives of New Music vol. 14, no. 2 and vol. 15, no. 1

136 (Spring-Summer/Fall-Winter 1976), 111 -54.


aliens on PROT. They form a group isomorphic to the twelve-tone group ofoperations on rows. Furthermore, when we identify rows with subsets L ofPROT, the sense of any row-operation coincides with the sense of the corre-sponding set-operation. E.g. if the set L corresponds to a certain row, thenthe set T3RI(L), as determined by the operations T3, R, and I on PROT, cor-responds to the row formed by inverting, retrograding, and T3-ing the givenrow.

Let J = 1̂ , the inversion operation that maps the pitch class A to thepitch class E. Then the small linear motive X, can be extracted from the rowJ(L,). We can see this by writing out the inverted row and italicizing the entriesE, A, and Bl> of the motive Xt as they come along in X,-order: J(Lj) =E-E\>-A-B-B\>-C-F%-.... Another way of expressing the phenomenonis to point out that all three of the member pairs of Xj are member pairs of theinverted row: E precedes A in X} and also in J(Lj); A precedes Bl> both in Xl

and in J(Lj); E precedes Bl> both in Xj and in J(Lj). That is, the three mem-bers of Xj, (E, A), (A, Bl>), and (E, Bl>), are also all members of J(Lr Wereformulate this observation once more: There are three members of Lj whoseJ-transforms are members of Xr And finally, we can reformulate the observa-tion into our present terminology most concisely: INJ(Lj, Xj)(J) = 3.

Are there other inversion operations I such that INJ(Lj, Xj)(I) = 3? Asit turns out here, there are not: J is the only one of the twelve inversionoperations with that property. Working backwards through the semanticequivalencies of our observations in the paragraph above, we can interpretour most recent observation as telling us that J(Lj) is the only inverted form ofLj that serially embeds the small linear motive Xr We might say that Xj has ahigh "signature value" for J(Lj) among the twelve inverted forms of Lt: If wesense that an inverted form is at hand and we intuit the three protocol pairs ofXj clearly, that is enough information, abstractly, to identify J(Lj) as thespecific form at hand.

This property of Xj was noted by Michael Cherlin in connection withevents near the opening of act 1, scene 2 in Moses und Aron.5 The sceneportrays the brothers meeting in the desert; it begins with a lot of Aron music,light-textured, grazioso, piano and pianissimo, scored for solo flute accom-panied by violins, harp and pianissimo horns. Then, just before Aron starts tosing, there is one measure of Moses music, scored for loud trombone andstring bass. The trombone plays the short linear motive Xp extracted from therow-form J(Lj). The "signature motive" Xj is here attached to Moses as hesteps forth on stage. Aron immediately thereafter begins to address Moses,singing the prime row-form combinatorial to J(Lj) and then J(L,) itself. Xl

sounds particularly powerful here because it rearranges the opening trichord

5. The Formal and Dramatic Organization of Schoenberg's Moses und Aron (Ph.D. diss.,Yale University, 1983). 137


of L! , a trichord which had a strong tonic character as a harmony during thepreceding (opening) scene of the drama.

Babbitt has published a discussion of just such small linear signaturemotives in his own music.6 He gives the row of his composition Reflections asL = C-B-D-A-Db-Bb-E-F-G-Eb-Gb-Ab. Then he points out that theordered trichord X = B-D-A is "uniquely characteristic [of the row] twithin the transpositional sub-array, and of [one inverted form] to within theinversional sub-array." In our terminology, INJ(L, X)(Tj) is less than 3 unlessi = 0; also INJ(L,X)(I) is less than 3 unless I is the one specific inversion Jwhich Babbitt singles out. He then goes on to discuss the ordered tetrachorY = B-D-A-Db. He notes that Y "is unique ... for the total array." In ourterminology, INJ(L, Y) (f) is less than 6 for the forty-seven twelve-tone oper-ations f other than f = T0; L itself is the only twelve-tone form of L thatcontains all of the six protocol pairs of Y. Y is a signature for L among itsforty-eight forms; X is a signature for L among its twelve transposed forms; Xis also a signature for J(L) among the twelve inverted forms of the row.

Now let us turn out attention back to X2 and L2 on figure 6.7. Thaggregate X2, considered as a subset of PROT, contains 12 member pairs.Hence INJ(L2, X2) (f) can be at most 12, iff is an operation. (In that camap N distinct members of L2 1-to-l into N distinct members of X2, so that Nmust be 12 or less.) The forty-eight specific transformations f that interest ushere are in fact operations. As it turns out, none of our forty-eight operations factually embed X2 in some form of L2, satisfying INJ(L2,X2)(f) = 12. How-ever there are operations f that do {£ of the job, satisfying INJ(L2,X2)(f) =11. These operations are f = T! , f = RT7, f = J, and f = RT6 J, where J is theinversion l£ . Since the row is its own retrograde at the tritone, these fouroperations generate only two forms of L2, namely T t(L2) = RT7(L2) andJ(L2) = RT6J(L2). Figure 6.8 shows how well the ordering of the aggregateX2 fits into each of these row-forms.

FIGURE 6.8

6. "Responses: A First Approximation," Perspectives of New Music vol. 14, no. 2 andvol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 3-23. The discussion is on page 10.138


The figure makes it visually clear how X2 fits '4i within" either row.Figure 6.8 (b) shows how only the pair (C#, F#) of X2 is not within the rowT1(L2); the row contains instead the protocol pair (F#, C#)- Figure 6.8(cshows how only the pair (Bb, F) of X2 is not within the row J(L2); the rowcontains instead the protocol pair (F, B|?). If only the tenor voice of X2 wentE-F#-C# instead of E-C#-F#, then the embedding of figure 6.8(b) wouldbe perfect. Or, if only the alto voice of X2 went G-F-Bj? instead of G-B[?-F,then the embedding of figure 6.8(c) would be perfect. Or, yet again, if onlythe tenth and eleventh notes of all the rows involved were exchanged, thenboth embeddings would be perfect. This urge to "make small adjustments"with one set or another will be further discussed later. It is a typical featureof situations in which an INJ function almost attains a theoretical maximumpossible value.

X2, as mentioned before, is the aggregate governing soprano, alto, tenor,and bass voices at the opening of Variation 3 in the Semi-Simple Variations.Various (12-tone) forms of X2 govern SATB relations throughout Variation3. SATB aggregates of similar format govern other variations, but none ofthose fit more than "{§ within" any form of the row L2. To put it in ourterminology, if X is an aggregate governing SATB anywhere in the piece out-side Variation 3, the INJ(L2,X)(f) is at most 10, for each operation f we areconsidering. Thus we can say that the SATB-aggregates of Variation 3 aremaximally compatible with forms of L2, compared to such aggregates fromother variations. Statements of this sort are very useful to express structuraldifferences among sections of a piece that sounds at first extremelgeneous in texture throughout. That is particularly so when the statementscan be backed up by precise measurements like -fj, y§, and the like.

INJ helps us pinpoint and explore precisely other structural differencesamong sections of the composition. For example, let V and V be SATB-aggregates from any one variation; then INJ(V, V) (T0) = either 0 or 4. Thatis, V and V will either have no common pairs or exactly 4 common pairs. LetV l 5 V2, ..., V5 be SATB-aggregates from the first, second, ..., fifth varia-tions; then with two exceptions INJ(Vm, Vn)(T0) is less than or equal to 2.That is, with two exceptions, aggregates Vn, and Vn from different variationswill have only 2 or fewer common pairs. Since two distinct aggregates in thisformat could theoretically share as many as 11 common pairs, we can say thatthe level of "ordering cross-talk" between variations is very low, half as low asthe level of cross-talk within each variation (INJ(V, V')(T0) = 4). Indeed thatlatter level (4 pairs out of a possible 11) is itself none too high. The twoexceptions are these: INJ(V5, V2)(T0) = 4 and INJ(V4, V2)(T0) = 5. Wecansay that Variations 5 and 4 thus "talk with" Variation 2, so far as SATB-aggregate ordering goes, at a level equalling or even surpassing the level ofcross-talk within each individual variation.

INJ(V4, V2)(T0) = 5 is a maximum compared with other values of 139


INJ(Vm, Vn)(T0). This observation suggests we devote special attention toaggregate-relations between V4 and V2. And when we do so, we shall notice afeature of the composition we might not quickly have come to notice other-wise. Whichever V4 and V2 we select to represent Variations 4 and 2 respec-tively, the two SATB-aggregates will share exactly one 3-note linear segmentfrom among the four segments D#-B-E, Ab-C-G, C#-F#-D, andB[?-F-A. If Vm and Vn come from variations other than 4 and 2, the twoSATB-aggregates will not share any 3-note linear segments.

Let us call the family of four segments listed above the "pivot aggregate."Figure 6.9 shows how the pivot aggregate controls the tenor and bass voices ofVariation 2, and the soprano and alto voices of Variation 4. The bar lines onthe figure mark off SATB-aggregates within each variation.

FIGURE 6.9

140

INJ numbers bring quickly and effortlessly to our attention the fact thatthe relationship of figure 6.9 between Variation 4 and Variation 2 is a uniquerelationship between variations in the piece; it is not a ubiquitous feature of alarge-scale design. As noted before, this sort of observation is very useful inbringing to our attention special discriminations within a composition thatsounds at first extremely homogeneous.

6.3 In this section we shall explore further the notion of "if-only adjust-ments" in connection with INJ. The abstract notion can be formulated asfollows. Suppose INJ(X, Y)(f) is near its theoretical possible maximum in a


certain situation. The number might be close to the cardinality of X forinstance, so that almost all of X is mapped into Y by f. Or f might be 1-to-l andthe injection number might be very close to the cardinality of Y, so that thetransformed set f(X) comes close to embedding Y. In such cases, a smalladjustment in X or in Y might enable us to remove the "almost" component ofthe situation, bringing the injection number up to its theoretical maximumvalue. The parts of X or Y that do not quite fit may come under pressure toconform, giving rise in the music to urges for generating new material.

We encountered an if-only situation in discussing the Semi-Simple Vari-ations. Figure 6.8 earlier showed how the SATB-aggregate X2 was almostembedded in the rows Ti(L2) and J(L2); more precisely "{i embedded."While discussing the figure, we mused about making if-only adjustments ineither the aggregate or the row, to make the embedding work completely. Sofar as I can tell, the speculative adjustments do not correspond with musicalpressures in Babbitt's piece.

But in the Schoenberg song we examined earlier, similar speculativeadjustments do correspond with strong musical pressures. In examining theprogression of the 3-note chord X = Angst to the 3-note chord Y = Hoffen,we observed that INJ(X, Y)(wE) = 2 out of a maximum possible 3, and thatINJ(X, Y)(I) also = 2 out of a maximum possible 3, "I" here meaning I| =

TjL

^Bb- "If only" the F|? of chord Y were adjusted to F natural, we noted, boththe injection values of 2 above would rise to the maximum 3. In this connec-tion the if-only speculation led to fruitful analytic ideas about the "missing Fnatural," the "missing I-partner," the missing lover, begehren, F[? as func-tional substitute for F natural, and so on.

The interested reader will find an extensive treatment of if-only adjust-ment and its analytic implications elsewhere in my writings, in connectionwith the opening two chords of Schoenberg's piano piece op. 19, no. 6.7 I callthe chords "rh" and "Ih" for right hand and left hand.

6.4 The cited article goes on to discuss "progressive" and "internal" trans-formations in connection with the succession rh-lh. "Progressive" transfor-mations make rh into something much like Ih; "internal" transformationsmake rh into something much like itself, or Ih into something much like itself,or both. The reader will recall that we used this nomenclature earlier, in con-nection with various transformations pertaining to harmony and melody inthe song "Angst und Hoffen."

We shall now extend the nomenclature and put it into a completely

7. "Transformational Techniques in Atonal and Other Music Theories," Perspectives ofNew Music, vol. 21, nos. 1-2 (Fall-Winter 1982/Spring-Summer 1983), 312-71, especially 336-42.1 am indebted to Michael Bushnell for having observed the rewards that this sort of approachbrings in analyzing the music. He worked with transposition operations only; I enlarge his auralfield to include inversions as well. 141


general theoretical setting, invoking only a family S of elements, certaintransformations f on S, and the INJ function. Given sets X and Y, suppose weare inspecting the values of INJ(X, Y)(f), MT(X, X)(f), and INJ(Y, Y)(f) as fvaries over a certain family INSPECT of transformations.

For certain transformations f within the family INSPECT, the value ofINJ(X, Y)(f) will be maximal, or at least relatively high subject to the con-straints of the situation. We shall call these transformations progressive. Theymap a lot of X into Y.

For certain transformations f within the family INSPECT, the value ofMT(X, X)(f) or MT(Y, Y)(f) will be high. We shall call these transformationsX-internal or Y-internal accordingly. A transformation which is both X-internal and Y-internal can be called "internal (for the progression X-Y)."An X-internal transformation maps a lot of X into X.

Intuitively, an X-internal transformation tends to extend/elaborate/develop/prolong X in the music, while a progressive transformation tends tourge X onwards, to become something else (like Y).

Progressive and internal transformations will tend to combine mathe-matically in certain interrelated ways, by their very natures. If I transform Xto be much like itself, and then transform the result to be much like itself, it islikely that the composition of the two gestures will make X much like itself.That is, the composition of two X-internal transformations will tend to beX-internal. Similarly, the inverse of an X-internal operation will tend to beX-internal. Similarly, an X-internal transformation followed by an X-Y-progressive transformation will tend to be an X-Y-progressive transfor-mation; and an X-Y-progressive transformation followed by a Y-internal trans-formation will tend to be X-Y-progressive.

As a result, when we inspect the families of progressive and internaltransformations pertinent to a given X-Y situation, we shall find thosefamilies tending to interrelate algebraically according to the considerationsjust surveyed.

We can introduce other useful nomenclature. An f such that INJ(X,X)(f) is minimal or at least relatively small, given the constraints of X andINSPECT, can be called X-external. Such an f maps X largely outside itself.We can also define a dispersive transformation to be one that maps X largelyoutside Y, makeing the value of INJ(X, Y)(f) minimal or relatively small.These definitions avoid mentioning the complements of the sets X and Y inS, which may not be "sets" according to our definition if S is infinite.

External and dispersive transformations tend to enter into typical al-gebraic relations with themselves, with each other, and with internal andprogressive transformations. An X-internal transformation followed by an X-external one will tend to be X-external; a progressive transformation followedby a Y-external one will tend to be dispersive; and so on.

A good example of dispersive transformations is furnished by measure 8142


of Schoenberg's op. 19, no. 6. This is the cryptic, very dense measure thatprecedes the final return of the chords rh and Ih in measure 9. Figure 6.10(a)reproduces measure 8.

FIGURE 6.10

Forte has noted that the music embeds many forms of the rh chord.8 Forour purposes, we can observe that the music embeds four transposed forms ofrh. Those are the four shown in figure 6.10(b): T2(rh), T5(rh), T7(rh), andT9(rh). Now if we let the interval i range from 0 through 11, we shall find thatthere are six values of i for which INJ(rh, Ih) (Tj) = 0; these values are i = 2,4,5, 7, 9, and 0. That is, T2, T4, T5, T7, T9, and T0 are the six dispersivetransposition operations, given the progression rh-to-lh. To put it anotherway, T2(rh), T4(rh), T5(rh), T7(rh), T9(rh), and T0(rh) = rh are the trans-posed forms of rh that have no common tones with Ih. As figure 6.10(b) showsus, four of these six forms are embedded within the music of figure 6.10(a).(The rest of that music does contain common tones with Ih.) And, as figure6.10(c) shows us, a fifth dispersive form, T0(rh) = rh, ensues immediatelythereafter, joining the parade of dispersive forms and thereby linking measure8 to the downbeat of the final reprise at measure 9.

The relevance of external transformations to traditional theory is illus-trated by the "semi-combinatorial hexachord." If X is a 6-note set of pitchclasses and I is an inversion operation that transforms X into its complementX, then I is an X-external transformation: INJ(X,X)(I) = 0.

This is a good place to think about exploring how the injection functionrelates with set-conplementation when S is finite. We shall soon carry out thatexploration, in section 6.6. Before that, it will be helpful to prove a theoremand a corollary about INJ(X, Y)(f) when f is an operation OP.

8. The Structure of Atonal Music, example 102 (p. 99). 143


6.5.1 THEOREM: If f is an operation OP, then INJ(X, Y)(OP) is the cardi-nality of OP(X) n Y, that is the number of common members shared by thesets OP(X) and Y.

We took note of this theorem informally in the commentary followingDefinition 6.2.1 earlier. The fact is important enough to warrant formalverification.

Proof: Let M = INJ(X, Y)(OP); let N = card(OP(X) n Y). Let x t , x 2 , . . . ,XM be the distinct members of X that map into Y via OP. Since OP is 1-to-l, theelements OP(Xi), OP(x2), ..., OP(xM) are distinct members of OP(X) n Y.Thus OP(X) n Y contains at least M distinct members; N is greater than orequal to M.

Now let y l 5 y2, ..., yN be the distinct members of OP(X)nY. ThenOPîX OP"1^)* .. . , OP"1^) are N distinct members of X, each ofwhich maps into Y under OP. So there are at least N distinct members of Xthat map into Y under OP; M is greater than or equal to N. q.e.d.

The injection function applied to operations thus generalizes Regener'sCommon-Note Function, which was developed for the special case in which Xand Y are sets of pitch classes and OP runs through the twelve transpositionoperations.9

6.5.2 COROLLARY: If f is an operation OP, then

INJ(Y,X)(OP) = INJP^YXOP"1).

Proof: An element z is a member of the set OP~1(X)n Y if an onlyif OP(z) is in X and z is in Y; this is the case if and only if OP(z) is in X andOP(z) is in OP(Y), that is if and only if OP(z) is a member of the set X nOP(Y). So the transformation OP maps the set (OP~1(X)n Y) 1-to-l ontothe set (XnOP(Y)). Therefore the two sets have the same cardinality:card(OP(Y)nX) = card(OP~1(X)nY). Applying Theorem 6.5.1 to bothsides of this equation, we infer the formula of the Corollary.

Now we are ready to explore set-complementation in connection withINJ. It will simplify matters greatly to restrict our attention to transforma-tions f that are operations OP in this context. We shall then be able to use thetheorem and the corollary we have just proved. We mustjestrict the abstractsetting by supposing S to be finite; then the complement X of a set X will be aformal "set" by our criterion, i.e. a finite subfamily of S.

6.6.1 THEOREM: Suppose S is finite. Given sets X and Y with complements Xand Y; given any operation OP; then formulas (A) through (E) below obtain.

9. Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music vol. 13,no. 1 (Fall-Winter 1974), 191-212. The Common-Note Function is defined on page 202.144


(A): INJ(X, Y)(OP) = cardX - INJ(X, Y)(OP).(B): INJ(X, Y) (OP) = cardY - INJ(X, Y) (OP).(C): INJ(X,_Y)(OP) = cardY - cardX + INJ(X, Y)(OP).(D): IfcardY_=cardX, then

INJ(X, Y)(OP) = INJ(X,Y)(OP).(E): (Generalized Babbitt Hexachord Theorem)

If cardX_= ^cardS, thenINJ(X,X)(OP) = INJ(X,X)(OP).

Proofs:_(A): The operation OP maps each member of X either into Y or into Y. So

cardX = INJ(X, Y)(OP) (the number of X-members mapped into Y) plusINJ(X, Y)(OP) (the number of X-members mapped into Y). The formulafollows. This argument works just as well for any transformation f.

(B): Here it is essential that OP be an operation, so that we can applyCorollary 6.5.2. We write INJ(X, Y)(OP) = INJ(Y,X)(OP-1), via 6.5.2.This, via formula (A) just proved, = cardY - INJ(Y, X)(OP-1). And, apply-ing 6.5.2 again, we infer that this number is indeed cardY — INJ(X, Y)(OP),as claimed.

(C): INJ(X,Y)(OP) = cardY = INJ(X,Y)(OP), via (B); this iscardY - (cardX - INJ(X, Y) (OP)), via (A); this is cardY - cardX +INJ(X,Y) (OP), as desired.

(D) is an obvious corollary of (C). And (E) is an obvious corollary of (D),setting Y = X. q.e.d.

The methods of proof I have used are essentially Regener's. I have calledformula (E) the Generalized Babbitt Hexachord Theorem because Babbitt'stheorem, somewhat disguised, is a special case of this formula. For readerswho may feel the disguise is perfect, I shall show the connection. Let X be a six-note pitch-class set; let OP be a transposition operation Tj. Theorem 6.6.1 (E)above tells us that INJ(X,X)(TJ = INJ(X,X)(Ti). Theorem 6.5.1 then tellsus that the cardinality of (Tj(X) n X) is the same as the cardinality of(T;(X) n X). Now the cardinality of (T;(X) n X) is the number of members ofX that lie the interval i from some member of X. In other words, the number isIFUNC(X, X) (i). Similarly, the cardinalitypf (T}(X) n X) is IFUNC(X, X) (i).We have shown, then, that IFUNC(X,X)(i) = IFUNC(X,X)(i) for everyi: the complementary hexachords contain the same number of i-dyads foreach i. This is Babbitt's theorem. One sees that 6.6.1(E) is a very broadgeneralization.

6.6.2 EXAMPLE: Let us see how the Generalized Hexachord Theorem appliesin another specific context. The reader will recall the space PROT of protocolpairs which we constructed earlier (6.2.4) in examining some topics from serial 145


theory. We noted that the various twelve-tone rows can be regarded as thosesubsets L of PROT that are linear orderings on the pitch classes. In this model,the retrograde of the row L corresponds to the set-theoretic complement L ofthe set L in PROT. For a pair (p, q) is in the complement of the set if and only ifp does not precede q in the row, which is the case if and only if p precedes q inthe retrograde of the row, which is the case if and only if (p, q) lies in thatportion of PROT corresponding to the retrograde row.

The family PROT has 132 = 12 times 11 members. And any row, as asubset of PROT, contains exactly 66 protocol pairs. To see that, suppose thepitch classes of the row come in the serial order p l 9 p2, ..., p12. Then thesubset L of PROT contains 11 pairs of form (Pi,pn), and 10 pairs of form(p2,pn), ..., and 1 pair of form (pu, pn). The cardinality of the set L is thus11 + 10 H + 1, which is 66.

We are thus in a setting to which 6.6.1(E) pertains. card(S) =card(PROT) = 132; card(X) = card(L) = 66 = |card(S). In this setting, arow and its retrograde (complement) play a set-theoretic role formally analo-gous to that of a hexachord and its complement in traditional atonal theory.

Here, 6.6.1 (E) tells us the following. Let OP be any operation on PROT.Given any row L with retrograde L, let N be the number of pairs (p, q) in Lsuch that the pair OP(p, q) is also a precedence relation in L; let N' be thenumber of pairs (p', q') in L such that OP(p', q') is also a precedence relation inL. Then N = N'.

In this connection, OP(L) need not itself be a row. Indeed L itself can bereplaced by any set of cardinality 66 within PROT and the theorem remainsture, "L" now being simply the set-theoretic complement of L. But theapplication is of particular interest when we interpret the complementary setsL-and-L as row-and-retrograde.

The twelve-tone operations Tj and I are induced on PROT by corre-sponding operations on individual pitch classes: Tj(p, q) = (Tj(p),Tj(q));I(p,q) = (I(p), I(q)). But in general there need not be any operation op onpitch classes such that OP(p, q) = (op(p), op(q)). There is no such op, forinstance, in the case of the retrograde operation R on PROT: R(p, q) = (q, p).Nor is there such an op for any of the operations to which R contributes, e.g.RTj and RI. One can also construct fancier operations on PROT not inducedby pitch-class operations op. For example: If p and q are in the same whole-tone scale, OP(p, q) = (p, q); if p and q are in opposite whole-tone scales,OP(p>q) = (T3(q),T9(p)).

When L is a row and OP(L) is also a row, then INJ(L, L) (OP) measuresthe size of the largest partial ordering on the pitch classes which can beembedded in both OP(L) and L, i.e. whose protocol pairs are compatible withboth those rows. That is because INJ(L, L)(OP) is the cardinality of

146 OP(L) n L, as we know from 6.5.1.


Now we shall show formally how INJ completely generalizes IFUNCwhen there is a GIS at hand.

6.7.1 THEOREM: Let (S, IVLS, int) be a GIS. Then for each interval i and forall sets X and Y,

IFUNC(X, Y)(i) = INJ(X, Y)(T,).

Proof: Let IFUNC(X, Y)(i) = M; let INJ(X, Y)(Tj) = N. We shall seethat N must be at least as big as M, and that M must be at least as big as N.

Since IFUNC(X,Y)(i) = M, there are M distinct pairs (Xj .yj ) ,(x2, y2), ..., (XM, yM) such that xm lies in X, ym lies in Y, and int(xm, ym) = i.For each such pair, ym = Ti(xm). For m and n distinct, xm and xn are distinctmembers of X. (Otherwise we would have ym = Ti(xm) = Tj(xn) = yn, whencethe pairs (xm, ym) and (xn, yn) would not be distinct, contrary to supposition.)Thus X has at least M distinct members whose i-transposes lie in Y. That is,INJ(X, Y)(Tj) is at least as big as M. Or: N is at least as big as M.

Now let Zi, z 2 , . . . , ZN be the N distinct members of X whose i-transposeslie within Y. (There are N such, since INJ(X, Y)(T;) = N.) For each suchzn,zn is in X, u = Tj(zn) is in Y, and int(zn, u) = i. Therefore the pair (zn,u)was counted as some (xm, ym) above. So every one of the N elements z1,..., ZN

is one of the M elements x l 5 . . . , XM. Hence M is at least as big as N. q.e.d.The logic of Theorem 6.7.1 can be visualized through the following aid.

Imagine X and Y as two finite configurations of points in the Euclidean plane.Suppose i is the vector (directed distance) "to the right and up 30 degrees for adistance of 5 inches." We can ask: "From points of X to points of Y, howmany distinct arrows can I draw that go to the right and up 30 degrees for adistance of 5 inches?" The answer to this question is IFUNC(X, Y)(i). We canalso ask: "If I move the whole X-configuration to the right and up 30 degreesfor a distance of 5 inches, how many points of the displaced configuration willthen coincide with points of Y?" The answer to that question isINJ(X, Y)(Tj). One intuits easily that the two questions are logicallyequivalent.

Next we shall explore what happens to INJ when the sets X and/or Y aretransformed by some operation A.

6.7.2 THEOREM: Given a family S of objects, given sets X and Y, given atransformation f on S and an operation A on S, then formulas (A), (B), and(C) below obtain.

(A): INJ(A(X),Y)(f) = INJ(X, Y)(fA).(B): INJ(X, A(Y))(f) = INJ(X, Y)(A~1f).(C): INJ(A(X), A(Y)) (f) = INJ(X, Y) (A'1 f A). 147


Proofs: (A): INJ(A(X), Y)(f) is the number of t within A(X) such thatf(t) is a member of Y. Set t = A(s); then the family of such t is in 1-to-lcorrespondence, via A, with the family of s in X such that fA(s) is a member ofY. And the number of such s is exactly INJ(X, Y)(fA).

(B): INJ(X, A(Y)) (f) is the number of s in X such that f (s) is a member ofA(Y). Now f (s) is a member of A(Y) if and only if A"1 f (s) is a member of Y. Sothe number at issue is the number of s in X such that A~1f(s) belongs to Y.And that is exactly INJ(X, YXA^f).

(C): The formula follows at once from (A) and (B). q.e.d.

Formula 6.7.2(C) is of particular abstract interest. We can imagine thatthe shift from X-and-Y to A(X)-and-A(Y) reflects a "modulation" of thesystem by the operation A. For instance, if we are in a GIS and A is atransposition or an inversion, we are transposing or inverting (the sets of) thesystem accordingly. It is natural to ask: "If we modulate the system by A, whateffect does that have on the INJ function?" At first one might supposethat INJ would remain unaffected by the modulation: INJ(A(X), A(Y)) =INJ(X, Y). But, as formula 6.7.2(C) shows, that is not in fact the case. Ratherthe INJ function is itself "modulated" according to the formula.

We noted a specific instance of this general phenomenon earlier, duringour analysis of "Angst und Hoffen." The wedge-structure of figure 6.3(b)converged on E; it was studied in connection with the transformations WE =wedging-to-E and I = inversion about E. The wedge-structure of figure6.3(c) converged on F#; it was studied in connection with the transformationsWF* = wedging-to-F# and J = inversion about F#. To get from the situationof (b) to the situation of (c), one "modulates the system by T2." The followingequations obtain: F# = T2(E); wF* = T2wET2"1; J = T2IT2"

1. The firstequation relates the focal points of the two wedges, also the bass notesof the Angst and Hoffen chords. The second equation leads, via 6.7.2(C),to the relationship INJ(T2(X),T2(Y))(wF«) = INJ(X, YXT^w1^) =INJ(X, Y) (WE). The third equation leads via the same formula to the relation-ship INJ(T2(X),T2(Y))(J) = INJ(X, Y)(T2

1JT2) = INJ(X, Y)(I). Thus,when we modulate the system from E-centricity to F#-centricity via the opera-tion T2, then the wedge WF* plays the role, with respect to the modulated setsT2(X) and T2(Y), that the wedge WE originally played with respect to the setsX and Y. Similarly, the inversion J plays the role, with respect to the modu-lated sets, that the inversion I originally played with respect to the unmodu-lated sets. The reader will recall, perhaps, our earlier remarks on this subjectby way of preparation (pp. 128-129).

Here is another, more abstract, example of system-modulation. LetX be an atonal hexachord that inverts into its complement via the inversionI = IB. Then INJ(X,X)(I) = 0. Suppose some music projecting X-and-its-complement "modulates" to a new section projecting T5(X)-and-//j-148


complement. Here I plays the role of "f" and T5 plays the role of "A" inFormula 6.7.2(C). We cannot suppose that INJ(T5(X), T5(X))(I) = 0: thenew hexachord T5(X) will not invert into its complement via the inversion I,B/C inversion. Rather, T5(X), the modulated hexachord, inverts into itscomplement by the inversion J = T5IT~^. Formula 6.7.2(C) tells us this:INJ(T5(X), T5(X))(J) = INJ(X, X)(T-i JT5) = INJ(X, X)(I) = 0.

Using formulas 3.5.6 (A) and (B), we can compute J = T5l£T7 =I|T7 = Ij:. Thus T5(X) inverts into its complement about E and F (or aboutB!> and B). The system having modulated by T5, the transformation J =T5IT~* now plays the role that the transformation I originally played.

Formula 6.7.2(C) tells us this sort of thing in great generality: When asystem modulates by an operation A, the transformation f = AfA"1 plays thestructural role in the modulated system that f played in the original system, inthe sense that INJ(modulated X, modulated Y(f) = MT(X, Y)(f).

6.7.3 Theorems 6.7.1 and 6.7.2 enable us to generalize the abstract ques-tions about IFUNC we asked earlier, toward the end of the section 5.1. Weasked, for instance, under what circumstances in a GIS we would haveIFUNCCXj, Xj) = IFUNC(X2, XJ. Via 6.7.1, we can rephrase the question,asking under what circumstances in a GIS we shall have INJ(Xj, Xj)(Tj) =INJ^Xj, XjXTj) for every transposition Tr And that question can easilybe generalized: Given any family S of objects and any family INSPECTof transformation on S, under what circumstances shall we haveINJ(Xj, Xj)(f) = IN](X2, XjXf) for every member f of INSPECT? We donot have to demand that S be in a GIS, or that INSPECT be a group; indeed,the question makes sense even if INSPECT is not a closed family (semigroup)of transformations.

