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A b str a c t
A Voicing-Centered Approach to Additive Harmony in Music in
France, 1889-1940
Damian Blattler 2013
Music written in France during La Belle Epoque and the interwar
period is
remarkable in part for its development of additive harmony, i.e.
its incorporation of novel
chords into conventional tonal contexts. The conventional
explanatory apparatus for this
harmonic phenomenon, the extended-triad model of ninth,
eleventh, and thirteenth
chords, poorly describes those features that allow certain novel
chords to participate in
tonal progressions; this study develops a model of additive
harmony that better accounts
for the structure and behavior of the additive chords found in
this repertoire. Chapter 1
examines the history of theories of additive harmony, tracing
three distinct strategies for
explaining simultaneities that are not triads or seventh chords:
adapting basic chord types,
identifying non-chord elements, and formulating new basic chord
types. Chapter 2
presents a model of additive-harmonic chord structure in which
voicing plays a
foundational role; chords are conceived of as pitch-space
voicings constrained by the
pragmatic considerations of tonal plausibility and chordability.
Chapter 3 investigates
the role voicing plays in the special additive-harmonic case of
polychordal polytonality.
The dissertation closes by discussing the connections between
the small details at the
musical surface that are the focus of this study and several
larger music-theoretical issues.
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A Voicing-Centered Approach to Additive Harmony Music in France,
1889-1940
A Dissertation Presented to the Faculty of the Graduate
School
ofYale University
in Candidacy for the Degree of Doctor of Philosophy
byDamian Joseph BlSttler
Dissertation Director: Daniel Harrison
December 2013
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UMI Number: 3578315
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2014 by Damian Blattler All rights reserved
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Contents
Examples, Tables, and Figures i
Acknowledgements viii
Introduction 1
Chapter 1 - Strategies in Additive Harmonic Theory 131.1
Additive Harmonic Thinking in Rameaus Traite 171.2 The
Modified-Basic-Types Strategy 261.3 The Non-Chord-Elements Strategy
511.4 The New-Basic-Types Strategy 68
Chapter 2 - A Vertical-Domain Model for Additive Harmony in
75Music in France, 1889-1940
2.1 Overview of the Model 752.2 Anchor Structures and Tonal
Plausibility 792.3 Chordability 992.4 Additive Chords and Non-Chord
Verticalities 123
Chapter 3 - The Special Case of Polychordal Polytonality 1323.1
Theories of Polychordal Polytonality 1343.2 Implications of
Chordability Restrictions and 149
Anchor Structures for Polychordal Polytonality
Conclusion 165
Appendix - C++ Program for Parsing All Verticalities in a 28-
169Semitone Span
Bibliography 171
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Illustrations
Figures
1.1 Reduction of the final cadence of Ravels L enfant et les
sortileges, R154-end.
1.2 Reproduction of example 27.1 from Roig-Francolis Harmony in
Context.
1.3 The ill fit of extended-triad labels when applied to certain
chords in the Parisian modernist repertoire.
1.4 Competing functional elements in the penultimate chord of L
'enfant et les sortileges.
1.5 Comparison of the impact of inversion upon an additive chord
and upon a triad.
1.6 Over-applicability of the extended-triad model in the
absence of guidelines for voicing.
1.1 Spectrum between competing ideals in additive harmonic
theory.
1.2 Realization of the chord types described in Rameaus Traite
other than the perfect chord and the seventh chord.
1.3 Rameaus example II.6; an irregular cadence.
1.4 Transcription of Rameaus example II. 11 - explanation of a
suspended fourth as a heteroclite eleventh chord.
1.5 Transcription and annotation of Rameaus examples II.41
and11.42; use of supposition to explain voice-leading
novelties.
1.6 Rameaus inability to explain ninths resolving over falling-
fifth bass motion.
1.7 Rameaus difficulty explaining suspensions in the bass
voice.
1.8 Heinichens use of invertible ninth chords to explain novel
dissonance treatment.
1.9 Set of extended chords from the Ist edition of Marpurgs
Handbuch.
1
5
7
8
9
11
15
18
20
21
23
24
25
27
28
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1.10 Marpurgs first-inversion thirteenth chord with fifth,
seventh, 29and ninth omitted.
1.11 James M. Martins transcription of Sorges Plate VI, examples
314 and 5; presentation of the dominant ninth chords and their
relationship to the leading-tone seventh chord.
1.12 Dehns ninth, eleventh, and thirteenth chords realized in C
major. 33
1.13 Dehns prohibited voicings of the ninth chord. 34
1.14 The harmonic-series justification for the extended-triad
model 3 7of additive harmony.
1.15 Lobes inversions of altered ii9 in A minor. 38
1.16 Transcription of Ziehns set of chromatic seventh chords.
39
1.17 Harmonic treatment of the whole-tone scale in Schoenbergs
41Harmonielehre.
1.18 Schoenbergs four- and five-note quartal chords, and their
42resolutions as substitute dominants.
1.19 Quartal chords in Schoenbergs Kammersymphonie. 42
1.20 Ottmans figure 10.18. 43
1.21 Pistons example 22-12; various inversions of a G
major-ninth 44chord.
1.22 Ulehlas voicing guidelines for second-inversion major-ninth
45chords.
1.23 Ulehlas analysis of the implications of chromatic
alteration to 46the thirteenth above a dominant seventh.
1.24 Prouts three fundamental chords. 48
1.25 Sixth-inversion thirteenth chord in Day. 48
1.26 Free treatment of the ninth in Prouts Harmony. 49
1.27 Days explanation of a ii7 chord as a third-inversion
51dominant eleventh.
ii
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1.28 Kimbergers Figure XXXVI - non-essential dissonances. 52
1.29 Kimbergers explanation for seventh chords that resolve
53upwards by step.
1.30 Catels chord formed by the suspension of the octave, the
sixth, 55and the fourth.
1.31 Catels chord formed by raising the chordal fifth and
lowering 55the chordal third.
1.32 Fetis concept of substitution. 56
1.33 Fetis presentation of Catels inversions of the so-called
57fundamental chord of the seventh.
1.34 Fetis analysis of Beethovens improper inversion of the
58leading-tone seventh.
1.35 Fetis derivation of the ii7 chord. 58
1.36 Ravels analysis of an unresolved appoggiatura in Les Grands
61Vents venus d 'Outre-mer.
1.37 Ravels analysis of an unresolved appoggiatura in Oiseaux
61tristes from Miroirs.
1.38 Ravels analysis of an unresolved appoggiatura in Vaises
nobles 62et sentimentales, vii.
1.39 Lenormands derivation of an unresolved appoggiatura chord.
64
1.40 Casellas set of Ravel s genuine appoggiature chords. 65
1.41 Ravel Sonatine/i, mm. 85-87; a final tonic chord with added
ninth. 66
1.42 Kitsons derivation of an unresolved appoggiatura chord.
67
1.43 Hindemiths example 62 - non-triadic chords. 68
1.44 Preferring a new-basic-type label to tenuous
modified-basic-types 69or non-chord-elements labels.
1.45 Persichettis example 4-4; the roots of quartal chords
determined 70by melodic motion.
iii
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1.46 Persichettis conditions for incorporating quartal chords
with triads.
1.47 Hindemiths chord groupings.
1.48 A Hindemith harmonic-fluctuation graph.
2.1 The pitch classes of the L 'enfant chord arranged as a
cluster, as an integrated chord, and as disassociated strata.
2.2 The relationship between tonally plausible additive chord
and common-practice chord modeled as a two-step process.
2.3 The set of common-practice chord types.
2.4 First-order anchor structures and the common-practice chord
types they can stand in for.
2.5 Ravel, Laideronette, imperatrice des pagodes from Ma mere I
Oye, mm. 16-24: |t|-anchored substitute supertonic.
2.6 Chords with distinct pitch-class content but a shared tonal
plausibility.
2.7 Different tonal plausibilities arising from distinct
voicings of a single pitch-class set.
2.8 Hulls example 194; an example of the benefit of abandoning a
stacked-thirds conception of additive chords.
2.9 Conflict between the chord identities projected by the bass
motion and those projected by the verticalities themselves
2.10 Use of seventh chords in parallel motion in the Parisian
modernist repertoire.
2.11 Ravel Jeux d eau/iii; |e|-anchored final tonic 92
2.12 Chromatically altered chordal third in Saties Gymnopedie
No. 1, mm. 36-39.
2.13 Omitted chordal third in Debussys La fille aux cheveux de
lin, mm. 18-19.
2.14 Second-order anchor structures.
iv
71
73
74
78
80
82
84
86
87
89
89
91
92
93
93
95
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2.15 Examples of second-order anchor structures. 96
2.16 Interaction between anchor structures of different orders.
97
2.17 Second-order anchor structure cadence in Stravinsky, L
'histoire 98du soldat, p. 6.
2.18 Interaction of anchor structures of the same order. 98
2.19 First-order anchor structures not disturbed by the presence
of |7|. 99
2.20 Unlikely anchor-structure voicings. 100
2.21 Ravel, Miroirs, Alborada del gracioso, mm. 130-134. 101
2.22 Ravel, Pavane de la Belle au bois dormant from Ma mere I
'Oye; 102first-order anchor-structure chord supporting ro-interval
1.
2.23 Ravel Vaises nobles et sentimentales i, mm. 1-2. 104
2.24 Jazz voicings from Levines The Jazz Piano Book. 105
2.25 Ravel, Pavane pour une infante defunte; use of ro-interval
1 to 107destabilze a previously stable cadence.
2.26 Marked use of ro-interval 1 in Chabriers Les Cigales.
109
2.27 Scarcity of adornment options for second-order anchor
structures. 113
2.28 Ravel, String Quartet, end of 1st movement. 114
2.29 Whole-step adjacencies in final tonics in the Parisian
modernist 118repertoire.
