The Conflict between Particularism and Generalism in Andrew Mead's "Introduction to the Music of Milton Babbitt"

Yale University Department of Music

The Conflict between Particularism and Generalism in Andrew Mead's "Introduction to theMusic of Milton Babbitt"An Introduction to the Music of Milton Babbitt by Andrew MeadReview by: Ciro ScottoJournal of Music Theory, Vol. 46, No. 1/2 (Spring - Autumn, 2002), pp. 285-345Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/4147682 .

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REVIEW ARTICLE

The Conflict Between Particularism and

Generalism in Andrew Mead's Introduction to

the Music of Milton Babbitt

Ciro Scotto

An Introduction to the Music of Milton Babbitt Andrew Mead Princeton, New Jersey: Princeton University Press, 1994 321 pp.

I. Webs about Webs

I doubt if any person on the planet, other than the composer himself, possesses a more intimate and thorough knowledge of Babbitt's works than Andrew Mead. The evidence of that knowledge is amply on display in his book An Introduction to the Music of Milton Babbitt. Consequently, Mead's book is an indispensable resource for anybody pursuing Babbitt scholarship. It is, for example, the required text for my own seminars on Babbitt's music. Its greatest strength is in its exploration of the music. Although Mead makes no attempt to formulate or explicate the theoretic group structures that underlie array construction, he should not be chided for his decision.' While knowledge of the group-theoretic properties of arrays is certainly an important and even necessary part of Babbitt scholarship,2 it is certainly not sufficient. After all, one only needs knowledge of common twelve-tone lore (sets, their properties, and the constraints

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determining certain types of arrays) to be able to construct pre-compositional arrays. Furthermore, to talk about the relationship between array structure and compositional realization, one only needs to invoke an analytic theory. Mead, however, does bring something extra to the table. An accomplished composer himself, he brings his years of experience gained by working within the twelve-tone system to bear on his analytical observations. Consequently, he is able to focus our attention on details that oth- erwise might escape our view, and he is able to draw out the implications of a compositional gesture, because he knows the various roadways open to a gesture for its development.

In any review, we want to investigate at least some of the usual questions. Some examples of these questions might be: who is the appropriate reader; what is the overall organization, focus, and approach; what is new about the approach; what is special about the book; what are the pro- duction issues, if any; how comprehensively is the topic explored; and are there any important analytic or theoretic issues raised in the course of the work? The last question is especially important in the present case, because the value of an analytic enterprise focused on a single composer's work does not solely depend on categories such as breadth of coverage. Sometimes a study is as important for the theoretic questions it raises, because those questions may stimulate readers to explore new and different analytical trajectories in addition to the trajectories followed in the work. Although the topics seem straightforward and discrete, that impres- sion is an illusion. The discussion of one category often impacts or overlaps the discussion of another and transforms any hierarchical organization of the topics into a web of related topics. The difficulty for any reviewer is to move through the web of issues without getting inextricably tangled in them.

To readers familiar with Babbitt's compositions, moving around a hypertexted space will be a familiar if not comfortable experience. For readers new to Babbitt's world, a hypertexted space could be as confus- ing as entering(?) the World Wide Web for the first time. One can easily find oneself asking questions such as "how did I get here, how do I get back to where I was, and can I can even remember where it was that I started?" Or one can find oneself overloaded with information trying to sort through the various levels of data for the data that applies to one's level of inquiry. An even more difficult task is trying to conceptualize the Web. Have you ever had to explain or define it to someone who did not already know what it is? The usual explanation, "it is a network of com- puters connected by telecommunications lines," does little if anything to relate the experience or content of the Web. Nevertheless, experienced surfers develop techniques and employ tools that both help them navigate and process the data on the Web. Browsers, for examples, have "Home" buttons that are a kind of fail-safe when you become hopelessly lost.

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They also maintain history files, which are records of your surfing activ- ity enabling you to retrace your steps should a casual visit to one site prove to be more important than you first realized. Consequently, what at first seems overwhelming becomes manageable, informative, and enjoyable.

In producing a book about Babbitt's compositional world, Mead attempts a task no less difficult than trying to conceptualize and navigate the entirety of the Web. It is not the nature of the task that makes it difficult, since many similar studies have been completed about other composers; it is the nature of the material. Babbitt is a composer who values the contextual, which underscores the fact that the twelve-tone "system" is much less prescriptive than the tonal system and twelve-tone compositions are more self-referential than their tonal brethren are. Conse- quently, finding meaningful specifications for traversing the dimensions of Babbittian space is hard. Another danger inherent in the task is letting the explicative power of the study be undermined by its being too close a reflection of its subject of investigation. That is, any composition potentially contains an enormous amount of information, and the explicative power of an analysis often rests on what is excluded as well as on what is included. In general, Mead navigates through the dangers, and readers new to Babbitt's world will find many useful tools to help them with their own explorations. Nevertheless, the study is not free of subtle problems that lead to confusion, at least for this reader. The safest node from which to launch our investigation is the book's organization.

II. Organization

Overall, the book can be divided into three main parts: part 1, Pro- logue and Chapter 1; part 2, Chapters 2 through 4 and the Epilogue; and part 3, several sections-some equivalent to appendices-"Catalog of Compositions by Milton Babbitt," "All-Partition Arrays," "Notes," "Bib- liography," "Discography," and "Index." Of course, one would think that some sections of part 3, such as "Notes" and "Index," almost need no comment, because they are part of every scholarly study. Nevertheless, the chapter notes require comment because they often exhibit a stylistic feature that hinders their immediate usefulness. They sometimes lead to citations that point a reader (who may not have the necessary background information to understand the implications of a concept) to a specific passage offering a more in-depth discussion of a point being made in the main text. For example, in the context of a discussion about aggregates and twelve-tone rows in chapter 1, Mead appends a footnote to the following statement:

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Twelve-tone rows may be thought of in two complementary ways-as strings of intervals or as strings of pitch classes. Both aspects of row structure are important in our hearing of twelve-tone music and reflect the duality discussed above. (p. 15)

The footnote says "See Babbitt 1987."3 Since the word "see" is included in the citation, one would assume that this is not just a general reference. Rather, the reference is intended to provide additional information supporting the claim that both aspects of row structure are important in our hearing of twelve-tone music. If the foregoing is a correct characteriza- tion, then the reference is to the entire book Words about Music. Conse- quently, the reader cannot quickly access some background material about the point at hand. However, in another place in the same chapter (footnote 23, a reference to the same book), the citation is more complete, "See chapter 1 of Babbitt 1987." This at least limits a reader's search to the first chapter. Of course, this could be a simple typographi- cal error; however, I found it a consistent feature of the notes.4 A note purporting to give some background on a technical matter often refers to its source broadly and imprecisely. Now this could be taken as a reviewer being overly fussy, and the point really not significant in the context of the larger issues at hand. Mead's inconsistency, however, in handling details and the lack of adequate background information may prove frus- trating for the reader who really does want an introduction to Babbitt's music. The general discussion of this issue, unfortunately, will have to be delayed, so that we do not lose the thread--organization-that brought us to this node (hit the back button).