Likewise, we earlier asked under what conditions in a GIS we wouldhave IFUNC(X,, XJ = IFUNC(Y,, Y2). We can generalize that questionanalogouly: Given any family INSPECT of transformations f on a familyS of objects, under what conditions shall we have INJ(X,, X^f) =MT(Y,, Y2)(f) for every member f of INSPECT?

The ideas of 6.7.2 enable us to expand that question even farther. Sup-pose we have a family S of objects, a family INSPECT of transformationsf on S, and a group MDLT of "modulating operations" A on S. Under whatconditions, given sets Xj, X^ Y,, and Y2, will there exist some modulationA such that when we modulate Yj and Y2 by A, obtaining Y', and Y'2, weshall have DSfJCX,, X^f) = INKY',, Y'2)(f) for every f in INSPECT?Via 6.7.2(C) this amounts to demanding that ESTJ(X,, X^f) and INJ(Yt,Y2(A~1fA) be equal numbers, for each member f of INSPECT. In the spe-cial case where we are in a GIS, where INSPECT is the family of trans-positions, and where MDLT is the group of transpositions, the questionasks under what conditions, given the four sets, there will exist some interval 149


j such that for every interval i, IFUNCCX^XjXi) = IFUNC(Tj(Y1),Tj(Y2))(i) = IFUNCCY^Y^Gij-1). (5.1.6 gives us the last equality. It isconsistent with 6.7.2(C) because T^-i = T^TjTji The group of transpo-sitions is aAm'-isomorphic to the group of intervals.)

Besides using 6.7.1 and 6.7.2 to generalize earlier questions, we can alsouse them to help out with computations we earlier found difficult to execute.The formula following is a good example.

6.7.4 THEOREM: In any GIS let I = !„. Fix any referential element and setj = LABEL(u). Then for any sets X and Y, and for each interval i,

IFUNC(I(X),I(Y))(i) = INJ(X, Y)(Pk)

where Pk is the interval-preserving operation labeled by k = ji"1]"1.Proof (optional): IFUNC(I(X), I(Y)) (i) = INJ(I(X), I(Y)) (Tj), via

6.7.1. This, via 6.7.2, = INJ(X, Y)^"1^!). Set I^T,! = OP. It now sufficesto prove that OP = Pk, where k = ji"1.)""1.

r1 = 1^(3.5.9). So OP = I^Tji;;. We can compute T^ by 3.5.6(A);the result is I*, where x = Tj(u). So OP = I"I*. And we can compute thecomposition of the two inversions by 3.5.8. It is PmTn, where m =LABEL(u)LABEL(x)-1 and n = LABELOO^LABELCv). Here n = e, sowe have computed that OP = Pm, where m = LABEL(u)LABEL(x)~1.Since x = T,(u), LABEL(x) = LABEL(u)-i(3.4.3). Replacing LABEL(u)by j, we then have m = j(ji)"1 = ji-1j-1 = k. Thus OP = Pk as desired.

6.8 To demonstrate further the generalizing powers of INJ, we shallnow use it to generalize Forte's K and Kh relations. For our generalization wesuppose only a family S of objects and a group of operations on S which weshall call "canonical" for reasons known only to us. We shall denote the groupby CANON.

The complement of the set X will be denoted X. If S is infinite, X will notbe finite, hence not a "set" in our terminology. Nevertheless we shall speak ofits "cardinality," understanding the value infinity for the expression cardX. IfS is finite, cardX will mean the finite cardinality of the set X.

We shall restrict our attention to sets X and Y such that cardX < cardXand cardY < cardY. We can do this because Forte's K and Kh relations arenot affected by the restriction. If cardX should be less than cardX, we cansimply exchange the roles of the sets X and X as they do or do not enter intoK or Kh relations with other sets.

Having made that restriction, we may also suppose that cardX < cardY.Otherwise, we can simply exchange the roles of the sets X and Y in thearguments coming up.

So our restrictions, in sum, are these: cardX < cardX, cardX < cardY <cardY. Given those restrictions, Forte's K and Kh relations hinge on the750


logical combination of two more primitive relations, which we can call K! andK2.

(Kx): Some (canonical) form of X is embedded in Y. (K2): Some form of X is disjoint from Y (and hence embedded in Y).X and Y, subject to our restrictions, enjoy Forte's K relation if they enjoy

either Kj or K2; they enjoy his Kh relation if they enjoy both K^ and K2. Nowthe relations Kj and K2 correspond to very natural aspects of INJ(X, Y), asthat function takes arguments from the canonical group. Specifically, K t andK2 are respectively equivalent to K\ and K2 below.

(K;): For some A in CANON, INJ(X, Y)(A) = cardX.(K'2): For some B in CANON, INJ(X, Y)(B) = 0.Now we can express Forte's relations, generalized, as follows:(K): INJ(X, Y) (A) takes on either its theoretical maximum value cardX(subject to our restrictions), or its theoretical minimum value 0, as Avaries over CANON.(Kh): INJ(X, Y)(A) takes on both its theoretical maximum and itstheoretical minimum values, as A varies over CANON.We can use the ideas of "progressive" and "dispersive" transformations

(6.4), to rephrase K\ and K2 yet once more, into the respective forms of K/{and K2 below.

(K'i): CANON contains some maximally progressive transformationwith regard to INJ(X, Y).(K2): CANON contains some maximally dispersive transformationwith regard to INJ(X,Y).

Let us return to relation K\ above. K't says there is some A in CANONsuch that INJ(X, Y)(A) = cardX, but it does not tell us how many such Athere are. In case CANON is infinite, that may or may not be a meaningfulquestion. When CANON is finite, the question is definitely interesting. Byasking it, we can distinguish multiplicities of K! -ness about the Kx relations ofvarious sets: Some pairs of sets are "more K^-related" than others, Likewise,when CANON is finite, we can distinguish multiplicities of K2-ness about K2

relations, counting how many members B of CANON satisfy condition K2

above. We can then attach pairs of multiplicity numbers to sets in the Khrelation, to indicate the multiplicities of K!-ness and K2-ness involved.

For a specific example, take S to be the twelve chromatic pitch classes,and take CANON to comprise both transpositions and inversions. Take Y tobe the black-note pentatonic scale. Take X to be the F#-major triad. Then wemay say that X and Y are Kh-related with multiplicity (2,6): There are twomembers A of CANON embedding A(X) in Y, and there are six members B ofCANON such that B(X) is disjoint from Y. Now, holding onto Y, take X to bethe set (Ajj, B|?, Dfc>). The new X is Kh-related to Y with multiplicity (4,8).

6.9 Coming back from the specific example, let us return to the general 757


situation with CANON finite, still subject to our conditions on the cardina-lities of X and Y. For each integer N between 0 and cardX inclusive, we canask how many members A of CANON satisfy the equation INJ(X, Y)(A) =N. Our multiplicity-values aoove asked that question for N = cardX and forN = 0, but we can just as well ask it for every N in between. The answer we get,i.e. how many such A there are in CANON, is the value for the argument NofRegener's Partition Function, here generalized.10 We shall denote that valuebyRGNPF(X,Y)(N).

In this special case (CANON being finite), the Partition Function enablesus to derive the EMB function by the formula below.

RGNPF(X.Y)(cardX)EMB(X, Y) - RGNpF(X)X)(cardx)-

That is, EMB(X, Y) is the multiplicity of Kj-ness about X-in-Y, dividedby the multiplicity of Kj-ness about X-in-itself. The formula is given withoutproof. If X is symmetrical in some way, there may be several distinct membersof CANON, other than the identity, that map X onto itself. If M members ofCANON map X onto itself, then there will be M times as many distinctoperations embedding X in Y, as there are forms of X embedded in Y. RGNPFcounts operations; EMB counts forms. This accounts for the denominator ofthe fraction in the formula above.

6.10 OPTIONAL: In all our work so far with set theory, we have supposed the"sets" under scrutiny to be finite. This section will outline briefly the workneeded to extend our results so that they can be applied to sets "of finitemeasure" in certain "measure-spaces." During this discussion we shall relaxthe use of the word "set" so as to conform to standard mathematical usage.Roughly speaking, that usage makes the word a synonym for "family" or"ensemble" of things.

Given a family S of objects, a family FLD of subsets of S is called a fieldofsubsets if it satisfies (1), (2), and (3) below.

(1): 0 (the empty set) and S are members of FLD. (2): If X is a member of FLD, then so is its complement X.(3): If X and Y are members of FLD, then so is X u Y, their set-theoreticunion.It follows deductively that whenever X and Y are in FLD, so are X n Y

(which is the complement of X u Y), Y — X (defined as Y n X), and so on. Afield of sets is called a sigma-field if (4) below is also tr

(4): Whenever Xj , X2 , . . . is a countable family of members of FLD, so isthe countable union X = Xi u X2 u — X here is the set of s such that sis a member of at least one of the sets Xn. "Countable" means "capable ofbeing put into 1-to-l correspondence with the natural integers."

752 10. Ibid., p. 206ff.


A measure on the field FLD is a function mes that assigns to each memberX of FLD a value which is either a non-negative real number or infinity, insuch fashion that mes(X u Y) = mesX + mesY whenever X and Y are dis-joint. If FLD is a sigma-field, one supposes mes to be sigma-finite unless it isspecified as not so. The condition for sigma-finiteness is that mes(UNIONXn) = SUM(mesXn) whenever X1? X2 , . . . is a countable family of mutuallydisjoint members of FLD. The countably infinite sum is understood in theusual sense, as the limit of its partial sums; that limit may be infinity.

We can deduce that if X is included in Y, then mesX < mesY. FormesX < mesX + mes(Y — X), mes being non-negative, and that number ismesY since X and (Y — X) are disjoint.

We shall need one more concept. Given a field FLD of subsets of a familyS, a transformation f on S is called measurable when, given any member Y ofFLD, the set of s mapped by f into Y is also a member of FLD.

Now we can generalize INJ. We suppose a family S, a field FLD ofsubsets, and a measure mes on FLD. By a "set of finite measure" we mean justthat, a member X of FLD such that mesX is finite. If X and Y are sets of finitemeasure, and if f is a measurable transformation on S, then the family F of ssuch that f (s) belongs to Y is a member of FLD, so the family X n F is also amember of FLD. Furthermore, mes(X n F) < mesX, so X n F is a set of finitmeasure. We then define INJ(X, Y) (f) = mes(X n F). This number measures"how much of X" (according to mes) is mapped into Y by f.

Using this definition for INJ and restricting X, Y, etc. to be sets of finitemeasure, we can generalize much of the machinery in chapter 6 as it stands.Theorem 6.5.1 and the formulas that follow from it must be restricted sothat the operations at hand are "measure-preserving," satisfying mesOP(X) =mesX. We could derive more flexible and much more complicated formulas byallowing "measure-scaling" operations; OP is such if for some constantnumber "scale," mesOP(X) = scale • mesX. Our work on the Partition Function (6.9) may not generalize easily, since CANON is not likely to be finite ifthe sets X, Y, etc. are not finite. But often it will be possible to establish a"good" field of subsets within CANON, and a "good" measure on that field;then for any real numbers a and b with 0 < a < b, we can defineRGNPF(X, Y)(a, b) to be the measure, within CANON, of the set of oper-ations A satisfying a < INJ(X, Y)(A) < b.

The mathematical ramifications of all this lie far beyond the scope of thepresent book. The work of chapter 6 up to the present is just one special case ofthe general system now being sketched. In that special case, FLD contains allsubsets of S, so all transformations fare measurable; mesX is the cardinality ofX, allowing infinity as a possible value. The sets of finite measure are exactlythe finite sets, what we have been calling "sets" up until now.

For another special case we can consider a non-musical setting. Let S be 753


a Euclidean plane containing a Seurat painting. We take FLD to consist of allregions in the plane with a well-defined total area, and we use area as ourmeasure. Let X be the region of the plane consisting of all animals in thepainting; let Y be the region consisting of all plants. Let f be the transforma-tion on S: move up and to the right at a 45-degree angle for 3 centimeters. ThenINJ(X, Y)(f)/areaX answers the question: To what extent are the animals ofthe painting to be found 3-cm.-below-and-at-a-45-degree-angle-to-the-left-ofplants?

We now adopt a new measure, redmes. This measure assigns to eachmember of FLD the number of red dots it contains completely.(red)INJ(X, Y) (f )/redmesX now answers the question: To what extent do reddots within the animals of the painting lie 3-cm.-below-and-at-a-45-degree-angle-to-the-left-of plants? If we want to shift our attention to yellow dots,then (yellow)INJ(X, Y) (f )/yellowmesX answers the analogous question withrespect to yellow dots.

For our next example suppose we are analyzing a certain piece and wewant to ask questions like this: Of the amount of time the violin is playingabove high C during the piece, how much of that time will the clarinet beplaying pianissimo or softer five seconds later? We can use a special case of thegeneral model at hand. Fixing a referential unit of time and a referential time-point zero, we take S = the real numbers, as modeling the continuum of time-points within which our piece occurs. For each real number a, let Za be the setof numbers s satisfying s > a, and let Za be the set of numbers s satisfyings < a. Take FLD to be the smallest sigma-field that contains every Za andevery Za. As it happens, there is essentially only one well-behaved measure onFLD that satisfies mes(Za n Zb) = (b — a) for every pair of numbers a < b.We shall use this measure; we can think of it as measuring the "amount oftime" in a set. Take X to be the set of time-points in the piece during which theviolin is playing above high C; take Y to be the set of time-points in the pieceduring which the clarinet is playing pianissimo or softer. Take f to be time-point transposition by five seconds: The time-point f (s) occurs five secondsafter s. Then the number we asked for above is INJ(X, Y)(f)/mesX.

For a final example we shall explore the space S of time spans. Werepresent S by the Euclidean half-plane of number-pairs (a, x), where x ispositive. FLD will be the smallest sigma-field containing all square-shapedregions within the half-plane. FLD contains all the sets we shall want todiscuss here; e.g. it would contain the region comprising all animals in theSeurat painting. There is a unique measure on FLD that makes the measure ofeach square region its area; we shall call this measure "area." We may or maynot want to use area as a measure for our musical purposes. There are othermeasures that have other desirable properties. For instance, in connectionwith our non-commutative GIS, there is an essentially unique measure thatmakes every interval-preserving operation measure-preserving: mesP(X) =154


mesX. We can call this measure "P-invariant measure." It is essentially uniquein this sense: If mes' is any other P-invariant measure, then there is somepositive number "scale" such that mes'(X) = scale • mesX for every memberX of FLD. There is also a different measure on S, also essentially unmakes every transposition-operation measure-preserving: mesT(X) = mesX.We can call this measure "T-invariant measure."

If we are analyzing a specific piece, we may also want to use measures likethe Seurat measure earlier that counted red dots. The following discussion willhelp develop a "Seurat model" for our analysis. Suppose the piece begins atthe time point BEGIN and extends for the duration EXTENT. Then everytime span that pertains to an event within the piece must satisfy the twoconstraints BEGIN < a and a + x < BEGIN + EXTENT. The set of timespans (a, x) satisfying those constraints forms a triangular region in the half-plane. This triangle plays a role in our model analogous to that of the rectangleon which Seurat painted, a bounded region within its plane.

In applying time-span analysis to the piece, we are hearing events withwell-articulated beginnings and durations, events whose temporal locationand extent can be sensibly modeled by time spans. We may then fill in thetriangle-of-the-piece by dots. A dot at the point (a, x) within the trianglemodels the location and extent of an event in the piece which begins at a andlasts x units of time. We can color certain dots red. Say these are the dotspertaining to stringed-instrument events. We can color certain dots yellow.Say these are the dots pertaining to events "above middle C." Orange dots willthen pertain to events involving some stringed instrument (s) playing abovemiddle C. Our model assumes the format of a triangular Seurat painting, andwe can apply the ideas of "red measure," and so on.

Given a time span (b, y), we can construct the family SHADOW(b, y) ofall time spans (a, x) that "happen within" the time of (b, y). We constructedthe shadow of (BEGIN, EXTENT) above; that was the triangle-of-the-piece.SHADOW (b, y) in general is the set of all spans (a, x) satisfying the twoconstraints b < a and a + x < b + y. This set forms a triangular region in thehalf-plane. Given two events in the piece, event 1 and event2, let (a, x) and(b, y) be the respectively pertinent time spans. Then (a, x) is in the SHADOWof (b,y) if and only if event 1 happens during the time of event2.

Let us imagine a certain section of the piece which begins at the timepoint BEGSEC and lasts for a duration of DURSEC time units. TakeX = SHADOW (BEGSEC, DURSEC). X is the set of all time spans pertain-ing to potential or actual events within the given section of the piece. LetBRASS be the family of time spans which articulate events played by brassinstruments. (Those spans might be the green dots on our "painting.") TakeY = SHADOW(BRASS). A time span belongs to the set Y if and only if anactual or potential event to which it pertains happens or would happencompletely during some brass event, e.g. a sustained brass note or chord. Take 755


f to be transposition-by-the-interval-(4,2) in our non-commutative GIS. Thusthe span f (a, x) begins 4 x-lengths after a, and lasts for a duration of 2x timeunits. We may then ask for the value of orange INJ(X, Y) (f). When we do sowe are asking the following question. How many string events pitched abovemiddle C are there within our given section of the piece such that if you start atthe event, counting "ONE," and then beat three more event-durations, count-ing "-two-three-four," and then beat two more event-durations yet, counting"ONE-two," your new count of "ONE-two" will measure a span of time lyingcompletely within the extent of some sustained brass note or chord?

156

Networks (1): Intervals andTranspositions

We saw in Theorem 6.7.1 how INJ generalizes IFUNC when there is a GIS athand: IFUNC(X, Y) (i) = INJ(X, YXT,). This relationship enables us notonly to generalize IFUNC, but also to replace entirely the concept of interval-in-a-GIS by the concept of transposition-operation-on-a-space. Instead ofthinking: "i is the intervallic distance from s to t," we can think: "Tj is theunique transposition operation on this space that maps s into t." We can evenshift our attention, if we wish, from the atomic "points" s and t to the one-element "Gestalts" X and Y, X being the set that contains the unique members and Y being the set that contains the unique member t; then there is a uniquemember T; of the transposition-group satisfying INJ(X, Y) (Tj) > 0; the label ifor this unique transformation is i = int(s, t).

7.1.1 By such thinking, we can replace the idea of GIS structure by the ideaof a space S together with a special sort of operation-group on S. This specialsort of group is what mathematicians call simply transitive on S. The groupSTRANS of operations on S is simply transitive when the following conditionis satisfied: Given any elements s and t of S, then there exists a unique memberOP of STRANS such that OP(s) = t.

Given any s and any t in the space of a GIS, then there is a uniquetransposition-operation T satisfying T(s) = t, namely T = Tint(M). So thegroup of transpositions is simply transitive on the space of any GIS. Con-versely, the following theorem is also true: Let S be a family of objects and letSTRANS be a simply transitive group of operations on S; then there existsa GIS having S for its space and STRANS for its group of transpositions.

We shall now prove that theorem. Given S and STRANS as described, letIVLS be an "index family," that is a family of elements i, j , . . . which can be putinto 1-to-l correspondence with the family STRANS. We shall write "OP;" 757

Transformation Graphs and7

7.7.2 Transformation Graphs and Networks (1)

for that operation within STRANS that corresponds to the member i of theindex family IVLS. Now we turn IVLS into a group by defining the binarycombination ij = k in IVLS when (OPj)(OPj) = OPk in STRANS. The groupIVLS, by this construction, is anti-isomorphic to the group STRANS.

We have a space S and a group IVLS; next we define a function int fromS x S into IVLS. Given r and s in the space S, we take int(r, s) to be that uniquemember i of IVLS such that OPj(r) = s. i is unique because STRANS is simplytransitive.

Holding onto r, s, and i above, suppose that int(s, t) = j in the same sense.That means OPj(s) = t. Then (OPj)(OPi)(r) = OPj(s) = t. By the groupstructure defined for IVLS, (OPj)(OP,) is OP(ij). Hence OP^r) = t. Then, bythe construction of the function int, int(r, t) = ij. Thus int(r, t) = int(r, s)int(s, t); Condition (A) of 2.3.1 is satisfied.

Now we shall show that Condition (B) of that definition is also satisfied,so that (S, IVLS, int) is a GIS. Given s and i, set t = OPj(s). By definition ofint, int(s,t) = i. Iff is any member of S satisfying int(s, t') = i, then OPj(s) =t'. But OPj(s) is precisely t. So in this case t' = t. We have shown that, given sand i, there is a unique t satisfying int(s, t) = i. So we have shown thatCondition 2.3.1(B) is satisfied.

Thus (S, IVLS, int) is a GIS. And STRANS is the group of transpositionsfor this GIS; indeed Tj = OPi for every i in IVLS. To see this, we recall thatT;(s), in any GIS, is the unique member of S that lies the interval i from theelement s. Now in this particular GIS, the member of S that lies the interval ifrom s is OPj(s). Hence in this GIS Tj(s) = OP^s). That being the case forevery sample s, T; = OPj as an operation on S. q.e.d.

7.1.2 By virtue of 7.1.1, all the work we have done with GIS structures sincechapter 2 can be regarded as a special branch of transformational theory,namely that branch in which we study a space S and a simply transitive groupSTRANS of operations on S. From a strictly mathematical point of view, thiswould have been a more elegant way to develop the material of chapters 2through 5. We could even have deferred the study of GIS structure until muchlater, after a more general exploration of transformations on musical spacesusing CANON, INJ, and other such constructions.

Yet there are also advantages to the order of presentation we haveadopted in this book. By starting with a study of GIS structure, we have built alink between the historically central concept of "interval" and our presenttransformational machinery. To some extent for cultural-historical reasons, itis easier for us to hear "intervals" between individual objects than to heartranspositional relations between them; we are more used to conceivingtranspositions as affecting Gestalts built up from individual objects. As thisway of talking suggests, we are very much under the influence of Cartesian

158 thinking in such matters. We tend to conceive the primary objects in our

Transformation Graphs and Networks (1) 7.1.2

musical spaces as atomic individual "elements" rather than contextuallyarticulated phenomena like sets, melodic series, and the like. And we tend toimagine ourselves in the position of observers when we theorize about musicalspace; the space is "out there," away from our dancing bodies or singingvoices. "The interval from s to t" is thereby conceived as modeling a relationof extension, observed in that space external to ourselves; we "see" it out therejust as we see distances between holes in a flute, or points along a stretchedstring. The reader may recall our touching on these matters in 2.1.5, where wepointed out how the historical development of harmonic theory has dependedon such a projection of our intuitions into a geometric space outside ourbodies, that is, the "line" of the stretched string, a space to which we can relateas detached observers.

In contrast, the transformational attitude is much less Cartesian. Givenlocations s and t in our space, this attitude does not ask for some observedmeasure of extension between reified "points"; rather it asks: "If I am at s andwish to get to t, what characteristic gesture (e.g. member of STRANS) shouldI perform in order to arrive there?" The question generalizes in severalimportant respects: "If I want to change Gestalt 1 into Gestalt 2 (as regardscontent, or location, or anything else), what sorts of admissible transforma-tions in my space (members of STRANS or otherwise) will do the best job?"Perhaps none will work completely, but "if only ...," etc. This attitude is byand large the attitude of someone inside the music, as idealized dancer and/orsinger. No external observer (analyst, listener) is needed.

In 7.1.1 above we sketched a mathematical dichotomy between intervalsin a GIS and transposition-operations on a space: Either can be generatedformally from the characteristic properties of the other. More significant thanthis dichotomy, I believe, is the generalizing power of the transformationalattitude: It enables us to subsume the theory of GIS structure, along with thetheory of simply transitive groups, into a broader theory of transformations.This enables us to consider intervals-between-things and transpositional-relations-between-Gestalts not as alternatives, but as the same phenomenonmanifested in different ways. Consider figure 7.1 in this connection.

759FIGURE 7.1

7.7.2 Transformation Graphs and Netw

The figure shows how the melodic motif of the "falling minor ninth(sixteenth)" is developed over Schoenberg's piano piece op. 19, no. 6, out fromthe intervallic structure of the opening chord in the right hand, a chord wehave earlier called "rh." Figure 7.1 (a) displays rh, along with the three fallingpitch-intervals that can be heard within it; they are notated on the figure as— 5, — 9, and —14 semitones. Figure 7.1(b) shows how the same network ofintervals governs the scheme by which the falling-ninth motif is transposedover the course of the piece. This is interval-language. Alternatively, wecould use transposition-language to put the matter as follows: The threetransposition-operations T_5, T_9, and T_14, which move the falling-ninthmotif forwards in time over figure 7.1(b), are exactly those members ofSTRANS which move the individual pitches of rh downwards in space, asshown on figure 7.1 (a). "Forwards in time" and "downwards in space" arephenomena that work together in many ways over the course of the piece.1

But we do not have to choose either interval-language or transposition-language; the generalizing power of transformational theory enables us toconsider them as two aspects of one phenomenon, manifest in two differentaspects of this musical composition: Intervals structure the referential sonor-ity rh as an Unterklang; transpositions make the falling-ninth motif moveforward through the piece. This, I think, is the sense in which we accept thesymbol " — 5" on figure 7.1 (a) and the symbol " — 5" on figure 7.1(b) alegitimately the same, finding the usage suggestive rather than confusing.The two symbols are pointing at the same phenomenon, not at differentphenomena.

Before we leave figure 7.1, let us note that the chord of (a) and the variousfalling ninths of (b) have only one common pitch-class. This emphasizes thatin comparing (a) to (b), we are talking about intervals and transpositions, notabout pitches and pitch-elaboration (diminution). If figure 7.1 (b) were to haveappeared five semitones lower, then one could argue that the basic pheno-menon involved was that of the pitches B, F#, and A in the chord, elaboratedto become BBt>, F#F, and AG# in the new figure 7.1(b). But this is not thecase; the phenomenon under discussion involves intervals and transpositions,not pitches or pitch-classes and their structural ornamentation.

The remainder of chapter 7 comprises further examples demonstratingvarious interrelationships of intervallic structures with transpositional pro-gressions in the manner of figure 7.1(a)-(b), over a variety of musical styles.All these examples will involve intervals among pitches or pitch-classes.The distinction between pitch or pitch-class elaboration and intervallic/transpositional interrelationship will not always be so clear, through theexamples, as it was in figure 7.1. We shall concentrate mainly upon the

1. This aspect of the music is brought out very well in the analysis by Robert Cogan and160 Pozzi Escot, Sonic Design (Englewood Cliffs, N. J.: Prentice-Hall, 1976), 50-59.

Transformation Graphs and Networks (1) 7.2

intervallic/transpositional interrelationships, for those are our current focusof study. But we shall also discuss from time to time how those interrelation-ships interact with pitch or pitch-class elaborations.

FIGURE 7.2

7.2 EXAMPLE: Figure 7.2 shows the beginning of the Zauber motive fromWagner's Parsifal, so far as pitch classes are concerned. The motive appearswith a variety of rhythms in the music drama. I shall call the serial network offigure 7.2 "Zauber" or "Z" for present purposes. Forms of the motive do notappear in the foreground of the music until Kundry's first-act entrance (ride);figure 7.2 shows Z as its pitch classes appear during the kiss in the middle ofact 2.

Figure 7.3 sketches melody and harmony for the phrase that introducesthe Motive of Faith in the Prelude to act 1. This is long before Z has appearedin the foreground. Yet the intervallic structure of the Z motive governs theplan of modulations for the phrase, as the arrows on the figure show us.

FIGURE 7.3 767

7.2 Transformation Graphs and Networks (1)

In figure 7.3 one hears not only the intervals of modulation but also thespecific pitch classes Ab-Cb-Ebb~Eb being tonicized; these are the pitchclasses for Z which were displayed in figure 7.2 above. Of course we hear themusic of figure 7.3 long before the music of figure 7.2 (second-act kiss). On theother hand, we do hear the local keys of figure 7.3 elaborating a pitch-classvariation on a structure related to Z, namely the Liebesmahl motive thatopened the opera: Ab-C-Eb-F-(etc). In this context, hearing the successivetonics Ab-Cb-Ebb-Eb of figure 7.3 as a variation on the Liebesmahl helps ushear Zauber itself, when it appears in the foreground later on, as a variant ofthe Liebesmahl.

762

Figure 7.4 shows the succession of local tonics during the transformationmusic of the first act. The principal local tonics are represented by opennoteheads; measure numbers above them indicate where the cadential tonici-zations discharge. Usually these discharges coincide with entries of importantmotives; the names of the motives appear on the figure preceding the measurenumbers: BELL, GRAIL, AG = Agony, and LM = Liebesmahl. Some filled-in noteheads also appear on the figure; these represent local tonics whichare subordinated to the principal tonics in various ways. The B tonic atmeasure 1084 is a structural dominant preparation for the E tonic at measure1092. The B tonic takes the Grail Motive, not the Bell Motive like other tonicsfrom measure 1074 to measure 1106. The C tonic at measure 1115 is supportedby no special motivic entry; it is a structural neighbor to the Db = C# tonicsthat surround it by open noteheads on the figure. After the neighboring eventand the enharmonic shift, the predominant motive on the figure changes, fromBELL to AG (measure 1123 and following). Then, after measure 1140, AGdisappears and BELL returns. The cadence at measure 1140 is deceptive. The


implied tonic is A|>, which appears as an open notehead on the figure; thesubstitute root that actually carries the Liebesmahl entry is F(?. The situationat measure 1148 is similar, in fact sequential.

Up to measure 1140 the open noteheads are organized by serial forms ofthe Zauber motive. These forms are beamed on figure 7.4 and labeled as Z t,Z2, Z3, and Z4, meaning the first Z-form, the second Z-form, and so on.2 EachZ-form is a retrograde inversion of the Z-form that precedes it, specificallythat RI form which uses for its opening two notes the final two notes of thepreceding form. Thus Z2 begins E(?-E-, taking its point of departure from theend of Z^-Eb-E. Likewise Z3 begins G-B|?-, taking its point of departurefrom the end of Z2,-G-B|?.

It does not much matter whether we call figure 7.4 up to measure 1140 apattern of "intervals" among tonics, or of "transpositions" among tonics, oreven of "modulations" among keys: We are talking about the motivic exfoli-ation of one phenomenon in various ways. "Modulations" is only awkward ifwe try to associate the term with Schenkerian notions of pitch or pitch-classprolongation. That is clearly not happening here (up to measure 1140), andsince Schenker himself rejected the word for his discourse, I feel free to use it inthis non-Schenkerian connection. Indeed I prefer it (for that reason) to theword "tonicizations."