2.30 Debussy, Les collines dAnacapri, mm. 1-8; use of
119verticalized consecutive whole-step adjacencies as a marked
event.
2.31 Debussy, Feuilles mortes, mm. 6-10; adjacent-whole-tone
120sonority treated as a passive musical object.
2.32 Reductive power of the models three voicing constraints.
123
2.33 Types of verticalities in the Parisian modernist
repertoire. 124
v
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2.34 Ravel, Vaises nobles et sentimentales, mm. 56-61;
non-chordable 125 verticalities interpreted as the coincidence of a
chordal backgroundand non-essential dissonance.
2.35 Debussy, La Puerta del Vino, mm. 5-12; viable additive
chords 127interpreted as passing tones.
2.36 Reduction of Poulenc, Rag Mazurka from Les biches, R89;
128suspension of a major tenth above the bass.
2.37 Prokofiev, Sonata for Flute and Piano, mm. 1-4. 129
2.38 Debussy, Feux dartifices, mm. 85-90. 131
3.1 Anchor-structure, stacked-thirds, and polychordal readings
of an 132extended triad.
3.2 Milhauds table of bitonal combinations. 136
3.3 Febre-Longerays monotonal derivation of all bichordal
138combinations of two major triads.
3.4 Koechlins context-dependent polychord. 139
3.5 Persichettis hierarchy of bichordal combinations. 140
3.6 Examples of Persichettis four types of permissible
trichords. 142
3.7 Persichettis demonstration of grouping as a requisite of
143polyharmony.
3.8 Ulehlas analysis of the assimilation of the upper parts by a
144dominant seventh chord.
3.9 An exception to the assimilation effect of Figure 3.8 - a
tonic 144triad above its dominant seventh.
3.10 Thompson and Mors Fig. 1 - excerpt from Theodore Duboiss
148Circus.
3.11 Bichordal superimpositions that run afoul of Chapter 2s
153chordability restrictions.
3.12 Analysis of bitonality in Milhauds Botafogo from Saudades
158do Brasil.
vi
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3.13 Polytonal effect generated by an (M/M, 6) chord
superimposition; 161mm. 34-46 of Ipanema from Milhauds Saudades do
Brasil.
3.14 Analysis of non-bitonal polychordal construction in Ravels
162Sonata for Violin and Cello.
3.15 Analysis of Milhauds Corcovado from Saudades do Brasil,
164mm. 1-8.
Tables
3.1 All potential bichordal combinations involving major triads,
minor 151triads, and dominant seventh chords.
3.2 The bichordal combinations of Figure 3.11 tabulated by
ro-interval 157distance between chord roots.
vii
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Acknowledgments
First and foremost, I would like to thank Daniel Harrison, the
advisor to this
dissertation, for all of his guidance, patience, and
encouragement during the last several
years; this work has benefitted enormously from his insight and
mentorship. I also
extend thanks to my committee members, Peter Kaminsky and
Patrick McCreless, who
have given generously of their time and knowledge to this
project.
Thanks go too to all of my graduate student colleagues, who have
provided
invaluable information, critique, and camaraderie through the
various stages of this
project. In particular I would like to thank the members of
Professor Harrisons lab
group - Chris Brody, Jennifer Darrell, Megan Kaes Long,
Elizabeth Medina-Gray, John
Muniz, Jay Summach, and Christopher White - for their collegial
consideration of my
work and for the inspiration and motivation their work provided
for me.
Finally, I am eternally grateful to my family and to my wife
Jackie for their faith
in my musical pursuits and for their love and support.
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Introduction
The music of composers working in France between 1889-1940 is
shot through
with harmonic moments like the one in Figure 1.1, the final
cadence of Ravels 1925
opera L enfant et les sortileges. These are moments where the
functional sense of a
Figure 1.1. Reduction of the final cadence of Ravel,Lenfant et
les sortileges, R154-end.
The Child
Ma - man!
Chorus
doux.est
oboes IPOrchestra PP z
strings
chord progression is clear, but the chords used in that
progression are unconventional by
common-practice standards. In the Ravel example, we hear the two
bracketed chords as
completing an authentic cadence; the chords follow a clear
tonic-to-subdominant-to-
dominant root progression of G-E-C-A-D that begins at R153. The
first bracketed chord,
however, is neither a triad nor a seventh chord (and instead
involves a scalar subset B-C-
D-E, with the pitches of the chorus evaporating upon the strings
entrance), while the
final G-major triad is accompanied by a dissonant F it.
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Jeremy Day-0Connell has termed this incorporation of novel
chords into tonal
contexts additive harmony.1 It is an innovative feature of the
music of this time and
place; while there is a lot of variety and stylistic change in
music in France between
1889-1940, there is also a common fascination with colorful
harmonies and their
potential tonal applications.2 This feature can be seen to stem
in part from a late-
nineteenth-century desire to establish a compositional style
distinct from the dense
counterpoint of German expressive maximalism, one that involved,
among other things, a
renewed focus on pleasure, sensation, and the sensuous potential
of chords.3 For the
purposes of this study, the 1889 Exposition Universelle - an
event which capped the
rehabilitation of French national pride after the humiliation of
the Franco-Prussian War4 -
serves as a symbolic start date; it is around this time that
Debussy and other composers of
his generation started to differentiate their style from German
music and the
Germanized French music of Franck and his contemporaries.5 The
additive-harmonic
innovations of this generation were taken up both by subsequent
generations of French
composers as well as by foreign-born composers who wrote for
Parisian audiences (e.g.
Stravinsky, Martinu, and Prokofiev at various points in their
careers); additive harmonic
practice developed in a relatively cohesive soundscape up until
the German occupation of
France during World War II (1940), after which the European
compositional mainstream
turned more comprehensively toward serial and atonal
composition. Since this
repertoires focus on the sensuous potential of novel chords
participates in French
1 Jeremy Day-OConnell, Pentatonicism from the Eighteenth Century
to Debussy (Rochester, New York: University o f Rochester Press,
2007), 147.2 Jim Samson, Music in Transition: a Study o f Tonal
Expansion and Atonality, 1900-1920 (New York: W.W. Norton, 1977),
35.3 Richard Taruskin, The Oxford History o f Western Music, vol. 4
(Oxford: Oxford Univeristy Press, 2005), 59-61.4 Martin Cooper,
French Music (London: Oxford University Press, 1951), 18-21 and
34-77.5 Samson, 34.
2
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modernisms general privileging of pleasure and beauty over
passion and the sublime,6
and in order to avoid confusing that music written by French
composers during 1889-
1940 with the overlapping-but-not-coterminous set of music
written for French audiences
at the same time (with the majority of notable composers of the
period working in Paris),
I shall refer to this repertoire as the Parisian modernist
repertoire.
Although additive harmony is commonplace in the Parisian
modernist repertoire,
it is poorly accounted for by modem music theory. This is due in
part to a simple lack of
attention. Most discussion of the development of tonal language
in the late-nineteenth
and early-twentieth centuries focuses on horizontal-domain
procedures; the most
common narrative is of how adherence to certain common-practice
voice-leading
principles (e.g. conjunct or parsimonious voice-leading, or a
circumscribed set of
teleological background stmctures) allowed for the incorporation
into tonal contexts of
new harmonic successions.7 In revealing these underlying tonal
logics, however, this
type of work often bypasses or reduces-out exactly that
phenomenon - additive
harmonys vertical-domain surface details - which interests this
study. (The
6 Taruskin, 69-72.7 The majority o f this work involves extended
Schenkerian procedures, and most o f this literature focuses on the
chromatic German repertoire o f the time period. A few studies have
applied these techniques to the repertoire in France that is the
focus o f this study: Matthew Brown, Tonality and Form in Debussys
Prelude a L Apres-midi d un faune,'" Music Theory Spectrum 15, no.
2 (Fall, 1993): 127-143; Sylvain Caron, Tonal composition and new
perspectives on Faures harmony, Canadian University Music Review
22, no. 2 (2002): 48-76; Eddie Chong, Extending Schenkers Neue
musikalische Theorien und P h a n ta s ie n Towards a Schenkerian
Model for the Analysis of Ravels Music (Ph.D. diss., Eastman School
o f Music, 2002); Charles Francis Navien, The harmonic language o f
L horizon chimerique by Gabriel Faure (Ph.D. diss., University o f
Connecticut, 1982); Adele Katz, Challenge to Musical Tradition (New
York: Alfred A. Knopf, 1945); Edward R. Phillips, Smoke, Mirrors,
and Prisms: Tonal Contradiction in Faure, Music Analysis 12, no. 1
(March 1993): 3-24; Boyd Pomeroy, Tales o f Two Tonics: Directional
Tonality in Debussys Orchestral Music, Music Theory Spectrum 26,
no. 1 (Spring,2004): 87-118; Felix Salzer, Structural Hearing:
Tonal Coherence in Music (New York: C. Boni, 1952); Jim Samson,
Music in Transition: A Study o f Tonal Expansion and Atonality,
1900-1920 (New York: Norton, 1977); Avo Somer, Chromatic
Third-Relations and Tonal Structures in the Songs o f Debussy,
Music Theory Spectrum 17, no. 2 (Autumn, 1995): 215-241; and Avo
Somer, Musical Syntax in the Sonatas o f Debussy: Phrase Structure
and Formal Function, Music Theory Spectrum 27, no. 1 (Spring,2005):
67-96.
3
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bypassing/reduction of additive chords often relies on the
conventional extended-triad
model of additive harmony, a model whose shortcomings will be
discussed presently.)