Although some of the other sections of part 3, "Catalog of Composi- tions by Milton Babbitt," "Bibliography," and "Discography," are also part of scholarly studies of this type, they are especially useful, appreciated, and well done with regard to Babbitt's music. Although Mead characterizes Babbitt as first and foremost a composer, the section of the bibliography devoted to Babbitt's writings is a graphic illustration of the fact that Babbitt is a composer who also has much to say about music. Any Bab- bitt scholar will find this resource invaluable, since many of the publi- cations are in journals where one might not expect to find writings by Babbitt.5 The other section of the bibliography, "Writings About Milton Babbitt and Related Topics," is comprehensive both for the completeness of its cataloging of analyses by other authors and for its choice of articles covering the theoretical machinery of Babbitt's music. Also, the catalog of compositions is especially useful for locating scores outside of the library environment. As well as the date of composition, title, and instrumentation, each listing also contains the publisher's name. Since Babbitt's music has been handled by no less than eight publishers, this information will be very useful in locating materials, especially with World Wide Web

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G AbE Eb E Db AD D F F Gb C B B BE

F# F Bb Bb C E B G# G# G# G C# D# D A\Bb

DA A G# Bb E E6l C C G GB C# C# F# F# F

C# FGB B Gb Gb Eb D Ab B AE E

A

BE EF F EAB A B D F# D DC C GAb

Db

F# C# C D Ab G G E B Eb Eb F Bb A

3223 4231 3241 5314 5231 43213 42312 5322

Example 1. First sub-array for Playing for Time

search engines. Likewise, the inclusion of recording company and record/ CD numbers in the discography will be very helpful in locating these extremely important "documents," many of which are out of circulation.6

The final section of part 3, "All-Partition Arrays," is an interesting and useful special feature of Mead's book, which requires a brief detour away from our topic of the book's organization. In this section, Mead includes the all-partition arrays associated with some of Babbitt's compositions.7 That is, each array in this section functions as the source pitch-class material for a particular composition. For reference purposes, Example 1 illustrates the first section of the array from Playing for Time, a solo piano work. Furthermore, as Mead points out, many of the included arrays (either as they appear in the appendix or transformed under Tn, TnI, TnM, or TnIM or with blocks reordered) are associated with more than a single composition.8 What makes this section special is the abstract presentation of the arrays. In the main body of the work focused on this material (chapters 3 and 4), the author discusses an array (or a portion of an array) as part of and perhaps inseparable from its compositional realization. So one wonders, what use could be served by including this material abstracted from its compositional realization?

A quick answer to this question is that the inclusion of these arrays could prove useful to readers who wish to go beyond Mead's exploration of the music to perhaps explicate and explore the meta-theoretic group structures that underlie array construction. Perhaps the most interesting

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way a reader could accomplish this goal would be to use this material in conjunction with Robert Morris's book Composition with Pitch-Classes, which is an exploration of the group structures underlying array construction. Babbitt's arrays are one form or one example of what Morris calls compositional designs. In fact, they are a subset of the more general set of arrays discussed by Morris. Consequently, this material, perhaps in conjunction with Morris's work, offers readers a unique opportunity to study Babbittian array structures and gain another perspective on both Babbitt's music and array composition in general, perhaps by using one of the arrays as the basis of a new composition. Finally, since these arrays can be the source of pitch-class material for more than one composition, they could aid readers in exploring the many facets of array composition in works not covered in the book, or in delving deeper into works that are introduced in the book. Nevertheless, presenting the array material abstractly does have implications that go beyond any of the reasons outlined above. In fact, the inclusion of the arrays gives us insight into the focus or approach that Mead is taking in his introduction to Babbitt's works-a point we return to later in the review.

The main body of the work is, of course, part 2 (Chapters 2, 3, and 4) where Mead examines, often in great detail, the compositional architecture of many of Babbitt's works. Essentially, the author divides the composer's output into three periods (1947-60, 1961-80, and 1981 to the present), where each period is explored in a separate chapter. Although at first glance the divisions appear to correspond to the conventional categories of early, middle, and late periods, they actually underscore a major technical shift or development in the way Babbitt composed music using twelve-tone arrays.9 Chapter 2 explores compositions using trichordal arrays as their source of pitch-class material (Example 2 illustrates a section of a trichordal array), while Chapter 3 explores compositions using the less uniform all-partition array.10 While the compositions discussed in Chapter 4 are not literally based on a new type of array, they do result from a new way of employing an array or more precisely multiple arrays. If an array can be characterized as a grid or matrix for the polyphonic presentation of twelve-tone rows where each column in the matrix contains partial orderings from the linear row forms that together preserve the total chromatic, then a "super-array," as Mead calls it, can be thought of as the polyphonic presentation of complete arrays. Although the temporal ratio between the columns of member arrays can be any proportion, the simplest ratio the columns can assume is one-to-one, a kind of first species array counterpoint. Even in this abstract presentation of the structure of a super-array, we can see that one of the basic premises of the Schoenbergian and Babbittian compositional universe, the total chromatic (i.e., aggregate), is no longer preserved when arrays are combined to form super-arrays. We will examine the consequences of this theoretic

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To(P) F A F G E G Db Bb D B E C

T7I(P) Db Bb D B Eb C F# AF G# EG

To(Q) GO EG F# AF B Eb C Db Bb D

T71(Q) B Eb C Db Bb D G# EG F# A F

Example 2. First half of the opening solo trichordal array from Composition for Four Instruments

issue for Mead's approach and what new territory they might lead us to explore later in the review.

III. Contextualization of the Analyses-Part I

Although the analyses in part 2 are arguably the most important contribution of this work, part 1 (Prologue and Chapter 1) might be equally as important, because the information presented in this section contextualizes the analyses. That is, the Prologue and Chapter 1 filter or color the reading and function of the analyses. Actually, the Prologue and Chapter 1 themselves consist of two parts that contextualize Chapters 2, 3, and 4 in different ways. The first division in part 1 (Prologue and the subsections of Chapter 1 titled Introduction and The Twelve-Tone System) colors the reading or function of the analyses, while the second division (the subsections titled Babbitt's Pitch Structures, and Babbitt's Rhythmic Structures) colors the reading or function of the array in Babbitt's music. In the context of hypertexted spaces, however, the filtering performed by the divisions of overlap so that the second division also colors our reading of the analyses and the first division also colors our reading of the function of the array in Babbitt's music. As well as contextualizing the function of the array, the second division of part 1 also functions as a tutorial focusing on the sets, the sets' properties, the constraints determining certain types of Babbittian arrays, and the compositional architectures that might result from array properties.