Continuing along figure 7.4, we note that the open noteheads followinmeasure 1140 build up to one beamed form of the BELL motive. It is verysignificant that this structural Bell Motive appears "in A|j." The deceptivecadence at measure 1140, featuring an entry of the Liebesmahl in the trom-bones, forcefully recalls the same event right at the curtain-rise of act 1. In thisconnection, pitch-class relations clearly are important. We are to understandthe Ab of measure 1140 as the tonic of the opera; we are to understand thestructural Bell Motive of measures 1140-50, beamed on figure 7.4, as aprolongation of that tonic; and we are thereby to understand the mammothlocal tonic C of measures 1150-62ff. as the third degree of that Ab. It is justthis section of act 1 that enables us (with the help of the Parsifal bells) to besure that the C tonic which ends the act is indeed a functional III of A(? (interalia).

To support this hearing, we can note that the four pitch-classes of the BellMotive in Ab, portrayed by the beamed noteheads of measures 1140-50 onfigure 7.4, are in fact a serial permutation of the four pitch-classes that beginthe Liebesmahl (and the opera): Ab-C-Eb-F. The pitch-class relation is verystriking, yet Wagner apparently goes to great lengths to conceal it. Thedeceptive cadences at measure 1140 and measure 1148 help to do so; so doesthe absence of the Bell Motive in Ab from the foreground of the music here(and so far as I can recall anywhere else in the music). Whatever private games

2. I am grateful to Thomas Christensen for calling Zj and Z3 to my attention. 163


Wagner may have been playing, it is safer for us as analysts to treat thepermutational relation between the Bell Motive and the Liebesmahl as inter-vallic, rather than pitch-class prolongational.

Figure 7.4 enables us to explore another motivic relationship involvingBELL. This motive has the same organizing function after measure 1140 thatZauber had before measure 1140, that is the function symbolized by groupingopen noteheads on the figure with beams. The compositional relationshipinvites us to explore the intervals to which those beams give rise in each case,and indeed we can hear an interesting intervallic relationship now that we areprimed to listen for it.

Figure 7.5 shows how we can hear BELL as an overall progression of - 3,elaborated by two subprogressions of 7. Compare this network of intervals tothat of figure 7.2, which displayed Zauber as an overall progesssion of 7,elaborated by two subprogressions of 3 (inter alia).

Let us now return our attention to the network of Z-forms displayed byfigure 7.4 up to measure 1140. We shall have much more to say later about theserial technique of Rl-chaining that links successive Z-forms. When thistechnique is applied to any given serial motive over and over, alternate formsof the motive-chain will be transposed forms of each other, the interval oftransposition depending upon the serial structure of the motive. In thisspecific case, Z3 = T10(Z1)andZ4 = T10(Z2). The repeated Rl-chaining thusgives rise to "structural sequencing" on figure 7.4: Measures 1096-1140 (withA[?) "sequence" measures 1074-1100 structurally. (The musical foregroundsof the two passages are not related sequentially.)

The sequencing-interval of 10 in this particular case is a dispersiveinterval for Z as an unordered set: INJ(Z, Z)(T10) = 0; T10(Z) has no com-mon notes with Z. The dispersive function is clear on figure 7.4, where thesequencing forms of Z fill up chromatic pitch-class space very diligently.Indeed, the open noteheads of the figure up through the A|? of measure 1140constitute a non-repeating ten-note series. (F# and B are missing. It isamusing, if far-fetched, to imagine them as representing the absent Klingsor.)

7.3 EXAMPLE: Figure 7.6 graphs an intervallic/transpositional structure164 which we shall call "the nuclear gesture with pickup." The nuclear gesture

FIGURE 7.5


FIGURE 7.6

comprises the pitch-class interval (transposition) 4, subarticulated into twointervals (transpositions) of 2. The pickup gesture consists in approachingsomething by pitch-class interval/transposition 5.

Figure 7.7 shows through a variety of examples how the nuclear gesturedominates intervallic/transpositional configurations over the last movementof Brahms's Horn Trio op. 40. It may seem awkward to be using pitch-classintervals in this context, since the piece is so highly structured registrally and isso diatonic. Our reasons will become clear by the end of this discussion. Forother purposes it would make more sense to use pitch intervals in semitonesup, or diatonic intervals in scale steps up, or degree intervals in scale steps upmodulo 7.

(a) of the figure displays the basic motive of the movement. The nucleargesture with pickup governs the first half of the motive; the second half isgoverned by what we shall call the "complementary gesture," which articu-lates 8 into 10 + 10.

(b) of the figure sketches the bass line for the opening phrase. We hearhow it is governed as a whole by the nuclear gesture. Now each stage of thatgesture is inflected by its own pickup. The octave leap on B|?, which appears inparentheses at the end of (b), is an important motive of the piece. Here itinterrelates with the octave B[? that delimits the ambitus of motive (a).

(c) of the figure shows the melody beginning to descend from its firstclimax. The complementary gesture governs the melodic structure as a whole;each stage of that gesture is inflected by a pickup. The pickup interval of 5 doesnot get complemented; it always remains 5, never becoming 7.

(d) shows how the network of (c) gets tightened rhythmically at theapproach to the last recurrence of motive (a) within the first group.

Over (a) through (d), the intervallic/transpositional interrelationshipscan be analyzed as fallout from the pitch and pitch-class motives. The nuclearand complementary gestures are always applied to Ejj-F-G and G-F-Efr.functioning as degrees 1-2-3 and 3-2-1 in E(? major. The pitch and pitch-class motives, with their degree functions, can claim priority here: In (a)-(b)-(c)-(d) we can deduce the intervallic relations of interest from stronger pitchand pitch-class relations, but we cannot run such deductions in the otherdirection. So a traditional Schenkerian approach to (a)-(b)-(c)-(d) revealsmore than does our intervallic/transpositional approach.

The same can still be said of (e), which shows the melody at the opening of 165


FIGURE 7.7

166


FIGURE 7.7 (continued) 757


an episode that begins toward the end of the second group and eventuallyleads to the closing theme. The principal pitch-classes that carry the structureof our (e)-network are still G and F, even though they are now degrees 6 and 5in Bk (The whole passage takes place over a dominant pedal in that key.)Intervallically and transpositionally, (e) shows how transposition-by-10, whichgoverns the sequence, exfoliates from the interval-of-10 between G and F;this interval of 10 develops the "subinterval of 10" from the complementarygesture, blowing it up rhythmically. Each stage of the 10-gesture is inflectedby a pickup.

In passages (f) through (h), the intervallic/transpositional networks takeon a life of their own. They become autonomous structures; no longer sub-ordinated to concomitant local pitch or pitch-class events, they rather interactwith such events or perhaps even determine them, (f) shows an extendedsequential passage from the development section. The model for the sequenceis sketched on (f) over measures' 103-05 "etc." Over measures 104-05, thepedal note B of the ostinato figure combines with the opening two notes of thelegato theme that follows, projecting the complementary gesture. (At "etc."the legato theme stops moving stepwise.) The pedal pitch-class B is inflectedby a pickup. To hear the gestures here, one must listen to pitch classes, notpitches, (f) continues on, showing how the sequence from measure 103 etc,through measure 113 etc., to measure 123 etc., projects the nuclear gesture.

The mobile harmony of (f) achieves as its goal a reattained dominant ofEl> major. Thereupon the horn launches an unusually extended solo passage,(g) shows first how the opening eight measures of the solo, measures 137-44,are structured by the complementary gesture with pickup. By projecting thatgesture at this pitch-class level, Brahms finally gets his "octave-Bt" idea tointeract with the system of nuclear and complementary gestures. (The octave-Bl> idea stood apart from those gestures in passages (a) and (b) above.) As aresult of this interaction Brahms generates a Gl> at the end of the complemen-tary gesture, a G!> standing in an 8-relation to the initial Bl> of that gesture.

We are aware of the 8-relation from B\> to Gl> as it carries over into themusic of measures 145-47. And that gives us a clue as to what is going on overmeasures 145-49 more broadly. Another clue is furnished by rhythmic aug-mentation: The pickup motive of measure 137 etc. is augmented rhythmicallyto measure 145 etc., and the complementary-gesture motive of measure 141 isblown up rhythmically to measure 149 etc. These clues explain the newintervallic/transpositional gesture graphed in (g) over measures 145-49 (withpickup): An overall interval of 4 is subarticulated into 8 + 8. The subinterval 8of the new gesture identifies with the overall interval 8 of the complementarygesture—both span Bl>-to-GI> during passage (g). And, just as rhythmic valueare multiplied by 2 in passing from measures 137-44 to measures 145-56, sointervallic values are also multiplied by 2. Instead of the complementary ges-ture 8 = 10 + 10, we now have an expanded gesture involving twice those168


numbers: 2 - 8 = 2-10 + 2-10or, modulo 12,4 = 8 + 8. The new gesture willtherefore be called "the complementary gesture times 2." Our ears follow theintervallic idea "times 2" here because of the rhythmic augmentation. Eachstage of the new gesture takes its own pickup.

(h) shows how the new gesture was already presented in the first move-ment, derived in a musical manner that was almost the equivalent of analgebraic demonstration. While the horn melody in the treble clef chains 10-intervals into suspension-patterns sequencing through 8-intervals, the con-comitant bass line indicates how those 8-intervals in turn chain up to form thecomplementary-gesture-times-2. Each stage of the times-2 gesture, in the bassof (h), takes its own pickup; we must use the fundamental bass in thisconnection for the first stage. I have chosen for example (h) the reprise-form ofthis passage rather than its form in the exposition. That is because the pitchclasses in the bass line of (h) as given make it easy for the reader to hear therelation of (h) to measures 145-49 in (g).

FIGURE 7.8

7.4 EXAMPLE: Figure 7.8(a) sketches the opening of the Minuet fromBeethoven's First Symphony. The C pedal bass is omitted from the firstcomplete measure, signifying that we are to understand an F root and a G rootfor the two harmonies of that measure in the present context. This is an old-fashioned way of hearing, especially at the usual tempo for the piece, but weshall find that the old-fashioned hearing is of interest here. Those who havetrouble hearing an F root, given the C bass, will be helped to do so bysummoning up the memory of the slow movement. That movement, the lastmusic we have heard before figure 7.8(a) begins, has just ended in F major.

Figure 7.8(b) aligns beneath (a) a sketch for the opening of the entire 769


symphony. In the deceptive cadence of measure 2, a C harmony is understoodas interpolated between the G7 and the A-minor harmonies. This too is an old-fashioned but consequential hearing in the context. The progression of G to(interpolated) C is bracketed. On (a) above, the repeated progression of G to Charmony is also bracketed.

Ignoring the repetition of the bracketed harmonic progression in (a),counting the theoretically interpolated C harmony in (b), and considering thefirst complete measure of (a) to contain F and G harmonies, we can then referboth passages to a common network governing the progression of roots, anetwork suggested by figure 7.9.

FIGURE 7.9

The intervals on the arrows are familiar ratios from fundamental basstheory. They are expressed here as fractions between 1/2 and 1, to capture the"falling" sense of the root progressions. The representation of chord-changesby "intervals" between roots is not quite adequate. We shall improve themodel in chapter 8; meanwhile the somewhat inadequate representation willserve. The root-intervals have an impressive tradition behind them, exemplify-ing the desire of earlier theorists to subsume intervals and transformations(here harmonic changes) into one general phenomenon (the fundamentalbass).

Traditional criticism would describe the relation between the Adagio andthe Minuet by saying that the latter parodies the former, as a passage in asatyr-play may parody a dramatic theme treated majestically in an earlierdrama of a tetralogy. Our "common network" of figure 7.9 enables us topropose a different relation between the passages, one that does not infer somuch structural priority for the Adagio from its temporal priority in thecomposition. Namely, we can conceive both (a) and (b) of figure 7.8 asdifferent realizations, in different environments, of one underlying abstractgesture, a gesture symbolized by figure 7.9. The mere possibility of this shift in170


critical stance is important, no matter which stance one prefers in viewing theparticular art-work at hand.

Consider this analogy. When one first encounters Mr. X, one sees him informal attire, discharging an important and solemn public professional duty.When one next encounters him, he is surrounded by screaming children andfriends in a park, his mouth filled with potato chips, rushing down a hillcarrying a football in one hand and an open can of beer in the other. Here wewould surely be reluctant to adopt a view analogous to that of the traditionalcriticism, claiming that the first X is a norm for X-ness and the second X is aparody of the first. Rather, our attitude would be analogous to the alternativeview. We might notice in X's body, features, movements, voice, and the likecertain things common to both occasions; from that we would infer a certainabstract "X-ness," and we would say that this X-ness was being manifested onboth occasions, albeit differently in different environments. A dramatist ornovelist might first introduce us to X either at the public occasion or in thepark. One could also imagine an open-form play or novel, in which the authorallowed either scene to precede the other at the choice of the performers or thereader. Traditional criticism would attack this idea on the grounds that wecannot separate our concept of X-ness from the particular way in which wehave built up that concept through time. And so on; having suggested thephilosophical and aesthetic issues our investigations engage, I shall not pursuethe matter farther here.

I do, nevertheless, want to work out some specific critical consequencesthat emerge when we take figure 7.9 as a norm for both figure 7.8(a) and figure7.8(b). First, let us analyze the "tail" on our normative figure 7.9, that goesfrom the C harmony to the A-minor harmony, as a secondary feature of thestructure. (The minor harmony is a "secondary triad.") Imagine the nodecontaining the A-minor harmony, then, as erased from figure 7.9, along withthe arrow labeled "5/6."

It is then possible to analyze the rest of the figure as a succession of threeV-I cadences moving along the circle of fifths. This interpretation of thegraph is projected by the music of figure 7.8(b), the opening Adagio. There wehear a motive, the motive repeated with harmonic variation, and the motiverepeated again with rhythmic variation. The local tonics for the motive-formsmove along nodes of figure 7.9 located at successive arrowheads, first F, thenC, then G. (We have dropped the tail from the figure.) This music, in sum,projects a "progressive" reading of figure 7.9-as-norm.

The opening of the Minuet, in contrast, projects a different way ofreading figure 7.9-as-norm. Here the figure is interpreted as manifesting asense of balance in its cadence structure, not a progressive chaining of tonics.To explore the cadential sense of balance, let us construct an intervallic graphcalled CADENCE, shown in figure 7.10.

Now let us apply CADENCE to the analysis of figure 7.9-as-norm, 171

7.4 Transformation Graphs and Networks (I)

always ignoring the tail. The leftmost side of figure 7.9 exhibits CADENCEformatting the subnetwork C-F; G-C. The rightmost side of figure 7.9exhibits CADENCE formatting the subnetwork G-C; D-G. This suggestshearing the entire normative network of figure 7.9 (without tail) as comprisingtwo CADENCES, the right side of the first CADENCE being elided into theleft side of the second CADENCE. According to this reading, figure 7.9without tail is to be understood as a contraction of figure 7.11.

772

This interpretation of figure 7.9 illuminates the repeated G-C pro-gression within the Minuet theme, marked by the brackets on figure 7.8(a):That projects the normative repeated G-C progression which we see on figure7.11.

The pitch classes that fill the nodes of figure 7.11 are the roots of variousharmonies. Some harmonies are (local) tonics; others are dominants. Thetonic pitch classes fill nodes that have arrowheads pointing to them. Figure7.12 isolates the tonics of figure 7.11 and organizes them in a network of theirown.

This network in fact fills the nodes of the CADENCE graph (figure 7.10)yet again. The relation is hard to see because the diagonal arrows of figure 7.10go to the right while those of figure 7.12 go to the left. But the CADENCE

FIGURE 7.10

FIGURE7.il


graph, as a configuration of nodes and labeled arrows, knows no "right" and"left." We are using those visual distinctions here to indicate musical chro-nology, not graph-structure as such. Musical chronology is naturally crucial.We may hear a dominant precede its tonic, symbolized e.g. by the right-pointing diagonal arrows on figure 7.10; we may hear a dominant follow andinflect its tonic, symbolized e.g. by the left-pointing arrows on figure 7.12.But in this particular system of root-relations, the normative graph itselfdisplays only a dominant node and a tonic node connected by a labeledarrow; chronological distinctions in this system function not as norms but asdifferent interpretations of the normative graphs. That is a feature of classicalfundamental-bass theory, not of our graphic structures per se. We have manysystems of rhythmic "intervals" at hand within which we could constructanalogous graphs that would enforce musical chronology.

The opening of the Minuet interprets the basic norm of figure 7.9 in themanner of figures 7.11 and 7.12. The music specifically reads the normativenetwork as a backwards-laid-out CADENCE of local tonics (figure 7.12),diminuted into two balanced fowards-laid-out CADENCEs of roots (figure7.11). We have already noted how the repeated G-C progression within theMinuet theme helps to bring out this balanced structure. The idea of balance isalso projected melodically to some extent: The first CADENCE of rootsharmonizes the tetrachord G-A-B-C in the principal melodic line; the sec-ond CADENCE harmonizes the answering similar tetrachord D-E-F#-G.Of course the irrepressible thrust of the rising scale works against this feelingof balance, along with other features of the passage. It is curious that figure7.8(a), so ebullient in its texture, should project a reading of our underlyingharmonic norm as balanced, while figure 7.8(b), so stately and measured in itstexture, should project a reading of the same norm as progressive.

Figure 7.13 continues from the end of figure 7.8, asserting a structuralcorrespondence between the next four measures of the Minuet, arriving at astructural dominant, and the continuation of the first movement all the wayup to the dominant which prepares the first big tonic tutti in the Allegro. Thecorrespondences displayed here are mostly melodic, serial, and motivic, invol-ving particularly the thematic leading-tone-to-tonic idea. Fundamental-bass 173

FIGURE 7.12


structure is not involved to the extent it was in connection with figure 7.8.Figure 7.13 is really not our business here, but I think the reader will be glad tohear it anyway. One would not easily conceive listening for its correspon-dences without having first noted the correspondences of figure 7.8. Then too,to the extent one assents to the analytic implications of figure 7.13, thatconfirms the propriety of figure 7.8 and our work that issued therefrom. Andof course figure 7.13 is interesting in its own right. It owes an enormous debt tothe theoretical ideas of Schenker.

7.5 The interested reader will find in an article by John Peel a sensitive andanalytically revealing use of small transposition-graphs and networks fordiscussing a passage from Schoenberg's String Trio.3

John Rahn has published a discussion of the theme from the secondmovement of Webern's Symphony op. 21 that bears very suggestively on ideaswe have been considering throughout this chapter.4 Casting his discussionin the form of an ear-training exercise, Rahn directs students' attention tonetworks of tritones and networks of semitones; he also directs their attentionto concomitant rhythmic structures. One could perhaps construct an appro-priate GIS for his discourse, a GIS involving pitch and rhythm together in adirect-product system. Aspects of his analysis might then be recast andextended, to involve networks of direct-product intervals.

3. John Peel, "On Some Celebrated Measures of the Schoenberg String Trio," Perspectivesof New Music vol. 14, no. 2 and vol. 15, no. 1 (Spring-Summer/Fall-Winter 1976), 260-79.

4. John Rahn, Basic Atonal Theory (New York: Longman, 1980), 4-17.

174

FIGURE 7.13

Transformation Graphs andNetworks (2): Non-IntervallicTransformations

Figure 7.9, the normative fundamental-bass network we constructed for theBeethoven symphony, exhibited a feature we discussed early in chapter 7. Onthe one hand, we could conceive of its arrows as signifying intervals betweenindividual elements, in this case roots. On the other hand, we could alsoconceive of the arrows as signifying transformational relations between Ge-stalts, in this case chords or harmonies, or even potential keys. Resolving theambiguity, we could also conceive of the arrows as denoting some tertiumquid, a phenomenon whose manifestations include both harmonic intervalsbetween pitch classes and transformational relations between chords.

To conceive such a tertium quid, in the form of the fundamental bass andits progression, was Rameau's supreme inspiration. And some of the prob-lems to which that conception gives rise are neatly pinpointed by figure 7.9 aswell. That is particularly the case as regards its A-minor "tail," the tail weconveniently docked before starting our earlier discussion. The interval of 5/6,which labeled that tail, does inform us correctly that the root A, as a pitch classin just intonation, is 5/6 of the root C (modulo the pertinent congruencerelation). But the numerical ratio does not tell us that the A harmony is minorrather than major. If we transpose a C-major chord by the interval 5/6 weobtain an A-major chord, not an A-minor chord. Thus, when we pass from theC-node to the A-node on figure 7.9, we are really applying some transfor-mation other than harmonic-transposition-by-(5/6), some transformationwhich is more than a synonym or isomorphic image for that interval.

8.1.1 We can solve this problem very elegantly by adopting and modifyingsome ideas from the function theories of Hugo Riemann. Our space willconsist not of pitch classes but of objects we shall call "Klangs" Each Klang 775

8


is an ordered pair (p, sign), where p is a pitch class and sign takes on the values+ and — for major and minor respectively. The Klang models a harmonicobject with p as root or tonic, an object whose modality is determined bythe sign. We can transpose a Klang by transposing its pitch class while pre-serving its sign; thus (C, +) transposed by 5/6 is (A, +). Rather than writingT(C, +) = (A, +) here, we shall write the symbol for the transformation tothe right of its argument: (C, +)T = (A, +). The reader will recall our havingdiscussed such "right orthography" a long time ago, in section 1.2.4. Rightorthography will conform much better than left orthography to our intuitionsin the contexts we shall be exploring just here. The one special thing we have towatch is that the order of composing transformations is reversed under rightorthography: If f and g are transformations, then (Klang)fg = ((Klang)f)gdenotes Klang-transformed-by-f, all transformed by g; this was denoted by"gf(Klang)" in left orthography.

We define the operation DOM on Klangs: DOM is transposition by theinverse of the dominant interval. Thus (p,sign)DOM = (q, sign), where q isthat pitch class of which p is the dominant. We can read this equation as tellingus that (p, sign) becomes the dominant 0/(q, sign). On a graph we could have a(C, +) node, an (F, +) node, and an arrow labeled DOM from the first to thesecond. Right orthography conforms nicely to the visual layout of the graphhere: Being at (C, +) and following an arrow labeled DOM we arrive at(F, +); that is, (C, +)DOM = (F, +).

In a similar spirit we define the operation MED: (p, sign) becomes themediant of its MED-transform. For example, (C, + )MED = (A, —), and(C, —)MED = (Ab, +). If we are at (C, + ) on a graph and follow an arrowlabeled MED, we arrive at (C, + )MED = (A, -).

Now we can rewrite the network of figure 7.9 as a network of Klang-transformations, rather than fundamental-bass intervals. Figure 8.1 is theresult.

The transformations DOM and MED drive the network of figure 8.1 in a

FIGURE 8.1176


natural musical way. In general, DOM and MED will always drive similarKlang-networks in the same way. That would happen even where an instanceof (F, +) occurred in some music followed by an instance of a dominant-related (C, +). We could use the visual layout of an analytic network to reflectthe musical chronology, having a DOM arrow pointing to the left from a(C, 4-) node to an (F, +) node. In that case we would be saying: "(C, +) refers(back), as dominant, to (the preceding) (F, +)." The DOM arrow, here asbefore, makes a dependent Klang point at the local tonic Klang to which itrefers. For the configuration of nodes and arrows as a configuration, it isimmaterial whether the DOM arrows point right or left (or up or down).

Our unusual definition of DOM is what makes the graphs move naturallyin this way. The orientation of all the DOM arrows would be reversed ifwe had chosen the more usual alternative idea, in defining a "dominant"Klang-transformation. The usual idea is represented by the transformationDOM', which takes a Klang and transforms it into its own dominant. E.g.(F, +)DOM' = (C, +): "Being at (F, + ), take its dominant and obtain(C, 4-)." Observe how poorly figure 8.1 would fit our kinetic intuitions aboutthe music under study if we reversed all the DOM and MED arrows, usingDOM' and the analogous MED' instead.

This elucidates one problem with the conceptual structure of Riemann'sfunction theories. His dominants, other than secondary dominants, do notpoint to their tonics via implicit DOM arrows. Rather the tonics point to theirdominants, generating them by implicit DOM' arrows. Then the dominantsjust sit around, not going anywhere. This conceptual flaw in Riemann'sapproach makes his musical analyses subject to inertia and lifelessness, sel-dom doing justice to the power and originality of his theoretical ideas.

An even more basic problem for Riemann was that he never quite workedthrough in his own mind the transformational character of his theories. He didnot quite ever realize that he was conceiving "dominant" (whether DOM orDOM') as something one does to a Klang, to obtain another Klang. Here, Iconjecture, he was unduly influenced by a desire to promote his notation as asubstitute for Roman-Numeral notation; I think it was this desire that led himto conceive "dominant" and the like as labels for Klangs in a key, rather thanas labels for transformations that generate Klangs from a local tonic (along thelines of DOM'), or that urge the Klangs of a key towards their tonics (alongthe lines of DOM).

We may continue to explore other transformations on the family ofKlangs, following or modifying Riemann. We can define SUBD, the formalinverse of DOM. "(F, +)SUBD = (C, +)" means that F major becomes thesubdominant of C major. Even though SUBD = DOM"1 in a group ofoperations, the arrows on the graphic format enable us to distinguish a SUBDarrow read forwards from a DOM arrow read backwards. We also have leftand right directionality at our disposal in this connection, to represent musicalchronology. Thus, analyzing a plagal cadence in C major, we would draw a 777


SUBD arrow pointing from (F, +) on the left to (C, +) on the right. Analyz-ing a tonic-dominant progression in F major, say an opening phrase terminat-ing with a half-cadence, we would still put (F, + ) on the left and (C, -1-) on theright, but now we would draw a DOM arrow pointing leftwards, from (C, +)to(F, +).

In similar vein, we can define and explore SUBM, the formal inverse ofMED. We can consider other sorts of transformations too. For example, wecan define REL, the operation that takes any Klang into its relative minor/major. (C, +)REL = (A, -); (C, -)REL = (Eb, +). REL is not the sameoperation as MED or SUBM: (C, -)REL = (Eb, +) but (C, -)MED =(Ab, +); (C, +)REL = (A, -) but (C, +)SUBM = (E, -). We can alsodefine PAR, the operation that takes any Klang into its parallel minor/major,(p, sign)PAR = (p, —sign). We can define Riemann's "leading-tone exchange"as an operation LT: (C, +)LT = (E, -); (E, -)LT = (C, +). We can alsodefine more exotic operations on Klangs. For instance we can define anoperation SLIDE that preserves the third of a triad while changing its mode:(F, +)SLIDE = (F#, -);(F#, -)SLIDE = (F, +). The SLIDE relations be-tween (F, +) and (F#, —) can be heard in the last movement of Beethoven'sEighth Symphony, where the F-major theme that begins on the note A, thethird of the triad, is transformed at measures 379-91 into F# minor, where itbegins on the same A; the theme slides back into F major at measure 392. ASLIDE relation between (C, +) and (C#, —) can be heard over measures103-10 in the slow movement of Schubert's posthumous Bb-Major PianoSonata. Over those measures, thematic material which we expect to hear in C#minor is presented in C major instead.

Using such transformations to label arrows, we can construct networksthat could not be conceived using only intervals-and-transpositions. Forexample, figure 8.2 displays interesting relations between a "Tarnhelm net-work" (a) and a "Valhalla network" (b).

The Tarnhelm network of (a) takes (B, +) as a tonic for the motive in itsown context; it asserts structural relaxation on that Klang. This seems legiti-mate; besides, Wagner not infrequently interprets the motive in a largercontext of B minor or B major, e.g. at the end of G otter dammerung I, or at theend of Tristan. (E, + ) on network (a) is bracketed to indicate that this Klangfunctions by implication only; (E, — ) substitutes for it in the music.

The Valhalla network, figure 8.2(b), asserts the indicated relationsamong the principal Klangs over the first presentation of the Valhalla themein Das Rheingold, during measures 1-20 of scene 2. We shall be concernedwith the "modulating" part of the network, the part that extends frommeasure 7 onwards. That is why the events of measures 1 -6 are in parentheses.

Graphs (a) and (b) make visually clear a strong functional relationshipbetween the Tarnhelm progression and the modulating portion of the Val-halla theme, a relationship which it is difficult to express in words.178


FIGURE 8.2

8.1.2 Obviously, we could not construct the graphs of figure 8.2 usingintervals-cum-transpositions. The fact is obvious if by "intervals" we meanintervals between the notes of a fundamental bass. However, might we not beable to think of "intervals" here in some more extended formal sense? Thematter bears theoretical examination, if only for the sake of review.

The reader will recall how we showed in section 7.1.1 that the entirenotion of a GIS can be developed formally from a given family S of objectsand a given simply transitive group STRANS of operations on S. When a GISis developed therefrom, STRANS becomes the group of transposition oper-ations for that GIS.

Let us take S to be a family comprising a given Klang and all other Klangsthat can be derived from it via any chains of DOM, MED, SUBD, and SUBMtransformations. Let STRANS be the group generated by the four citedoperations. Since SUBD = DOM"1 and SUBM = MED"1, STRANS isgenerated by DOM and MED. The reader can now verify that DOM itselfis generable by MED: DOM = (MED)(MED). That is, given any Klang,if it becomes the mediant Klang of a second Klang which in turn becomesthe mediant Klang of a third Klang, then the first Klang is the dominantof the third Klang. E.g. ((C, +)MED)MED = (A, -)MED = (F, +) = 779


(C, +)DOM;((C, -)MED)MED = (Ab, + )MED = (F, -) = (C, -)DOM.Since DOM = MED2, the group STRANS here is generated by MED

alone: The operations of STRANS are precisely the powers of MED, includ-ing MED° = IDENT and MED~n = (MED"1)" = SUBMn. Once we havethis insight into the structure of our group STRANS here, it is not difficultto show that the group is in fact simply transitive on the defined family ofKlangs, whatever the system of intonation we are using. Then we can regardthe powers of MED as formal "intervals" on that family, in the sense of 7.1.1.We may then regard figure 8.1 as a formal "intervallic" network amongKlangs, since it involves among its transformations only powers of MED.

In this system, the Klang (C, —) is derived from (C, +) by 7 applicationsof MED: (C, +)MED7 = (A, -)MED6 = (F, +)MED5 = (Bb, +)MED3 =(Eb, +)MED = (C, —). Thus MED7, applied to a major Klang in somesystems of intonation, will yield the parallel minor Klang. However, MED7

applied to a minor Klang will not yield the parallel major Klang, no matterwhat the system of intonation. E.g. (C, -)MED7 = (F, -)MED5 =(Bb, -)MED3 = (Eb, -)MED = (Cb, +). Thus MED7 is not the sameoperation as PAR on the family of Klangs at hand.

Now the graphs of figure 8.2(a) and (b) reference both the operationsSUBM and PAR. It follows that we shall not be able to find a simply transitivegroup on a suitable family of Klangs that enables us to consider figure 8.2(a)and (b) as formally "intervallic" graphs. Our group would have to includeboth SUBM and PAR; containing SUBM, it would contain MED =SUBM'1; then the group would contain both MED7 and PAR. Then thegroup could not be simply transitive. E.g. given elements (C, +) and (C, —) inour family of elements, there would not be one unique member of the grouptransforming the former Klang into the latter; both MED7 and PAR woulddo the job. We could only salvage the formalities of this situation by cleavingso firmly to just intonation that we were willing to admit an infinite number ofdistinct Klangs (C0, +), (Clf +), (C2, +),..., (C0, -), (C,, -), (C2, -),...and so on, whose roots lay syntonic commas apart. Then (C0, +)MED7 and(C0, +)PAR would have different formal values: The former would be(C_1? —) and the latter would be (C0, —). We would end up with a gameboard like that of figure 2.2 earlier, only with yet a third dimension reversingthe modality of each Klang on the board. Clearly, this model would be at leastawkward for analyzing Wagner's music. (It does engage some of the perfor-mance problems, especially for the singers.)