There is an analogous difference of focus in work that analyzes
the Parisian modernist
repertoire in terms of systematizable properties of pitch
collections.8 While this work is
eminently valuable in developing alternate lenses with which to
view this repertoire and
in detailing the sonic resources in play, it does not directly
address activity at the musical
surface - activity whose structures and tonal implications are
essential components in this
repertoires aesthetic effect.9
The explanation of additive harmony is therefore left to harmony
textbooks (and
generally then to back-of-the-book chapters in common-practice
harmony texts). The
majority of these textbooks account for additive harmony with
the extended-triad model
8 The vanguard work in this area was done by Richard S. Parks
(Richard S. Parks, Pitch Organisation in Debussy: Unordered Sets in
Brouillards, Music Theory Spectrum 2 [Spring, 1980]: 119-134;
Richard S. Parks, Tonal Analogues as Atonal Resources and Their
Relation to Form in Debussys Chromatic Etude, Journal o f Music
Theory 29, vol. 1 [Spring, 1985]: 33-60; Richard S. Parks, The
Music o f Claude Debussy [New Haven: Yale University Press, 1990]).
Believing traditional tonal theory unable to adequately analyze
Debussys works, Parks instead made analytic use o f an adapted form
o f Allen Fortes pitch-class-set genera; deep structure in Debussys
music - the generator o f its surface detail - is conceived o f as
recurring pitch collections and scales. Another study examining
abstracted pitch-class collections in Debussys music is Mark
McFarland, Transpositional Combination and Aggregate Formation in
Debussy, Music Theory Spectrum 27 (2005), 187-220.
Many o f the nexus set-classes found by Parks (such as 5-35,
6-35, and 8-28) are readily conceived o f as scales, and it is this
thread o f pitch-collection analysis that has been most followed
since Parks initial work. Studies that deal with scales in the
French 1889-1940 repertoire include: Steven Baur, Ravels Russian
Period: Octatonicism in His Early Works, 1893-1908, Journal o f the
American Musicological Society 52, no. 3 (Autumn, 1999), 531-592;
Day-OConnell; Allen Forte, Debussy and the Octatonic, Music
Analysis 10, no. 1/2 (March-July 1991), 125-169; David Kopp,
Pentatonic Organization in Two Piano Pieces by Debussy, Journal o f
Music Theory 41, no. 2 (Autumn, 1997): 261-287; the chapter on Feux
dartifice in David Lewin, Musical Form and Transformation (New
Haven: Yale University Press, 1993): 97-159; Dmitri Tymoczko, The
Consecutive-Semitone Constraint on Scalar Structure: A Link between
Impressionism and Jazz, Integral 11 (1997): 135-179; and Dmitri
Tymoczko, Scalar Networks and Debussy, Journal o f Music Theory 48,
no. 2 (Autumn, 2004): 219-294.9 The case for retaining a focus on
tonal structures in analysis o f Debussys music in particular, made
in reaction to the Parks-led atonal analysis project, is also
argued by Avo Somer and Douglass M. Green. Somer writes Despite
pervasive and often radical chromaticism that may well demand a
fresh evaluation o f its tonal coherence, Debussy's musical
language can be fully understood only in the light o f its
allegiance to tonal centricity and its transformations o f
classical harmonic practices (Somer, Chromatic Third- Relations and
Tonal Structure in the Songs o f Debussy, p. 215); Green more
simply states that How Debussys music is perceived is a knotty
problem that is not helped by ignoring tonal associations in his
works (Douglass M. Green, Review o f The Music o f Claude Debussy
by Richard S. Parks, Music Theory Spectrum 14, no. 2 [Autumn,
1992]: 214-222, p. 217).
4
-
of ninth, eleventh, and thirteenth chords.10 While the details
differ from presentation to
presentation, the basic logic of the model is that the seventh
chord can accommodate
further dissonances arranged in a stack of thirds upwards from
the chordal root. Figure
1.2, a reproduction of example 27.1 from Miguel A.
Roig-Francolis Harmony in Context,
is one example of this model; Roig-Francoli writes of these
chords that the ninth chord
Figure 1.2. Reproduction of example 27.1 from Roig- Francoli,
Harmony in Context, extended tertian chords generated by stacking
thirds on top of the seventh chord.
(ffl *--------- & ----- ........8 ..... ........ ^
;................. :------f t
------0 -----
f t f t -t*
7DM: V v ? v ' i V 11
is used as an independent, nonlinear chord which results from
adding one more third on
top of a seventh chord . . . If we add one more third on top of
the ninth chord, we will
have an eleventh chord, and yet one more third will produce a
thirteenth chord.11 These
extended triads can then substitute for the seventh chord,
allowing for new dissonant
10 Post-WWII university-level textbooks which discuss additive
harmony (however briefly) using the extended-triad model include
Edward Aldwell and Carl Schachter, Harmony and Voice Leading, 3rd
edition (Belmont, CA: Wadsworth Group, 2003); Thomas Benjamin,
Michael Horvit, and Robert Nelson, Techniques and Materials o f
Tonal Music: From the Common Practice Period to the Twentieth
Century, 6lh edition (Belmont, CA: Thomson-Wadsworth, 2003); Leon
Dallin, Techniques o f Twentieth Century Composition (Dubuque, I A:
Wm. C. Brown Company, 1957); Allen Forte, Tonal Harmony in Concept
and Practice, 3rd edition (New York: Holt, Rinehart and Winston,
1979); Robert Gauldin, Harmony Practice in Tonal Music, 2nd edition
(New York: W.W. Norton and Company, 2004); Stefan Kostka, Materials
and Techniques o f Twentieth Century Music 3rd (Columbus, OH:
Prentice Hall College Division, 1989); Stefan Kostka and Dorothy
Payne, Tonal Harmony, 4Ih edition (Boston: McGraw-Hill, 2000);
Robert W. Ottman, Advanced Harmony, 5th edition (New Jersey:
Prentice Hall, 2000); Vincent Persichetti, Twentieth-Century
Harmony (New York: W.W. Norton and Company, 1961); Walter Piston,
Harmony, 4th edition, revised and expanded by Mark DeVoto (New
York: W.W. Norton and Company, 1978); Miguel A. Roig-Francoli,
Harmony in Context (New York: McGraw Hill, 2003); Peter Spencer,
The Practice o f Harmony, 5th edition (Englewood Cliffs, NJ:
Prentice Hall, 2003); Greg A. Steinke, Bridge to Twentieth-Century
Music: A Programmed Course, revised edition (Needham Heights, MA:
Allyn and Bacon, 1999); and Ludmila Ulehla, Contemporary Harmony:
Romanticism through the Twelve-Tone Row (New York: Free Press,
1966).11 Roig-Francoli, 786-787.
5
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chords to participate in tonal progressions. (The nuances of
different versions of the
extended-triad model will be explored in chapter 1.)
While justification for the extended-triad model is occasionally
provided either by
locating chord extensions in the overtone series or by
attributing generative significance
to the stacking of thirds, the main features that recommend the
model are its simplicity
and its facile compatibility with common-practice theories of
harmony. The fact that the
models chord types can be arranged in sequence - triads are
followed by seventh chords,
seventh chords by ninth chords, and so on - has been used both
as a pedagogical
sequence and as a narrative about the development of chord
types.12 By positing that
taller chords are built upon seventh chords, the extended-triad
model can lean on
common-practice ideas about dissonance resolution, the origin of
dissonance, and
dissonances typical role in harmonic progression; the model can
therefore explain novel
verticalities without having to generate too many new principles
of chord structure and/or
behavior.
For all its sequential neatness and adaptability, however, the
extended triadic
model struggles to adequately describe both the range of
additive harmonies found in the
Parisian modernist repertoire and the features of those
harmonies that allow for their
participation in tonal progressions. For one, there are many
chords in this repertoire that
can only awkwardly be accounted for by the extended-triad model,
either as gapped taller
chords or as tertian chords voiced in a non-tertian manner.
Figure 1.3 provides two
examples; (a) shows how the extended triad model has no better
option than labeling the
12 A particularly clear-cut narrative claim is made by Alfredo
Casella: Assuredly the chord o f the major ninth, introduced by
Weber, gave a totally different complexion to the entire musical
language o f the 19th century. Nor is it less evident that the
exploitation o f this chord reaches its culminating point in D
ebussy.. . . The following harmonic concept [the augmented 11th
chord] . . . it is only in Ravel that the new chord is finally used
in a constant, conscious, and spontaneous manner. Alfredo Casella,
Ravels Harmony, The Musical Times 67, no. 996 (February 1, 1926):
125.
6
-
penultimate chord of L enfant et les sortileges as a D 13th
chord with missing third, fifth,
and eleventh, while (b) shows how the model might read the final
tonic chord of
Koechlins rondel Leau as a C 13th chord with missing seventh and
eleventh in which
the ninth is voiced below the third.13 The ill fit of these
labels calls into question the
Figure 1.3. The ill fit of extended-triad labels when applied to
certain chords in the Parisian modernist repertoire.
a) reduction of the final two chords of Lenfant et les
sortileges and an extended-triad interpretation of the highlighted
chord
------------------ = a h 1------ f n *------.....^
......m..........
------------- *:--- >------
o
.' ....w .................................. ^ ** 3 13th chord
missing3rd, 5th, and 11th?
b) reduction of Koechlins Leau from his 9 Rondels, op. 14, mm.
42-44, and an extended-triad interpretation of the highlighted
chord
8...................................
a J J J J S I J J n J j j j ^ 1
rs-e-t H
rr t r r ~ if ....................r\ 1 *-----1-------
r ~----\I -------------------------
1L*---------------------------1
.^..' J........C 13th chord missing7th and 11th, with the 9th
voiced below the 3rd?
centrality of thirds-stacking in a model of additive harmony.