As well as suggesting how the analyses could aid interested parties in approaching Babbitt's music, the Prologue of part 1 outlines the book's breadth of coverage. The goal of the volume is "a celebration of Milton Babbitt the composer and of his music ... and [it] is conceived as a guide to a more informed hearing of Babbitt's work" (p. 3). As the topics covered so far imply, Mead clearly takes a structuralist approach towards achieving the goal of a more informed hearing of Babbitt's music. The

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structuralist viewpoint also acts as a filter that limits the volume's breadth of coverage:

In keeping to this particular goal, I have had to omit a great deal about Babbitt. This is not a biographical study, nor does it attempt to place Bab- bitt in a historical context, except, incidentally, as an heir to the insights of Arnold Schoenberg and Anton Webern, among others. Although Mil- ton Babbitt was a major figure in the development of electronic music in the United States, I have chosen not to emphasize his work with the synthesizer in a special way, because the music he composed for that medium raises essentially the same problems of comprehension for the listener as his compositions for more conventional ensembles. (p. 3)

IV. My Views on Structuralism and Post-modernism

With the influence post-modernist thinking has had on at least raising the question of what the scope and method of theoretical inquiry should be, it might seem anachronistic to take a purely structuralist approach. However, I do not believe that one should view any work as incomplete, flawed, or old fashioned, because it does not view structure through the contextual lenses of biography, history, and culture. Unlike Nicholas Cook, I do not believe that a structuralist approach that takes as its foundation an abstract structure, such as the array, and then demonstrates how the structure is realized in a composition necessarily invokes a Platonic view of composition."

A composition viewed as a set of interrelated elements that form a unity'2 can consist of any number of systemically related sub- or supra- systems. Ozbekhan's view of Polanyi's concept of a systematically organized text demonstrates the idea:

... Polanyi views language, or rather the composition and articulation of a literary text, as a hierarchically organized system. First, we have the generation of letters-an alphabet, which is an initial system possessing its own laws, concepts, and operations. Second, we have the rule-gov- erned grouping of letters that form words; this, too, is a separate system that has its own structure (vocabulary), etc. Third, words become organized into sentences-another system, and one capable of expressing statements in accordance with a particular grammar, syntax and semantics. Fourth, we have a text whose value, or literary worth, is judged by aesthetic values that prevail in the language, culture, fashion, etc., current in the specific environment wherein all the foregoing has occurred, and this is yet another system that operates in relation to its own elements and in its particular terms... The important things to note in this.., .example are: (i) that we have one system formed by four systems that could be seen

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as independent; (ii) that the system is hierarchical inasmuch as the creation of letters does not and cannot govern the rules of the system above it, which is a vocabulary; nor is a vocabulary, by itself, capable of defin- ing the structure of a grammar, a syntax or semantics, and finally, neither of these can directly suggest the values whereby the final outcome will be judged as a work of art. However, if we go the other way around we find that, given a system of aesthetics, we can find out how the syntax and semantics ought to be used to satisfy it; that given a grammar, a syntax and semantics we will know how to use a vocabulary, and that given a vocabulary, we can develop an alphabet. Thus, although the component systems can, in some sense, be considered as independent from each other, they are, nevertheless, hierarchically organized when viewed in relation to the end: a text that can be valued as a literary composition. Hence, the end-system is the organizing system of the whole, while the others either organize themselves and/or the system that is immediately below them (and are, of course, organizationally influenced by the system(s) above them-from which they proceed); (iii) this does not mean that each independent system does not have its own internal hierarchy, and its own rules, objectives, goals, etc.; (iv) nor does it mean that the system of values-that which organizes the whole-could have any meaning apart from the others, any more that any one of [the] component systems could have meaning apart from the systems that are subordinate to them.13

The super-system modeling the composition and articulation of a literary text could serve equally well as a model for the construction and articulation of a musical text, a composition.'4 The musical model would also consist of a number of systems that can be seen as both independent and interdependent. For example, compositional systems or composing theories and analytic systems or analytic theories would be equivalent to the third level in the literary model (i. e., a particular grammar, syntax and semantics), and level-three subsystems can have their own internal hierarchy, their own rules, objectives, and goals. As in the literary model, the fourth level in the musical enterprise is a composition (the end- or organizing system) whose value can be judged by the aesthetic values that prevail in the current cultural environment. The system of values that com- prise the fourth level, like the other levels in the super-system, operates in relation to its own rules, objectives, and goals. Although each component system can be analyzed independently, a broader context for analysis is created by viewing component systems as interrelated and forming a super-system. In this broader context, the significance of each component system is illustrated in relation to the systems that are subordinate to them.

If we view Mead's approach as simply focusing our attention on the somewhat independent third-level systems (i.e., a particular grammar and

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syntax) so we may better understand those systems, then we do not have to view the analyses as incomplete because they do not place Babbitt's work in a biographical, historical, or cultural context. In fact, in keeping with the introductory nature of the book, Mead's work could be seen as a first step towards a more contextually complete analysis of Babbitt's work. Consequently, other theorists can build on this firm foundation and discover how third-level systems interact with the cultural codes revealed by postmodernist analyses of fourth-level systems.

Although the approach is decidedly structuralist, Mead does offer us a glimpse of at least one way third- and fourth-level systems could interact. The author of the literary model says that given a system of aesthetics, it should be possible to discover how the syntax and grammar satisfy the goals of the system of aesthetics. In a subsection of Chapter 1, Mead introduces the idea of maximal diversity, which can be interpreted as an aesthetic principle guiding the construction of Babbitt's compositional system and the composition and articulation of a piece of music in his system (p. 19). Maximal diversity, as a guiding principle, simply means a composer takes full advantage of the associations and relationships possible within a given compositional system:

A row class contains the maximum number of different ways to transform a given ordering under a certain set of constraints. Babbitt has extended this idea to virtually every conceivable dimension in myriad ways throughout his compositional career. All sorts of aspects of Babbitt's music involve the disposition of all possible ways of doing something within certain constraints.