8.2.1 We have seen that non-intervallic transformation-networks can berevealing in connection with Klangs. They can also be revealing in connectionwith serial transformations. Let us consider a series s of pitches or pitch-classes s1? s2, ..., SN. We can apply to s the Rl-chaining operation RICH.RICH(s) is that retrograde-inverted form of s whose first two elements areSN-! and SN, in that order. Thus if s = A-C-Eb-E (a form of Zauber), then180


RICH(s) is Eb-E-G-Bb, and RICH(RICH(s)) is G-Bb-Db-D. The RICHtransform of RICH(s), being a retrograde-inverted form of a retrograde-inverted form of s, must always be some transposed form of s. In the Zauberexample above, the interval of transposition is 10: G-Bb-Db-D is the 10-transpose of A-C-Eb-E.

For another example, let us examine s = Eb-B-Bb~D-C#-C-F#-E-G-F-A-G#. This is the prime row of Webern's Piano Variations op. 27.RICH(s) is A-G#-C-Bb-Qf-B-F-E-Eb-G-F#-D, and RICH(RICH(s))is then F#-D-Db~F- etc. As before, the RICH transform of RICH(s) is atransposed form of s, but now the specific interval of transposition is different.F#-D-Db-F-etc. is the 3-transpose, not the 10-transpose, of Webern's row.

In general, when we RICH the RICH-transform of an abstract pitch orpitch-class series s = s t, s2 , . . . , SN, the transposed form of s that results willbe TJ(S), where the interval of transposition is i = int(s l5sN) + int(s2,sN_1).(We are supposing that N > 2. The formula is given without proof.) Thusthe interval of transposition for the Zauber series A-C-Eb-E was 10 = 7 +3 = int(A, E) + int(C, Eb), while the interval of transposition for Webern'srow Eb-B A-G# was 3 = 5 + 10 = int(Eb,G#) + int(B, A).

It follows: When we define the operation TCH as (RICH) (RICH), thenTCH(s) is always some transposed form of s, but just which transposed formdepends on the internal structure of any given argument s upon which TCH isoperating. Specifically, if i = int(s l5sN) + int(s2,sN_1), then TCH(s) = Tj(s).We shall call i here "the TCH-interval for s." The TCH interval for a retro-grade or an inverted form of s will be the negative (group inverse) of this i; theTCH interval for a retrograde-inverted form of s will be the same as the TCHinterval for s itself. (These facts follow from the formula defining i above.)

8.2.2 EXAMPLE: Let us turn our attention once more to figure 7.4, the earlierexample from Parsifal, inspecting the first, second, third, and fourth forms ofZauber that are beamed thereon and labeled as Z t, Z2, Z3, and Z4. Consider-ing Z as a serial motive, we can graph Wagner's transformational techniquehere by figure 8.3.

181FIGURE 8.3


We have already discussed how the TCH operation makes Z3 the 10-transpose of Zx and Z4 the 10-transpose of Z2. During our earlier discussionof the passage, we pointed out how the chaining technique creates structuralsequencing as a result. We can "hear" the structural sequence in the noteheads of figure 7.4, even though the foreground events of the music overmeasures 1074-1100 are quite different from those of the music over measures1096-1140.

8.2.3 EXAMPLE: Webern is fond of using RICH and TCH, especially as row-transformations in his serial works. Figure 8.4 graphs two instances from thePiano Variations, (a) of the figure coincides with the thematic middle sectionof the first movement; (b) coincides with the frantically syncopated variationin the last movement.

The nodes on figure 8.4 are understood to contain various forms of therow. On (a) of the figure, T5 is not TCH. The row-form that fills both theleftmost and the rightmost nodes of (b) is the prime row-form cited earlier, theform which opens the third movement. The same form also fills the rightmostnode of (a). This is the first occurrence of the prime form in the piece; after

182 FIGURE 8.4


measure 37, the music continues to project the prime form of the row as itlaunches the big thematic reprise of the movement. The prime form that fillsthe rightmost node of (b) launches the coda of the third movement and theentire piece. The total rhythm of this coda matches that of the first-movementtheme reprised at 1,37: In both cases we hear a rhythmic ostinato whoserepeated units project four attacks and a rest in steady note-values.

8.2.4 If we excise any four consecutive nodes from figure 8.4(b), along withthe arrows that connect them, we shall have essentially the same graph as thatof figure 8.3. The same transformations are arranged and combined in thesame structure of nodes and arrows, even though the contents of the nodes areWagnerian in one case and Webernian in the other. We shall say that the twonetworks-of-series are isographic. The isography would not obtain if we wrote"T10" for TCH on figure 8.3 and "T3" for TCH on figure 8.4(b): T10 and T3

are not the same transformation.

8.2.5 EXAMPLE: Let us define another operation on series, an operationcalled MUCH. MUCH(s) is that retrograde-inverted form of s whose begin-ning overlaps the ending of s to the maximum possible extent. Figure 8.5(a)shows how Bach chains MUCH and RICH in the first Two-Part Invention.

Figure 8.5(b) shows Bach's transformational technique in a graph whichis very similar to the earlier graphs involving RICH. OP is the transformationRICH-after-MUCH; OP' is the transformation MUCH-after-RICH. MUCHand RICH do not commute on the family of all series, but the OP-interval-of-transposition for any given s is always the same as the OP'-interval for the

FIGURE 8.5 183


retrograde-inversion of s. In the specific example at hand, that interval is 2-diatonic-steps-down. Bach's foreground sequence is constructed by a methodvery similar to that used by Wagner and Webern, in building their structuralsequences. Figure 8.5 shows how artfully Bach's transformational techniqueuses the characteristics of his motive to fit his meter.

8.2.6 EXAMPLE: Wagner also uses RICH in the foreground. Figure 8.6sketches the opening of the "Todesverkiindigung," Die Walkure, act 2, scene4, starting from three measures before the scene begins.

FIGURE 8.6

184

The scene is dominated by the Fate motive, whose melodic componentFATE is bracketed over measures 1-2 on the figure. FATE functions both asa 3-note series and as an intervallic network, 2 = (— 1) + 3 semitones. Theinterval 2 of that network is originally spanned from A to B, and the thematicpitch-idea of "A to B" thereby goes along with the prime form of the FATEmotive/series/network. Locally, "A to B" helps us hear FATE as related to the


(head of the) LOVE motive, which has just been repeated over and over in themelody preceding measure 1.

The repeated LOVE fragment also projects an overall sense of "A to B";as it repeats, it inflects A by a pickup F# and thereby defines a total ambitus ofF#-to-B. These features enable us to extract the intervallic network of figure8.7(a).

Figure 8.7(b) displays the FATE network for purposes of comparison. A3-arrow and a 2-arrow are common to both networks. Another way ofrelating the motivic fragments is to regard LOVE as an essential A-to-Belaborated by an F# pickup 3 below A, while regarding FATE as an essentialA-to-B elaborated by a G# pickup 3 below B. This view attributes emphaticpriority to pitch relations and even pitch hierarchies, at the expense of inter-vallic relationships per se. We shall see that the intervallic structures of the"Todesverkiindigung," like those of the Brahms Horn Trio studied earlier,take on a more autonomous role as the music develops.

We can hear another strong relation of LOVE and FATE in the musicthrough the harmonization of LOVE that immediately precedes FATE: A-to-B within LOVE is supported by C-to-B in the bass; the total matrix therebyprojects a pitch-class form of FATE. Figure 8.6 indicates the pitch-classintervals of the FATE network on the LOVE matrix accordingly. A 2-arrowconnects A to B within the melody of LOVE; that 2-arrow is subarticulatedinto a "3"-arrow, from A to the C below, and a (— l)-arrow, from C in the bassto B in the bass. The bass B supports and doubles the B of the LOVE melodyas a pitch class.

Supposing a fundamental bass for the A minor harmony of LOVE, an Athat goes under and hence conceptually "before" the C of the basso continue,we can even identify a serial form of FATE embedded in this LOVE music,namely A-C-B. This is the unique form of the FATE series, other than theprime form, that embeds the essential A-to-B gesture.

Starting from that A-C-B, we can hear a RICH chain of FATE formsproceeding along the bass line of figure 8.6. A-C-B before measure 1 chains 185

FIGURE 8.7

#.2.6 Transformation Graphs and Networks C2)

into C-B-D across measure 1; C-B-D in turn chains into B-D-C#, whichcarries the harmony through the bass of the FATE motive proper. Thechaining continues on in the bass line, through D-C#-E, C#-E-D#, andeven farther (across the DEATH music). We recognize the sequencing of theFATE bass as a typical TCH sequence, created by Rl-chainmg. The intervalof the sequence, that is the TCH interval for the FATE bass, is 2. Computingthe interval specifically from the series A-C-B, our point of departure in thebass, we have i = int(A, B) + int(C, C) = int(A, B), which is 2. So we see that2-as-TCH-interval, governing the sequence, is the "same" 2 as the 2 that spansA-to-B, that is int(A, B). We do not need such heavy transformational ma-chinery, just to hear that the 2-sequences of the music are related to the A-Bgesture of the motive. But the transformational machinery clarifies just howthe relation is worked out, and it attributes thereby a special and characteristicformal function to the compositionally prominent interval in this serialcontext.

The FATE series is Rl-chained in the melody, too. Just as the bass chainbegan with A-C-B, one form of FATE that embeds A-to-B, so the melodicchain begins with A-G#-B (measures 1-2), the other form of FATE thatembeds A-to-B. Again the TCH interval is 2 = int(A, B) + int(G#, G#) =int(A, B). Again the musical sequence from measures 1-2 to measures 5-6 iscarried by a TCH sequence, now in the melody. The melodic chain of measures1-8, covered by a slur on figure 8.6, is recalled in summary at the end of theDEATH motive, covered by another slur over measures 10-12 on the figure.The last stage of FATE under the slur, B-A#-C#, recurs yet again at measure13, to launch the transposed return of all the FATE music, which starts thechaining underway all over again. Via the relationships just mentioned, wecan hear that the melodic B-A#-C# of measures 13-14 is in the TCH-relationto the A-G#-B of measures 1-2. Hence the large sequence, the sequence thatcarries all of measures 1-12 into all of measures 13-24, is itself a TCH-sequence, a product of Rl-chaining.

It would be possible and even legitimate to draw many 2-arrows uponfigure 8.6, e.g. transposing measures 1-4 into measures 5-8, transposingmeasures 10-llj of the melody into measures 11-12 of the melody, andtransposing all of measures 1-12 into all of measures 13-24. We are certainlystrongly aware of the interval 2 in these connections. The resulting graphwould be inadequate, though, so far as it suggested that the relations involvedwere only transpositional, the results of expanding the interval 2 = int(A, B)into T2-relations between larger Gestalts. We would miss the function of 2 asa TCH interval in two independent FATE chains, one in the bass and one inthe melody, each chain launched by one of the two FATE-forms that embedA-to-B. We would miss hearing how the sequences in the music hang on thepertinent stages of RI chaining in the outer voices.

Figure 8.8 shows how a motive I shall call FATE' is Rl-chained over186


measures 56 and following. This is where Siegmund "looks into the eyesof the Valkyrie," as Brunnhilde puts it later on. The subinterval 3 of theFATE network 2 = (—1) + 3 now becomes the overall interval of the FATE'network 3 = (— 1) + 4; ( — 1 ) remains a subinterval of FATE'. The pitchclass A of FATE' is bereft of its FATE-partner B, just as Siegmund will belonely in Valhalla, bereft of his sister/wife. The comparison is suggestivebecause the F# and A of FATE' recall the F# and A of LOVE, displayedearlier in figure 8.7(a). Interval 3, the TCH-interval for FATE', is preciselyint(F#, A)(= int(F#, A) + int(E#, E#)). On figure 8.8, the FATE' series goesthrough four TCH-sequences. Its course is therefore isographic to the bass-line chain of figure 8.6, which also went through four TCH-sequences. RICHand TCH thus enable us to hear a way in which the bass line of figure 8.6 andall of figure 8.8 project the same overall transformational gesture. Obviously wecould not hear such a relation using T2 in one case and T3 in the other; thoseare different transformations.

After Siegmund has looked into Brimnhilde's eyes he goes into an ex-tended Wagnerian stichomythy, asking a series of questions about Valhallawhich Brunnhilde answers in turn. What really concerns Siegmund, we dis-cover, is the idea that Sieglinde may not be able to accompany him to Valhalla.His concern is well founded. As he asks each question, he sings or sings alongwith an entry of the DEATH motive. Figure 8.9 tabulates his questions inbrief, along with the keys in which the DEATH melody enters therewith,during measures 70-133.

FIGURE 8.9 187

FIGURE 8.8


We are intensely aware of Siegmund's concern for Sieglinde even beforehis final questions, because the keys of figure 8.9 and the intervals among theirtonics project the pitch classes and intervals of the LOVE motive, as dia-grammed in figure 8.7 (a) earlier. During the repeated LOVE music just beforethe scene change, Siegmund had calmed Sieglinde, and she is lying asleep in hisarms during the whole exchange with Brunnhilde.

8.3.1 EXAMPLE: RICH and TCH have provided good examples of serialtransformations that are not intervallic/transpositional but are nonethelessvery suggestive in connection with node/arrow analytic graphs. The retro-grade operation on series is another such example. So are the various oper-ations RT and RI, T being some transposition and I some inversion. (For agiven u and a given v, RI* and RICH are not the same operation, as theyoperate upon a variety of series.)

Webern's piece for string quartet, op. 5, no. 4, demonstrates othersuggestive serial operations. The operation TLAST transposes a series by itslast interval; the operation TFIRST transposes a series by its first interval.Hence TFIRST"1 transposes a series by the complement of its first interval.TLAST makes the last note of a given series the next-to-last note of thetransformed series; TFIRST"1 has a sort of "dual" effect, in that it makes thefirst note of a given series the second note of the transformed series. Figure8.10 shows three series of pitch classes, arranged in a network that involvesTFIRST'1 and TLAST in this "dual" relationship.

FIGURE 8,10

The three series cited by the figure are projected by the three appearancesof a prominent unaccompanied rising motive FLYAWAY in Webern's piece.Each of these unaccompanied appearances ends a major section of the work.The order in which figure 8.10 cites the three series is not the chronologicalorder in which the music presents the forms of FLYAWAY. In the music, theform that begins on A(7 occurs last; indeed it is the last event of the piece. Thevisual "centrality" of this form on figure 8.10 portrays a cadential functionfor the event, transformationally "balanced" as it is between the other twoforms. Just so, the formal "centrality" of a tonic Klang, balanced between its188


dominant and subdominant, is often projected temporally by a final cadencein which the tonic is the last event. While it would be perfectly possible to labelthe arrows of figure 8.10 as "T8" and "T3" rather than TFIRST'1 andTLAST, the transpositional labels would conceal, not reveal, the balancingcentrality of the form beginning on A|?, the form that ends the piece. Theintervals of transposition are of course very important in other connections.

8.3.2 EXAMPLE: We define operations FLIPEND and FLIPSTART on seriesof three pitches or three pitch-classes. FLIPEND transforms the seriesS1-s2-s3 into the series s^Sj-a, where a is the inversion-about-s3 of s2.(a = Is*(s2); int(s3,a) = int(s2, s3).) The inverse operation FLIPEND"1 thentransforms the series ti-t2-t3 into the series ti-b-t^ where b is the inversion-about-t2 of t3. (To verify this, given t l s t2, and t3, set Si = t l 5 s2 = b-as-b-is-defined, and s3 = t2. Then observe that FLIPEND transforms the s series intothe given t series.)

Dually, FLIPSTART transforms S1-s2-s3 into a-s1-s3, where a is theinversion-about-Si of s2; then FLIPSTART"1 transforms ti~t2-^-3 intot2-b-t3, where b is the inversion-about-t2 of tl. Figure 8.11 shows whathappens when FLIPEND and FLIPSTART"1 are chained in alternation,starting with one series of three pitches in (a) and another in (b). The arrowsabove the staff indicate applications of FLIPEND; the arrows below the staffindicate applications of FLIPSTART"1.

FLIPEND and FLIPSTART are my own names for transformationsused by Jonathan W. Bernard in studying how Varese's music expands,contracts, and displaces registral space.1

8.4 The serial transformations just studied, i.e. RICH, TCH, MUCH,TLAST, TFIRST, FLIPEND, and FLIPSTART, are all easily generalized tooperate on series whose elements are members of an abstract commutativeGIS. In the non-commutative case, it is not clear just how some of theoperations are to be defined; different possibilities are equally plausible. Forinstance, given the abstract series s = s l5 s2, ..., a, b within some abstractGIS, let us try to define RICH(s) abstractly. We can consider three possiblydifferent series, t, u, and v, as plausible candidates for "RICH(s)." t is theretrograde of I(s), where I is (a/b)-inversion l£. u is the retrograde of J(s),where J is (b/a)-inversion I£ = (I*)"1, v is a series constructed as follows. Letthe serial intervals of s be it = int(s1, s2), i2 = int(s2, s3),..., iN_j = int(a, b).The series v starts with the element a, and then proceeds according to thesuccession of intervals iN_1} iN_ 2,... ,i2,i!. Elements a and b will be the first andsecond elements of all three series t, u, and v. t and u are both retrograde-

1. Jonathan W. Bernard, The Music of Edgard Varese (New Haven and London: YaleUniversity Press, forthcoming). 189


inverted forms of s. The serial intervals of v are the same as the intervals of s inreverse order. If the GIS is commutative, t, u, and v will all be the same series. Ifthe GIS is not commutative, t, u, and v may be three distinct series.

8.5 In section 8.1 we studied Klang-transformations as potential labels forarrows on graphs, arrows that could not be analyzed adequately by intervallic/transpositional ideas alone. In sections 8.2 and 8.3 we studied a number ofserial transformations in the same connection. Inversional relations betweenelements in a GIS, even a well-behaved commutative GIS, may also give riseto non-transpositional arrows on graphs, that is I-arrows. Consider figures8.12 (a) and (b).

The pitch classes on the networks stand for the pitches presented at theopening of the second movement in Webern's Piano Variations. There, thepitches come in pairs articulated by a rhythmic motive that remains essentiallyconstant over the events portrayed in the figure. Graph (a) analyzes the pitches190

FIGURE 8.11


FIGURE 8.12797


of the phrase in an intervallic/transpositional network. One could add moreinterval-arrows connecting certain pitches by criteria other than immediatetemporal succession. Graph (b) analyzes the same pitches using not only someintervallic/transpositional arrows (shown as curved) but also some inversionalarrows (shown as straight). "I" is inversion about the pitch A4.

Graph (b) reflects the rhythmic motif of the music by a visual motif, thevertical I-arrow. It represents by a visual symmetry the mirror symmetry of thepitch registration. It makes manifest the row-structure of the passage, and theway in which that structure interacts with the dux/comes structure of themusical canon. Graph (b) thus reflects more clearly than graph (a) not only thecompositional structure of the passage but also our foreground perception ofits shapes.

Let us denote by IPAIR the graph consisting of two nodes connected bya two-way I-arrow. Now "I" will mean pitch-class inversion about A. As aconfiguration of pitch classes, network 8.12(b) has four subnetworks mani-festing the graph IPAIR. Those are the subnetworks relating Bt?-and-G#,A-and-A, F-and-C#, and D-and-E; the subnetworks are all isographic. Thenetwork of row-forms being used is also isographic: Two row-forms are beingused, each of which is the I-inversion of the other. Also manifesting IPAIRisographically is a network interrelating the "antecedent" and "consequent"row forms that respectively control the first twelve and the second twelvenotes of the third movement. Antecedent and consequent forms there areI-inversions, each of the other. The consequent is presented in the musicisorhythmically to the antecedent. IPAIR, in short, is a thematic graph.

792

Transformation Graphs andNetworks (3): Formalities

It is time now to become formal about what we are doing when we draw nodes,arrows connecting some pairs of nodes, and names of transformations label-ing those arrows, sometimes also putting pitches, Klangs, series, row-forms,or other objects "into" the nodes. I have already worked out most of theseformalities elsewhere.1 Their treatment here differs somewhat, mainly in thatthe present exposition is more general.

9.1.1 DEFINITION: By a node I arrow system we shall mean an ordered pair(NODES, ARROW), where NODES is a family (i.e. set in the mathematicalsense), and ARROW is a subfamily of NODES x NODES, i.e. a collectioncontaining some ordered pairs (Nl9 N2) of NODES. We say that nodes Nt

and N2 are "in the arrow relation" if the pair (N l5N2) is a member of thecollection ARROW. For present purposes, we shall stipulate that every nodeis in the arrow relation with itself. That is, we assume that (N, N) is a memberof ARROW for every node N.

On figure 9.1, the nodes M t and M2 are not in the arrow relation; neitherare M2 and Mj. Nodes M4 and M5 are in the arrow relation; so are M5 andM4. Nodes M5 and M6 are in the arrow relation; M6 and M5 are not. Arrowsfrom M! to M! , from M2 to M2, and so on, are all understood on the figure.

9.1.2 DEFINITION: Nodes N and N' in a node/arrow system communicate if

1. David Lewin, "Transformational Techniques in Atonal and Other Music Theories,"Perspectives of New Music vol. 21, nos. 1-2 (Fall-Winter 1982/Spring-Summer 1983), 312-71.The formalism there is confined to pages 360-66. The article consists mostly of analyses, togetherwith exemplary graphs and networks, that will interest the reader who has enjoyed chapters 7 and8 of the present book. The music studied there overlaps that studied here only slightly.

9

193


FIGURE 9.1

there exist nodes N0, N t, ..., Nj which satisfy criteria (A), (B), and (C)following.

(A): N0 = N.(B): For each j between 1 and J inclusive, either (Nj.^Nj) is in theARROW relation, or else (NJ5 Nj.J is.(C): Nj = N'.

The criteria demand a finite unbroken path of forwards-or-backwardsarrows which starts at N and ends at N'. The nodes M t and M2 of figure 9.1communicate. But neither Mj nor M2 communicates with M4.

The communication relation among nodes is easily proved to be reflexive,symmetric, and transitive. Hence "communication" is an equivalence relationon the NODES of a node/arrow system. As figure 9.1 suggests, the nodeswithin any equivalence class all communicate, each with any other, whilenodes in different equivalence classes do not communicate.

9.1.3 DEFINITION: A node/arrow system is connected if any two nodescommunicate.

The system displayed in figure 9.1 is not connected. It can be analyzedinto two component subsystems, each of which is connected; there is nocommunication between any node of one subsystem and any node of theother. This sort of structure is typical of disconnected node/arrow systems.Any disconnected system can be analyzed into component connected sub-systems that do not communicate with one another. Each such subsystemis a pair (NODESa, ARROWJ, where NODESa is an equivalence-class ofNODES under the communication relation, and ARROWa is that subcollec-tion of ARROW comprising pairs of nodes from NODEa.

As we proceed, we shall find the following construct necessary and useful.

9.1.4 DEFINITION: An arrow chain from node N to node N' in a node/arrowsystem is a finite series of nodes N0, N x , . . . , N, satisfying criteria (A), (B), and(C) below.194


(A): NO = N.(B): For each j between 1 and J inclusive, (N^, Nj) is in the ARROWrelation.(C): Nj = N'.

The criteria demand a finite unbroken path of forwards-oriented arrows,starting at N and ending at N'.

We are now ready to "label" our arrows formally with symbolic trans-formations. At this stage of our work, the nodes do not yet "contain" anyobjects to be transformed, so we shall represent the eventual transformationsabstractly, as members of some abstract semigroup.

9.2.1 DEFINITION: A transformation graph is an ordered quadruple (NODES,ARROW, SGP, TRANSIT) satisfying criteria (A), (B), (C), and (D) below.

(A): (NODES, ARROW) is a node/arrow system.(B): SGP is a semigroup.(C): TRANSIT is a function mapping ARROW into SGP.(D): Given nodes N and N', suppose that N0, N1? ..., Nj is an arrowchain from N to N'. Suppose that M0, M1 } . . . , MK is also an arrowchain from N to N'. For each j between 1 and J inclusive, let Xj =TRANSIT(Nj.j^Nj). For each k between 1 and K inclusive, let yk =TRANSIT(Mk_1, Mk). Then the semigroup product X j . . . x2

xi is equalto the semigroup product yK ... y2yi-

FIGURE 9.2

The setting in which criterion (D) holds sway is illustrated by figure 9.2.Symbolically, the eventual "contents" of N' will be the Xj-transform of the...of the x2-transform of the x^transform of the eventual "contents" of N. Thatis, the contents of N' will be the (Xj . . . x2x ̂ -transform of the contents of N, inthe sense of our left-orthographic convention. Likewise, the contents of N'must also be the (yK ... yjy^-transform of the contents of N. To ensure thatno contradiction can possibly arise, we must ensure that the semigroupproducts are equal. And that is what Criterion (D) of the definition does.

9.2.2 OPTIONAL: Criterion (D) also enables us to prove that for each node N, 195


TRANSIT(N, N) must be some idempotent member of SGP. Tp prove this,consider the following two formal arrow-chains from N to N: N0 = Nt =Nj = N; M0 = Mj = M2 = MK = N. Set Xj = TRANSITCN^NO, yt =TRANSITCM^Mi), y2 = TRANSIT^, M2). Then, via the criterion,xi = V2vi- But x1? y1} and y2 are all the same member of SGP, namelyTRANSIT (N, N). So that element is idempotent, as claimed.

Criterion (D) then ensures that whenever M0, M l 5 . . . , MK is an arrowchain from N back to N, the product yK ... y^ (as in 9.2.1) is equal to theidempotent TRANSIT(N, N).

The only idempotent member of a group is the identity element. (If zz = z,then zzz"1 = zz"1 and z = e.) Hence, when SGP is a group, TRANSIT (N, N)must be the identity.

9.2.3 DEFINITION: An operation graph is a transformation graph in whichSGP is a group.

Now we shall render formal the idea of "putting objects into" the nodesof a transformation graph, and transforming the objects about as indicated bythe graph and by some appropriate semigroup of transformations. That is theidea of a transformation network, which we shall now define, distinguishing itformally from the transformation graph whose nodes its objects fill.

9.3.1 DEFINITION: A transformation network is an ordered sextuple (S,NODES, ARROW, SGP, TRANSIT, CONTENTS) having the features (A)through (D) below.

(A): S is a family of objects (that are to be transformed in various ways).(B): (NODES, ARROW, SGP, TRANSIT) is a transformation graphsuch that SGP is a semigroup of transformations on S.(C): CONTENTS is a function mapping NODES into S."CONTENTS(N)" can be read: "the contents of node N."(D): Given nodes Nt and N2 in the ARROW relation; if f =TRANSIT(N1,N2), then f (CONTENTS^)) = CONTENTS(N2).

To see how feature (D) of the construction reflects our intuition, we caninspect figure 9.3.

The figure depicts nodes Nt and N2 in the arrow relation. On thegraph, the member f of SGP labels the arrow from N! to N2; that is, f =

FIGURE 9.3196

Transformation Graphs and Networks (3)

TRANSIT(N1,N2). f is some transformation on S, according to feature (B)of 9.3.1. Inspecting figure 9.3 further, we see that $! is the CONTENTS ofN! and s2 is the CONTENTS of N2. In this situation, (D) of the definitionassures us that we shall have f^) = s2.

9.3.2 DEFINITION: An operation network is a transformation network forwhich SGP is a group of operations on S.

9.3.3 THEOREM: Let S be a family of objects. Let GP be a group of operationson S. Let (NODES, ARROW, GP, TRANSIT) be an operation graph whosenode/arrow system is connected. Let N0 be a node; let s0 be a member of S.

Then there exists a unique operation-network having S for its objectsand (NODES, ARROW, GP, TRANSIT) for its graph such that s0 is theCONTENTS of N0.

The theorem is given without proof. It says that all the contents of all thenodes in a connected operation-network are uniquely determined, once weknow the contents of any one node. Intuitively, we can follow some path offorwards-or-backwards arrows from the given N0 to any other node, since thesystem is connected. As we go along that path, we can fill in the CONTENTSof each node by applying the indicated TRANSIT operation when we traversean arrow forwards, or the inverse (NB) of that operation when we traversean arrow backwards. Criterion (D) of 9.2.1 enables us to infer that it doesnot matter which path we might follow from N0 to a specific other node inthis connection; the end result will be the same as regards the necessaryCONTENTS of that other node. To get some sense of why this works, let uslook at the graph displayed in figure 9.4.

FIGURE 9.4

Here, we are supposing A, B, C, and D to be operations upon some familyS. Fixing some member s0 of S, let us construct an operation-network having Sfor its objects and figure 9.4 for its graph, such that s0 is the CONTENTS ofnode N0 on the figure. Since s0 is the CONTENTS of N0 and A is the 797

9.3.3


TRANSIT-operation from N0 to N3, the CONTENTS of N3 on the figurewill have to be A(s0). (We know this by 9.3.1(D).) Let us write "s3" for "thenecessary CONTENTS of node N3." Then we shall have to have s3 = A(s0).Now what about s l5 the necessary CONTENTS of node Nj? Since B is theTRANSIT-operation from Nx to N3, we shall have to have s3 = 6(5!). Sincewe have already determined what s3 must be, and we are searching for s1, wecan infer that the Si we are looking for is derived as st = B"1 (s3). Note how wehave leaned on the fact that B is an operation at this point in our construction.When we write "Sj = B"1^)," we are using the idea that a unique st is welldefined by the relation s3 = B^); that idea in turn rests on the suppositionthat B is 1-to-l and onto.

So, given our initial s0, we have derived the necessary CONTENTS s3

and Sj for the nodes N3 and Nt in the operation network we are constructing:s3 = A(s0); Si = B'^SS). What about s2, the necessary CONTENTS of N2?Here, it seems at first that we have two different choices for s2. Since an arrowlabeled D points from Nx to N2, we must have s2 = D^). But also, since anarrow labeled C points from N2 to N3, we must have s2 = C"1^). It is justhere that 9.2.1 (D) comes to our rescue. The criterion tells us in this situationthat, in fact, D(SJ) = C~1(s3). Therefore it only seems that we have "two"choices for S2; in fact the value of s2 is well determined.

Let us see just how we can infer from Criterion 9.2.1(D) that D(SJ) =C~1(s3). The Criterion notices an arrow-chain going from Nx directly to N3;it also notices an arrow-chain going from Nx through N2 to N3. It informs usthat a certain algebraic relation must therefore obtain among the TRANSIT-operations B, D, and C that link the nodes along those arrow-chains. Therelation must obtain, that is, for us to have spoken at all of a well-formed"operation graph." The relation given by the Criterion here is B = CD.

Now Si was chosen to satisfy the relation B(Si) = s3. So we can infer thatCD(S!) = s3. And thence we can infer that D^J = C"1^), which is justwhat we wanted to show.

Obviously, the sort of situation we have just examined in connectionwith the arrow-chains of figure 9.4 can get very complicated on a generaloperation-graph. The way in which Criterion 9.2.1 (D) helps us out is basicallythe same, though.