Already for triads and
seventh chords the stacking of thirds is better understood as a
convenient descriptor
rather than as the generator of those chord structures. With
taller chords, not only are
13 A common alternative explanation for the Koechlin chord would
be to label it a tonic chord with added sixth and ninth ; this
necessarily puts it in a separate category (chord with added notes)
to the Ravel chord (extended chords). One o f the aims o f this
dissertation is to be able to discuss these two chords as instances
o f the same phenomenon - vertically enriched tonal
progression.
7
-
generative claims more dubious - would-be eleventh and
thirteenth chords appear more
commonly with gaps than without - but many of the labels
generated by the extended-
triad model, such as those in Figure 1.3, are inelegant at
best.
Another flaw of the extended-triad model is that it inadequately
addresses the
crucial impact voicing has on the identity of additive chords.
While voicing is already
significant for common-practice textures, the characteristic of
additive chords that makes
voicing particularly crucial is that many of them include pitch
classes from two or more
functional categories; Figure 1.4 shows that the penultimate
chord of L enfant et les
sortileges contains a dominant base in the lowest voice as well
as powerful functional
agents from both the subdominant and tonic functions.14 With
these competing
Figure 1.4. Competing functional elem ents in the penultimate
chord of Lenfant et les sortileges.
Tonic agent
Subdominant base & agent
Dominant base
functional elements present, the functional character of a chord
progression is highly
dependent on the vertical ordering of these pitch classes. This
is why the progression in
Figure 1.5a does more violence to the character of the original
progression in Figure 1.1
than the progression in Figure 1.5c does to that in Figure 1.5b.
In Figure 1.5a, the pitch
classes of the chord are inverted to create a root-position C
major-ninth chord with
14 The language o f functional bases and agents is taken from
Daniel Harrison, Harmonic Function in Chromatic Music: A Renewed
Dualist Theory and an Account o f its Precedents (Chicago and
London: University o f Chicago Press, 1994).
X T
8
-
Figure 1.5. Comparison of the impact of inversion upon an
additive chord and upon a triad.
a) inversion of the penultimate chord of Figure 1.1
tr?
b) replacement of the penultimate chord in Figure 1.5a with a V
chord
h
c) inversion of penultimate chord of Figure i.5b
fm
f ?
missing fifth; this replaces the authentic character of the
original progression with a
plagal one. In Figure I.5b the penultimate chord of L 'enfant et
les sortileges is replaced
with a dominant triad, which is then inverted in Figure 1.5c.
Because the scale degrees in
the new penultimate chord in both Figure I.5b and Figure I.5c
are all dominant-
functional, Figure 1.5c changes the degree of finality of Figure
1.5b but not its basic
functional character.15
Another example o f the functional importance o f the vertical
ordering o f pitch classes is the fourth inversion o f the
major-mode Iadd6 chord. Henry Martin has noted that placing the
added note 6 in the bass prevents the chord from serving as a
major-mode tonic; the resulting vi7 chord cannot be a secure
9
-
This brief discussion suggests that pitch-class content alone is
an inadequate
marker of additive chords identity and behavior, and suggests
instead that voicing plays
a crucial role (an idea that will be explored further in Chapter
2). The extended-triad
model, however, struggles to account for this fact; since the
model is designed to be
easily compatible with readily invertible common-practice chord
types, it accommodates
voicing information only as a cumbersome add-on. The following
passage is one small
example of this type of after-the-fact list of voicing
restrictions, taken from Ludmila
Ulehlas book Contemporary Harmony, one of the more thorough
treatments of the
extended-triad model:
Due to the upward climb o f thirds, the thirteenth rightfully
demands the highest position. Occasionally, it will permit the
ninth to preside above, with the thirteenth directly beneath. The
seventh o f the chord must not be placed next to the thirteenth . .
. With inversions, the root may be placed in the uppermost voice..
. Nestled next to the seventh, [the thirteenth] becomes unclear and
permits the upper tones to predominate.16
Although such rules are cumbersome (Ulehla is forced to provide
equivalent rules
for each type of extended triad and each of their chromatic
alterations), forgoing any
discussion of voicing/inversion means that the extended-triad
model forfeits its ability to
draw connections between chord structure and chord behavior. The
consequence of
allowing both chord-tone omissions and free invertibility is
that any pitch combination
can have any step class as a potential root. Figure 1.6
demonstrates: on the basis of pitch-
class content alone, the L 'enfant chord naively could be
labeled a C ninth chord with
missing fifth in fourth inversion (chord a), an E thirteenth
chord with missing third, ninth,
tonic-function chord, as the elevated subdominant influence
created by the presence o f scale degree 6 in the bass suppresses
the influence o f the upper voice tonic elements. Henry Martin,
From Classical Dissonance to Jazz Consonance: The Added Sixth
Chord, unpublished draft o f June 8,2005 (distributed in the course
Theory and Analysis o f Contemporary Tonality, Yale University,
Fall 2007): 40.16 Ulehla, 100-101.
10
-
and eleventh in third inversion (chord b), a B eleventh chord
missing fifth and seventh in
first inversion (chord c), and so on. There is no reason
inherent in the extended-triad
Figure 1.6. Over-applicability of the extended-triad model in
the absence of guidelines for voicing.
a)-c) potential alternate readings of the penultimate chord of
Lenfant et les sortileges.
v &(*) (b) (c)
d)-e) implausible chord labels available with the extended-
triad model
_fl i . |o ilM
(d) (e)
model as to why one reading is preferable to another.
Furthermore, the fact that all
diatonic step classes are present in a thirteenth chord means
that, again allowing for
invertibility and chord-tone omissions, any combination of tones
could be considered
some type of extended triad. The chromatic cluster in Figure
I.6d could ostensibly be
analyzed as a Bb raised-thirteenth chord missing third, fifth,
ninth, and eleventh in sixth
inversion (Figure I.6e), even though the verticality in (d) is
unlikely to be found
participating in conventional tonal progressions.
Unable then to describe the full range of additive harmonies or
to effectively
discuss voicings crucial role in those chords structure and
behavior, the extended-triad
model is not a consistently applicable analytic tool for the
Parisian modernist repertoire.
This dissertation endeavors to develop a more effective model of
additive harmony that,
by making voicing a central concern, better accounts for the
range of verticalities found
11
-
in this repertoire and for how and why they work. Chapter 1
examines the history of
additive harmonic theory, tracing three distinct strategies for
explaining simultaneities
that are not triads or seventh chords: adapting basic chord
types, identifying non
harmonic tones, and formulating new basic chord types. Chapter 2
presents a model of
additive-harmonic chord structure in which chords are conceived
of as pitch-space
voicings constrained by the considerations of tonal plausibility
and chordability; this
pragmatic figured-bass approach to additive harmony explains how
a chords voicing
relates to its most frequent behaviors and to its suitability
for tonal use. Tertian structure
is shown to be one of many potential adornments of an additive
chord, and not an
essential feature in its identity. Chapter 3 investigates the
role voicing plays in the
special additive-harmonic case of polychordal polytonality,
wherein tertian structure is an
essential feature but the desired effect is not the production
of a single composite chord,
but rather the separation of the music into distinct auditory
streams. I close by drawing
bigger-picture parallels between vertical-domain innovations in
music in France and
horizontal-domain innovations in Germanic music, and suggest
that paying attention to
the intricate surface textures of this repertoire adds a new
dimension to our understanding
of the development of tonal language in the late-nineteenth and
early-twentieth centuries.
12
-
Chapter 1
Strategies in Additive Harmonic Theory
While this dissertation develops a model of additive harmony for
the harmonic
writing of the Parisian modernist repertoire, the fundamental
question of additive
harmonic theory - how can one explain those musical moments that
do not immediately
look like a tonal system's basic chord types? - is valid for all
tonal theory. That question
is a necessary byproduct of the reductive impulse in tonal
theory. The central conceit of
tonal theory is that music is organized and impelled by its
chordal successions; that claim
is more easily sustained when the tonal system involves a
limited set of objects and
behaviors. (The case that a certain chord plays a determinative
role in musical succession
is most convincing when that chords use consistently produces
one or a few possible
outcomes, rather than a wide range of potential events.) Hence
the impulse to reduce the
number of objects in a theorized tonal system: the fewer objects
there are, the fewer
outcomes/behaviors there are that need to be explained and the
more tightly formulated
and compelling theories of chord progression can be.
Such a reduction of types enabled the construction of the first
comprehensive
tonal theory, Jean-Phillipe Rameaus 1722 Traite de I harmonie;
using a theory of chord
inversion, Rameau pares the wide variety of figures found in
earlier thoroughbass
treatises down into only two chords - the triad and the seventh
chord - which can appear
on every scale degree. The tremendous power of this theoretical
move has been hinted at
above and is discussed in depth elsewhere, and the privileged
central status of invertible
triads and seventh chords has been foundational for the majority
of tonal theory since
13
-
Rameau.1 Additive harmonic theory is a necessary byproduct of
this reductive move.
Musical practice throws up many verticalities that are not (or
do not immediately appear
to be) triads and seventh chords (or simple passing/neighbor
figurations thereof); additive
harmonic theory is required to explain how these moments fit
into the tonal system.
This chapter defines three foundational strategies for
explaining non-standard
verticalities - the modified-basic-types strategy, the
non-chord-elements strategy, and the
new-basic-types strategy - and uses them to examine the history
of additive harmonic
theory. The modified-basic-types strategy explains novel
verticalities as
modified/expanded versions of conventional chords (as in the
extended-triad model of
ninth, eleventh, and thirteenth chords discussed in the
introduction). By positing that
these modified/expanded chords and their progenitors behave
similarly (e.g. that a C9
chord can substitute for the C7 chord from which it is dervied),
this strategy allows for an
expanded range of verticalities to participate in conventional
progressions.