Besides offering analyses of specific compositions, the main body of the work (Chapters 2, 3, and 4) successfully demonstrates how the syntax and grammar of Babbitt's compositions satisfy the principle of maximal diversity. For instance, Mead's Example 3.11 and the accompanying text illus- trate that Babbitt uses every possible sub-grouping of the Arie da Capo quintet into soloist, trio, and quartet. The various groupings are generated by assigning one or more array lynes of the six-lyne array to particular instruments for a given array block (p. 142).15 Through similar examples of relating array structure to many musical dimensions, Mead leads us to the conclusion that a Babbitt work offers a listener the possibility of exploring an intricately detailed, maximally varied, and ever-changing landscape. Building on his foundation, I would say the maximalist aesthetic extends itself to the way one could explore such a landscape. Rather than a single (structural or cultural) pathway through its sights, a Babbitt work offers a traveler numerous trajectories through its sequence of events, numerous opportunities for discovering cultural tropes, and numerous interactions between the two that lead to "a cumulative contain-

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ment, a successive subsumption," from which a work is "entified as a unified totality."'6

In his Prologue, Mead gives an interesting nod to post-modernist concerns. As if to allay fears that the Apollonian and technical nature of his work lacks any connection to the Dionysian and to human nature, Mead makes a very personal declaration:

Make no bones about it, I enjoy this music, and it is my hope that the following remarks will increase others' enjoyment as well ... The pleasure I derive from Babbitt's music is not merely some intellectual satisfaction at teasing out a complex puzzle. Nevertheless, the strong emotional and expressive charge of his music is deeply rooted in the ways notes and rhythms work together to create webs of association and connection over ever-larger spans of time. (p. 3)

This seemingly innocent statement is extremely important, because it connects Babbitt's music not just to the technical tradition of Western art music; it connects it to the cultural tradition that tells us that music has power that "taps emotions and desires at depths beyond the reach of any order.""17 I would like to add my voice to Mead's in celebrating the emo- tive power of this music. First, if music can act as a conduit through which we explore our own emotional terrain, then Babbitt's works have led me to reexamine familiar territory and reach out into unfamiliar territory, and I believe I am a better person for having taken the trip. Second, as a per- former of Babbitt's works, I have felt an emotional charge equal to the charge I have felt from performing other works from both the Western and non-Western traditions.18 I can only hope that Mead's emphasis on the technical side of Babbitt's work (it is an intriguing world to get lost in after all) does not become a barrier to simply enjoying the sound of these works. I also hope that the technical path Mead has chosen does not foster the belief that it is the only technical path leading to modeling structure in Babbitt's music.19 Nevertheless, for Mead the expressive power of a musical work is just as inextricably tied to its syntax as that power may be tied to a cultural trope for some post-modernists.20 This claim gives future theorists (and musicologists) another starting point from which they can investigate the interaction between third- and fourth-level systems.

V. Contextualization of the Analyses-Part II

One omission, however, veering from the structuralist path (and damp- ening the celebration somewhat) is Mead's decision to omit any discussion of Babbitt's electronic works:

Although Milton Babbitt was a major figure in the development of electronic music in the United States, I have chosen not to emphasize his work

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with the synthesizer in a special way, because the music he composed for that medium raises essentially the same problems of comprehension for the listener as his compositions for more conventional ensembles. (p. 3)

One of Babbitt's major theoretical works and compositional techniques (the timepoint system) was written and developed specifically for the electronic medium.21 Therefore, it could be argued that many of the techniques associated with the compositional realization of the array, such as timbral differentiation, might have originated from his work with the RCA synthesizer. After all, at the level Babbitt was working at, instruments and instrumental timbre are not a given. One must design those instruments and instrumental timbres perhaps in conjunction with and in relation to the pitch material those instruments will play. Furthermore, I believe the electronic works are some of his most compelling compositions, because the electronic medium allows us to experience Babbitt's compositional world in a context where historical and cultural associations very mini- mally filter the experience. Therefore, listeners new to Babbitt's music may actually find the electronic compositions easier to appreciate.

As noted above, however, Mead claims that the problems of comprehension are the same for Babbitt's electronic works and non-electronic works. Although I agree that the electronic works may essentially raise the same problems of comprehension, I do not agree that each medium offers the same solution to those problems. The electronic medium offers composers greater control over the quantifiable elements of music, such as rhythmic and durational patterns, dynamic contrast, and differentiation of timbre, among others. Therefore, the electronic works may also offer listeners a better chance of comprehending the structures Mead regards as essential to our appreciation of Babbitt's music, because they may set those structures into greater relief. Even if this proves not to be the case, the electronic compositions may still offer us insights into Bab- bitt's compositional world that his works for conventional ensemble do not offer us.22 Consequently, their omission leaves a gap in the examina- tion of Babbitt's compositional architecture.

Raising the issue of comprehension gives us further insight into how division 1 of part I contextualizes the analyses and the underlying analytic model supporting the analyses. They not only reveal the structure of the compositions; the model and the analyses are, as the following quote from part 1 informs us, a listening model as well. That is, they should lead to the goal of achieving a more informed hearing of Babbitt's music:

And while his musical surfaces revel in great sensuous beauty, they can only grant us incidental gratification unless we attempt to hear the ways they reveal the underlying long-range motion through the background structure that forms the lasting emotional drama of his compositions ... Our appreciation of the range and subtlety of Babbitt's music, however,

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depends on our ability to perceive the underlying structure behind the local details, to follow events below the surface. Once we can recognize the significance of various changes in intervallic patterns, rhythms, or instrumentation, we can begin to assimilate the wealth of surface detail as the most immediate manifestation of larger patterns of transformation and recurrence that add up to our sense of the overall unfolding of a composition. This is what leads us to the vital center of his music, the source that animates the farthest reaches and ramifications of the sounding surface ... a lack of comprehension or a superficial acquaintance can level the greatest distinctions and render even the most emphatic expression obscure ... Its dynamic qualities depend on a series of dialectics between the surface moments of a piece and their source in its underlying structures ... (pp. 4, 5, and 8)

Some of Mead's guidelines, in my hermeneutic reading of the passage, appear even stronger than suggesting how we could use the analyses to inform our hearing, because they imply we cannot achieve a more informed hearing of these works without knowledge of the underlying structure. That is, the quoted passage appears to imply that surface structures only become comprehensible in relation to their more complete counterparts in the background. The background, in other words, appears to validate the surface. The emphasis on the background as the source of comprehension could lead to the conclusion that the surface has no structural integrity of its own. This interpretation of the relationship established between the surface and background yields another perspective in which the surface may contain independent structures, but only those structures that lead to uncovering the background can lead to a sense of the overall unfolding of a composition. Finally, since the surface is comprehensible only by hearing it as a manifestation of the background, the structural relationship of these levels appears to be hierarchic.

The suggested method of acquiring knowledge of the background appears to be through analytical listening from the surface to the background. That is, we hear surface details (changing intervallic patterns, rhythms, and instrumentation) as local articulations of transformations taking place over longer musical spans, and relating the local events produces or leads to a hearing of the underlying long-range motion through the background structure. And finally, our assimilation of both local details and long-range transformations lead us to our sense of the overall unfolding of a composition. Anyone familiar with Schenker's work will, of course, recognize the similarities between the analytical models.

The three major divisions of Schenker's work Derfreie Satz outline his model of background structure and the transformations that relate foreground structures to the background through the middleground. The presentation of Mead's analytic model, however, is not as straightforward.