During the musical discussions of chapters 7 and 8 we invoked from timeto time the concept of "isography." It would seem that we can now define thatconcept rigorously: Two transformation networks are isographic if they"have the same graph." But that definition is not yet formal enough. Forsuppose we want to assert an isography between the networks (S, NODES,ARROW, SGP, TRANSIT, CONTENTS) and (S', NODES, ARROW,SGP', TRANSIT', CONTENTS'), where S and S' are different families ofobjects. Those two networks have the same node/arrow system, but they198


cannot have the same graph. The graph of one is (NODES, ARROW, SGP,TRANSIT); the graph of the other is (NODES, ARROW, SGF, TRANSIT').SGP', a semigroup of transformations on S', cannot be "the same as" SGP, asemigroup of transformations on S. And therefore TRANSIT', a functiontaking on values in SGP', cannot be "the same as" TRANSIT, a functiontaking on values in SGP.

Moreover, the concept of isography is particularly suggestive exactlywhen S' and S are different families of objects, as above. S for example mightbe the twelve chromatic pitch classes, while S' might be a family of set-forms,motive-forms, or row-forms; SGP could comprise transpositions and inver-sions on pitch classes, while SGP' comprised transpositions and inversions ofset/motive/row forms.

To make our intuitions about isography work out formally here, we needthe concept of isomorphism between one transformation graph and another.We can then define two networks to be isographic if their graphs are isomor-phic. To define the isomorphism of graphs, we shall in turn have to define theisomorphism of node/arrow systems.

9.4.1 DEFINITION: Two node/arrow systems, (NODES, ARROW) and(NODES', ARROW), are isomorphic if there exists a 1-to-l map NODEMAPof NODES onto NODES' such that for every pair (Ni,N2) of NODES,(N1}N2) is in the ARROW relation if and only if (NODEMAP(N1),NODEMAP(N2)) is in the ARROW' relation.

A function NODEMAP having the indicated property will be called anisomorphism of (NODES, ARROW) with (NODES', ARROW').

9.4.2 DEFINITION: Given two transformation graphs (NODES, ARROW,SGP, TRANSIT) and (NODES', ARROW, SGP', TRANSIT'), the two willbe called isomorphic if there exists a pair (NODEMAP, SGMAP) havingproperties (A), (B), and (C) following.

(A): NODEMAP is an isomorphism of (NODES, ARROW) with(NODES', ARROW').(B) SGMAP is an isomorphism of SGP with SGP'.(C): For every pair (Nx, N2) in ARROW,TRANSIT'(NODEMAP(N1), NODEMAP(N2)) =SGMAP(TRANSIT(Nt, N2)).The pair (NODEMAP, SGMAP) will be called an isomorphism of the

first graph with the second.

9.4.3 DEFINITION: The transformation networks (S, NODES, ARROW,SGP, TRANSIT, CONTENTS) and (S', NODES', ARROW, SGP',TRANSIT', CONTENTS') are isographic if the transformation graphs(NODES, ARROW, SGP, TRANSIT) and (NODES', ARROW, SGP', 799


TRANSIT') are isomorphic. If (NODEMAP, SGMAP) is an isomorphismof the first graph with the second, then that pair is an isography of the firstnetwork with the second.

9.4.4 EXAMPLE: Figure 9.5 shows some transformation networks, all ofwhich are isographic. (a), (b), and (c) will be familiar from our recent dis-cussion (in section 8.5) of Webern's Piano Variations.

Let us work out the isographies of the figure very formally. We can fix thenode/arrow system that underlies all the graphs and networks: NODES is atwo-element family; every pair of NODES is in the ARROW relation.

Graph (a) has as its SGP the group of operations on pitch classes thatcontains the identity E and the operation I = l£. (This is a group, since II =E.) Graph (a) has as its TRANSIT function the function TRANSIT^, Nx) =TRANSIT(N2, N2) = E; TRANSIT^, N2) = TRANSIT(N2, Nj) = I.Graph (b) has the same SGP and the same TRANSIT function as graph (a).So in this special case, graph (b) is in fact literally "the same as" graph (a).

Graph (c) however is "different" (though isomorphic). Its semigroupcomprises two operations on twelve-tone rows, not two operations on pitchclasses. The row-operations are E (which leaves any row alone) and I (whichinverts any row about the pitch class A). The TRANSIT function for graph (c)maps ARROW into this new semigroup of row-operations. The semigroup ofrow-operations, while "new," is isomorphic with the old semigroup of pitch-class operations, under the correspondence of pitch-class-operation E withrow-operation E and pitch-class-operation I with row-operation I. We cantake this map of the old semigroup into the new one as our formal SGMAP.And we can take as NODEMAP the identity map of our fixed NODES ontoitself. The (NODEMAP, SGMAP) is a formal isomorphism of graph (a) (orgraph (b)) with graph (c). Therefore (NODEMAP, SGMAP) is a formalisography of network (a) (or network (b)) with network (c).200

FIGURE 9.5


Networks (a) and (b) are derived from the opening of the second move-ment in Webern's piece. Suppose we played the music a semitone higher;then we could derive networks (d) and (e) of the figure instead. Here J isinversion-about-B)?. For graphs (d) and (e) the semigroup consists of theidentity operation E on pitch classes and the inversion-operation J on pitchclasses. The new semigroup is isomorphic with semigroup (a) under the mapSGMAP(E) = E; SGMAP(I) = J. Using this SGMAP and the identity mapon NODES as NODEM AP, we establish a formal isomorphism of graph (a)(or graph (b)) with graph (d) (or graph (e)). Networks (a) (or (b)) and (d) (or(e)) are thereby isographic.

The notions of isomorphism we have just been exploring can be extendedsuggestively to more general notions of "homomorphism."

9.5.1 DEFINITION: Given node/arrow systems (NODES, ARROW) and(NODES', ARROW), a mapping NODEMAP of NODES into NODES'is a homomorphism of the first system into the second if (NODEMAP^j),NODEMAP(N2)) is in the ARROW' relation whenever (N l5N2) is inthe ARROW relation. NODEMAP is a homomorphism onto if it mapsNODES onto NODES' in a special way: Whenever N't and N'2 are in theARROW relation, there exist Nt and N2 in the ARROW relation such thatN\ = NODEMAP(Ni) and N'2 = NODEMAP(N2). A homomorphismNODEMAP is 1-to-l as a homomorphism between systems if it is 1-to-l as amap of NODES into NODES'.

Under these definitions, a 1-to-l homomorphism of one system ontoanother is an isomorphism in the sense of 9.4.1, and an isomorphism in thatsense is a 1-to-l homomorphism of the first system onto the second. Here, thespecial definition of "homomorphism onto" in 9.5.1 above is crucial. It ispossible for NODEMAP to be a homomorphism of (NODES, ARROW) into(NODES', ARROW) and also a 1-to-l map of the family NODES onto thefamily NODES', without being an isomorphism of the two systems. That is sobecause the second system may "have more arrows." For example we couldsimply take NODES' = NODES and add more arrows to ARROW forARROW'. Then the identity map of NODES into NODES' is a homomor-phism of (NODES, ARROW) into (NODES', ARROW') and also a 1-to-lmap of NODES onto NODES', but it is clearly not an isomorphism of thetwo systems. It is not a homomorphism of the first system onto the second, inthe full sense of 9.5.1.

9.5.2 DEFINITION: Given transformation graphs (NODES, ARROW, SGP,TRANSIT) and (NODES', ARROW', SGP', TRANSIT'), a homomorphismof the first graph into/onto the second is a pair (NODEMAP, SGMAP)

having features (A), (B), and (C) below. 207


(A): NODEMAP is a homomorphism of the node/arrow system(NODES, ARROW) into/onto the system (NODES', ARROW).(B): SGMAP is a homomorphism of the semigroup SGP into/onto thesemigroup SGP'.(C): For every pair of nodes (N l5N2) in the ARROW relation,TRANSIT'(NODEMAP(N1), NODEMAP(N2)) =SGMAP(TRANSIT(N15 N2)).The graph homomorphism (NODEMAP, SGMAP) is defined to be 1-to-

1 if both NODEMAP and SGMAP are 1-to-l maps, of NODES into NODES'and SGP into SGP' respectively. According to these definitions, an isomor-phism of the first graph with the second (in the sense of 9.4.2 earlier) isprecisely a 1-to-l homomorphism of the first graph onto the second.

9.5.3 EXAMPLE: Earlier (section 7.3), we studied graphs (a) and (b) of figure9.6 in connection with Brahms's Horn Trio. We called graph (a) the "comple-mentary gesture," and graph (b) the "complementary gesture times 2." Theintervallic augmentation that transforms graph (a) into graph (b) is in fact aformal homomorphism.

To verify this, let us begin by attaching the name (NODES, ARROW) tothe three-node node/arrow system common for both graphs. Take NODEMAPto be the identity map on NODES. NODEMAP is then, trivially, an isomor-phism of the node/arrow systems involved for the two graphs.

Take SGPa, the semigroup for graph (a), to be the group of the twelvechromatic pitch-class intervals. Take SGPb, the semigroup for graph (b), to be(provisionally) the same group. The values of TRANSITa and TRANSIT,, areas indicated on graphs (a) and (b) of the figure. Take SGMAP to be themapping of the interval i into the interval 2i, a map that transforms SGPa

into SGPb. That is, take SGMAP(i) = 2i for each interval i. Then SGMAPis a homomorphism of SGPa into SGPb: SGMAP (i + j) = SGMAP(i) +SGMAP(j) (mod 12). As defined, SGMAP is neither 1-to-l nor onto.

Requirements (A) and (B) for Definition 9.5.2 are now verified, as regardsa potential homomorphism of graph (a) into graph (b). It remains to verifyrequirement (C) of the definition. This is easily done by inspecting the

FIGURE 9.6

202


numbers labeling the three arrows on each graph: If interval i labels anyarrow on graph (a), then interval 2i labels the corresponding arrow on graph(b). That is, if i = TRANSIT^, N2), then 2i = TRANSIT'(N^Nj). Or,yet more formally, TRANSIT'(NODEMAPCNO, NODEMAP(N2)) =TRANSIT'(N15 N2) = 2i = SGMAP(i) = SGMAP(TRANSIT(Nl5 N2)), asdemanded by requirement (C) of the definition. The requirement is also satis-fied in case N2 = Nt = N: TRANSIT'(NODEMAP(N), NODEMAP(N)) =TRANSIT'(N, N) = 0 = 2-0 = SGMAP(O) = SGM AP(TRANSIT(N, N)).The case N! = N2 = N is trivial here, as it always will be when the semigroupsinvolved are groups; when the semigroups are not groups, TRANSIT'(N', N')and TRANSIT(N, N) will be idempotents within semigroups that may havemany idempotents, and the requirement of the definition is not triviallysatisfied.

To make our graph homomorphism here a "homomorphism onto," weneed only redefine SGPb as the group of all even intervals. Then SGMAP takesSGPa onto SGPb, and the graph homomorphism thereby becomes "onto" inthe sense of 9.5.2. It is not, of course, 1-to-l. (SGMAP is still not 1-to-l.)

9.5.4 EXAMPLE: For this example we shall use the word "tritone" to mean acollection of two pitch-classes spanning that interval. There are six tritones inthat sense: (C, F#), (C#, G), (D, Ab), (E[>, A), (E, B[?), and (F, B). Transposinga tritone by a pitch-class interval i has the same effect on the set as transposingit by interval i + 6. So for instance transposing (C,F#) by 5 yields (F, B);transposing (C, F#) by 11 also yields the unordered set (F, B). Accordingly, wecan define six formal "transposition operations" on the family of tritones. Weshall call the six operations O-or-6, l-or-7,2-or-8,3-or-9,4-or-10, and 5-or-l 1.The six operations form a simply transitive group on the family of tritones. Wecan therefore construct a GIS having the tritones for its objects and the sixoperations for its formal intervals. We write "int((C, F#), (F, B)) = 5-or-l 1,"and so on.

Consider the map SGMAP that takes pitch-class interval i into tritone-interval i-or-(i + 6). This map is a homomorphism from the group of pitch-class intervals, onto the group of tritone intervals. That is, if we transpose agiven tritone by i-or-(i + 6), and then transpose the resulting tritone by j-or-(j + 6), we shall end up having transposed the original tritone by (i + j)-or-(i + j + 6), all mod 12 of course.

SGMAP is part of a homomorphism that transforms the graph ofnetwork (a), in figure 9.7, onto the graph of network (b).

We imagine arrows labeled "0" from each node of (a) to itself, and arrowslabeled "O-or-6" from each node of (b) to itself. The NODEMAP for thegraph homomorphism takes the two top nodes of (a) into the top node of (b),and the two bottom nodes of (a) into the bottom node of (b). The CONTENTSof various nodes on networks(a) and (b) in the figure help us see why 205


NODEMAP is a musically plausible function here. But the homomorphism ofthe graphs does not depend formally upon the CONTENTS with which thenodes are filled in the networks. The graphs as such know nothing of thesecontents; they know only the node/arrow systems, the semigroups involved inlabeling the arrows, and the TRANSIT functions that provide those labels.

9.5.5 EXAMPLE: Figure 9.8(a) transcribes the opening phrase from the firstexample in the Scholica Enchiriadis that shows the Symphony of the Diates-seron.2 (b) of the figure graphs the melody "Nos qui vivimus"; the numbersmeasure steps up or down in the mode, (c) of the figure is a network whosenode/arrow system is disconnected; the network exhibits the vocal lines ofPrincipalis and Organalis separately. Graph (b) is a homomorphic image ofgraph (c). The homomorphism works as follows: NODEMAP takes the firstPrincipalis node of (c) and the first Organalis node of (c) both into the firstnode of (b); NODEMAP takes the second Principalis node and the secondOrganalis node of (c) both into the second node of (b); and so on; SGMAP isthe identity map of SGP onto itself, where SGP is the pertinent group ofintervals, that is the group of distances in steps up a scale.

Graph (b) is however not a homomorphic image of graph (d) under theanalogous NODEMAP. For there is no possible SGMAP, mapping our inter-vals homomorphically into themselves, that satisfies both SGMAP(l) = 1 andSGMAP(3) = 0. Any homomorphism SGMAP that satisfies SGMAP(l) =1 must satisfy SGMAP(3) = SGMAP(1 + 1 + 1) = SGMAP(l) +SGMAP(l) 4- SGMAP(l) = 1 + 1 + 1=3 .

The graph of (d) can be constructed as a formal "product" of graph (b)with graph (e). But that is quite another matter. Network (d) must be distin-

2. The transcription is taken from Oliver Strunk, Source Readings in Music History (NewYork: W. W. Norton, 1950), p. 130.

FIGURE 9.7

204


FIGURE 9.8 205


guished not only from (c) but also from (f) and (g). (f) is a network whosegraph is (b); each node of (f) contains a network whose graph is (e). (f) is thusa network-of-networks; the arrows on (f) labeled 1, — 1, and 0 transposeentire (e)-networks. (f) models the thought, "We are singing (the graph of)'Nos qui vivimus,' singing diatessera ((e)-networks) as we go."

(g) is a network whose graph is (e); each node of (g) contains a networkwhose graph is (b). (g) is thus, like (f), a network-of-networks. It reflects a wayin which Organalis might think: "Principalis is singing 'Nos qui vivimus'; I tooam singing 'Nos qui vivimus,' and my relation to Principalis is governed by theSymphony of the Diatesseron (the symbol '3' on the graph)."

Our ability to form the "product network" of (d), and the networks-of-networks displayed in (f) and (g), is heavily dependent on the followingaspects of the situation: The transformations involved in (b), that is T15 T_ l 5

and T0, are all operations; also the transformation T3 involved in (e) is anoperation; also T3 commutes with T l9 with T_15 and with T0. We shall notpursue the abstract theory of such matters any farther here.3

Earlier, in connection with the vocal line of the song "Angst und Hoffen"(figure 6.4), and again in connection with our study of twelve-tone rows asfamilies of "protocol pairs," (section 6.2.4), we discussed various techniquesfor modeling series of pitches, pitch classes, or other objects. Figure 9.8(b)suggests yet another technique: We can regard a series as a certain type oftransformation-network. Just what type is a matter we shall clarify and makeformal later on (in section 9.7.7). The terminology we shall develop therewill tell us that a network can model a series if the node/arrow system is"precedence-ordered and linearly ordered under that ordering."

The three examples we have just studied, 9.5.3, 9.5.4, and 9.5.5, show usthat there is a lot of variety in the forms that graph-homomorphisms canassume. In 9.5.3, NODEMAP was an isomorphism of the node/arrow systemand SGMAP was a proper homomorphism of the semigroup, i.e. not anisomorphism. In 9.5.4, NODEMAP was onto but not 1-to-l, while SGMAPwas again a proper homomorphism. In 9.5.5, NODEMAP was not 1-to-l,while SGMAP was an isomorphism.

We now turn away from isographies, isomorphisms, and homomor-phisms, to explore some different matters. When we use a transformation-

3. T3, a gesture which we can call "climb three rungs up on the modal ladder," correspondsto the concept of dia + tesseron, given the difference in the manner of counting rungs. T3 is not thesame gesture as RISE(4/3), meaning "rise so as to get higher in the harmonic pitch-ratio of 4-to-3." RISE(4/3) does not commute with the transformations TI and T_j of the example. That is ofcourse a salient problem of the style. And that is how our machinery views the problem. Ourmachinery also provides us with the formally different models of (d), (f), and (g), which give usinterestingly different ways of thinking about what is going on in the Symphony.206


network to discuss events in a musical passage, the configuration of itsnode/arrow system may allow us to isolate and discuss formal properties ofcertain nodes that implicitly assert corresponding formal functions for theCONTENTS of those nodes. In this connection, it is interesting to study"input nodes" and "output nodes" for node/arrow systems, and hence forgraphs and networks that involve those systems.

9.6.1 DEFINITION: An input node for a node/arrow system is a node IN towhich no proper arrows point. That is, if (N, IN) is in the ARROW relation,we must have N = IN. Analogously, an output node is a node OUT fromwhich no proper arrows issue. That is, if (OUT, N) is in the ARROW relation,we must have N = OUT.

9.6.2 EXAMPLE: In the network of figure 9.9, the node on the left is an inputnode and the node on the right is an output node.

FIGURE 9.10

FIGURE 9.9

The reader will recognize the graph as the "complementary gesture" fromthe Brahms Horn Trio. In discussing figure 7.7(g) earlier, we noted how themotif of the Bb octave-leap, when led into the complementary gesture,generates the pitch class G[?. And we explored some consequences of thatgeneration. Here we can note that our intuitions about "putting in" Bj? and"getting out" G[? correspond nicely to the formal input and output functionsof the left hand and right hand nodes on figure 9.9.

9.6.3 EXAMPLE: Figure 9.10 shows a network of Klangs whose graph we

207


earlier called CADENCE, when we were studying passages from Beethoven'sFirst Symphony (in section 7.4).

The Klang (C, +) that fills the left node of the figure is the same asthe Klang that fills the right node. But the function of the Klang asCONTENTS(left node) is different from '^function as CONTENTS (rightnode). In the former capacity, the Klang is an input; in the latter capacity it isan output.

The input and output functions for (C, +) in the network reflect verywell in this setting two of the three principal ideas about tonicity that havegoverned most theories of tonality since the eighteenth century. The inputfunction reflects the idea of tonic-as-generator, a tonic that asserts itself in thevery act of sounding a tone, setting a musical process in action, a tonic whichgenerates other tones through that action. The output function reflects theidea of tonic-as-goal, a tonic that appears at the end of a completed gesture asa point of repose towards which events have been moving. The third principalidea asserts as tonic a center of balance in a well-balanced structure. That idea,too, is manifest in the visual aspect of figure 9.10: The figure balances aboutthe two nodes containing (C, +).

We cannot completely translate out input/output formalities into ideasabout tonicity. For instance we do not want to assert (G, +) or (F, +) onfigure 9.10 as "tonics," even though the former Klang fills an input node andthe latter Klang an output node. Nevertheless the input/output formalities aresuggestive in connection with tonal theory.

9.6.4 EXAMPLE: Figure 9.11 shows a network whose nodes are filled by formsof the FATE motive from the opening of Die Walkiire, act 2, scene 4, the"Todesverkiindigung." We discussed this network in connection with figure8.6 earlier.

The horizontal arrows on the figure are labeled by the TRANSIT-operation RICH; the curved arrows are labeled by TCH. The transformation

FIGURE9.il208

9.6.4


BIND which labels the diagonal arrows takes a series of pitch classes andtransforms it into that one of its retrograde-inverted forms which has the samefirst and last notes. BIND commutes with TCH.

Figure 9.11 has precisely one input node, the node indicated at the lowerleft. The input node is filled by the FATE-form A-C-B, which therebyacquires a special generative function for the network. This formal status ofA-C-B as network-generator corresponds very well with the musical prioritywe assigned the motive-form in our earlier analysis. There we noted how theentire opening of the "Todesverkiindigung" grows out of the measures im-mediately preceding the scene, measures in which the A-B gesture of theLOVE motive is harmonized with C-B in the bass, A functioning also asfundamental bass below C and hence implicitly asserting a relation of A-before-C. The three protocol pairs (A, B), (C, B), and (A, C) determine themotive A-C-B as a partial ordering. Dramatically, the (A, B), (C, B) and(A, C) protocols in the music just before scene 4 represent Siegmund's relationto the sleeping Sieglinde, a relation which metaphorically lies under the entireSiegmund/Brunnhilde scene just as Sieglinde lies under it literally.

The input and output functions we have been exploring are essentiallyrhythmic aspects of node/arrow systems, in a certain structural sense: Inputnodes "happen before" other nodes with which they communicate; outputnodes "happen after" others with which they communicate. More generally,one observes that the arrows of any node/arrow system have a formalrhythmic structure of their own, a structure which can engage musical rhythmin varied and sometimes complicated ways. Our practice of laying out graphsvisually so that most arrows go from left to right on the page has made it easyfor us to put off investigating the issues that arise when we try to match theinternal arrow-flows of a network with the temporal flow of the music uponwhich the network comments. Here, now, we shall attempt to explore some ofthose issues, though we can hardly do them justice in one section of onechapter. We recall the definition of an "arrow chain" from node N to node N'in a node/arrow system; it is a finite series of nodes N0, N t , . . . , Nj such thatN0 = N, Nj = N', and (N^, N3) is in the ARROW relation for each jbetween 1 and J inclusive (9.1.4).

9.7.1 DEFINITION: An arrow chain (as above) is proper if there is at least onej between 1 and J inclusive such that (Njs N^) is not in the ARROW relation.

Intuitively, the definition demands at least one "one-way arrow" alongthe chain. (The implicit arrow between any node and itself counts as "two-way" in this connection.) Another way of intuiting the definition is to think ofa proper arrow-chain as one that cannot be "walked backwards." 209


9.7.2 DEFINITION: In a node/arrow system, node N precedes node N', and N'follows N, if there exists some proper arrow-chain from N to N'.

One must be careful to distinguish the relation "N precedes N'" from therelation "N is in the ARROW relation to N'."

On figure 9.12, for instance, M t precedes M3 because M1-M2-M3 is anarrow-chain from M^ to M3 that involves a one-way arrow (from M2 to M3).But M! is not in the ARROW relation to M3. M t is, on the other hand, in theARROW relation to M2. But Mt does not precede M2: There is no arrow-chain from M! to M2 which involves any one-way arrow.

9.7.3 DEFINITION: A node/arrow system is precedence-ordered if there is nopair of nodes (N, N') such that N both precedes and follows N'.

The reason why we speak of a precedence-ordered system as "ordered" isimplicit in the following theorem.

9.7.4 THEOREM: Let (NODES, ARROW) be a precedence-ordered node/arrow system. Let PRECEDENCE be the family of node-pairs (Ni, N2) suchthat N: precedes N2. Then PRECEDENCE is a (strict) partial ordering onNODES. That is, PRECEDENCE satisfies conditions (PO1) and (PO2)below.

(PO1): There is no pair (N l9N2) such that both (N1,N2) and (N^NJare members of PRECEDENCE.(PO2): If (N! , N2) and (N2, N3) are both members of PRECEDENCE,then so is (N1?N3).Proof: (PO1) for PRECEDENCE is equivalent to the condition of 9.7.3,

which is true in a precedence-ordered system. (PO2) is obvious: If there is a"good" arrow-chain from Nx to N2 and a "good" arrow-chain from N2 toN3, then there will be a "good" arrow-chain from N! to N3. q.e.d.

The reader will recall (PO1) and (PO2) from section 6.2.4 earlier, wherewe invoked them to characterize collections of protocol-pairs that were (strict)partial orderings on the twelve pitch-classes. Here we use the same mathemat-ical conditions to characterize (strict) partial orderings on NODES.

A precedence-ordered system is at least potentially compatible with ournaive sense of chronology. When used for analytic purposes, that system will

FIGURE 9.12

270


not have to assert that one musical event both "precedes" and "follows"another, in the strictly formal sense of 9.7.2. There is nothing intrinsicallycorrect or good about avoiding such assertions, but it is useful to have at handa formal criterion that characterizes those particular node/arrow systemswhich enable us to avoid them.4

9.7.5 OPTIONAL: We can be quite precise mathematically about what wemean in saying that a precedence-ordered system is "potentially compatiblewith our naive sense of chronology." Readers who do not care can move onfrom here to section 9.7.6.

Given the family of NODES, let us review what we mean when we speakof a (strict) partial ordering on NODES. We mean a collection P of node-pairs, a collection that satisfies conditions (PO1) and (PO2) following. (PO1):There is no pair (Nt, N2) such that both (Nt, N2) and (N2, NJ are membersof P. (PO2): If (N l5N2) and (N2,N3) are both members or P, then so is(Ni, N3). Theorem 9.7.4 told us that P = PRECEDENCE is a (strict) partialordering on NODES in a precedence-ordered node/arrow system.

A partial ordering L on NODES is "linear" or "simple" when, given anydistinct nodes N and N', either (N, N') or (N', N) is a member of L.

The following theorem can be proved: If NODES is finite, containing Jnodes, and if L is a linear ordering on NODES, then the members of NODEScan be arranged in a series Nj,, N2 , . . . , N, such that (NJ5 Nk) is a member of Lif and only if j is less than k.

The partial ordering P is "weaker than" the partial ordering Q, and Q is"stronger than" P, when every node-pair that is a member of P is a member ofQ, and Q contains some pair that P does not contain. That is, this relationbetween P and Q obtains when P is strictly included in Q as a set of node-pairs.A partial ordering P is "maximally strong" when there is no partial ordering Qstronger than P.

The following theorem can be proved: Every maximally strong partialordering is linear, and every linear ordering is maximally strong. WhenNODES is finite, the following theorem can also be proved: Given any partialordering P, there exists some maximally strong (i.e. linear) ordering L whichis either equal to P or stronger than P. When NODES is not finite, the sametheorem can be proved if one makes an additional logical assumption whichneed not concern us here.

We can apply all these theorems as follows: Given a precedence-orderednode/arrow system, there exists a linear ordering L of NODES which is either

4. Jonathan Kramer develops a very interesting sense in which he claims that the firstmovement of Beethoven's F-major String Quartet op. 135 begins with its ending. Kramer doesthis in the article, "Multiple and Non-Linear Time in Beethoven's Opus 135," Perspectives of NewMusic vol. 11, no. 2 (Spring-Summer 1973), 122-45. The last movement of Haydn's D-majorString Quartet op. 76, no. 5, might serve as another example. 277


stronger than PRECEDENCE or equal to PRECEDENCE. Assuming thatNODES is finite (which we shall assume from now on), we can then L-orderthe notes of this precedence-ordered system as a series Nj, N2, ... , N;, in suchwise that (N, Nk) is a pair within L if and only if j is less than k. SincePRECEDENCE is weaker than or equal to L, it follows that j must be lessthan k whenever N. precedes Nk in the system. So if we imagine a chronologyin which Nj "happens first," N2 "happens second, ". . . , and N; "happenslast," this chronology cannot be violated by the precedence relation of thesystem. That is, whenever N. formally precedes Nk, N. will "happen before"Nk in the L-chronology.

If PRECEDENCE is not itself linear, there will be more than one linearordering stronger than PRECEDENCE; accordingly there will be more thanone "linear chronology" of the above sort with which the precedence relationis compatible.

For example, in the precedence-ordered system of figure 9.13 we can takeeither of the two left-hand nodes as "N," and the other one as "N2," inimposing a linear chronology; we can similarly take either of the two right-hand nodes as "N3" and the other as "N4." This reflects the structure ofPRECEDENCE here, which makes each left-hand node precede each right-hand node, while neither left-hand node precedes the other and neither right-hand node precedes the other. The groupings of segments within the variouspossible linear chronologies are typical.

9.7.6 The gist of section 9.7.5 may be summarized as follows: When a finitenode/arrow system is precedence-ordered, its J nodes can be labeled by the num-bers 1 through J in such fashion that when j is less than k, it is possible forthe j* node N. to precede the k* node Nk, but impossible for Nk to precede N.This means that we can always display the system visually on a page (intheory) using a format in which all one-way arrows go from left to right.

We must be very careful to recognize that the words "precedence" and"precede" in the paragraph above refer to formal aspects of the node/arrowconfiguration, and not necessarily to the musical chronology of any passageupon which a network using that node/arrow system may be commenting.Even when the node/arrow system is precedence-ordered, it is perfectly pos-

FIGURE 9.13

212


sible for a node N to precede a node N' in a network, while the contents of Nare heard after the contents of N' in the pertinent music. Figure 9.14 will helpus explore the possibility.

FIGURE 9.14

(a) of the figure shows a network of Klangs. All the Klangs are under-stood to be major, except for the lower-case e|?-Klang, which is minor. Thenetwork models the harmonic progression at the opening of the slow move-ment in Beethoven's Appassionato Sonata. The Ej?-major Klang is bracketedto indicate that the Klang is not actually sounded but is theoretically under-stood. The fourth sonority heard in the music is modeled by two Klangs. It isfirst understood as a G(?-major Klang (with added sixth); then it is understoodas an eb-minor Klang (with minor seventh, inverted). This is Rameau's doubleemploi. The arrow goes only one way, from G[? to e(? but not back. Theoperation REL takes a Klang into its relative minor/major.

The left-to-right format of (a) arranges the nodes in an order thatcorresponds to the order of events in the music. This order is not compatiblewith the precedence relation of the node/arrow system, even though that 273

9.7.6


system is precedence-ordered. We see this incompatibility on (a) in the form ofsome one-way arrows that point from right to left, (b) of the figure rearrangesthe nodes on the page, in a new visual format compatible with the precedence-ordering; on (b), all one-way arrows point from left to right. We shall discusslater the box containing the word START and the arrow issuing from thatbox; the reader should ignore them for the time being.