The non-chord-elements strategy explains novel verticalities as
the coincidence
of conventional chords and non-chord-elements. This approach is
more theoretically
parsimonious than the modified-basic-types strategy; by labeling
certain pitches in a
verticality as being outside the tonal system, this strategy
leaves the core objects of the
system untouched.
The new-basic-types strategy accounts for novel verticalities by
defining new
chord types that must then be worked into the tonal system; the
strategy folds novel
verticalities into the system as entities unto themselves (i.e.
not as derivatives as triads
1 A fine discussion o f the historical origins and theoretical
basis for Rameaus inversion theory in the Traite, and o f how that
theory allowed Rameau to be the first theorist to truly examine
harmonic succession, is in Joel Lester, Compositional Theory in the
Eighteenth Century (Cambridge, Massachusetts: Harvard University
Press, 1992), 100-108 and 115-122.
14
-
and seventh chords). This back to the drawing board approach
reexamines the basic
primacy of triads and seventh chords established by Rameau; a
measure of theoretical
parsimony is sacrificed so that other chord types can be
recognized as foundational.
While this strategy is rarely used as basis for a complete
system of harmony (i.e. not
every vertical formation is allowed to be a chord unto itself),
discussion of alternative
chord types (e.g. quartal, quintal, or secundal chords) is found
sprinkled through
twentieth-century harmony texts.
Figure 1.1. Spectrum between competing ideals in additive
harmonic theory.
Modified-basic-typesstrategy
WWnmgi ng A I I I ^ Thsoreticalappticabity | | parsimony
New-basic-types Non-chord-elementsstrategy strategy
These three strategies can be understood as marking different
positions on a
continuum between two theoretical ideals (Figure l.l).2 On one
side of the continuum
2 These strategies share ideas with but are not identical to
(and indeed were developed independently from) the four general
strategies for modeling dissonance discussed by Richard Cohn in
Chapter 7 o f his book Audacious Euphony: Chromatic Harmony and the
Triads Second Nature (New York: Oxford University Press), 139-168.
Cohn defines four general strategies for modeling dissonance:
suppression (reducing dissonant chords away from deeper levels o f
structure), substitution (analyzing dissonant chords as
modifications-by-substitution o f other dissonant chords, e.g.
analyzing the diminished seventh chord as substituting i> 6 for
5 in a dominant-seventh chord), reduction to a subset (analyzing
dissonant tetrads as one o f their consonant triadic subsets, i.e.
analyzing an F#0 7 chord as an A-minor triad), and combination to a
superset (analyzing dissonant tetrads as the combination o f two
triads).
The main difference between my categories and Cohns is that his
describe the various ways dissonant tetrads are folded into
theories o f progression for nineteenth-century German music, while
mine describe the ways o f accounting for any sort o f novel
verticality in theory ranging from Rameau to the present. This
difference in aim accounts for the ways in which the two sets o f
strategies interact. Reduction-to-a-subset and substitution are
both forms o f the modified-basic-types strategy o f additive
harmonic theory, in that both try to explain a novel formation as a
modified form o f a more common quantity. However, as will be
explored in this chapter, the modified-basic-types strategy has
more manifestations than the reduction-to-a-subset and substitution
applications found in nineteenth-century
15
-
lies the ideal of wide-ranging applicability - the desire for a
non-reduced theoretical
system that can account for the entire variety of found
verticalities. Opposite it lies the
ideal of theoretical parsimony - the desire for an elegant and
concisely formulated
system. The three strategies strike different balances between
the two ideals. The
modified-basic-types strategy is a centrist position; it
stretches/loosens the definitions of
some objects in the system so that the system can accommodate
new sonorities without
significantly expanding its basic set of objects. The
non-chord-elements strategy is more
right-leaning in that it leaves the basic objects and mechanics
of the system undisturbed
- novel verticalities are not admitted into the system as
integrated objects. In examining
these two strategies in detail below, we will see how the
non-chord-elements strategy
interfaces with the modified-basic-types strategy. One approach
ends where the other
begins - a pitch must be either a chord tone or a non-harmonic
tone - and the position of
the line that separates what is inside the system from what is
outside the system is a key
issue for additive harmonic theory. The new-basic-types strategy
is more liberal than the
modified-basic-types strategy in that it allows a greater range
of found verticalities to be
considered basic chords. Incorporating those new chord types
into the tonal system,
though, requires formulating additional rules of progression
that can dilute the predictive
power of the tonal model. (Figure 1.1. shows these strategies
only in the abstract - the
work of individual theorists may utilize one or more of these
strategies. While some
writers pursue one strategy and suppress or belittle others,
other writers keep two
German theory, and so I collapse those two categories into one
broader one. Similarly, combination-to-a- superset is but one o f
the novel types of chord structures devised in order to explain the
broader range o f non-standard verticalities found in the Parisian
modernist repertoire. And because I am interested in the novel
verticalities at the musical surface, I do not deal with the
strategy of suppression or other aspects o f theory that deal with
deeper tonal structure beyond the explanations they give for why
surface detail is suppressable. So while Cohn and I do share a
similar approach to categorizing families o f theories, the
differences between our two sets o f categories are significant
enough that I will retain the use o f my own terms in this
chapter.
16
-
approaches in productive equilibrium or foreground one while
using others to paper over
any cracks. Individual theories, then, can be mapped anywhere on
the continuum or
indeed across several points on the line, and not just at the
nodes of Figure 1.1.)
Reading the history of additive harmonic theory in terms of
these three strategies
clarifies what is at stake in the often confusing and/or
trivial-seeming rules about the
invertibility of ninth chords or debates about the viability of
the unresolved appoggiatura
as a type of non-harmonic tone. In play is an issue fundamental
to theorizing musical
systems - the challenge of developing a system that both a) has
efficient and consistent
internal mechanics and b) is applicable to a range of real
musical situations. Viewing the
history of additive harmonic theory in terms of these three
strategies also allows
connections to be drawn between theories that work with
different repertoires. In the
sections that follow, I investigate each of the strategies in
turn and examine how their
ideas adapt to evolving musical styles, and also explore moments
of interaction between
strategies. First, though, I will examine additive harmonic
thinking in Rameaus Traite in
order to shed light on the interrelated genesis of the
modified-basic-types and non-chord-
elements strategies.
1.1 - Additive Harmonic Thinking in Rameaus Traite
As the first theorist to develop a fully formed theory of
tonality, Rameau is also
the first to grapple with additive harmonic theory. While
Rameaus system is built on the
primacy of the perfect chord (triad) and the seventh chord and
the interactions between
them, he does discuss the generation and behavior of three
additional chord types,
realized here in Figure 1.2: the ninth chord (chord al), the
eleventh chord (bl) and the
chord of the large sixth (c). All three of these additional
chord types are derived from the
17
-
Figure 1.2. Realization of the chord types described in Ram eaus
Traite other than the perfect chord and the seventh chord.
(Locations of Ram eaus written descriptions are given in the
in-text footnotes.)
Open note-heads comprise the initial basic chord; black note
heads are the added/subposed notes. The fundamental bass of any
chord is the lowest open note- head.
& 1 --- 1 8 *i f - 1L(al )
JL .... - i | -
_J--- -----m------(bl )
- m 1
fcjl --- ----i(c)
gg___ .. . - -... ... .....W~ -..... i
-
chords are generated by subposed bass tones, the fundamental
bass of the chord remains
the fundamental of the initial seventh chord. This is the
interpretation shown in Figure
1.2 for chords al and bl.4 In these two different derivations of
the same chords, we are
already exposed to a tension in additive harmonic theorizing
between, on the one hand,
generating modified chord types by using a structural principle
consistent with that used
to generate the systems basic chord types, and on the other,
having a chords identity be
defined by how it behaves in context.
Both the ninth and eleventh chords can be altered to involve the
raised leading
tone of the minor mode, producing the chord of the augmented
fifth (Figure 1.2, chord
a2) and chord of the augmented seventh (b2) respectively. Both
chords can also appear
in stripped-down forms; Rameau writes that the ninth chord
always appears without
seventh above the tonic note (a3, in C major), while the
eleventh can appear without the
third above the fundamental, the fifth above the fundamental, or
both (this last scenario is
shown as b3). Rameau calls this last form abnormal (heteroclite)
since it is not divided
[by fifths and thirds] as are the other chords.5
The chord of the large sixth (chord c in Figure 1.2) is also
produced from the
primary chord, specifically via the addition of a sixth above
the triads fundamental.
While Rameau recognizes that this chord might be interpreted as
an inversion of the
seventh chord built on the minor perfect chord, he stresses that
the context of an irregular
cadence (Figure 1.3) demonstrates that the chord of the large
sixth is distinct from a
4 Ibid., 88-89.5 Ibid., 108-109, 240, and 91.
19
-
minor seventh chord.6 Were D the fundamental of the first chord
in Figure 1.3, not only
Figure 1.3. R am eau's example II.6; an irregular cadence.
Major thirdAdded sixth
Fifth
Fourth note Fundamental bass Tonic note
There is a dissonance of a second between these two parts.
Irregular cadence in the major mode
would the fundamental bass progression be weakened (the
fundamental bass would not
be progressing by descending fifth, descending third, or
ascending fifth), but the C5 in
the alto voice would need to resolve as a dissonance (since the
dissonant sevenths of
seventh chords must resolve down by step). Instead, reading the
chord in Figure 1.3 as a
chord of the large sixth, with a fundamental of F, produces a
stronger progression in the
fundamental bass and means that the D in the upper voice is now
the added dissonance,
explaining its upward motion out of the C-D second.