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It is not so much that parts of the theory (i.e., its axioms, theorems, and assumptions) are missing, because they are scattered throughout the text. The problem arises from the shotgun presentation of the theory's components. In other words, we are given a great wealth of information with respect to individual trees, but we are left wondering about the image of the forest. Consequently, developing a clear picture of how the various parts of the analytical model fit and function in relation to each other is difficult, which is especially problematic for a model that Mead believes plays such crucial role in achieving a more informed hearing of these works. Furthermore, the diffuse presentation of the theory's components makes testing its results and consistency more difficult. The problem is not in the overall organization of the book, because the structure of Chap- ter 1 is designed to be a clear exposition of the components of a twelve- tone theory and Mead's theory of Babbitt's twelve-tone structures. The problem, at least for this reader, is that we are not sure how the parts presented in the exposition function in relationship to each other within the context of the theory.

In Chapter 1, for example, Mead introduces the notion of mosaics, which are partitions of the aggregate whose components are unordered collections, and mosaic classes, which are partitions of all mosaics into equivalence classes under transposition and inversion. Partitions generated by the array and mosaics differ in only one respect: the components of an array partition contain partial orderings and the components of mosaics contain unordered segments. How do these two similar structures function together? The question comes up in Chapter 3 in the analysis of Post-Partitions, where the reoccurrence of a 26 partition in the work's opening section is actually the reoccurrence of a mosaic, not a repeating array partition. Since the piece uses an all-partition array, the mosaic structure is superimposed on the partitions of the array. Mead's discussion centers on the mosaics, which implies that the mosaic structure is the underlying structure that makes sense of the surface, not the all-partition array. Several questions arise from the relationship between the mosaic and array. Is the mosaic a middleground structure that leads to the actual underlying structure, the array? If the mosaic is the underlying structure, what function does the array perform?23 The dichotomy between array partition and mosaic does not become apparent until the analysis in Chapter 3. In Chapter 1, they are simply puzzle pieces that do not give a clear indication of the picture that might emerge from their union.

The competition between mosaic and array for the status of underlying structure leads to a general observation about the analytic theory. It is difficult to relate statements such as "our appreciation of the range and subtlety of Babbitt's music.. . depends on our ability to perceive the

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underlying structure behind the local details" to the theory, because Mead never explicitly states what constitutes the underlying structure. He presents several candidates, such as arrays, aggregates, and rows, without overtly stating which structure occupies the top node in the hierarchy.24 To be more precise, the top node of the hierarchy appears to be context sensitive, because the structure that occupies it and its importance to our perception of the music shifts in relation to the pressure applied by various perceptual and compositional criteria. In the following passage, aggregates appear to achieve the status of underlying structure:

A twelve-tone row, whether thought of as a particular ordering of the twelve pitch classes or as a chain of intervals, is an abstraction, because the dimension in which the order is manifested is not defined, nor is the order of other dimensions determined. It is fundamentally important to realize that any compositional representation of a row contains a great deal more musical structure than is specified by the row itself: musical realities are aggregates, with their identifying distributions of pitch classes; rows are compositional tools used to control the structure of aggregates. The fact that the abstract ordering, the row, is manifested in only one musical dimension at a time, for the most part, allows its compositional realization to possess differentiations articulated in counterpoint against its recurring intervallic and collectional patterns. These differentiations can create linkages with other aggregates here and there in the composition, based in various ways upon the structure of the rows. Twelve-tone composition involves marshaling these various connections into strategies both local and long range to create the dramatic accretion that is a piece of music; to a large extent, the study of twelve-tone music is the study of the myriad ways this is done. [italics mine]

In my reading of the passage, the row as the underlying structure appears to be de-emphasized in favor of aggregates. That is, musical realities are not pitch realizations of rows; they are the sets formed by partial orderings taken from polyphonically presented pitch-realized rows, or aggregates. Rows primarily function as the generators of source components that provide aggregates their shape. Consequently, assembling the partial orderings into row forms is not as important as using the partial orderings to create linkages across aggregates. Since aggregates are composed of pitch-interpreted rows, they contain distributions of pitch classes (the partial orderings) that allow distinctions and connections to be made among aggregates and the distribution and content of the partial orderings provide each aggregate with an identity, a partitional shape. In the section on all-partition arrays, however, the perceptual consequences of an array composed of non-repeating partitions appear to blur the shape of partitions:

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Before discussing how all-partition arrays are transformed, it is worth making a few remarks on their perceptual implications. At first blush it might seem that using all partitions in an array is an intellectual conceit, that no listener could be expected to count up a list of partitions, recog- nizing what has yet to be heard. Indeed, this is not the point of the all-partition array, just as counting to twelve is not the point of hearing aggregates. The desired goal is the construction of a long string of aggregates, each with a different partitional shape.

In this passage, Mead appears to back away from aggregates as uniquely shaped and identifiable, which has the consequence of not being able to construct the array from the accretion of aggregates. Granted nobody could be expected recognize what partitions have yet to be heard, but if the shape of each partition is as distinct as this and the previous passage lead me to believe they should be, then I do not see why one should not be able to list the partitions already heard and reconstruct the array from listening. In fact, our ability to perceive the underlying structure (whether it is aggregates and their contents or the entire array) behind the local details seems dependent upon our ability to identify and distinguish aggregates, not just establish contour relationships between them. If a partition's shape is blurry, some of its pitch classes cannot be segmented into its generating partial orderings. If an aggregate's pitch classes cannot be segmented into its generating partial orderings, then the source of surface moments in the underlying structure will remain opaque. In the chapter on superarray compositions, Mead identifies the underlying structure as the array, but our perception of the underlying structure so crucial to our understanding of the local details is, at this point, indirect at best:

... earlier works have at least some passage where the underlying material is relatively near the surface, but here the underlying array emerges only by inference. This helps to reinforce a critical point about hearing Babbitt's music, we are not listening merely to strip away the surface details to reveal the underlying array. The effect of the underlying array, like the effect of the row itself, is one that comes to permeate the work at many levels, creating a palpable resonance between specific moments and the piece as a whole.

While a theory that expands to incorporate new data may not be a problem, a theory that changes in a way that undermines its own premises is a problem. As I will demonstrate, the source of these inconsistencies is the result of Mead's attempt to construct a single theory for all of Bab- bitt's music (in much the same way that Schenker attempted to construct a single theory for all tonal music). The theory's context-sensitive component, however, undermines its effectiveness and skews our view of the music. While context sensitivity may undermine a general analytic the-

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ory applicable to all of Babbitt's music, it can also open a door leading to many alternative views-a point I will explore later. Therefore, in order to evaluate Mead's analytic theory effectively and uncover the source of its problems, I believe it is necessary to reconstruct and comment on what I believe to be its essential features.

VI. Models of Underlying Structure

As the above quotes and the discussion of the book's organization inti- mates (especially the basis for the divisions underlying Chapters 2, 3, and 4, and the special features in part 3), the twelve-tone Ursatz supporting Mead's model must be the array. If "musical realities are aggregates, with their identifying distributions of pitch classes," then arrays generate those middleground realities (p. 15). His focus or analytical approach is also similar to Schenker's, since it relies on demonstrating how the array (i.e., the background) "animates the farthest reaches and ramifications of the sounding surface" (p. 5). Another way to understand an array-based analytical method is to view the array as a filter giving us insights into the significance of surface combinations. In other words, what at first hearing may seem unintelligible or unconnected becomes comprehensible and unified, if we can perceive the underlying structure behind the local details. Time differentiation among the array's structural divisions appears to be the basis for Mead's distinction between background and surface.