The left-to-right format of (b), while respecting the one-way arrows,violates the musical chronology of the passage. A special aspect of thisviolation is the clarity with which it accents the input function of the two Gbnodes. The input function violates musical chronology: Db, not Gb, is whatgets this music under way. The input function also violates our sense ofstructural "priority": Db, not Gb, is the Klang structurally prior to all othershere. The Gb Klangs should be manifest as inflecting such a Db "point ofdeparture" for the tonal structure; the Gb Klangs should not themselvesappear as structural "points of departure," which figure 9.14(b) seems tomake them. These concerns are very well addressed by Schenkerian theory,which provides us with an apparatus of hierarchical levels, voice-leadingevents, and subordinate Klangs harmonizing voice-leading events. With sucha model in mind, we can easily see that the configuration of Klangs displayedin figure 9.14((a) or (b)) is not an adequate representation for the way tonalitycontrols this passage. The representation fails to model the middlegroundprogression Db-Ab-Db supporting a sustained melodic fifth degree; it failsto model the inflection of that fifth degree by its upper neighbor twice, oncewithin the opening Db Klang of the middleground and once on the way fromDb to Ab Klangs within the middleground; it fails to show how the Gb Klangssupport that neighboring inflection of the principal tone 5. Later on, we shallconstruct a "Schenkerian network" that can address these matters within ournetwork format. Meanwhile, we can observe that figure 9.14, incomplete as itis for analytic purposes, still does represent a foreground configu-ration of Klangs that engages a valid part of our musical experience. It wouldimpoverish, not refine, that experience to explain away the peripatetics of theconfiguration as "merely" incidental, as "naught but" the harmonization ofvoice leading, as "only" preparing the eventual dominant, and the like.

More specifically, even though the formal Gb inputs on figure 9.14(b)must yield somehow to overall D(? priority in a complete analysis, the formalGb inputs still reflect an interesting feature of the music, a feature it would beeasy to overlook if one were to plow through the foreground to a middle-ground level prematurely. One hears this feature upon singing over the musicmentally while looking at network-format (b). The input Gb nodes are thensensed strongly as "carriage returns," especially the second one. This feelingof "carriage return" on the Gb harmonies, sensed as one reads figure 9.14(b)in the musical chronology, interacts very effectively with the phrasing of thepassage.

9.7.6

214


It would be hard to express the carriage-return function so precisely inany other theoretical vocabulary. The function marks those precise momentsin the listening experience at which we shoot back from right to left on thefigure, violating the sense of the one-way arrows. More formally: Thecarriage-return moments are precisely those moments in the listening chro-nology at which that chronology violates precedence-ordering. At all othermoments, listening chronology is compatible with precedence-ordering. Byusing the expression "precedence-ordering," we are implicitly supposing thatthe node/arrow system to which we are applying this concept is precedence-ordered, with all that this entails about left-to-right, and so on. Our theoreticalmachinery enables us to pinpoint carriage-return moments, define them pre-cisely, and attribute a special theoretical function to them.

All this duly noted, we have still not resolved the theoretical problemsraised by the formal priority our model assigns to Gj? nodes over D[? nodes as"input" to the flow of events, and to the tonal structure. One line of attack onthe problems is suggested by the box on figure 9.14(b) containing the wordSTART, and by the arrow from that box to the indicated D[? node. We mayformally adjoin the box and the arrow to the node/arrow system, and thecontents START to the network, so as to help ourselves along. The STARTnode is an input node, and we can declare a formal convention that itsupersedes all other input nodes in function. When we start at the STARTnode, we cannot reach the G|? nodes without traversing some arrows back-wards. We might use just that feature of the system as a formal criterion forassigning a special sort of subordinate status to those nodes. There is noproblem walking arrows backwards here because our transformations are alloperations. Our analytic criterion for pointing the START arrow at theindicated D[? node could be diachronic (because the music starts there) arsynchronic (because that node begins the foreground elaboration of a higher-level tonic function).

Another formal device for modeling a discrepancy between precedence-ordering and musical chronology is indicated by figure 9.15. The figure

215

9.7.6

FIGURE 9.15


attaches to each node of figure 9.14(b) a certain time span; the musical eventcorresponding to the contents of that node occurs over that time span.Formally, we are constructing a new apparatus which we might call a"time-spanning network." That is a transformation network together with afunction TIMESPAN that maps each node into a certain time span. In thecase of a network whose contents are already time spans, we could takeTIMESPAN(N) = CONTENTS(N). A time-spanning network could modelvia TIMESPAN the exact time spans over which its events occur; that is thecase with figure 9.15. Or TIMESPAN(N) could model a certain range of timeduring which CONTENTS (N) might occur.

Instead of attaching time spans to the nodes of a network in this way,we could also attach time spans to the contents of those nodes. On figure9.15, for example, instead of a node N with CONTENTS(N) = Gb andTIMESPAN(N) = (3.5, .5), we could have a node N whose CONTENTSare the ordered pair (Gb,(3.5, .5)). On the revised figure 9.15, the familyof transformations would have to be more complicated. Each graph-transformation would be an ordered pair comprising both a Klang-transformation and a time-span-transformation.

Yet another formal device at our disposal is to incorporate Schenkeriantransformations into a network format. Figure 9.16 indicates one way inwhich this might be done.

The contents of the nodes in this network are ordered triples (Klang,degree, level). Thus (Ab, 5, 2) denotes an Ab-major Klang supporting a fifthdegree in the structural melodic voice at level 2. The operation PROJ + incre-

216 FIGURE 9.16

9.7.6


ments the level of its operand, transforming (Kng, deg, lev) into (Kng, deg,lev + 1). Klang and degree are thereby PROJected one level closer to theforeground. The operation PROJ— is the inverse of PROJ + ; it decrementsthe level of (Kng, deg, lev), transforming it into (Kng, deg, lev — 1). Sincethe objects (Kng, deg, lev) are purely formal, we can always reference levels"lev + 1" and "lev — 1" formally, even when they have no analytic pertinenceto a given situation. That is desirable here, in order to make PROJ-f andPROJ — well-behaved context-free operations. To save space on figure9.16, the PROJ arrows have all been drawn as two-way, signifying PROJ +or PROJ — as appropriate.

Within each level on the figure, each transformation is specified by a pair(Klangtrans, degtrans). Klangtrans is the pertinent Klang transformation,and degtrans is the pertinent degree transformation. Thus an arrow labeled(DOM, SUST) from (Kng, deg, lev) to (Kng', deg', lev) indicates that theKlang Kng is the dominant of the Klang Kng', while degree deg sustains tobecome degree deg' = deg. An arrow labeled (SUED, N +) from (Kng, deg,lev) to (Kng', deg', lev) indicates that Kng is the subdominant of Kng', whiledegree deg is the upper neighbor to degree deg'.

Distinguishing levels in the manner of figure 9.16 enables us to makeinput terminology conform better to our intuitions. The Gb nodes of figure9.16 are indeed still input nodes, but we can now say that they are "input atlevel 3. In the same sense, we can say that the node containing (Ab, 5,2) is"input at level 2," distinguishing it in this capacity from the node containing(Ab, 5,3). The Db nodes of level 2 are both output nodes at level 2; the Dbnodes of level 3 are all output nodes at level 3. The Db node of level 1 is bothinput and output at that level, according to our definition (9.6.1).

Figure 9.16 may be made to engage rhythmic mensuration by attachingtime spans to its nodes along one of the lines suggested earlier. Determiningwhere to end the time span for (Db, 5,2) and where to begin the time span for(A)?, 5,2) is an interesting methodological and phenomenological problem.5 anumber of assertions seem plausible. My own preference is to carry the Dbtime span right up to the Ab Klang, and also to begin the At? time span rightafter the Db Klang stops sounding. The two time spans would then overlap onlevel 2, and the Gb-eb-Eb part of level 3 would all occur during the time spanof the overlap. This satisfies my hearing, and it is also an elegant way toelaborate the theoretical idea behind the double emploi.

Figure 9.16 as it stands is not equivalent to a Schenkerian reading, whichwould devote less attention to Klangs that do not project Stufen, moreattention to the bass line and to the essential counterpoint between the outer

5. Fred Lerdahl and Ray Jackendoff discuss pertinent matters at length in their importantbook, A Generative Theory of Tonal Music (Cambridge, Mass, and London: MIT Press, 1983).The interested reader can explore various ways in which their tonal tree-structures resemble anddiffer from transformation-networks of the sort displayed by figure 9.16. 217

9.7.6


FIGURE 9.17

voices.6 I think it likely, though I am not certain, that actual Schenkeriangraphs could be represented in network formats of the sort under presentconsideration, when suitably extended.

9.7.7 A short time ago, in connection with our study of "Nos qui vivimus,"we observed that a series of objects could be modeled by a certain type oftransformation network. Now we are in a position to specify formally just

6. A good Schenkerian analysis of the theme as a whole is presented by Allen Forte andSteven E. Gilbert in their Introduction to Schenkerian Analysis (New York: W. W. Norton, 1982),154-56.

9.7.7

218


what type. Specifically, a formal "melody" can be defined as a transforma-tion network whose node/arrow system is precedence-ordered, and linearlyordered under that ordering. According to our work so far, that means therewill be one and only one way of labeling the J nodes with the numbers 1through J so as to be compatible with the one-way arrows of the system.

This concept of "melody" is very elaborate, for it carries within it the ideaof transforming earlier events to later ones, along the arrows of the network,by transformations from a specified semigroup. That idea was not implicit inour earlier models for series.

Different configurations of arrows can give rise to one and the sameprecedence-ordering on a given family of NODES. When that ordering islinear, corresponding networks whose nodes have the same contents willnevertheless be formally different "melodies" by our definition. We shouldfind a better word than "melody" if we want to continue to work along theselines. For instance, the two node/arrow systems of figure 9.17(a) and (b) giverise to one and the same precedence-ordering. Networks (a) and (b) areformally different "melodies."7

7. "Melody," however, is exactly the proper term if we want to follow and extend the usageof Ernst Kurth, when he claims that "the basis of melody is, in the psychological sense, not asuccession of tones... but rather the impetus of transition between the tones." Figures 9.17 (a) and(b) depict different transition-structures. The cited text appears in Grundlagen des linearenKontrapunkts (Bern: Drechsel, 1917), p. 2. ("Der Grundinhalt des Melodischen ist im psycho-logischen Sinne nicht eine Folge von Tonen... sondern das Moment des tfbergangs zwischen denTonen.") 279

9.7.7

Transformation Graphs andNetworks (4): Some FurtherAnalyses

220

10.1 EXAMPLE: Figure 10.1 sketches some motivic work from the passageopening the development section in the last movement of Mozart's G-MinorSymphony, K.550. Up to measure 133, the entire orchestra except horns isplaying the indicated line, allegro assai and essentially staccato. There is oneexception: The quarter note B which appears on the figure at the end ofmeasure 128 is actually an eighth rest and a sixteenth triplet, filling in the

FIGURE 10.1

10

diminished fourth scalewise under a slur. I am supposing that it is legitimatefor us to consider that gesture a variation on the motivic model of the figure,inter alia.

From the pickup of measure 127 until measure 133, the passage projectsforms of a pitch motive PM, comprising a diminished fourth up followed by adiminished seventh down. This motive is Rl-chained in the manner familiarto us by now; the transformation RICH takes us along the chain of PM-forms (E-Ab-B), (Ab-B-Eb), (B-Eb-Ffl), (Eb-Ffr-Bb), (FJ-Bb-Ctf),(Bb-Ctf-F), and (C#-F-G#). The TCH interval is a falling fourth. On figure10.1, the four prime forms of PM are indicated by the brackets numbered 1,2,3, and 4a/b. The form (C#-F-G#) is bracketed twice, by brackets 4a and 4b.This reflects an interesting ambiguity about its rhythmic location, an ambi-guity we shall soon investigate.

Beneath the staff on the figure a series of numerical durations appears.The numbers label the distances in quarter-note beats between the time pointsat which successive notes are attacked. I am supposing that we hear an ictus atthe barline of measure 128. (Later on, we shall hear how the music of measures125-27 prepares this ictus.) The barline of measure 128 thereby articulates theduration of five quarters, between the two B naturals, into (2 + 3) quarters.

The three attacks within PM, together with the ictus, define the dura-tional series 1 + 2 + 2 (quarters) as a rhythmic setting for the pitch idea. Weshall call 1 + 2 + 2 the "durational motive" DM. On the figure, bracket 1 isplaced around PM so as to articulate DM; the corresponding duration-numbers 1, 2, and 2 below the staff are also bracketed.

Bracket 2 also articulates DM, now as the rhythmic setting for the nextTCH-form of PM. Bracket 3 is placed around the next TCH-form of the pitchmotive, that is around F#-Bb-C#. Here the rhythmic setting is no longer DMitself, but rather an augmented (rhythmically transposed) form of DM: 2 +4 + 4. The rhythmic transposition will be denoted as T2, multiplying alldurations by 2.

The next TCH stage of the pitch motive is C#-F-G#. Bracket 4a givesthis stage the durational setting 4 + 4 + 2; bracket 4b gives it the setting 4 +2 + 2. Both these durational settings are serial transformations of DM. Theseries of 4a retrogrades the elements of series 3: 4 + 4 + 2 retrogrades 2 +4 + 4. The series of 4b inverts the elements of series 3: 4 + 2 + 2 inverts 2 +4 + 4. We can regard the inversion as multiplicative, about the numericalproduct 8: 8 divided by 2,4, and 4 (series 3) yields 4,2, and 2 (series 4b). Or wecan regard the inversion as additive, about the numerical sum 6:6 take away 2,4, and 4 (series 3) yields 4,2, and 2 (series 4b). Whichever way we think of theinversion, durational series 4b is a retrograde-inversion of durational series4a. In fact, series 4b is precisely RICH(series 4a).

Thus the transformational motif of Rl-chaining, very audible in the pitchstructure of the passage, is also projected in the durational structure. The 227

10.1Transformation Graphs and Networks (4)


musical effect is bewildering on first hearing, because both the durationalseries involved in the RICH relation, that is both 4a and 4b, are heard asalternative rhythmic settings for one and the same pitch motive. And thiseffect strikes our ears just as we are beginning to adapt to a heavy reliance onmotivic listening, having temporarily lost the local tonic.

Bracket 5 shows how the play of durational motives continues onthrough the change of texture at measure 133. The durational series forbracket 5, 2 + 2 + 4, is a retrograde-inversion (multiplicative or additive) ofseries 3, just as series 4b was a retrograde-inversion of series 4a. Since series3 was a multiplicative transposition of DM, durational series 5 is also amultiplicative retrograde-inversion of DM itself. In fact series 5 is preciselythe RICH-transform of DM, using multiplicative inversion: (multiplicative)RICH(1 +2 + 2 ) - 2 + 2 +4.

FIGURE 10.2

Figure 10.2 summarizes in a network the transformational interrelationsof the durational motive-forms so far surveyed. On the figure, T is multiplica-tive T2,1 is the pertinent multiplicative inversion, and RICH is considered tobe defined multiplicatively also.

The RICH-arrow between the node marked "DM = 1; 2" and the nodemarked "5" suggests some aural explorations. Going back to figure 10.1, wecan hear the rhythmic RICH-relation between bracket 1 and bracket 5 byfocusing on a linking rhythmic element, that is, the rhythmic identity ofmeasure 127 (at the end of bracket 1) with measure 132 (at the beginning ofbracket 5). This is exactly the linking aspect of the Rl-chain involved. We hearthat the high Ab of measure 127 and the low G# of measure 132 are theregistral boundaries for figure 10.1. The enharmonically equivalent A(? andG# are tied together, too, by the diminished-seventh harmony implied overmeasure 127; this harmony implicitly recurs under bracket 5.

We can hear the rhythmic RICH-relation between bracket 2 and bracket5 by focusing on measures 129-30 as an intermediating stage. That pair ofmeasures spans the same durational series as series 5: 2 + 2 + 4. And in

10.1

222


measures 129-30 the series 2 + 2 + 4 is explicitly linked to the end of bracket2, as the RICH-transform of series 2. This rhythmic RICH-transformation iseasy to pick up aurally because it coincides with a RICH-transformation ofthe pitch motives that correspond: B-E|?-F#, with rhythm 1 + 2 + 2, istransformed into RICH(B-Eb-F#) = Eb~F#-Bb, with rhythm RICH(1 +2 + 2) = 2 + 2 + 4. It remains then only to hear that measures 132-33, underbracket 5, reproduce the attack rhythm of measures 129-30, the intermediat-ing stage. And this is quite possible if one hears measures 127-29 and mea-sures 130-32 as a pair of three-measure groups, or if one hears measures129-31 and 132-34 as such a pair.

Let us return now to brackets 6 and 7 on figure 10.1. Bracket 6 applies tothe winds only, whose durational series is 4 + 2 -I- 2 over this span. The seriesrecapitulates the series of 4b. Despite the radical change of texture, we areaided in hearing this relation by the boundary tones of measures 133-35,specifically by the opening high F of measure 133, the final C# at measure 135,and the low G# at the end of measure 134. Those three notes keep alive inpermuted order the pitch classes C#, F, and G# from the PM-form ofmeasures 131-32, spanned by bracket 4b.

Bracket 7 shows how the rhythm of the winds during bracket 6 becomesdiminuted approaching the barline of measure 135, into the diminuted series2 + 1 + 1. This diminution (multiplicative transposition by ̂ ) undoes the effectof the earlier augmentation (transposition by 2), the augmentation we under-went in passing from bracket 2 to bracket 3. Accordingly, durational series 7bears to series 1 = 2 = DM the same relation that series 6 = 4b, the augmen-tation of series 7, bore to series 3, the augmentation of series 1 = 2 = DM.Thus series 7 inverts series 1 either multiplicatively or additively: 2 divided by1,2, and 2 (series 1) yields 2,1, and 1 (series 7); alternatively, 3 take away 1,2,and 2 also yields 2, 1, and 1.

Figure 10.3 extends figure 10.2 to display some of the new relationshipsinvolving series 6 and series 7. J is the pertinent multiplicative inversion \\.

Our study has shown us how the straightforward Rl-chaining in onedimension of this music both suggests and conceals a very elaborate transfor-mational network, also involving RICH-relations, in another dimension. Thewhole discussion is grossly oversimplified as regards both dimensions. Wehave not, for instance, considered the rhythmic implications of the PM-formthat goes from the G# of measure 132 to the F of measure 133 in the bassoon(an octave below figure 10.1), and thence through the stepwise filler down tothe C# of measure 135 (again an octave below the figure). This form retro-grades in register the PM-form of measures 131-32. The stepwise filling-in ofthe diminished fourth F-to-C# is a huge rhythmic expansion, in retrograde, ofthe tiny sixteenth-note triplet figure that filled in from B to Eb in measures128-29. We have also not considered the crucial way in which the music getsto figure 10.1. The Rl-chaining of PM actually begins earlier; it involves the 223

10.1


all-important high D|? of the rocket theme, the main theme of the movementand the source of the diminished-seventh leaps. Figure 10.4 helps us explorethis a bit.

FIGURE 10.4

The beamed arrows on the figure pick out a PM-form that extends the RIchain of pitch motives one TCH stage backwards from our earlier analysis.The rhythmic setting of the PM-form A-D[?-E is not related to DM by anystandard serial transformation. It does, however, interact with the ictus at thebarline of measure 128. Figure 10.4 allows us to hear the source of the 3quarters' duration between the ictus and the following B. That duration is inrhythmic sequence with the 3 quarters' duration between the high D[? and theE natural of measure 126. The 3 quarters' sequence causes the rhythmicidentity of bracket 1 with bracket 2 to extend backwards. Not only do E-AJ7-B-ictus and B-Eb-Fft-Bb demarcate the same durational series DM =1 + 2 + 2, but also Dj?-E-A[7-B-ictus and ictus-B-Eb-F#-Bb demarcate

10.1

FIGURE 10.3

224


the same series 3 + 1 + 2 + 2. In fact, the rhythmic identity extends backeven farther: A-Db-E-Ab-B-ictus and B-ictus-B-Eb-F#-Bb demarcatethe same series 2 + 3 + 1+2 + 2. Only after the Bb of measure 130 does therhythmic parallelism break off: After that Bb the first duration of 4 appears,launching us into the transformational complexities of figure 10.2.

10.2 EXAMPLE: Figure 10.5 sketches some structural aspects of Bartok's"Syncopation," no. 133 from Mikrokosmos, vol. 5. Our ability to hear pro-longation in this sort of context is the subject of an important study by RoyTravis.1 Travis tries to hear (GBD) + (Eb F#) as a tonic chord for the piece. Itworks better to hear (G) + (A#C#D#F#) in that capacity. This is the finalsonority of the piece and the only complete harmony attacked by both handssimultaneously, an event which occurs at the final barline. That is, it is the onlyharmony which does not involve a "Syncopation." The syncopations of themusic bring out a broad variety of structural contrasts between the two hands.The left hand plays white notes; the right hand plays black notes. The left handplays "down" both metrically and sotto; the right hand plays "up" bothmetrically and sopra (with its black notes). The white notes in the left handarrange themselves into triadic formations within a G Mixolydian mode; theblack notes in the right hand arrange themselves within a pentatonic F# mode.The left hand's chord of reference is a G-major |, a chord with G on thebottom; the right hand's chord of reference is an F# added-sixth harmony in 3inversion, a chord with F# on the top.

Over measures 1-10 of figure 10.5, one sees how these elements contendduring the opening section of the piece. The black notes win out. The opening(D#F#) of the figure is spelled (EbF#) by the composer, to emphasize its localdependency on the left-hand G tonic. But as the section progresses, (EbF#)becomes (EbGb) and shows that it can move down an octave to displace the

1. "Toward a New Concept of Tonality?" Journal of Music Theory, vol. 3, no. 2 (November1959), 257-84. 225

FIGURE 10.5

70.2


left-hand G at measure 9. The black dyad celebrates the success of its incursionby bobbing triumphantly back up to its original register, with a crescendo tosf; now the composer spells it (D#F#), reflecting its forthcoming orien-tation towards the right-hand tonic F#.

The next section of the piece, measures 11-25, shows (D#F#) continuingto exercise its registral mobility: It moves up an octave over that span of thepiece, along with C# and eventually A#. The octave transfer, represented by aslur on figure 10.5, plows through the climactic D of measure 18. This D, asfifth of G, shows that left-hand, G-oriented material can also become regis-trally mobile. Ds and Gs continue to move downwards in register from measure25 to the end, while the right-hand chord of measure 25 moves up anotheroctave during that time. Bartok puts (D#F#) back into its original spelling, as(E|?F#), from measure 25 on. In the right-hand scheme of things, as displayedon figure 10.5, the climactic D5 of measure 18 is apparently "only" a chroma-tic passing tone. But D5 is also the pitch about which the sonority A#4, C#5,D#5, F#5 is inversionally symmetrical. That sonority, as it appears with astructural downbeat at measure 25, projects the chord-of-reference for theright hand. Also, the pitch class D is a center of inversion for both the black-note collection and the white-note collection.

In both the left hand and the right hand over measures 11-25, RI-chaining provides the means of harmonic progression. The left hand producesthe Rl-chain of registrally ordered trichords A-C-E, C-E-G, E-G-B,G-B-D. There is even a hint that this process continues on in register, as theright hand takes over the climactic D5 at measure 18 and leads it on up to F#5.But the black tonic F# is too clearly foreign to the white-note chain, for thishint to develop further. Indeed the left hand emphasizes just that point, atmeasure 25, by providing the white F natural (not the black F#) as a chordalthird above D, within the sonority G-B-D-F. The F natural breaks the RI-chain of the left hand, which cannot pass the color barrier.

Just as the Rl-chain of white trichords in the left hand appears beamedover measures 11-25 on figure 10.5, so do two analogous Rl-chains of blacktrichords in the right hand. The first chain breaks at F#-G#-B; rather thancontinuing on to the next RICH-stage G#-B-C#, the right hand substitutes adifferent form of the same trichord, namely G#-A$-C#, and then starts a newRl-chain therefrom. The color barrier is again involved: Once the white B hasbeen generated by the Rl-chain in the right hand, it must be replaced by theblack A# = Bj?. So, just when the Rl-chaining is about to carry F#-G#-B onto G#-B-C#, A# substitutes for B and the next trichord is G#-A#-C#instead. The substitution preserves the outer voices of the trichord involved,and also preserves its set-class. The replacement of B by A# recapitulates in asetting of F# tonicity the replacement of B by B[? that we heard earlier, atmeasure 3, in a setting of G tonicity.

The double Rl-chaining of the right hand over measures 11 -25 is also226

10.2


FIGURE 10.6

involved with a cadential proportioning (balance) of the trichord forms.Figure 10.6 displays this proportioning by a pertinent transformation-network.

The straight diagonal arrows are RICH transformations. Among itsinput and output chords, figure 10.6 manifests a classical "triple proportion":C#-D#-F# (input) is to Ftf-G#-B (output) as G#-A#-C# (input) is toC#-D#-F# (output). The triple proportion also involves an isography of theleft side, on figure 10.6, with the right side, on the same figure. That is, therelation of the material grouped by the two right-hand beams, over measures11-25 on figure 10.5, is an isography. Intervals of 5 span the trichords and(therefore) measure the TCH-transpositions. They make the triple proportionhere sound almost like I-IV-V-I, especially since the pitch class F#, appear-ing at the bottom of the first "I" and at the top of the final "I" in theI-IV-V-I, has a certain tonicity about it as regards the pitch structure of theright hand. Perhaps this sound led Travis to hear a functional I-IV-V-Igoverning the tonal structure of the piece as a whole.

10.3 EXAMPLE: Figure 10.7 sketches aspects of the opening seven measuresfrom the first of Prokofieff's Melodies op. 35.1 am indebted to Neil Minturnfor bringing this passage to my attention. The four-fiat signature on the figureis mine; Prokofieff writes no signature.

The harmony in the music is far from traditional. Yet it is diatonicenough, and the outer voices are diatonic enough, so that some harmonicevents stand out as "strange." Foremost among these are the cadence har-monies, E minor over the last half of measure 2 and Eb minor over the last halfof measure 5. At both these cadences we expect Eb-major harmony.

We can analyze the cadential substitutions by using the terminology ofKlangs and Klang-transformations. The network of figure 10.8 (a) does so,and also brings the D-major Klang of measure 6 into the picture: D major, theSLIDE transform of Eb minor, bears to the latter Klang the same relation 227

10.3


which E|j major, a Klang heard in the upper voices of measure 1 and expectedat measure 2^, bears to E minor, the Klang actually heard at measure 2j.

Figure 10.8 (a) does not attempt to engage the C-major and F-minortriads of measure 6 in its Klang-network. The proportion among the four

FIGURE 10.8

10.3

228


triads of measure 6 (with pickup) is clear enough aurally, supported as it is by arhythm-and-contour motive: E[? minor is to C major as F minor is to D major.But the proportion seems hard to portray by Klang-relations that mesh withour aural intuitions. Formally, we can of course note that F minor is the minorsubdominant of C major, and that C major is the major dominant of F minor:(F, -) (PAR) (SUED) = (C,+) and (C, +) (DOM) (PAR) = (F, -). Buthow functional are these relations in a context that emphasizes the proportion(E(? minor)-is-to-(C major) as (F minor)-is-to-(D major), all after a cadencein Et>? And what sort of Klang relation might we hear in this tempo andrhythm governing the terms of that proportion? Can we hear, for instance,(Eb, -)(PAR)(REL)(PAR) = (C, +) here? Our attempt to hear any func-tional Klang-relation is bound to be hindered by the parallel voice-leadingof the chords involved here.

It seems more fruitful to analyze the triadic formations of measure 6using the discourse of pitch-class sets and pitch-class transformations. Indeedwe can fruitfully study the triadic formations of the entire passage using thatdiscourse. Figure 10.8(b) shows the result. To save space, major and minortriads are denoted there by upper- and lower-case letter names. The newanalysis describes the triadic structure of the passage as follows, ignoringsevenths and added sixths. First, E minor substitutes for an expected Efc> majorin the cadence at measure 2^; the transformation involved is inversion aboutG, as depicted by the leftmost vertical arrow on figure 10.8(b). (Since we arenow talking about pitch-class sets and not Klangs, IQ is a pertinent transfor-mation while SLIDE, which inverts a Klang about its mediant, is no longerpertinent.) The pitch class G is a plausible center of inversion; it is highlyaccented by its boundary functions within the opening melodic phrase ofmeasures 1-3. In particular, the substitution of E-minor harmony for E[?major occurs exactly when the melody reaches its climax G, and the E-minorharmony remains around while the melody drops back to the low G, where itsits until its next phrase begins in measure 4 (on that G).

The next vertical arrow on figure 10.8(b) shows inversion-about-F#-and-G relating the E|?-major harmony we expect to the Ej?-minor harmony wehear, when the next cadence arrives at measure 5j. This inversion, whichreplaces the pitch class G of the E(?-major harmony by the pitch class F# =G[? of Eb minor, thereby develops the chromatic relation of the pitch classes Gand F#, the relation which governed the first chromatic event of the melody(and of the music as a whole), when F# appeared on the second beat ofmeasure 1.

The structuring power of that F# is then developed as shown by the nextvertical arrows on figure 10.8(b). F# becomes a new center of inversion:Inversion-about-F# relates E^-minor harmony to D-major harmony, andalso (NB) C-major harmony to F-minor harmony, all during the new thematicmaterial of measure 6 and following. E[? -minor and D-major harmonies mark 229

10.3


the temporal boundaries of the little phrase, while C-major and F-minorharmonies mark its registral boundaries. The repeated rhythm-and-contourmotif within the phrase (of measure 6 with pickup) groups E[j minor with Cmajor, and F minor with D major. Since all the minor/major relations offigure 10.8(b) so far have been analyzed as pitch-class inversions, the figurealso explores hearing E[? minor become C major by inversion (about F), andhearing F minor become D major by inversion (about G). Those inversionsare depicted by horizontal arrows on the figure. Ip and I % thereby inflect thecentral inversion, Ip"j|, for the little phrase of measure 6. The structuralcentrality of F# between F and G in this arrangement recalls the melodicposition of F# in measure 1, where it mediated between a preceding melodic Gand a subsequent melodic F.

In connection with the vertical arrows of figure 10.8(b), we have notedhow the progressing centers of inversion develop the pitch classes G and F#which are so characteristic of the incipit motive in the melody. We can alsoadopt a purely intervallic stance toward the progression of inversional centers,and toward the structure of the incipit motive. Figure 10.9 elaborates thatidea.

FIGURE 10.9

(a) of the figure is a network whose nodes contain the inversion oper-ations associated with the vertical arrows of figure 10.8(b). The TM arrowsmean Tn Ig = Ip^ and Tn Ip# = Ip*. The family S of operands here is thefamily of inversion-operations, and SGP is the group of left-multiplications-by-transposition-operations, as that group operates on the stipulated S.

Figure 10.9(b) is essentially extracted from the accompaniment of mea-sure 1, where it supports the thematic G-F#-F gesture of the melody. Thenetwork implied by figure 10.9(b) is isographic to the network of figure10.9(a).

The networks discussed in connection with figures 10.8 and 10.9 aremusically compelling at least to the extent that one would want to show how

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230


they relate to other aspects of the music, as one goes on to explore those otheraspects. These topics, for instance, would repay such a study: the "minor-thirdroot relations" within measure 6 (with pickup); the chromatic line, rising inparallel minor thirds, that starts with Eb-and-Gb at the pickup to measure 6and ends with F#-and-A in the middle of measure 6; the tonal structure of thebass line, either in Ab major or in a Mixolydian Eb.

10.4 EXAMPLE: The opening section of Debussy's Reflets dans I'eau, up to thereprise at measure 35, is rich in interrelations between transformationalnetworks and other sorts of musical structures. A sketch for the passageappears as figure 10.10.