Rameau devotes individual chapters to the ninth, eleventh, and
large-sixth chords,
and these adapted chord types do important work in relating
non-standard verticalities to
the central triads-and-sevenths machinery of his theory. Ninth
and eleventh chords are
used to explain various forms of dissonance treatment, including
the behavior of
6 The chord formed by adding a sixth to the perfect chord is
called the chord o f the large sixth. Although this chord may be
derived naturally from the seventh chord, here it should be
regarded as an original. Ibid., 75.
20
-
suspended fourths at cadences. In Figure 1.4, the resolution of
D4 to C#4 over a sounding
static bass is explained as the downward resolution of the
seventh in a progression of the
fundamental bass from E-A. This means that the bass-voice A of
the second chord is a
subposed bass note, and that the chord is an A-E-D heteroclite
form of the eleventh
chord.7
Figure 1.4. Transcription of R am eaus example 11.11 -
explanation of a suspended fourth as a heteroclite eleventh
chord.
The five upper parts are sounding voices; the fundamental bass
is R am eau's analytic gloss.
p ........ = 1
k ........................................ = 1
.................................................. J
$ ..... .................. ..... ....
# = = = = ^
u
f t O -
Tenor*TWt-----------------------------O-----------------------
.......................( K ...............................
O O -----------------------1
""U.................................................................
r - & --------------------------------------------------
-
BassA
Fundamentalbass
4 7 i
7 7s
* The tenor represents the fundamental bass in chords by
supposition. A This is the bass o f chords by supposition.
There is an additional wrinkle to Rameaus presentation of ninth
and eleventh
chords, beyond his generating ninth and eleventh chords by
subposition rather than
7 Ibid., 90.
21
-
upwards third-stacking, that differentiates it from modern-day
extended-triad models.
Rameaus ninth and eleventh chords are generated from the primary
chords using the
constituent intervals of those chords, and so their structure
represents an application of
the modified-basic-types strategy for dealing with novel
verticalities. Rameaus
treatment of these chords in context, however, involves the
non-chord-elements strategy.
The non-chord elements are the subposed bass notes themselves.
Rameau frequently
describes these bass notes as supernumerary, and they do not
participate in the action
of the fundamental bass nor can they be moved from the lowest
voice; in Rameaus
system, ninth and eleventh chords can not be inverted as primary
chords can. The
subposed bass tones are coloristic rather than functional, and
are therefore outside of the
system of chord progression. By way of contrast, the added sixth
plays an integral role in
the chord of the large sixth - the dissonance it creates with
the chordal fifth ensures that
the final chord of Figure 1.3 is understood as an arrived-at
tonic, and not as the dominant
of the first chord with the fundamental of F.8 This means that,
in analytical application,
Rameaus conception of ninth, eleventh, and large-sixth chords is
the reverse of that
found in most modem textbooks, in which elevenths and especially
ninths are freely
moveable chord members while the added sixth is a
voicing-dependent ornamental tone.
The main benefit of conceiving of ninth and eleventh chords as
the
combination of fundamental seventh chords with non-chord
subposed basses is that it
simplifies the explanation of dissonance: if the ninth and the
eleventh above the subposed
bass are both actually sevenths above the fundamental bass, then
a) the rules for their
treatment are clear and b) the seventh is still the source of
all dissonance, a fact central to
8 Ibid., 75.
22
-
Rameaus mechanistic conception of the tonal system.9 This allows
Rameau to elegantly
explain unusual voice-leading structures like that in Figure
1.5a. The apparent
transgression of the F2-E3 seventh resolving to a D2-D3 octave
in similar motion can
instead interpreted as a fifth above the true fundamental bass
of A2 that correctly
resolving to the octave; the F2 is then interpreted as a
subposed bass note (Figure 1.5b).
Locating the fundamental bass at A2 also means that the ninth in
(a) is the true seventh,
which explains why it resolves downward by step to a
third.10
Figure 1.5. Transcription and annotation of Ram eaus exam ples
11.41 and II.42; use of supposition to explain voice-leading
novelties.
(a) (b)
3 9 3 3 7 3
Apparent transgression - F-E seventh resolving Rameaus
explanation: the F in the progression on the leftto the D-D octave
in similar motion is a subposed bass note; the real fundamental o f
the middle
chord is A
But because the lowest tone of a ninth chord cannot participate
in progressions of
the fundamental bass, Rameau is at pains to explain the
dominant-to-tonic falling-fifth
bass-motion progression from the second to the third chord of
Figure 1.6a. If ninths
above the bass are interpreted as sevenths above the fundamental
bass, they must resolve
as sevenths, either to an octave above a first-inversion chord
(Figure 1.6b) or to a third
above a root position chord (Figure 1.6c); the second of these
acceptable resolutions
appears in the progression from the third to the fourth chord of
Figure 1.6a. Motion
9 Ibid., 112. Although Rameau talks o f both major and minor
dissonances, the minor dissonance o f theseventh is prior, in that
its presence above a perfect major chord is what turns the third o
f that triad into themajor dissonance.10 Ibid., 132-133.
23
-
Figure 1.6. R am eau's inability to explain ninths resolving
over falling-fifth bass motion.
a) Ram eaus example III.87; improper resolution of the ninth
chord above F3.
The ninth resolved by the fifth
Potential proper resolutions for the ninth chord above F3
(derived from Ram eaus example III.86).
b) resolves the seventh above the fundamental b ass to an
octave
c) resolves that seventh to a third.
(b)
Bassocontinuo
Fundamentalbass
l=i(C)
from the ninth to a fifth above the bass, as between the second
and third chords 1.6a, is an
improper resolution. Rameau seems to acknowledge that this is an
instance where his
tightly constructed theoretical system fails to account for a
relatively common
progression, writing We might further wish to resolve the ninth
by the fifth, with the
bass ascending a fourth. The harmony arising from this, however,
is improper and so we
leave this matter to the discretion of men of good taste.11
11 Ibid., 295-296 .
24
-
Because ninths and elevenths are not full chord tones and
therefore cannot
participate in inversion, Rameau also cannot use them to explain
dissonance treatment in
the bass voice; Rameau cannot draw a connection between the
suspended A3-G3 bass
motion of Figure 1.7a and the suspended middle voice of Figure
1.4. The motion of A3-
Figure 1.7. Ram eaus difficulty explaining suspensions in the
bass voice.
a) Ram eaus example IV.34
Same
or
b) what Ram eaus interpretation would be were the bass voice of
(a) moved into an inner voice; E would then be the subposed bass
voice
I ................7 4
Fundamental < 6) : 1-----
c) Ram eaus interpretation of (a); the A3 is a subposed bass
note
X T65
FundamentalBass m
G3 in Figure 1.7a cannot be a resolution of the seventh of a
chord with a fundamental
bass of B to the third of chord with a fundamental bass of E
(the interpretation presented
25
-
in Figure 1.7b) because that would make the E4 of the suspension
chord in Figure 1.7a a
subposed bass note that was not appearing in the lower voice.
Instead, Rameau is forced
to label the A3 as a subposed bass note (his interpretation is
presented as Figure 1.7c), an
interpretation that provides no explanation for that notes
downward-resolving tendency.
Rameaus Traite, then, uses both the modified-basic-types and
non-chord-
elements strategies to explain novel formations. The use of both
stategies demonstrates
the care Rameau takes in considering non-standard verticalities
within the context of his
tonal system, and each of the two strategies is further explored
by Rameaus successors.
1.2 - The Modified-Basic-Tvpes Strategy
In German theory of the eighteenth century, Rameaus ideas
combine with what
David A. Sheldon has described as a German compulsion to
categorize and explain
striking departures from the theoretical norm to produce
analytically more flexible
models of extended triads.12 One way of enabling extended triads
to account for a
broader range of novel formations is to allow the bass notes of
extended triads to
participate in inversion. Figure 1.8, a realization of a written
description from Johann
David Heinichens 1728 Der General-Bass in der Composition,
demonstrates this type of
usage.13 Heinichen explains the 7/4/2 and 7/5/2 chords in (b) as
inverted forms of the
9/5/3 and 9/6/3 chords in (a). The tendency for D to resolve to
C, present between D5
and C5 in the foundational forms, is preserved in inversion;
this explains why D3
resolves to C3 in the lowest voice of the inverted chord forms,
when sevenths above the
bass (found between D3 and C5 in the inverted forms) would
generally be resolved by
12 David A. Sheldon, The Ninth Chord in German Theory, Journal o
f Music Theory 26, no. 1 (Spring 1982), 61.13 Johann David
Heinichen, Der General-Bass in der Composition (Dresden: the
author, 1728), 208-209, quoted in Sheldon, 64.
26
-
Figure 1.8. Heinichen's use of invertible ninth chords to
explain novel dissonance treatment.
a) foundational ninth chords; the C3-D5 ninth is resolved by the
upper voice moving down by step
b) inversion of the chords in (a); D still resolves downward by
step to C
8.....-.-......-1 m= :::.=.=.
-------------------------------1.....r =\ t r ^
c) Heinichens alternate explanation of the 7/4/2 chord as being
generated by an anticipated resolution in the upper parts
anticipated resolution normal
$
t
the upper voice moving downwards by step. Heinichens explanation
requires a
conception of ninth chords as entities unto themselves - objects
well-defined enough to
retain their identity in inversion.14 Heinichens further
discussion of the 7/4/2 chord also
demonstrates the cheek-by-jowl relationship of the
modified-basic-types and non-chord-
14 It is worth noting here that, unlike in modem conceptions o f
chord-vs-non-harmonic tone, verticalities that resolve over the
same bass note were commonly considered distinct chords in much
eighteenth-century theory. (Lester, 209-210.) Examples include
Rameaus chords o f the augmented fifth and chords o f the augmented
seventh.