As stated earlier, an array is an abstract structure of intersecting rows and columns where the rows of the array are the members of a 12-tone row's row class and the columns contain row segments (partial orderings). Like the rows of the array, each column contains the complete chromatic formed by the 12-tone row segments generated by the intersection of the array's rows and columns. The important thing to realize about a line-plus-column array is that it can be thought of as an interpreted form of perhaps an even more abstract structure, the polyphonic combination of row forms with no columnar divisions.25 The row forms themselves, of course, create an abstract division between successive polyphonic groups of row forms. Besides its unique pattern of ordered pitch-class intervals, each twelve-tone row contains the complete twelve pitch-class chromatic or aggregate. If the aggregate property of a twelve- tone row that creates an abstract division between linear row forms is extended to the polyphonic presentation of rows (i.e., each row in the polyphonic group contributes some of its pitch-class material until an aggregate is formed), then there is a basis for the columnar divisions of the array, each should contain the aggregate (see Example 1).

The distinction between columnar aggregates and linear row forms becomes the basis for the distinction between background and surface. Like twelve-tone rows, the array is implicitly interpreted in time in a left-

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A B I I I I

GAb6Eb Eb Db AD D F F GbC Bb B BE

F# F Bb Bb E B G# G# G# G C# D# DA

BI_ I L B A

Example 3. Mid-point temporal aggregates from first two lynes of the array in Example 1

to-right manner. Consequently, since the progression through the columnar aggregates will take place more quickly than the progression through pitch classes of the linear row forms, the more quickly moving columnar aggregates become the middleground and the slower progression through the linear row forms becomes the background. It should be noted that, in spite of Mead limiting the function of rows to generators of content for aggregates, the perception of the background/middleground division depends on perceiving the more quickly unfolding columnar aggregates against the more slowly unfolding linear row forms. In other words, one must be able to perceive the linear aggregates as well as the columnar aggregates to perceive a temporal difference between them. The partial orderings contained in each aggregate become the "surface details" as Mead calls them. If one function of the background is to reveal the significance of combinations on the surface, then relating the partial orderings in the aggregate columns to the linear row forms reveals the significance of the partial orderings in relationship to the larger patterns of transformation and recurrence, the linear row forms. More precisely, the interval patterns of the partial orderings reveal their relationship to the basic interval pattern of the work, the row. That is one way we can hear the surface details as the most immediate manifestation of larger patterns of transformation and recurrence.

Mead states that the array also contains a midpoint temporal aggregate between the background and surface. He informs us that the classic hexachordal combinatorial pair of row forms used by Schoenberg underlies Babbitt's arrays. Since each adjacent pair of rows in an array form a hexachordally combinatorial pair from one of the row's "harmonic regions," each row pair will also generate aggregates at the hexachordal divide.26 (See Example 3.) The hexachordal aggregates could possibly afford a listener an intermediate level of association between the atomistic level of the columnar aggregate's partial orderings and the global level of the lin-

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ear row forms. The addition of a structural junction produced at the hexachordal divide of the combinatorial pairs whose union produces an array gives Mead's structural interpretation three temporally defined levels of aggregate where each temporal division is associated with a structural level. Again, in order to translate these divisions from the abstract boundaries of the array to the musical realities of a listening filter that leads us to the vital center of the music, we must be able to hear aggregates. Real- izing the importance the aggregate plays in the aural perception of the model's levels of relatedness, Mead asks a crucial question:

Can we hear aggregates-and if so, how? ... given a collection of a large number of different pitch classes, each represented once, we can recognize-although we are not able vividly to determine what pitch classes we have not yet heard-whether or not any additional note represents a new pitch class. By interpreting the recurrence of a pitch class as a signal that we have crossed a boundary, we can parse a highly chromatic undifferentiated musical surface into a discrete series of large bundles of pitch classes that we might call perceptual aggregates. Perceptual aggregates may or may not contain all twelve pitch classes, but because of their size this will not be a particularly vivid aspect of our hearing. Their pitch class content is not vivid, but our awareness of their boundaries will be ... The pro- ceeding is predicated on our recognition of pitch class repetition and its use to indicate boundaries between aggregates. (pp. 12-13)

To summarize, awareness of pitch-class repetition translates into awareness of aggregate boundaries. That is, upon hearing a repeated pitch class a mental boundary, equivalent to a columnar division in the array, should be drawn between the preceding group of pitch classes and the pitch classes that follow. Therefore, the first step in parsing the surface and gaining access to the underlying structure is dividing the surface into discrete bundles of notes based on the repetition of pitch classes.

Once the division between aggregates is established, comparisons between aggregates can be made on the basis of their set-class equiva- lencies of subset content, intervallic patterns, and invariance of pitch-class content. The changing internal structure in a succession of aggregates could also form the basis for establishing a sense of progression through the music as a progression through a succession of changing states. Each aggregate presents the same basic material (the complete chromatic) in a new arrangement similar to the manner in which a kaleidoscope generates new patterns of light by reordering the same bits of glass. In his discussion of Example 1.1, Mead demonstrates the variety of associations possible between four aggregates that would simultaneously form the basis for a sense of change and continuity (see Mead Example 1.1, p. 14). For example, the unordered pitch-class contents of the lines in A and B are identical, but reordering the dyads of the tenor line produces a differ-

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ent group of vertical sonorities. Between B and C the unordered pitch- class contents of the soprano and tenor lines are exchanged. Although the exchange maintains the unordered pitch-class identity of the individual lines, the ordered dyads of each line are exchanged and in some cases retrograded. The second exchange operation produces a new set of vertical sonorities, but maintains the pitch-class content of the hexachords formed by the first pair of trichords and the last and penultimate trichord of the progression. Finally, the vertical trichords of A become the lines of aggregate D.