The opening motive X plays a strong generative role. Particularly full ofimport are the transpositions T and T' that respectively take Db to Eb and Abto Eb within the motive.

Figure 10.11 (a) isolates this structure for study. Depending on variouscontexts to come, we sometimes hear the interval associated with T as onediatonic step up, sometimes as a major second up, sometimes as two semitonesup, and sometimes as the pitch-class interval 2. The interval associated with T'varies similarly depending on the GIS supplied by the context.

Figure 10.11 (b) shows how motive Y, which follows X in the music, canbe derived from X. Ab and Eb, connected by the T'-arrow, remain within Y.The T-relation of 10.11 (a), between Db and Eb within X, is "folded in" bothtemporally (serially) and in registral space, to become the T"1 relation dis-played in 10.1 l(b), between F and Eb within Y.2

The mathematical logic of the X-to-Y transformation has the followingimplication: If X should become T-transposed into T(X), then the new T-relation between the pitches in order positions 1 and 3 of T(X) will specificallyengage the relation Eb-to-F, thereby retrograding the last two pitches of Y.And in fact X is T-transposed into T(X) right after Y has sounded. The readercan follow these events along on figure 10.10. The new pitch Bb4 of T(X) isthereby generated in the principal melodic line. Within T(X), Bb-to-F is in theT'-relation. And Bb, as the high point of T(X), is the T-transpose of Ab, thehigh point of X.

Measures 5-8 repeat measures 1-4. As a result of the repeat, we hearT(X) return to X via T"1; Bb also returns to Ab via T"1; and so on. Thus thechange from T to T"1, a prominent characteristic of the change from theinternal structure of X to the internal structure of Y, now characterizes on alarger rhythmic level the change from the progression X-T(X) (measures1-3), to the progression T(X)-X (measures 3-5).

2. The reader will recall our discussing earlier just such kinds of "folding" transformationsin connection with serial trichords. That was in section 8.3.2, where we examined the transforma-tions FLIPEND and FLIPSTART. The transformation taking X to Y here is neither of those, butit is of the same genre. 231

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232

FIGURE 10.10

10.4

Transformation Graphs and Networks (4) 10,4

FIGURE 10.11

At measure 9 the new motive Zj appears in the melody. The way in whichZj pulls together the pitches and intervals of X, Y, T(X), and the repeat ofthose cells is aurally clear. The transformational situation is already so com-plex that it would require an inordinate amount of discussion to describe thissynthesis adequately in words. Let us just explore a few of its features, asdisplayed by figure 10.12.

FIGURE 10.12

The T"1 relation from F to E|>, which we heard within Y, is recapitulatedat the opening of Zt. The T"1 relation from Bfc> to At?, which we heard whenT(X) returned to X at measure 5, is recapitulated by the last two notes of Z t.And the relation between those two T"1 relations, within Zj, is an inverse T-relation. Figure 10.12 shows how this recalls the T'-features of X and T(X).

Beneath Zj in the accompaniment, a new idea makes its appearance. Thisidea involves a continuous chromatic rise of a certain object (here a certain3-note chord), over a total span of three semitones. The graph of this idea willbe called CHR; it is depicted in figure 10.13.

FIGURE 10.13 233


On Figure 10.10 we see how a CHR network fits homophonicallybeneath Z^ The first, second, and third of the resulting 4-note chords inmeasure 9 all have different structures. The fourth and last 4-note chord hasthe same structure as the first. That is because the CHR network in theaccompaniment has risen 3 semitones overall, while Zl in the melody has alsorisen 3 semitones overall, from its opening F to its closing Ab- The homo-phony thus identifies the 3-arrow on the CHR graph with the interval thatspans F-to-Ab across all of Zx . Because of the ways in which Zt synthesizesearlier motives, the F-to-Ab is heard in turn as part of a permuted Y embeddedwithin Zj. So the 3-arrow of the CHR graph can ultimately be traced back tothe Ab-F dyad within Y. The homophony between Zj^ and its CHR accom-paniment is marked by a crescendo. The crescendo is to become a significantthematic element.

In measure 10 the music of measure 9 is repeated and extended. Thecrescendo recurs. In the melody the repetition gives rise to a rotated form ofZ l5 marked "rot Z:" on figure 10.10. Rot Zl is Bb-Ab~F-Eb; it embedsserially the original form of Y, Ab-F-Eb, and precedes this Y by its overlap-ping inverse-RI-chained form Bb-Ab-F. (Bb-Ab-F is RICH'Âb-F-Eb).)This relationship is more or less inherent in the derivations of X, Y, T(X), theirrepetitions, and Z:.

When the music of measure 9 is extended during measure 10, a newmotive Z2 arises as shown on figure 10.10. Z2 is articulated, and associatedwith Z l s by its contour and its rhythm. Figure 10.12 earlier analyzed Zx as apair of T"1-related dyads in an inverse-T' relation. Z2 can be similarly ana-lyzed as a pair of T"1-related dyads in a T-relation. Figure 10.14 displays thatanalysis.

234

FIGURE 10.14

The accompaniment below Z2 projects a new network whose graph isCHR, a new network isographic to the network of measure 9 in the accom-paniment. The new network is the T-transpose of the old, as indicated onfigure 10.10 by the T arrow leading from under measure 9 to under measure10. The extended accompaniment within measure 10 rises 5 semitones from itspoint of departure, in contrast to the accompaniment of measure 9, which rose

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FIGURE 10.15

only 3 semitones from the same point of departure. Figure 10.15(a) is a graphconveying this idea.

Graph (b) of Figure 10.15 is a homomorphic image of graph (a). The twolower nodes of (a) each map into the one low node of (b), and the semigroups(groups) of intervals for the graphs are isomorphic. Figure 10.15(c) is anetwork whose graph is (b). We recognize that the pitches of (c) build a serialform of Y in the precedence ordering, namely the series F4-AJ?4-8^4. This isY inverted about F4-and-A|?4. The retrograde of the form was Rl-chainedinto Y within the rotated Zl motive we recently examined. We earlier notedthat F4 and A[?4 within Y were identified with the 3-semitone rise of thechromatic accompaniment during measure 9. During measure 10, the F4 andB[?4 of inverted-Y (that is, of figure 10.15(c)) are similarly identified with the5-semitone rise of the extended chromatic accompaniment. The inverted Y-form F4-A(74-B|?4, and the network of figure 10.15(c), are brought out by thehomophony in the same way as was the F-A[> dyad during measure 9: F4,Aj?4, and Bfc>4 are the three notes of the melody in measure 10 that aresupported by "dominant-thirteenth" harmonies; no other harmony appearsmore than twice during measure 10.

Figure 10.16 is a "product network." It adjoins beneath a copy of figure10.15(c) the bass notes of the dominant-thirteenth chords in an isographicnetwork. These are exactly the bass notes of measures 9-10 which are in pitch-class relation 3 to the melody above them. Beyond participating in thatrelation, the pitches A|?2 and D|?3 on figure 10.16 also function as temporaland registral boundaries for the bass line over measures 9-10. The tonality ofthe piece further reinforces their structural significance. 235

10.4


Figure 10.16 shows how the pitch and pitch-class interval 3 is proliferat-ing as a constructive element of the composition. The interval began as asecondary phenomenon; its complement spanned Ab-to-F within Y, whichappeared as the difference between the T'-related Ab-to-Eb and the T"1-related F-to-Eb- F-to-Ab then assumed greater prominence in the melody asthe temporal boundary for Zx, supported by the concomitant 3-semitone riseof CHR in the accompaniment. Figure 10.16 shows how the melodic F-to-Abis verticalized in the harmony, and how the interplay of horizontal and vertical3-intervals next generates a structural Cb in the bass. That Cb is the first"middleground" chromaticism of the piece; it will take on formidable propor-tions hereafter, especially after measure 18. In that connection, Cb will oftenbe heard in conjunction with F and Ab, following its prototypic generation onfigure 10.16.

Returning again to figure 10.10, let us now examine the large structure ofthe principal melodic line over measures 1-17. The pitch C5 of measure 10 isthe climax of this line, which has been using the rising T and T'-inversetransformations to ascend up to that point from the initial Db4. Theline is completely diatonic, so the leading tone C5 makes a strong effect asa provisional climax. The effect is somewhat concealed by the chromaticharmonization and by the possibility of hearing C5 as a neighbor to Bb4 ona subordinate level, even though the harmony does not support the neigh-boring function. Still, as one listens to the top staff of figure 10.10 byitself through measure 17, it is clear that there is unfinished business forthe principal melodic voice in its upper register. That business will not befully discharged until well beyond measure 35, where our present analysiswill stop. Nevertheless, we shall hear before measure 35 further importantdevelopments engaging C5 and Db5 in the principal melodic voice.

After the provisional climax on C5 in measure 10, the principal melody236

10.4

FIGURE 10.16


sits for a while on B|?4 and then returns over measure 14 back down to Dj?4,its original point of departure. The melodic descent embeds rotated Z l f

Bb-Ab-F-Eb, on the strong sixteenths of measure 14. Figure 10.17(a) showsthis and also shows how a rotated form of T(X) within the figuration proceedsback down via T"1 to the correspondingly rotated form of X.

The sixteenth-note appoggiaturas, all by descending whole tones, give theT"1 idea a heavy workout in the forefront of the melodic texture. Withinmeasure 14, the second half of the melody, F-Eb-Eb-Db, is in T'-relation tothe first half of the melody, Bb-Ab~Ab~Gb. The T' relations of Bb to F, andof Ab to Eb, are very familiar by now. Figure 10.17(b) shows a further T'-relation, one that involves the retrograde CHR gestures in the alto voice of thesame measure. Indeed, the entire second half of measure 14 is in T'-relation tothe entire first half of the measure. Figure 10.10 brackets a pentachord calledMAGIC which it asserts as controlling the first half of measure 14; T'(MAGIC) then controls the second half.

Tonally, the music has progressed from the tonic pedal of measures 1-8,through the dominant that opens measure 9, to the dominant-of-the-sub-dominant that ends measure 10 under the melodic Bb4. After the peripateticharmonies that prolong the melodic Bb4, the harmony discharges its sub-dominant obligation with T' (MAGIC) in the second half of measure 14, andthe tonal idea that carries measures 15-17 is a plagal cadence supporting aprolonged Db4 in the melody. Figure 10.10 shows how T' (MAGIC) leads intothat cadence. I forego with great reluctance analyzing the music hereabouts ingreater detail, particularly the ingenious chaining of Y, rotated X, and rotatedZl series in the outer voices of measures 16-17, and the set-theoretic relationsof those formations to the vertical sonorities there.

At measure 17^ and following in the music (represented as measure 18 onfigure 10.10), the cadential F of the plagal 4-3 gesture is confirmed by itsown verticalized Y-form F-D(natural)-C. The first interval of Y thereby ex-pands in structural power, transposing Y = Ab4-F4-Eb4 into T_3(Y) =F4-D4-C4. Earlier, figure 10.16 showed us how the original Ab-F dyad 237

FIGURE 10.17

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within Y generated Cb, the minor third above Ab; just so, the same Ab-F nowgenerates D natural, the minor third below F.

Figure 10.18 shows how this idea is expressed very precisely by the threeverticalized Y-forms within measure 18. The music ties T_3(Y) and its Dnatural together with Y and with J(Y) = A\?-C\?-D\>, the inverted formwhich originally generated Cb in the bass of figure 10.16. J(Y) appeared thereas T3I(Y), where I is inversion about F-and-Ab: J(Y) appeared in the bass offigure 10.16 coupled at the 3-interval beneath a structural melodic I(Y) =F-Ab-Bb.

Within the soprano line of measure 18 there are also references topermuted prime forms of melodic Y and melodic X, as indicated on figure10.10. These recollections help get the new large section of the piece underway,by recalling material associated with the opening.

Figure 10.10 also draws attention to the chromatic voice-leadings D-Eband C-Cb during measure 18; when the material repeats the voice leadings arereversed and then repeated. The chromatic notes D natural and Cb involved inthe voice-leading gestures arise as already discussed in connection with figure10.18.

I shall call the characteristic rhythm and contour that govern the secondhalf of measure 18 the "ruffling motive"; here the wind first ruffles the surfaceof the pond. The ruffling motive is bound together with the Cb events we havejust explored, including the C-Cb voice leading, the vertical Y of figure 10.18,and the vertical J(Y) of the same figure. The motive arpeggiates a Tristanchord upwards and then partially arpeggiates a £5 harmony downwards, allwithin the registral confines of the bass F3 and the upper note Ab4. TheTristan chord is in the correct spacing at the right pitch-level, once thedoubling Ab3 of the ascending ruffle is removed.

Over measures 20-21, the Tristan-harmony-cum-ruffle-motive moves upquasi cadenza in literal sequences, 3 semitones per stage, until it gets essentiallytwo octaves higher. The Tristan chord no longer contains any doublings.Figure 10.19 graphs the beginning of the cadenza sequence.

Now that D natural, as well as Cb, is on the scene along with F and Ab,the minor third or 3-semitone interval can be fully unleashed; the sequence athand unleashes it. Figure 10.19 has striking features in common with figure10.16 earlier, where the 3-interval first began to flex its muscles in a chromatic238

FIGURE 10.18

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context. Both the figures start with their outer voices on the basic 3-dyad F-and-Ab. In both, the outer voices then move by the melodic 3-interval, toproject Ab-and-Cb (G#-and-B) in the homophony. Both figures project formsof the Y motive: In figure 10.161(Y) = F-Ab-Bb above is coupled to J(Y) =T3I(Y) = Ab-Cb-Db below; figure 10.19 embeds Y and J(Y) subnetworks inthe ruffling as shown, and then sequences all Y-forms indefinitely.

I have no idea what private commentary on Wagner Debussy may haveintended by his use of the Tristan chord and his continued sequencing of it by 3semitones, a sequence which Wagner artfully only suggests in the first-actPrelude. Debussy's sequential cadenza, unleashing the hitherto restrained 3-interval, makes perfect sense in his own composition, of course. The firsttransposition of the Tristan chord, which Debussy spells as G#-D-F#-B, is asubset of the MAGIC pentachord D-Cb~Gb-(Bb)-Ab.

Ruffle-cum-Tristan sequences eight times in the music, arriving at mea-sure 22 essentially two octaves above its point of departure. Figure 10.10shows the Tristan chord at measure 22 only one octave higher, with its uppervoices rearranged. That is to accommodate the subsequent voice-leading intomeasure 24, and thence into measure 27 and measure 30; those events arepretty clearly in the correct structural registers where figure 10.10 portraysthem. 239

10.4

FIGURE 10.19


For the music between measure 22 and measure 24, where the cadenzaagain becomes mesure and a new theme appears, I hear two possible readings.One of them, marked "CHR and ret CHR?" on figure 10.10, is reflected inother notational aspects of the figure hereabouts. This reading takes thedescending minor third from F to D, first introduced on the figure at measure18, and fills it in by a retrograde CHR line; in counterpoint to this, an as-cending (prime) CHR line connects the Ej? of Tristan to the G|? of MAGIC;MAGIC then saturates the music from measure 23^ to measure 27. Theascending CHR line of this reading, from E|? to G^-within-MAGIC,retrogrades the opening gesture of figure 10.17(b) earlier, which showed aretrograde CHR line proceeding from Gb-within-MAGIC to E[?. On figure10.10, brackets demarcate the CHR groups of four events into which the voiceleading is articulated by this reading. Since the earlier CHR networks ofmeasures 9 and 10 were associated with thematic crescendi, and since theretrograde CHR events of measure 14 (figure 10.17(b)) could easily take adiminuendo, it would be useful to explore in connection with this reading howbest to structure the "poco a poco cresc. e stringendo" that begins at measure20. Does it stop at measure 22? Does it intensify there and continue on rightup to the arrival of the D natural at measure 23|? Does it continue even pastthat, right up to the (subito?) ppp mesure at measure 24, where the low Abpedal comes in? I have put parentheses and a question mark on figure 10.10,together with the crescendo sign under measure 22 there, to suggest andemphasize these questions.

The other possible reading I hear is indicated on the figure by theannotation "Voice-exchange Eb/F?" According to this reading, the basicgrouping is not of four CHR events, followed by ppp mesure; rather the basicgrouping is of three events within the Tristan harmony, followed by the arrivalof MAGIC harmony at the D natural that launches the uwmeasured section ofthe cadenza, at measure 23^. This is the point where the progressively moreagitated ripples of the water turn into turbulence; at the (subito) ppp mesureof measure 24 we presume that the burst of wind abruptly stops. The idea ofputting the structural downbeat for the MAGIC arrival not at the obviousmeasure 24, but rather just at the moment where periodic wave motion turnsto "aperiodic" turbulence, in measure 23^, is extraordinarily poetic. I wouldenjoy trying to play the passage this way for a small group of close friends; Iam not sure how well I could project it to a large public audience. Again, itwould be very helpful to have more indications for tempo and dynamicsbetween "pp, poco a poco cresc. e stringendo" at measure 20 and "(subito?)ppp, mesure" at measure 24. As it is, the pianist's decisions about tempo anddynamics will very much affect the sense of our alternate readings, and vice-versa. I write "(subito?) ppp" because I can conceive that Debussy heard adiminuendo from a MAGIC arrival at measure 23^ to ppp at measure 24, eventhough he did not write such a diminuendo. In that case, one imagines the240

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wind dying down more gradually, making the turbulence subside graduallyinto regular ripples again.

The rippling figuration over measures 24-29 ornaments a basic suc-cession of two harmonies: The MAGIC harmony just under discussion movesto a dominant-ninth structure at measure 27, as shown on figure 10.10. Thefigure also shows, by notations under the staff, how the progression involvesprecisely the two chromatic voice-leading gestures already exposed duringmeasures 18-19, that is D-to-Eb and Ct>-to-C. Within the overall progression,each of the two harmonies is prolonged by (or accompanies) a form of a newmotive I call V, marked in parentheses on figure 10.10. V stands for "vari-able"; the motive always proceeds by step up, third-leap up, step down, stepdown, and step down, ending on the same note with which it started, but thesizes of the "steps" and the "third" vary considerably, sometimes even withinone V-statement, and the rhythm of the motive is also extremely plastic. Thesefeatures of V presumably represent the perpetual mutations of things seenthrough reflets dans I'eau. (One thinks of Monet's pond at Giverny.) Thedescending part of the second V-statement here hooks up with the ripplingfiguration just before measure 30, to project the descending hexachordal lineBJ7-A[7-G(?; F-Et?-D[?. We have heard this descending hexachord before,namely in the melody of measure 14; there the B|?-A[7-Gb segment was alsolaunched by MAGIC harmony.

At measure 30 the V motive carries the final Dfr of the hexachord ondown to C5. We noted earlier that C5 was an interesting climax for theprincipal melodic line over measures 1-17, and that the melody seemed tohave unfinished business in its upper register. Debussy is now, at measure 30,turning his attention to some of that business. All at once, we hear burstingforth from the ripples a Z-form in a principal melodic line, a CHR-relatednetwork of trichords in parallel motion, and a crescendo; these were all aspectsof the music during its earlier rise over measures 9-10.

The Z-form Z3, articulated like Z2 earlier by rhythm and contour, circlesaround the critical pitch C5. It does touch D[?5, but there is some questionas to how essential D[?5 sounds in the melodic line. Hitherto, all motivic T-relations have been both by one diatonic scale degree and by two semitones (orby major second). Now we have to respond motivically to a relation that is andis not the same, between C5 and D|?5. This relation is quintessential^ by onediatonic scale degree, but it is just as quintessentially by only one semitone (orby minor second). Is this, or is this not, a bona-fide "T-relation?"

The answer, of course, is "Yes and no." We shall not get far arguing thequestion in that form; but it will be very much worth our while to explore theways in which the answer is ambivalent. More generally, let us explore furtherways (beyond the issue of T-relation) in which we feel ambivalent about theidea that the line might rise from C5 to D|?5 here.

The sense of structural rise seems strongly contradicted by the retrograde 241

10.4


forms for Z3 and CHR. Since we associated prime Z-forms with rising, overmeasures 9-10, we naturally feel that a retrograde Z-form is falling. And ofcourse the retrograde CHR network is falling very decisively.

But the crescendo is motivically important too. No matter how much theretrograde motives sound falling, the music is still rising in dynamic, like themusic of measures 9-10 that first led the melody up to C5. Debussy wouldsurely have written a diminuendo from mf, not a crescendo, had he heard acompletely subsiding effect here. It is difficult but crucial for the pianist tohusband the dynamics scrupulously, so the crescendo can proceed past mf andon to f without actually attaining ff. (Think of narrow-bore brass!)

To support further the idea that the line might be rising to D[?5, we canhear that D[?5 gets strong, if fleeting, harmonic support within the Z3-form.D[?5 is specifically supported by T' (MAGIC) harmony, continuing to followthe precedent of measure 14. Figure 10.20 shows what I mean.

Now the Dj? at the end of measure 14, the model for the D|?5 of figure10.20 in the relation just pointed out, is (or becomes) a very strong cadencetone; its function as an essential tone is not in doubt. By analogy, the D^S ofmeasure 30 sounds that much more essential.

But now a new consideration arises. The very strength of the fit betweenfigure 10.20 and measure 14 reminds us that everything shown on figure 10.10from measure 22 through measure 30 has in effect been transferred up anoctave from the Tristan chord of measure 20, which served as a launching pad.Figure 10.20, in particular, is still "an octave up" from measure 14. Accordingto this reading, the retrograde Z3 at measure 30 is an octave transfer of a formthat actually belongs an octave lower in the large melodic structure. C5 andDt? 5 within ret Z3 are to that extent not in the structural climax register at all;they are octave transfers of C4 and D[?4, from the cadence register. That viewis afforded support by the melodic doublings hereabouts: Both the second V-statement and the retrograde Z3 motive are doubled in the music an octavebelow where they appear on figure 10.10.242

FIGURE 10.20

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FIGURE 10.21

The tune is supported by a cadential progression in Efr major over a tonicpedal. The dynamic, ff and crescendo, is uniquely climactic for the compo-sition. (The unexpected bursting forth of Eb-major harmony a measure beforefigure 10.21 is only f, with a hairpin. Again the pianist must husband thedynamics very meticulously.) Here B[?4 and C5, that could not rise farther at 243

Still, no matter what the pianist does with the balance of these doublings,there has been no written dynamic heretofore as loud as mf, let alone mf with acrescendo to f, and that aspect of measure 30 by itself makes it hard to denythat we should hear C5 and D[?5 as climactic events to some extent, and notjust as doublings or transfers of C4 and D|?4.

In sum, the C5 and D(?5 of the retrograde Z3 at measure 30 are ambiva-lent in many ways: It is unclear to what extent they address the issue ofunfinished business in the climax register of the principal line, and to whatextent they represent C4 and D[?4 transferred temporarily up an octave; it isalso unclear to what extent ret Z3 is "rising," and to what extent it is "falling."The music plays with these ambiguities in the immediate sequel, via the littledescending arpeggios that alternate C with Db first in register 6, then inregister 5, and then in register 4. (The arpeggios are not shown on figure10.10.) Eventually the ret Z3 figure returns down the octave, as shown onfigure 10.10. Or one could put it that the doubling in the upper octave dropsaway. In any case, the registral issue is temporarily resolved in favor of thelower register. We become particularly convinced of this at measure 35, wherethe reprise begins. There we recognize the notes of Z3 in the lower register asthe notes of an alto voice that fits underneath the X motive of the soprano. Wemight say that Z3 belonged in the lower octave all the time; it was only tossedup an octave higher at measure 30 by the agitated activity of the waters, so tospeak, as a rare and curious submarine object that flashed momentarily intoview and then sank again. (One supposes that D|?4 is the surface of the pond.)As Z3 subsides under the surface at the reprise, the unfinished business in theclimax register remains unfinished.

But the discussion of figure 10.10 has ended. I should discuss, if onlybriefly, significant later events in the climactic register of C5 and above. Thekey to this analysis is the passage sketched in figure 10.21.

10.4


measure 10 and rose to Db5 only problematically at measure 30, finally maketheir breakthrough. Indeed, they get as far as £[7 5, with no question about it.Db 5, significantly, is not on the scene now that we are in the key of Eb. And C5is not a leading tone in the new key.

The "breakthrough" form of V at measures 57-58 directly follows two V-essays of a very different character. Debussy tells us that the Bb4 and C5 of thebreakthrough are to be compared with the same pitches in measure 9-10,by his use of the retrograde T(ZX) motive. Whereas in measure 10 only theBb-Ab of Zx moved up to T(Bb-Ab) = C-Bb within Z2, now all of Zj =F-Eb-Bb-Ab has moved up to T(Z1) = G-F-C-Bb, which is then retro-graded in measures 58-59 so as to descend from the climactic V-statement.Further, the whole tonality of the piece has temporarily moved up from Db toEb = T(Db), to accommodate and support the T-transformation of Zl. Thisis indeed an extraordinary expansion of the original T-relation which, thereader will recall, obtained precisely between the notes Db and Eb = T(Db),the notes which started and ended the initial motive X. The potential for thechange of tonality was perhaps already latent in the transformation of X toT(X) during measure 3. To the extent that T means "two semitones," T(X)would have had to be supported by Eb harmony in measure 3 to have the same"meaning" as X in measure 1.

We suggested in discussing measure 30 that the rise from C5 to Db5within a Z-form could not be completely convincing because the minor second(or distance of one semitone) is not completely T-ish in character. Now, overmeasures 56 and following, the change of tonality from Db to Eb provides anappropriate and enormous T-ish boost, enabling the melody to rise defini-tively beyond C5.3

3. For a sensitive appreciation of dynamics and contours rising and falling over this pieceon a large scale, and also for a convincing view of the E|? major breakthrough as it articulates alarge rhythmic design, the reader is referred to Roy Howat, Debussy in Proportion (Cambridge,England: Cambridge University Press, 1983), 23-29.244

10.4

Melodic and harmonic GISStructures; Some Notes on theHistory of Tonal Theory

Chapter 2 surveyed a variety of musical spaces pertinent to theories ofWestern tonality. In some of these spaces, our intuitions of directed distanceor motion from one position to another were measured by steps along somemelodic scale, diatonic or chromatic, linear or modular. In other spaces,our intuitions were measured by numbers reflecting harmonic relationshipsof various kinds, or by moves on a game board derived from harmonicrelationships.

The richness of tonal music, and of music in related idioms, is muchenhanced by the ways in which a variety of such intuitions come into play. Wecan review in this connection the Protean meanings of "the interval from F toAt?" in Reflets dans I'eau. We sense a harmonic interval within the Dt?-majortriad and the Tristan Chord; we sense also a melodic interval moving twosteps along a diatonic scale in D(? major, or along a diatonic hexachord(the cadential descending hexachord BÂbGbFEbDj?); to some extent wecan even hear F-to-At? as one melodic step along the pentatonic scaleDbE^FA^Bt?, as we listen to the melody at the beginning of the piece; finallywe also hear F-to-Aj? as spanning three semitones along a chromatic scale,once the CHROM figure comes onto the scene. A transformational approachenabled us to sidestep these ambiguities in chapter 10, referring there totransformations T and T' that mapped D[> to E|? and Aj? to £[7 respectively; wecould conceive T and T' as transpositions by "intervals" in any-or-all of theconceptual spaces involved; then we could compute a corresponding trans-formation T-1T' which mapped A(? to F in any-or-all of the spaces, a trans-formation worked out musically in the change from motive X to motive Y.

Such transformational discourse is particularly useful to discuss archi-tectural features of Reflets that obtain no matter what sorts of intervallic 245

Appendix A:

intuitions one considers. For instance, the tonic of the climactic fortissimo E|?major is in a T-relation to the tonic of D[? major no matter what kind ofinterval, in what kind of melodic or harmonic space, we consider T to betransposing D(? by. On the other hand, transformational discourse is corre-spondingly impoverished when it conies to exploring the varieties of spatialand intervallic intuitions at hand, and the ways in which the music bringsthose intuitions into play, each with the others. In connection with the bigclimax of Reflets, for instance, the major mode of the fortissimo music in E[?favors certain kinds of intervallic intuitions over others, when we hear Ej?major, not E|? minor, in relation to D[? major. The reader may also recall ourdiscussion of the high C-Db in the principal melodic line, and our question:"Is this, or is this not, a T-relation?" That question implicitly involves verybroad questions about the premises of the composition: To what extent is thepiece diatonic-melodic, so that one scale-step is one scale-step, regardless of itsacoustical size? To what extent is the melos chromatic, so that one semitone issomething necessarily very different from two semitones, even if both arespanning one diatonic step? To what extent is "the interval" attached to Theard in a harmonic context that gives it a size somewhere between 10/9 and9/8, but not as small as 16/15? To what extent can techniques of melodicmotivic transformation, involving the rhythm and contour of the Z motive inparticular, alter our impressions in any or all of these respects? and so on.Exactly these ambiguities must be appreciated, if we are adequately to appre-ciate the conceptual tensions of the local climax involving C5 and D|?5, beyondits high register and relatively high dynamic level.

We return, then, to the variety of intervallic intuitions surveyed in chap-ter 2. Pertinent syntheses of these intuitions are essential not only for manyoccasions in critical listening and analysis, but also for many abstract theoret-ical purposes. Indeed, such syntheses are among the greatest triumphs in thehistory of Western music theory, and their neglect or failure has led to some ofthe more embarrassing moments in that history. Among the latter, we maycite Rameau's argument that the harmonic intervals of 5/4 and 6/5 may beexchanged in relative register within the harmonic triad, so as to derive theminor triad from the major. This may be allowed, he says, since 5/4 and 6/5 areboth "thirds."* But at the time he says this, he has not as yet presented us withany scale along which we can measure distances of "three" degrees, and he hasassured us very strongly that melody is in any case thoroughly subordinate toharmony.2

1. Jean-Philippe Rameau, Traite de I'harmonie reduite a ses principes naturels (Paris:Ballard, 1722). "... la difference du majeur au mineur qui s'y rencontre n'en cause aucune dansle genre de 1'intervale qui est toujours une Tierce de part & d'autre;..." (p. 13).