27
i
-
elements strategies. Figure 1.8c displays Heinichens description
of how the same 7/4/2
chord can be understood as arising from an anticipated
resolution in the upper parts.15
While the musical structures of Figures 1.8b and 1.8c are
similar, their tales of origin are
different; in 1.8b, the melodic motion occurs because of the
energies inherent in a ninth
chord, while in 1.8c the 7/4/2 chord coincidentally appears as
the byproduct of a melodic
process.
Another way of expanding the analytic reach of extended triads
is to read a thirds-
stacking process into the structure of triads and seventh
chords, and then to extend that
process to produce not only the ninth chords and gapped eleventh
chords found in the
Traite, but full eleventh and thirteenth chords as well. This
variety of the extended-triad
model is developed in the 1755 first edition of Friedrich
Wilhelm Marpurgs Handbuch
bey dem Generalbasse und der Composition. Figure 1.9 reproduces
James Martins
Figure 1.9. Set of extended chords from the 18' edition of
Marpurgs Handbuch. Fundamentals of the foundational seventh chords
are underlined.
ninth chorda eleventh chorda th irteen th ohordedb &
db
rd
r td d
t&
at
ga g %g#a
b
b & / g
db
s 0 Q c e0
a ae o
a*1c ga
o a g
transcription of Marpurgs complete set of dissonant chords.16
The chords are created by
15 Heinichen, 187-189, reproduced in Sheldon, 66. A similar
temporal explanation o f the 7/4/2 chord is found in the 1755 first
edition o f Friedrich Wilhelm Marpurgs Handbuch bey dem
Generalbasse und der Composition, with the sole difference being
that the explanatory mechanism is delay o f the bass rather than
anticipation o f the upper parts. (David Sheldon, M arpurgs
Thoroughbass and Composition Handbook: A Narrative Translation and
Critical Study [Stuyvesant, NY: Pendragon Press, 1989]: 20.)16
James M. Martin, The Compendium Harmonicum (1760) o f Georg Andreas
Sorge: A Translation and Critical Commentary (Ph.D. dissertation,
The Catholic University o f America, 1981): 45
28
-
consecutively stacking subposed thirds beneath the fundamental
of a seventh chord.17
Unlike Rameaus ninth and eleventh chords, these extended triads
are true chords; their
subposed pitches are full chord tones in that they can
participate in inversions.
In the first edition of the Handbuch, Marpurg does not restrict
how these extended
triads operate; he states that the chords can appear in any
inversion, and that various
combinations of chord members can be omitted in reduced-voice
textures.18 This
combination of maximally extended stacks of thirds, unrestricted
invertibility, and
potential omissions lends Marpurgs extended-triad model
tremendous analytic
flexibility. Marpurg can provide a chordal explanation for
almost any dissonant
formation; for instance, the resolution of the 9/6/4 chords
double suspension in Figure
1.10 can be explained as the downward resolution of the eleventh
and thirteenth of a first-
inversion thirteenth chord on D with fifth, seventh, and ninth
omitted.19
Figure 1.10. Marpurgs first-inversion thirteenth chord with
fifth, seventh, and ninth omitted.
On the bottom staff, fundamental bass tones are shown with open
note heads, and omitted chord tones with filled- in note heads.
Fundamental bass : omitted chord tones
Sheldon, M arpurgs Thoroughbass and Composition Handbook, 1.18
J. Martin, 46-47.19 Ibid., 15.
29
-
One of the drawbacks to such a liberal conception of extended
chords, however, is
that many of the available pitch combinations a) rarely appear
in practice and b) are
harshly dissonant. Marpurgs contemporaries Georg Andreas Sorge
and Johann Philipp
Kimberger aired these criticisms in print, and in response
Marpurg altered his extended-
triad model in the 1762 second edition of the Handbuch; most of
the complete forms of
eleventh and thirteenth chords are omitted, while the number of
potential inversions of
the taller chords is restricted.20 And instead of subposing
continuous chains of thirds
beneath foundational seventh chords, Marpurgs 1762 edition
reverts to Rameaus
gapped method of subposition: the eleventh chord is now
generated by subposing a fifth
beneath a seventh chord (i.e. with no intervening third), and
the thirteenth chord by
subposing a seventh beneath a seventh chord.21 These
modifications both address
practical considerations (i.e. the rarity of full eleventh
chords in practice) and highlight
the connection between Marpurgs theories and those of Rameau.22
The modifications to
the 1762 edition of the Handbuch did not stop criticism of
Marpurgs work - he held
another full-blown public debate with Kimberger in the 1770s -
but they do reveal
awareness on Marpurgs part of the potential pitfalls of overly
liberal models of extended
triads.23
Another way to modify the extedend-triad model to account for
certain types of
unusual dissonance treatment is to extend the stacking of thirds
upwards rather than
downwards from the foundational seventh chord. This is first
done in Sorges
20 Ibid., 10. For detailed discussion o f Sorge and Kimberger's
criticisms o f Marpurg, see Lester, 196-197 and 247-249.21 Sheldon,
The Ninth Chord in German Theory, 69.22 Marpurg valued this latter
feature because positioning himself as the true heir to Rameau's
ideas - a stance Joel Lester has shown generally to be false and
based on poor transmission o f Rameaus work in Germany - was a
central component in his defense against his critics. (Lester,
150.)23 For discussion o f the second Marpurg-Kimberger debates,
see Lester, 250-256.
30
-
Compendium Harmonicum of 1760, where the dominant ninth chord is
presented as the
upward extension of a dominant seventh.24 This allows Sorge to
connect the behavior of
the leading-tone seventh chord and the dominant seventh chord,
with the latter being seen
as the generator of the former. In Figure 1.11, example 4 shows
how Sorge considers
leading-tone seventh chords to be rootless inversions of
root-position dominant ninth
chords.25 The connection between the leading-tone seventh chords
and the dominant
ninth chord is then further emphasized in example 5, which
demonstrates how the
seventh of the leading-tone seventh chord, in any inversion and
in both the minor and
major modes, often resolves to the root of the generating
dominant chord.
Figure 1.11. Jam es M. Martins transcription of Sorges Plate VI,
exam ples 4 and 5; presentation of the dominant ninth chords and
their relationship to the leading-tone seventh chord.
Generating extended triads in the upward direction gives the
ninth chord a firmer
acoustic foundation than does generating them by subposition:
Sorge can extract the
24 Sheldon, "The Ninth Chord in German Theory, 73-75.25 J.
Martin, 204 and 109.
-
dominant ninth from the harmonic series by tempering the fourth,
fifth, sixth, seventh,
and ninth overtones of the fundamental,26 whereas Marpurgs
subposition model is
dependent on a tenuous conception of the undertone series and
combination tones
imported from Rameau.27 This solid acoustic foundation means
that Sorges dominant
ninth chord is analytically more flexible; because it an
independent vertical entity - the
ninth is free and unsuspended (i.e. what Kimberger will label an
essential
dissonance) - the ninth chord is not limited to any specific
preparatory context.28
The dominant ninth chord is the only extended triad Sorge admits
into his theory
- all other dissonances are explained as produced by inverted
sevenths, suspended
dissonances, or some combination of the two. When combined with
Johann Philipp
Kimbergers clear formulation of the distinction between
essential and non-essential
dissonances, Sorges model of additive harmony - with the
upward-generated ninth as
the sole true extended chord, and all other dissonances
explained as non-chord elements -
becomes the baseline for much nineteenth-century theory,
including the entire
Conservatoire tradition running from Charles-Simon Catel in 1802
to Theodore Dubois
texts in the early twentieth century29
One theorist in the nineteenth century who does discuss extended
triads beyond
the ninth chord is Siegfried Dehn. In his 1840 Harmonielehre,
Dehn, like Sorge, limits
his discussion of extended triads to chords built around the
dominant, but expands the
26 J. Martin, 37-38 and 44.27 Heinrich Christoph Kochs 1782
Versuch einer Anleitung zur Composition attempts to reconcile the
subposition model with the overtone series. As in Marpurgs late
theory, Koch produces ninth, eleventh, and thirteenth chords by
subposing thirds, fifths, and sevenths beneath seventh chords. In
these constructions, however, Koch acknowledges that the lowest
note these chords is now the acoustic generator o f the seventh
chords fundamental; the bass note is called the eigentlich
fundamental, while the upper note is mitklingend. (Sheldon, The
Ninth Chord in German Theory, 79.)28 J. Martin, 204.29 Mildred
Freeman Rieder, Traite d harmonie by Theodore Dubois in the Context
o f ^ -C e n tu ry French Harmonic Theory and Pedagogy, (M.M.
thesis, The University o f Western Ontario, 1995): 21-24.
32
-
system to include eleventh and thirteenth chords. Ninth chords
are generated not upward
from the root of the dominant triad, but by placing a third both
above and below the
diminished triad of a key (Figure 1.12a; the added pitches are
shown with filled-in note-
heads); eleventh and thirteenth chords are generated by placing
the tonic note beneath the
dominant seventh and ninth chords respectively (Figure 1.12b and
c).30 The focus on the
dominant allows Dehn to explain why the thirds are absent in the
eleventh and thirteenth
chords - they are omitted because it is the tone of resolution
of the upper-voice dominant
chords sevenths.31
Figure 1.12. Dehns ninth, eleventh, and thirteenth
chordsrealized in C major.