As well as the ability to hear aggregates, an aggregate's identity depends on the ability to discern the parts of an aggregate and compare and differentiate the internal structure of one aggregate from another aggregate. The undifferentiated pitch classes of the aggregate, in other words, must be parsed into discrete bundles of pitch classes, since, as Mead cor- rectly informs us "as [unordered] collections of pitch classes, aggregates are indistinguishable" (p. 11). An unstated lemma of the model is that the ability to discern the parts of an aggregate also depends on the compositional realization of the array's partial orderings in pitch space as much as it depends on the array's abstract partitioning of the linear row forms into partial orderings that form aggregates in pitch-class space. Mead, for example, calls the four aggregates in his Example 1.1 sketches, which we can read as meaning compositional sketches, because as realizations of an abstract array structure, they invoke pitch-space compositional criteria to parse the pitch classes of the aggregate into discrete bundles of pitches. As obvious as it may seem, the four trichord simultaneities in sketch A are generated by the note-against-note counterpoint of three registrally distinct lines and the unique temporal position each group of pitches occupies. Neither of these properties is necessarily determined by the array's structure.27

Since the rows that form the lynes of an array are the source of the partial orderings in the array's aggregate columns, and since surface details are the most immediate manifestation of larger patterns of transformation, the linear dimension in Mead's model perhaps assumes a higher structural priority than other dimensions. That is, the structure of the linear dimension is the model and source for the structures that appear in other dimensions.28 After all, the fundamental structural unit in Babbitt's music is a twelve-tone row interpreted in time. The trichord simultaneities in sketch A, for example, are identical to sketch D's partial orderings. Between the two dimensions, however, it is the partial orderings in sketch D that perhaps afford greater insight into the fundamental structure-the row-and the underlying structure-the array-since they preserve the ordering associated with the row, not just the set types of the row's subset content. The trichord simultaneities in sketch A, as the most immediate manifestation of the partial orderings in the linear dimension,

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call attention to or perhaps direct attention to the linear dimension, and focusing on that dimension will eventually reveal the underlying long- range motion through the background structure. Therefore, perceiving the underlying structure depends on the ability to discern the partial orderings of row forms generated by an aggregate of an array.

Furthermore, the ability to discern the partial orderings of an aggregate, like the ability to discern the row, itself depends on the compositional realization of the array's partial orderings in pitch space as much as it depends on the array's abstract partitioning of the linear row forms into partial orderings that form aggregates in pitch-class space.29 Again, the partial orderings in sketch D are discernible as discrete segments that are members of different polyphonic strands, because they each occupy their own registral strand in pitch.30 The aggregates of an uninterpreted array, however, are as abstract as the rows it contains, because the dimension in which the partial orderings are manifested is not defined, nor is the order of other dimensions determined. Consequently, in the abstract, aggregates are not any more musical realities than rows. Both entities achieve musical reality through the compositional process, which means the perception of these realities is dependent on how well the compositional realization of the array's partial orderings in pitch space articulates those structures.

Although the partial orderings of a given aggregate may be parsed into discrete bundles in pitch space, without criteria for and without access to the criteria determining which groupings of pitch classes are structural and what functional roles those groupings might possibly perform, Mead is correct in stating that "a lack of comprehension or a superficial acquaintance can level the greatest distinctions and render even the most emphatic expression obscure" (p. 5). In Japanese Noh plays for example, very slow precise movements, such as the repositioning of a foot or a hand holding a prop, often have structural significance in the context of a play, such as indicating a change of state or a change from one world to another. With- out a knowledge base and access to the knowledge that indicates what gestures are significant, when those gestures are significant, and what those gestures signify, like aggregates, foot or hand movements will be indistinguishable from one another, and the most emphatic structural shifts will remain opaque.

VII. Transforming Compositional Knowledge into Analytic Knowledge

In spite of the emphasis on hearing aggregates and their internal structure, and in spite of the emphasis this places on the array as a listening model, I think the array as foundation for analysis can also serve the function of being a basis for differentiating structure. The array's struc-

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ture and knowledge of its potential for creating new structures through the various combinations of its partial orderings (as in Mead's Example 1.1) can provide insights into the structures that are potentially available for analysis through listening.31 Mead stresses the acquisition of technical knowledge as an essential component of fully comprehending Bab- bitt's works:

The trick to this, of course, is learning what to listen for and how to understand it in context ... By understanding the principles guiding the growth of trees or the interactions of wind and water we can better appreciate the dialectic of singularity within the totality, the moment in the flow of time. So to it is with Babbitt's music. Its dynamic qualities depend on a series of dialectics between the surface moments of a piece and their source in its underlying structures, between a structure's compositional interpretation and its abstract properties, between particular abstract structures and Babbitt's habitual comers of the chromatic universe, and ultimately between Babbitt's preferred perspective of the chromatic universe and the chromatic universe itself. [Italics mine] (pp. 4 and 8)

One method of gaining insights about those structures is to view structure from the composer's point of view. That is, knowledge of the theory guiding the generation of musical structures can be incorporated into an analytical theory about musical structures.32 To paraphrase Benjamin Boretz, a composing theory formulated from the array's structure may provide insights into "all the 'things' that things can be," and, in turn, [it may] constrain (by implication) "all the things that can be a 'thing."'33 In this light, another perspective on the associations that Mead highlights in Example 1.1 is not only how we might listen to array-based aggregate music, but how we might compose it as well. The underlying structure, therefore, can be the foundation for two functions, analytic listening and composition (i.e., generating musical structure). Of course, since the activities of listening and composing are interdependent, learning to compose array-based music is also learning how to listen to that same music.

The second division of part 1, which contains four subsections and begins with a section titled "Babbitt's Pitch Structures" (Chapter 1, pp. 20-37), could be viewed as fulfilling the function of a compositional tutorial that focuses on the sets, the properties of sets, the constraints determining certain types of Babbittian arrays, and the compositional architectures that might or might not result from array properties (as well as functioning as an analytical tutorial that focuses on the structures that are potentially available for analysis through listening). I believe, in fact, the compositional subtext of this book is one of its more important features. In the same manner that the study of Bach Chorales forms the foundation of harmonic training, Mead simultaneously uses Babbitt's array practices to formulate a stimulating theory of how one could compose with arrays.

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That is, I see no reason why his analytic model could not function as a composing theory. The problem arises from the conflation of composition with analysis, and the conflation of what might be with what is in the compositions that serve as the foundation of the model.

The other three major subsections of this division [titled "Trichordal Arrays" (p. 25), "All-Partition Arrays" (p. 31), and "Superarrays" (p. 37)], of course, explore the structural differences and structural uniqueness of each array type, and, more importantly, they mirror the major technical shift or development in the way Babbitt composed music using twelve- tone arrays that underscores the chapter divisions that follow. Therefore, abstractly exploring the structural differences and commonalities among the types of arrays lays a foundation for discussing the differences and commonalities amongst compositions based on each type of array. Fur- thermore, reviewing those structural differences and commonalities is extremely important, because it lays a foundation for determining the efficacy of the model as an analytical tool for exploring the space between a structure's compositional interpretation (its musical realities) and its abstract properties (the uninterpreted array). In other words, the structures Mead explores in these sections play a crucial role in articulating the divisions in his model of underlying structure.

The first development Mead explores is Babbitt's generalization of Schoenberg's hexachordally combinatorial row pairs. He outlines how Schoenberg created a type of twelve-tone counterpoint in which a row form is combined with its inversion to generate two additional aggregates (besides the rows themselves) formed from the combination of the row pair's first and second hexachords. We then learn how Babbitt extends the combinatorial property to include rows related to the original row by transformations other than TnI. He outlines Babbitt's discovery that only a select group of hexachords, the all-combinatorial hexachords (see Mead's Example 1.6, p. 23), can generate combinatorial pairs by combining a row with any Tn, TnI, RTn, or RTnI transformation of the row.34 Finally, by using the combinatorial rows from the opening of Three Com- positions for Piano (see Mead's Example 1.7, p. 24), he explores how Babbitt's compositional thinking incorporates combinatorial properties.