2. Ibid. "On divise ordinairement la Musique en Harmonic & en Melodic, quoique celle-cyne soil qu'une partie de 1'autre, & qu'il suffise de connoitre I'Harmonie, pour etre parfaitementinstruit de toutes les proprietez de la Musique,..." (p. 1).246

Appendix A

One might try to replace Rameau's implicitly melodic argument aboutthe "thirds" by a suitable harmonic argument: The intervals 5/4 and 6/5 areadjacent within the senario; they divide the interval 3/2 = 6/4 harmonically,and so arithmetically when reversed; therefore such a reversal is logical. Butone could argue in exactly the same fashion for the intervals 4/3 and 5/4, asthey divide 5/3. Would Rameau, then, have accepted the argument that weought to consider the harmony G4-C5-E5 as functionally equivalent to theharmony G4-B4-E5, by the analogous reasoning? Obviously, he would nothave; such reasoning there would violate the principle of the FundamentalBass. Helmholtz, however, might have been willing to entertain the argument;indeed he actually asserts harmonic equivalence of a certain sort between thesix-four position of the major triad and the six-three position of the minortriad. Both those positions comprise the highly consonant verticalities of afourth, a major third, and a major sixth; having the same vertical-intervalcontent, they are thereby the "most consonant" close positions for theirrespective pitch-class sets.3

Among the triumphal syntheses mentioned earlier a high position mustbe reserved for Zarlino's Istitutioni harmoniche.* Book 1 discusses intervals asphenomena in a harmonic space. Book 3 discusses intervals all over again asphenomena in melodic space, and synthesizes that approach with the math-ematical ideas of book K Abstract harmonic ratios are accessible to ourperception (as well as our intellect) because they can be filled in by notes of adiatonic series in melodic space; conversely, articulated segments of a uni-directional diatonic series make sense to our understanding (as well as ourperception) because of the harmonic relations obtaining between the bound-aries of the segments. This way of interrelating harmonic and melodic spacehas much in common with central aspects of Schenker's theories, in particularwith Schenker's understanding of the Zug, and even specifically of the Urlinie.Schenker's mature theory contains another triumphal synthesis of harmonicand melodic space, understood now in the context of functional tonality.5

Even Zarlino has an embarrassing moment, confusing melodic withharmonic space, when he comes to discuss the minor sixth. He wants toanalyze the major and the minor sixths as analogous structures. Specifically,he says that they "are composed ... from the fourth plus the major third, or

3. Hermann Helmholtz, Die Lehre von den Tonempflndungen als physiologische Grundlagefur die Theorie der Musik, 2d ed. (Brunswick: Friedrich Vieweg und Sohn, 1865). "..., so folgthieraus, dass die Quartsextenlage des Disaccords wohllautender ist als die fundamental, unddiese besser als die Sextenlage. Umgekehrt ist die Sextenlage beim Mollaccord besser als diefundamentale, und diese besser als die Quartsextenlage." (p. 325).

4. Gioseffo Zarlino, Istitutioni harmoniche, 2d ed. (Venice: Senese, 1573). Facsimile re-publication (Ridgewood, N. J.: Gregg Press, 1966).

5. Heinrich Schenker, Neue musikalische Theorien und Phantasien, vol. 3, Derfreie Satz(Vienna: Universal Edition, 1935).

Appendix A

247

the minor third." 6 He might have continued: So, in a major mode such asIonian, the fourth G3-C4 plus the modal major third C4-E4 yields the majorsixth G3-E4, while in a minor mode such as Dorian, the fourth A3-D4 plusthe modal minor third D4-F4 yields the minor sixth A3-F4. The modal ideais perfectly clear. Elsewhere, too, Zarlino makes a great point of the modalrelation between major and minor thirds in harmonic contexts; indeed he evenpoints to this as a specific resource for harmonic variety, beyond the resourcesof the senario itself: Some chords have a major third or tenth over the bass,others a minor third or tenth.7 One wishes, then, that he would have producedG3-(C4)-E4 and A3-(D4)-F4, to illustrate a modal analogy between the twosixths as being "composed ... from the fourth plus the major third, or theminor third."

Unfortunately he does not do so. Probably he was not as sensitive as weare to the thirds above the modal tonics C4 and D4 in the harmonic structuresabove; those thirds are not over the bass notes of the structures. Whatever hismotivation, he attempts to realize the analogy of the sixths as a feature of hisharmonic space rather than his modal theory, and that leads him into confu-sion. He has to adjoin the number 8 to the senario in order to get the harmonicratio 8:5 at hand for the minor sixth, and then he has to argue that theproportion 8:6:5 is somehow analogous to the proportion 5:4:3 in hisharmonic world. He even claims that 6 is a "harmonic mean term" between 8and 5; this is simply false if "harmonic" is to mean anything at all in thecontext.8 We may fairly put his argument into modern dress by regarding it asan attempt to draw a direct analogy between the major sixth G3-E4, asdivided by C4, and the minor sixth E4-C5, as divided by G5. Of course thisdoes not work. In particular, the conjuction of the minor third E4-G4 belowwith the fourth G4-C5 above is not at all the same thing as the conjunction ofsome fourth below with some minor third above, as in the Dorian modal sixthA3-(D4)-F4.

Zarlino could also, of course, have analyzed the sixths as arising byinversion from the thirds. But this approach would have been foreign to hispurpose, for then the sixths would not have been primary features of hisharmonic space, somehow embedded within the senario. Besides, the sixthsthat arise from inverting thirds have a very different modal character from thesixths that interest (or should interest) Zarlino. That is exactly the problem

6. Zarlino, 1st. harm., book 3, chapter 21. "... sono composte ... della Diatesseron et delDitono, over del Semiditono;..." (p. 193).

7. Ibid.,book3,chapter31. "... lavarietadell'Harmonia...nonconsistesolamentenellavarieta delle Consonanze, che si trova tra due parti; ma nella varieta anco delle Harmonic, la qualeconsiste nella positione della chorda, che fa la Terza, over la Decima sopra la parte grave ...,overo che sono minori . . .; overo sono maggiori..." (p. 210).

8. Ibid., book 1, chapter 16. "... tal proportione tra 8 & 5 termini son capaci di unmezzanotermine harmonico, che e il 6;..." (p. 33).248

Appendix A

with his "minor" sixth 8:5 (E4-C5). To the extent that we hear it in a "C-major" modal context as third-degree-to-octave, inverting tonic-to-third-degree = C4-E4 = 5:4, a major third, the sixth itself has a "major" modalcharacter about it, despite its small size. Contrast that with the modal charac-ter of A3-F4 in a D-Dorian context, as fifth-degree-(through-tonic-)to-third-degree: This sixth, in its context, has a "minor" modal character as well as aminor absolute size. Similarly, F4-D5 in D-Dorian, inverting D4-F4, has a"minor" modal character despite its large size, while G3-E4 in C-Ionian hasboth a "major" modal character and a large size.

Zarlino had no Stufen theory that could enable him to make suchdiscriminations. And yet it is quite possible that, even if one had been availableto him, he might have rejected it. He would have been uncomfortable makingthe meaning of his harmonic intervals so dependent on the contextual assign-ment of a modal tonic. For him this would have weakened the context-freeuniversality of his harmonic theory. Schenker, quite willing to assign struc-tural priority to contextual modal tonics inter alia, uses his Stufen theory topowerful effect in related connections. On the other hand, he finesses certainproblems about the universality of minor harmonic structures which Zarlinoattempts to confront, and succeeds in confronting to a remarkable extent.

Hindemith makes an interesting synthesis of melodic and harmonicspaces.9 He tries to show that a chromatic scale from C2 to C3 is filled by thosepitches, and only those pitches, which lie in "closest" harmonic relation to C2within a certain harmonic space. We ignore the overtones of C2; then G2,within the desired scale-segment, is harmonically "close" to C2 because thesecond partial of G2 is the third partial of C2. F2, within the desired scale-segment, is "close" to C2 since the third partial of F2 is the fourth partial ofC2. And so on, casting away harmonic octave-replicates of pitches alreadygenerated (which, happily, do not lie within the desired scale-segment). Forthe most part, this works quite well, though a bit of strain is perceptible in theconstruction of certain secondary relationships. The essence of Hindemith'sachievement was not just to find pitch classes that can be represented bypitches within a chromatic scale. After all, Zarlino and his forerunners coulddo that well enough and better. Rather, the achievement was to have shownhow pitches within a melodically well-packed region, a chromatic scale fromC2 to C3, could be regarded as pitches also within a harmonically well-packedregion around the tonic, harmonically well-packed according to Hindemith'sspecial algorithms for generating harmonic pitch-space. The one pitch withwhich Hindemith has trouble is A[?2, the minor sixth above the tonic C2.Curiously enough, his troubles resemble Zarlino's troubles with the minorsixth. Hindemith can generate E2, E[?2, and A2 without using partials of those

9. Paul Hindemith, Unterweisung im Tonsatz: Theoretischer Teil (Mainz: B. Schott'sSonne, 1937), 47-61. 249

Aooendix A

pitches, or of C2, that involve numbers greater than 6. E.g.: The fifth partial ofEb2 is the sixth partial of C2; the third partial of A2 is the fifth partial of C2.But in order to find A[?2 by this method, he would have to have used an eighth-partial relationship: The fifth partial of A|?2 is the eighth partial of C2. Thisrelation would presumably have made A(?2 too "remote" in harmonic space;besides, it might have given rise to awkward questions about seventh-partialrelationships. (Zarlino has to deal with the analogs of such questions, when headjoins the number 8, but not the number 7, to his senario.) Presumably forreasons of these sorts, Hindemith produces not A[?2 but Afc> 1 by his algorithm;A)?! is a unique pitch which he generates in this way outside the octaveC2-C3. Then, without much explanation, he brings A|? 1 up an octave, so thatit will lie within his desired scale-segment.10

The foregoing discussion of ways in which some theorists have attemptedto integrate harmonic and melodic tonal spaces, or have failed to integratethem, is not meant to be exhaustive or even representative. It is rather intendedto show that we do not really have one intuition of something called "musicalspace." Instead, we intuit several or many musical spaces at once. GISstructures and transformational systems can help us to explore each one ofthese intuitions, and to investigate the ways in which they interact, bothlogically and inside specific musical compositions.

10. Ibid. "The frequency ... of the fourth overtone [of C2] is now divided by 5 ... and sogenerates ... Ab 1 ..., whose second overtone ... is inserted in our store of pitches. (DieSchwingungzahl ... des vierten Obertones c1 wird nunmehr noch durch 5 ... geteilt ... underzeugt so ... das lAs ..., dessen zweiter Oberton ... in unseren Tonvorrat eingereiht wird.)"(p. 54).250

Appendix A

Non-Commutative OctatonicGIS Structures; More on

Simply Transitive Groups

Let S be the octatonic family of pitch classes comprising C, Cft, Dft, E, Fft, G,A, and Aft. Eight of the standard "atonal" operations on the twelve pitch-classes transform S into itself; these operations are T0, T3, T6, T9, l£*, IG, 1°»and l£*. The eight operations form a group on the twelve pitch-classes andtherefore, mapping S into itself, induce a group of corresponding operationson S; we shall call those corresponding operations RO, R3, R6, R9, K, L, M,and N respectively.

It is not hard to verify that the latter group is simply transitive on S: Givenmembers s and t of S, there is a unique OP, among the eight cited operationson S, satisfying OP(s) = t. (If t is in the same diminished-seventh chord as s,OP will be RO, R3, R6, or R9; if t is in the opposite diminished-seventh chordfrom s, OP will be K, L, M, or N.) We shall call this simply transitive group ofoperations STRANS1. The operations RO, R3, R6, and R9 may be thought ofas "rotations," to justify the use of the letter R in their names.

We can define another group of operations on S, STRANS2, as follows.RO and R6 (as above) are members of the group; so are two "queer" oper-ations Q3 and Q9. Q3 rotates each of the diminished-seventh chords within S,but in opposite directions; it maps C to Dft, Dft to Fft, Fft to A, A to C, andalso Cft to Aft (not to E), Aft to G, G to E, and E to Cft. Q9 is the inverseoperation to Q3; it maps C to A, . . . , Dft to C, and also Cft to E,. . . , and Aftto Cft.

Besides RO, Q3, R6, and Q9, STRANS2 also contains four "exchanging"operations XI, X2, X4, and X5. XI exchanges pitch classes within S that lieone semitone apart; it thus maps C to Cft, Cft to C, Dft to E, E to Dft, Fft to G,G to Fft, A to A#, and Aft to A. X2 exchanges pitch classes that lie twosemitones apart; it maps C to Aft, Aft to C, Cft to Dft, Fft to E, and so on. X4 257

Appendix B:

exchanges pitch classes that lie four semitones apart; it maps F# to A#, E to C,G to D#, and so on. X5 exchanges pitch classes that lie five semitones apart; itmaps A to E, D# to A#, F# to C#, and so on. It can be verified that STRANS2is a group of operations on S, and that it is simply transitive.

Using the method discussed in 7.1.1, we can develop a GIS structure for Sin which the members of STRANS1 are exactly the formal transpositionoperations. We can call this structure GIS1 = (S, IVLS1, intl). In GIS1, then,applying any one of the operations RO, R3, R6, R9, K, L, M, or N to amember s of S amounts formally precisely to "transposing" the given s by asuitable corresponding interval of IVLS1. We must be careful to distinguishthe operations K, L, M, and N, which are "GIS 1-transpositions" under thisformalism, from the operations l£* etc. that gave rise to them; !£* etc. areinversion-operations in a different GIS, a GIS involving a different family of(twelve not eight) objects, a different group of (twelve not eight) formalintervals, and a different function int. Likewise, and more subtly, we mustdistinguish the octatonic GIS 1-transpositions RO, R3, R6, and R9 from thedodecaphonic atonal-GIS-transpositions T0, T3, T6, and T9.

As it turns out, the members of STRANS2 are exactly the interval-preserving operations for GIS1. Every member of STRANS2 commutes withevery member of STRANS1. In fact, the members of STRANS2 are preciselythose transformations on S that commute with every member of STRANS1.

Using the method of 7.1.1, we can develop another GIS involving thefamily S, a GIS for which the members of STRANS2 are exactly the formaltransposition operations. We can call this structure GIS2 = (S, IVLS2, int2).In this GIS, applying any of the operations RO, Q3, R6, Q9, XI, X2, X4, or X5to a member s of S amounts precisely to transposing s, formally, by a suitablecorresponding interval of GIS2. The interval-preserving operations for GIS2are exactly the members of STRANS1; those are in fact precisely the trans-formations on S that commute with every member of STRANS2.

Either GIS1, or GIS2, or both, might lead to results of interest inanalyzing a variety of octatonic musics. STRANS2 and STRANS1, whichfigure as groups of interval-preserving operations in those respective GISstructures, are thereby also likely candidates for CANONical groups ofoperations in a variety of set-theoretical studies. The STRANS1-forms of a setwithin S are exactly the dodecaphonically transposed and inverted forms ofthe set that lie within S. The STRANS2-forms of a set within S are in generala more novel sort of family. Taking (C, E, G), for instance, we apply to itin turn the operations RO, Q3, R6, Q9, XI, X2, X4, and X5; its STRANS2-forms are thereby computed as RO(C, E, G) = (C, E, G), Q3(C, E, G) = (D#,Qf, E), R6(C, E, G) = (F#, A*, C|), Q9(C, E, G) = (A, G, AJ), XI (C, E, G)= (Cfl,DJF,FJF)f X2(C,E,G) = (A#, Ffl, A), X4(C,E,G) = (E,C,D#), andX5(C, E, G) = (G, A, C). If Y is any one of those eight sets, and Y' is any otherone, and f is any one of the eight operations in STR ANSI, then the number of252

Appendix B

members of Y whose f-transforms lie within Y is the same as the numberof members of Y' whose f-transforms lie within Y': JNJ(Y, Y)(f) =INJ(Y', Y')(f). More generally, if f is any one of the eight operations inSTRANS 1, and A is any one of the eight operations in STRANS2, and Y andZ are any sets whatsoever within S, then INJ(Y, Z)(f) = INJ(A(Y), A(Z))(f):the number of members of Y whose f-transforms lie within Z is the same as thenumber of members of A(Y) whose f-transforms lie within A(Z).

As an exercise, the reader may consider a new family of pitch classes, S= (C, C|, E, F, Gf, A), and develop on the new S two analogous simplytransitive groups of operations.

More generally, suppose now that S is any family of objects and thatSTRANS is any simply transitive group of operations on S. Consider thefamily STRANS' of transformations f on S such that f commutes with everymember of the given group STRANS. It can be proved that STRANS' is itselfa simply transitive group of operations on S, and that every transformation Awhich commutes with every member of STRANS' is (already) a member of thegiven group STRANS. When S is considered as a GIS whose formal transpo-sitions are the members of STRANS, then the members of STRANS' will bethe interval-preserving operations. Dually, when S is considered as a GISwhose formal transpositions are the members of STRANS', then the membersof STRANS will be the interval-preserving operations. If STRANS is com-mutative, then STRANS' will be precisely STRANS itself.

253

Appendix B


Index

ADJOIN, 121Anti-homomorphism, 14Anti-isomorphism: 14; i-to-Tj, 46, 47, 150AOD: 113-15; and interval-preserving oper-

ations, 121-22Argument, of function, 1ARROW. See Node/arrow systemArrow chain: 194-95; proper, 209Associative: law, 5; binary composition, 5Attack-ordered dyad, 113, 121-22. See also

AODAttack ordering, 117

Babbitt, Milton: and beat classes, 23; Semi-Simple Variations, 136, 138-40, 141, figs.6.7-6.9; Reflections, 138; hexachord theo-rem, 145

Bach, Johann Sebastian, Two-Part Invention#1, 183-84, fig. 8.5

Bart6k, B61a, "Syncopation," 225-27, figs.10.5-10.6

Beat class, 23Beethoven, Ludwig van: First Symphony,

169-74, figs. 7.8-7.13, 8.1; Quartet op.135, 2lln; Sonata Appassionata, 213-17,figs. 9.14-9.16

Bernard, Jonathan W, 62/», 189Binary composition, 5BIND. See Serial transformationsBoundary tones, 94, 95, 98, 102-03, 235Brahms, Johannes: G-Minor Rhapsody,

119-21, figs. 5.14-5.15; Horn Trio, 165-69, 202-03, 207, figs. 7.6-7.7, 9.6

Bresnick, Martin, 67n5Bushnell, Michael, 90, 141n

Cann, Richard, 85n7Canonical: group, 104, 106, 111, 113, 150-

52; equivalence, 104, 113, 121-22; forms,105, 119

Carriage return, 214-15Carter, Elliott, First String Quartet, 23n3,

62, 67-74, figs. 4.2-4.3Cartesian: product, 1; thinking, 158

Central: defined, 7; interval, 49, 52-55 pas-sim

Cherlin, Michael, 137Chopin, Fr£d6ric, B-Flat-Minor Sonata, 85-

87, 114, figs. 4.6, 5.11Christensen, Thomas, 163nChronology: 173, 177; and precedence,

210-18. See also PrecedenceClosed, family of transformations, 4Cogan, Robert, and Escot, Pozzi, 160nCollection, of objects, defined, 1Coltrane, John, 62Common-note function, 144Commutative: defined, 6; GIS, 50, 53, 56,

58Complement, of set, 144-45Composition: of functions, 2; binary, 5Cone, Edward X, 39nCongruence, defined, 10CONTENTS. See Network, transformationCOV, 122

Debussy, Claude, Reflets dans I'eau, 231-44,245-46, figs. 10.10-10.21

Diatesseron, symphony of, 204-06, fig. 9.8Direct product: of semigroups, 15; of

groups, 15, 61; of intervals, 93-94; ofGISs, 37-46, 174

Dispersive. See TransformationsDOM, 176-78, 229Double emploi. See RameauDuration classes, 24, 25

EMB: defined for sets, 105; for set classes,106; as probability, 107, 111; properties,107-08; topological model, 108-11, figs.5.9-5.10; and Partition Function, 152

Embedding number. See EMBEnvironment, of a set, 97-98Equivalence classes: defined, 8; canonical,

104Equivalence relation: defined, 7; arising

from function, 7; and partitioning, 8; 255

Index

256

Equivalence relation (continued)examples, 9; in quotient GIS, 33-35;canonical, 104

External. See Transformations

Family: of objects, denned, 1; finite, 144-45

FLIPEND. See Serial transformationsFLIPSTART. See Serial transformationsForms, of a set, 105Forte, Allen: 9n, 103, 104, 143, 150; and

Gilbert, Steven E., 218/1Function: defined, 1; onto, 3; one-to-one, 3;

inverse of, 3; and equivalence relation, 7Functional: orthography, 2; equality, 2;

equations, 2-3Fundamental bass, 169, 170, 173, 175, 185,

209, 247

Generalized Interval System (GIS): definedand discussed, 26-28; review of prelimi-nary examples, 28-30; and intuitions,250. See also Commutative; Direct prod-uct; IFUNC; Interval-preserving opera-tions; Interval-reversing transformations;Inversions; LABEL; Non-commutativeGIS; Octatonic; Quotient; Simply transi-tive group; Timbral GIS; Time spans;Transpositions

Graph, 171, 190-92; transformation, de-fined, 195; operation, defined, 196; iso-morphism, 199; homomorphism, 201-06;product, 204, fig. 9.8(d)

Greer, Taylor, 95nGroup: of operations, 4; abstract, 6; direct

product, 15; homomorphism, 14-15; quo-tient, 15; of intervals, 25-26; of transpo-sitions, 46-47; of interval-preservingoperations, 48; locally compact, 103; can-onical, defined, 104. See also Canonical;Simply transitive group

Harvey, Jonathan, 24nHasty, Christopher, 43nHaydn, Joseph, Quartet op. 76, no. 5, 211/jHelmholtz, Hermann, 247Hexachord: semi-combinatorial, 143, 148-

49; Babbitt Theorem, 145Hindemith, Paul, 75, 249-50Homomorphism: of semigroup, 13; and nat-

ural map, 14; of group, 14-15; of node/arrow system, 201; of graph, 201-06, 235

Howat, Roy, 244n

Ictus, 42-44, figs. 3.4-3.6Identity: operation, 4; element for semi-

group, 5

If-only adjustments, 140-41IFUNC: defined, 90; examples, 89-99, figs.

5.1-5.7; maximum values, 94, 95; proper-ties, 99-101; as probability, 101; and Z-sets, 103; questions about, 103-04; un-rolling, 120; generalized by INJ, 147; ef-fect of inversion on, 150

Induced map, of quotient family, 10INJ: defined, 124; free of GIS structure,

134; and operations, 144-45; and set-complements, 144-45; as generalizedIFUNC, 147; for transformed X and Y,147-49; questions about, 149-50; general-izing K and Kh, 150-51; for infinite sets,153, 154, 156

Injection function, 95, 99. See also INJInjection number. See INJInput: node, 207, fig. 9.9; Klang, 208, 215,

fig. 9.10; motive, 209; trichords, 227Int, 26Internal. See TransformationsInterval: intuitions about, xi-xii, 16, 17-20,

25-26, 74-75, 245-46, figs. 0.1, 4.4; re-placed conceptually by transposition, xiii,157-60, 245-46; scarce, 102-03; forwards(backwards) oriented in time-span GIS,113-14; between roots, 170; for TCH,181; varying in Debussy, 231, 245-46; inRameau, 246-47; in Zarlino, 247-49. Seealso Central; GIS; IFUNC; Vector

Interval-preserving operations: andLABEL, 47; defined, 47-48; form agroup, 48; characteristic behavior, 48-49;are sometimes transpositions, 49; com-mute with transpositions, 50; combinedwith inversions, 55; effect on IFUNC, 99;in time-span GIS, 111-13; and AODs,121-24; octatonic, 252

Interval-reversing transformations, 58-59Inverse: function, 3; element of semigroup,

6; operation, 56Inversions: defined in a GIS, 50-51, fig.

3.7; and LABEL, 51; when functionallyequal, 52, 53; combined with transposi-tions, 54; combined with interval-preserv-ing operations, 55; combined one withanother, 56; inverses of, 56; effect onIFUNC, 101, 150; as graph transforma-tions, 190-92, fig. 8.12

Isography: informally mentioned, 183, 187,192; defined, 198-200; further examples,227, 230, 234

Isomorphism: of semigroups, 13; i-to-Pj, 48;of node/arrow systems and graphs, 199

IVLS, 26

Jones, James Rives, 39«

Index

K, and Kh, 150-51Klang: 175-81, 213-14, 216-17, 227-29; in-

put and output, 207-08, fig. 9.10Kramer, Jonathan, 211nKurth, Ernst, 219/t

LABEL: defined and discussed, 31-32; forT,(s), 47; for Pi(s), 47; for Itfs), 51

Lerdahl, Fred, and Jackendoff, Ray, 217«Lewin, David, 32n, 42«, 60n, 88n, 89n,

120n, 128n, 141n, 193nLigeti, Gyorgy, Poeme symphonique, 23/i3,

67Linear ordering: 135,146; and precedence,

211-12, 219LT, 178

Mapping: defined, 1; of a set, 88Maximum values: of IFUNC, 94; of INJ,

140-41, 142Measurable transformations, 153Measure: abstract, 153; on time-span GIS,

154-55MED, 176-77Melody, models for, 133, 219, fig. 9.17Minturn, Neil, 227Modular musical spaces, 17, 20-25 passim,

36. See also Quotient, GISModulation of transformational system, 129,

148-49Moorer, James A., and Grey, John, 84nMorris, Robert D., 104n, 105/jMozart, Wolfgang Amadeus, G-Minor Sym-

phony, 220-25, figs. 10.1-10.4MUCH. See Serial transformations

Nancarrow, Conlon, Studies for PlayerPiano, 23n3, 66-67

Natural map: onto quotient family, 8; ontoquotient semigroup, 12

Network, 164, 165, 168, 170, 172, 176, 177,178, 192; transformation, defined, 196;connected operation, 197; operation de-fined, 197; isography, 198-200; product,206, 235, figs. 9.8(d), 10.16; as model forseries, 206, 218; Schenkerian, 214, 216-18, fig. 9.16; and time spans, 215-16,217, fig. 9.15; and melody, 219, fig. 9.17

Node/arrow system: defined, 193; communi-cation in, 193-94; connected, 194; arrowchain in, 194-95; isomorphism of, 199;homomorphism of, 201. See also Prece-dence

NODEMAP. See Node/arrow system, iso-morphism of, homomorphism of

NODES. See Node/arrow system

Non-commutative GIS, 50, 58. See alsoOctatonic; Time spans

Octatonic: scale, 17; non-commutativeGISs, 251-53

One-to-one. See FunctionOnto. See Function; Graph, homomorphismOperations: defined, 3; group of, 4; on a

GIS, 46-59; and INJ, 144-45. See alsoDOM; Graph; LT; MED; PAR; REL;Serial transformations; SLIDE; SUED;SUBM

Ordering. See Attack ordering; Linear or-dering; Partial ordering; Precedence; Re-lease ordering

Orthography, left and right, 2, 176Output: node, 207, fig. 9.9; Klang, 208, fig.

9.10; trichords, 227

PAR, 178, 229Partial ordering: of pitch classes, 135-40

passim, figs. 6.7-6.9; of nodes, 209, 211Partition Function, 152Peel, John, 174Pitch notation, xiiiPrecedence: relation, 210; ordering, 210,

219; and chronology, 210-18Probability: and IFUNC, 101; and EMB,

107, 111Product: of graphs, 204, fig. 9.8(d); net-

work, 206, 235, figs. 9.8(d), 10.16. Seealso Cartesian; Direct product

Progressive. See TransformationsProkofieff, Serge, Melodies op. 35, 227, figs.

10.7-10.9PROT: defined, 134; and rows, 134-35, 146Protocol pairs, 134, 211. See also PROT

Quotient: family, 8; semigroup, 10-12;group, 15, 29; GIS, 29, 32-37

Rahn,John, 174Rameau, Jean-Philippe: 73, 175, 246-47;

double emploi, 213, 217Reflexive property, 7Regener, Eric, 103n, 144, 145, 152REL, 178, 213, 229Release ordering, 117RICH. See Serial transformationsRl-chaining: 164, 235; in both pitch and

rhythm, 221-22; in 1. and r. hands, 226-27. See also Serial transformations, RICHand TCH

Riemann, Hugo, 22n, 73, 175, 177Row, as set in PROT, 134-35 257

Index

Schenker, Heinrich: 165, 174, 214, 247, 249;and networks, 216-18, fig. 9.16

Schoenberg, Arnold: Violin Fantasy, 101-03, fig. 5.8; "Angst und Hoffen," 125-33,148, figs. 6.1-6.5; "Die Kreuze," 133-34,fig. 6.6; Moses und Aron, 136, 137-38,fig. 6.7; Piano Piece op. 19, no. 6, 141,143, 160, figs. 6.10, 7.1

Scholica Enchiriadis, 204Semigroup: of transformations, 4; abstract,

5; multiplicative notation, 5; congruencesand quotients, 10-12; homomorphism andisomorphism, 13-15; direct product, 15

Serial transformations: RICH and TCH,180-88 passim, 221-24, 226-27; MUCH,183, fig. 8.5; RT and RI, 188; TLASTand TFIRST, 188-89, fig. 8.10;FLIPEND and FLIPSTART, 189, fig.8.11; and commutative GIS, 189-90; I,192; BIND, 208-09

Set: mathematical, 1; in a GIS, 88; environ-ments of, 97-98; class, 105; in a family S,124; transitivity, 129; theory in infinitecase, 152-56

SGMAP. See Graph, isomorphism, homo-morphism

SGP. See Graph, transformationSignature motive, 137-38Simple ordering. See Linear orderingSimply transitive group, 157; as transposi-

tion group, 157-58; on Klangs, 179-80;on octatonic family, 251-53; and its com-muting group, 253

Slawson, Wayne, 85SLIDE, 178, 227SNDW, 121Stockhausen, Karlheinz: rhythmic theory,

24n; Gruppen, 24n; Aus den siebenTagen, 62; Klavierstuck XI, 66

Strunk, Oliver, 2Q4nSUED, 177-78, 229SUBM, 178Symmetric property, 7

TCH. See Serial transformationsTFIRST. See Serial transformationsTimbral GIS, 82-84, 84-85, fig. 4.5Time spans: defined, 60; commutative GIS

for, 61; non-commutative GIS for, 74-81,

112, 154-56, fig. 4.4; and networks, 215-16, 217, fig. 9.15

Time unit: discussed, 61; contextual, 67TLAST. See Serial transformationsTMSPS, 60Transformations: and intuition of musical

space, xiii; defined, 3; internal and pro-gressive, 126, 132, 134, 141, fig. 6.6; ex-ternal and dispersive, 142-43, 164;modulated, 148-49; measurable, 153. Seealso Graph; Network; Operations; Serialtransformations

Transitive property, 7Transpositions: can replace intervals con-

ceptually, xiii, 157-60, 245-46; defined inGIS, 46; form a group, 46-47; andLABEL, 47; sometimes preserve inter-vals, 49; commute with interval-preserv-ing operations, 50; combined withinversions, 54; effect on IFUNC, 100;form simply transitive group, 157; inoctatonic GISs, 252

Travis, Roy, 225, 227Tristan chord, 238-40, 242

Unfolding. See Unrolling; VectorUnrolling: interval vector, 116-19; EMB,

119-20; IFUNC, 120; INJ, 131

Value, of function, 1Varese, Edgard, 189Vector: unfolding interval, 44; interval, 98,

104, 114, fig. 5.12; M-class, 106-07

Wagner, Richard: Parsifal, 161-64, 181-82,figs. 7.2-7.5, 8.3; Tarnhelm and Valhallanetworks, 179, fig. 8.2; Todesverkiindi-gung, 183-88, 208-09, figs. 8.6-8.9, 9.11.See also Tristan chord

Webern, Anton: Piano Variations, 38-44,181, 182-83, 190-92, 200, figs. 3.1-3.6,8.4, 8.12, 9.5; Pieces for Violin andPiano, no. 3, 90-99, figs. 5.2-5.7; Piecesfor String Quartet, op. 5, no. 4, 188-89,fig. 8.10

Wedging, 124-32 passim, figs. 6.1-6.4, 6.6Wintle, Christopher, 136n

Zarlino, Gioseffo, 73, 247-49, 250Zero time point, 63Z-sets: generalized by IFUNC, 103-04;

generalized further by INJ, 149-50

258

Generalized Musical Intervals and Transformations

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