(a) (b) (c)
Dehn limits his application of these extended triads more than
either Marpurg or
Sorge - for Dehn these chords cannot appear in inversion nor can
they sustain omissions
of their roots, uppermost tones, or leading tones - and
justifies his restrictions on
practical grounds. The extended triads cannot be inverted
because their identity is
dependent on a specific voicing. The chords do not belong to the
family of invertible
Stammakkorden because the closest possible arrangement of their
component pitch
classes does not form a stack of thirds.32 The extended triads
therefore require a specific
30 Siegfried Dehn, Theoretisch-praktische Harmonielehre mil
angefugten Generalbassbeispielen (Berlin: Verlag von Wilhelm Thome,
1840): 119-123.31 Ibid., 122.32 Ibid., 119-120: Zu den
Stammakkorden kann er aber nicht gezShlt werden, weil seine Tone,
wenn sie nach der natfirlichen Reihenfolge er diatonischen
Tonleiter von C dur oder C moll geordnet wtirden, nicht terzenweise
ilber einander zu liegen kommen, sondern in folgender Gestalt
[figure showing G-A-B-D-F in major, G-Ab-B-D-F in minor].
33
-
voicing to produce a properly tertian chord structure, and so
inversions of the ninth,
eleventh, and thirteenth chords that obscure that tertian
structure are prohibited; the same
principle rules out voicings such as those in Figure 1.13, in
which the would-be chordal
ninth is adjacent to the chord root.33 Omission of chord tones
is dismissed for the simple
Figure 1.13. Dehns prohibited voicings of the ninth chord, with
ninths placed directly above the chord roots.
reason that these extended triads cease to be themselves when
tones of the ninth,
eleventh, or thirteenth are omitted, and because Dehn furnishes
other labels for the chords
resulting from those omissions (e.g. vii07 for a dominant major
ninth chord missing its
root).34
While it does elevate extended chords to the status of
key-delineating
Hauptakkorde (I, V, or chords containing the 7 - 4 tritone),
tethering extended chords so
Although not Stammakkorde, ninth, eleventh, and thirteenth
chords are still true chords in Dehns system (and not the
coincidence o f Stammakkorde and non-harmonic tones, as ninth and
eleventh chords are theorized in Rameaus Traite)-, all three
extended triads belong to the privileged set o f key-defining
Hauptakkorde: I, vii, vii 7, vii7, V7, V9, V7/ I and V9/ 1. (Ibid.,
121-122).33 Ibid., 223.34 Ibid., 120-121. Dehn is quite severe his
denouncement o f the practice labeling chords based on absent
notes, calling into question whether or not theory that involves
such practices is worthy o f the term system: Wie schon weiter oben
bemerkt wurde, konnen nur solche Intervalle eines Akkords
weggelassen werde, die ihn nicht wesentlich von andem Akkorden
unterscheiden. Nun unterscheidet sich der Nonenakkord eben durch
seine None von den Vierklangen, die nach ihrem Sussersten Intervall
Septimenakkorde genannt werden; wenn man daher von einem
Nonenakkord den Basston fortlasst, so bleibt er kein Nonenakkord
mehr, sondem die iibrigbleibenden T8ne bilden ihrer terzenweisen
Lage entweder einen kleinen oder einen verminderten Septimenakkord;
ISsst man hingegen die Oberstimme des Nonenakkords fort, so bleibt
ein Dominantenakkord, dessen Umkehrungen man eben so wenig, wie die
Umkehrungen der erst genannten Septimenakkorder, fur Umkehrungen
eines Nonenakkords mit weggelassener Ober- order Unterstimme
ansehen kann. Solche Annahme k6nnte aber auch leicht dahin
verleiten, z. B. den Dreiklang h d f, oder den Dreiklang g h d,
oder endlich den Dreiklang d f a, von Nonenakkord g h d f a
abzuleiten, indem man von diesem entweder die Tftne g und a, oder f
und a, oder endlich die T8ne g und h fortlSsst. - Von einer
systematischen Construction der Akkorde kann bei einem solchen
Verfahren gar nicht die Rede sein, oder man mOsste die
Grundbedeutung des ursprlinglich griechischen Wortes Systema ganz
ausser Acht lSssen.
34
-
closely to the diminished triad limits their explanatory power.
If extended chords are
neither invertible nor transposable to other scale degrees, the
model can only account for
a handful of progressions: V9 resolves to I and cadences with
multiple suspended upper
voices.35 Dehns model, then, leans toward the opposite pole of
Figure 1.1 than does
Marpurgs 1755 model; whereas Marpurgs chords are endlessly
adaptable but often
theoretically dubious, Dehns chords are more ontologically
secure but less analytically
adaptable.
Marpurg, Sorge, and Dehn have presented three different ways of
and
justifications for generating extended triads: Marpurg generates
them by subposing the
various intervals of a dominant seventh chords below a
fundamental-bass carrying
dominant seventh, Sorge stacks thirds upwards from a chordal
root by drawing pitches
from the overtone series, and Dehn places thirds on either side
of a diminished triad with
the practical aim of categorizing the extended triads as
Hauptakkorde. Even though we
have not yet looked at theorists who apply modified-basic-types
thinking to the Parisian
modernist repertoire, in the analytic implications of these
three ways of generating
extended triads, we see the range of issues that any application
of modified-basic-types
strategy must contend with: What justifies/motivates the
modification of the basic chord
types? How much modification can the basic chord types withstand
(either in terms of
extension or chromatic alteration) before their identity is
compromised? How flexible are
the resulting modified basic types (i.e. are they freely
invertible, and on what scale
degrees/in what contexts can they occur)? These parameters will
structure the following
35 Dehn specifically notes that the ninth chord is also not
suitable for deceptive cadential progressions, as the necessary
resolution o f the ninth ((die Auslosung der Terzdecime der Tonart)
would produce a seventh chord (Nebenseptimenakkord) above the
submediant. (Ibid., 223.)
35
-
discussion of late-nineteenth-, twentieth-, and
twenty-first-century applications of the
modified-basic-types strategy.
Issues o f Generation
After Dehn, justification for extended-triad manifestations of
the modified-basic-
types strategy is sought either in pragmatic appeal to a
stacked-thirds approach or in the
overtone series. A pithy justification of the stacked-thirds
approach is made by Robert
Ottman: The principle of chord construction by adding thirds can
be continued past the
triad and the seventh chord to include chords of the ninth, the
eleventh, and the thirteenth.
. . No further additions are possible, since another third
duplicates the root two octaves
higher;36 similarly unadorned statements are found in the work
of Emile Durand,
Theodore Dubois, and Leon Dallin.37 While this justification is
parsimonioius and
pedagogically useful, it does not necessarily illuminate the
true nature and behavior of
extended triads, a fact alluded to by Stefan Kostka and Dorothy
Payne in their textbook
Tonal Harmony: Just as superimposed 3rds produce triads and
seventh chords,
continuation of that process yields ninth, eleventh, and
thirteenth chords (which is not to
say that this is the manner in which these sonorities evolved
historically). The
introduction to this dissertation has also suggested that
considering stacked thirds as
36 Robert W. Ottman, Advanced Harmony: Theory and Practice, 5th
ed. (New Jersey: Prentice Hall, 2000): 292. Vincent Persichetti
demonstrates how the stack o f thirds can be extended higher if the
would-be- repeat tone is chromatically altered; he does admit,
though, that these chords are so unwieldy as to be of limited use,
mostly in parallel harmony or as harmonic punctuation. Vincent
Persichetti, Twentieth- Century Harmony: Creative Aspects and
Practice (New York: W.W. Norton and Co., 1961): 85-87.37 Emile
Durand, Traite complet d harmonie theorique et pratique (Paris:
Leduc, 1881 ):71; Theodore Dubois, Traite d harmonie theorique et
pratique (Paris: Heugel, 1921): 23;; Leon Dallin, Techniques o f
Twentieth Century Composition (Dubuque, I A: Wm. C. Brown Company,
1957): 60. Dallin adds a little more justification to the
stacked-thirds model o f additive harmony by claiming it produces
chords that are most similar to common-practice chords:
Contemporary chords built in thirds have the closest relation to
conventional harmonies since they continue the process by which
triads and seventh chords are constructed, superimposition o f
thirds. (Dallin, 60).38 Stefan Kostka and Dorothy Payne, Tonal
Harmony, 4th ed. (Boston: McGraw-Hill, 2000): 436.
36
-
foundational for additive harmonies often fails to produce
either elegant or pragmatic
labels for many additive chords in the Parisian modernist
repertoire.39
A acoustic basis for extended triads is argued because, past the
triad, the new
pitches that appear in the overtone series arrive in a
stack-of-thirds order - the 6th partial
forms the (out-of-tune) interval of a compound minor 7th above
the fundamental, the 8th
partial forms a compound major 9th, the 10th partial forms a
compound augmented 11th,
and the 12th partial forms a compound major 13th (Figure 1.14).
This theory has been put
forward by A. Eaglefield Hull, Ludmila Ulehla, and Rene
Lenormand among others.40
Figure 1.14. The harmonic-series justification for the
extended-triad model of additive harmony.
Issues o f Modification
Creating extended triads out of basic chords is not the only way
to modify basic
chord types in order to account for novel surface formations; a
different application of the
modified-basic-types strategy is to chromatically alter chord
tones in order to create new
verticalities whose identity and behavior is still related to a
basic chord. This approach
39 Ludmila Ulehla locates the dissolution o f the triadic model
at the 11th chord, where the inclusion o f the natural 11th above
the root can necessitate the omission o f the chordal third (a
situation Ulehla insists is explained by the desire not to have a
suspensions projected tone o f resolution already present in
another voice, not the presence o f a minor 9th between chordal
third and 11th): The natural eleventh becomes a chord tone in Modem
styles in which it frequently replaces the third, and in so doing,
abandons the former Classical concept o f triadic sound. (Ludmila
Ulehla, Contemporary Harmony [New York: The Free Press, 1966]:
90.)40 Ren6 Lenormand, A Study o f Modern Harmony, trans. Herbert
Antcliffe (London: Joseph Williams, Ltd., 1915): 7;; A. Eaglefield
Hull, Modern Harmony: Its Explication and Application (London:
Augner, Ltd., 1915): 94; and Ulehla, 59. Dallin a