Following the maximalist principle, Mead demonstrates the effect using all four types of combinatoriality has on the pitch-class intervals and pitch-class invariances produced by the one-to-one counterpoint of lynes in a hexachordally combinatorial row pair. Although the lyne pair related by T6 in Example 1.7 generates only one ordered pitch-class interval, 6, between members occupying the same order number, lyne pairs related by inversion and/or retrogression produce a more variegated pattern of intervals. Pairs of hexachords determine the pattern, so each columnar aggregate formed by the lynes contain the same distribution of intervals. In the RTnI or R type combinatorial row pairs, the ordering of

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the intervals in the second columnar aggregate is a retrograde version of the pattern in the first columnar aggregate. As well as preserving intervals, the TnI and RTn combinatorial row pairs also preserve the pitch-class content of their dyads. Furthermore, in the RTn combinatorial pair the invariant pitch-class dyads are coupled to the invariant interval, so the second columnar aggregate presents the retrograde of both the intervals and pitch-class dyads from the first columnar aggregate. All of these properties determined by hexachordal combinatoriality can be exploited compositionally to generate boundaries between columnar aggregates and distinguish one combinatorial row pair, or block, from any other block.

Another type of invariance also can play an important role in the composition of arrays. The all-combinatorial hexachords produce more than one row pair that holds the pitch-class content of its hexachords invariant. All the row pairs in Example 1.7 are from the same harmonic area, so the pitch-class content of To(P)'s hexachords is preserved in the other seven row forms in the example. Mead informs us that this is another manifestation of Babbitt's maximalist principle, since it increases "the number of interval patterns that a fixed collection of six pitch classes may project as part of the row class" (p. 22). The invariance of hexachordal pitch-class content in each harmonic area is a property of hexachordal combinatoriality that can be exploited compositionally to generate boundaries between columnar aggregates and distinguish one area from another.35

Mead also informs us that another way of generating boundaries between aggregates is to exploit another property of the all-combinatorial hexachords, excluded intervals. Each of the all-combinatorial hexachords excludes from one to three interval classes and transposing a hexachord by that interval class maps a hexachord onto its complement. Since that interval can only occur between the hexachords of a row not within the hexachord, the excluded interval can function as a boundary demarcating the division between aggregates.36 The B first-order all- combinatorial hexachord (6-8 [023457]) generates the row in Example 1.7, and it excludes interval class 6 from its repertoire of intervals. The ordering of the row places the excluded interval right at the row's hexachordal divide. In each of the two-lyne arrays the only instance of interval class 6 formed by pitch-class adjacencies in the row occur at the boundary between columnar aggregates. Therefore, one function interval class 6 can perform is to signal the boundary between aggregates.

Mead explores the foundational role hexachordal combinatoriality plays in generating almost all of Babbitt's other array structures. The union of two hexachordally combinatorial row pairs from different harmonic areas, for example, can produce a trichordal array (see Mead's Example 1.8, p. 26). He also explores the new compositional terrain each array type opens up. The natural emphasis trichordal arrays place on tri-

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chords as a fundamental compositional unit leads to both new methods of constructing these arrays and new compositional strategies derived from applying the serial operations to trichords. Since all trichordal set classes except for 3-10[036] can generate an aggregate by applying one or more of the four serial operations to the seed trichord, a single row form through its trichordal subsets can embody the serial transformation that relate members of the row class.37 The trichord and its transformations are the fundamental unit in this compositional environment. That is, rather than viewing the row as fundamental and generating subsets, the subset is fundamental and it generates rows.

This point of view is reflected in the process that Babbitt uses to construct trichordal arrays from the four serial transformations of one member of a trichordal set class. Mead provides the Latin Square that serves as the template for the construction of this type of trichordal array (see Mead's Example 1.9, p. 27). Each column and each row contains an aggregate and all four transformations of the seed trichord. Mead informs us that each of the square's four quadrants also contains all four transformations of the generating trichord. To get this level of nesting places some interesting restrictions on the rows in each of the array's lynes. For example, although each pair of lynes (the pair beginning with transformation A and C and the pair beginning with transformation B and D) are hexachordally combinatorial row pairs, all four rows are not members of the same row class, nor can they be if each quadrant is to contain all four transformations of the generating trichord (see Example 2).38 Therefore, it appears as if the combinations of trichords generate the row forms, rather than that the row forms generate the trichords. Furthermore, trichordal arrays containing more than one row form are probably the reason Mead focuses attention in his model on the aggregate and array rather than the row as a composition's underlying structure.

Mead informs us that this new array property leads to a new compositional strategy that uses trichordal segments to generate different hexachords from the all-combinatorial family. In Example 2, the generating trichord is a member of set class 3-3[014]. Combining the first trichord of the To(P) form, pitch-class set { 6, 9, 5}, with its TII transformation produces a member of the A hexachordal type, set class 6-1 [012345]. The transformational combination based on the A hexachord generates all the row forms in the array. Combining the first trichord of the To(P) form with its T7I, transformation produces a member of the E hexachordal type, set class 6-20[014589]. All the hexachords produced by combining trichords from lyne pairs that are members of the same row class in each of the aggregate columns are members of the E hexachordal type.39 Like the tonal system's orthogonal distribution of intervals in the vertical and linear dimensions, one trichord can combine with its transformations to generate hexachordal set classes that differentiate the vertical and linear

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dimensions in twelve-tone music. Furthermore, since each trichordal set class (with the exception of 3-10[036], 3-5[016], and 3-8[026]) can generate more than one member of the all-combinatorial hexachords, the hexachordal set classes can function as conduits leading to new sections based on different trichords. Substituting trichordal set class 3-2[013] for 3-3[014] in Example 2, for example, could produce an array that would still have the A hexachord generating all the row forms in the array, but the hexachords produced by combining trichords from lyne pairs that are members of the same row class in each of the aggregate columns would be members of the B hexachordal type, set class 6-8[023457].

Although hexachordally combinatorial lyne pairs still form the foundation of the all-partition array, the cardinality of the all-partition array's partial orderings is no longer determined by presenting all the serial transformations of a single trichord or by dividing the array's lynes (i.e., the rows in the array) by two or three.40 Since all-partition arrays do not present all the serial transformations of a single trichord, the array contains only members of a single row class. Each aggregate of the new array type presents a unique segmentation into subsets of the pitch-class material contributed by the array's rows. The number of rows forming the array determines the maximum number of subsets contained in each aggregate. There are eleven ways, for example, the rows in arrays combining six rows can contribute a subset of their pitch-class material to form an aggregate. If one row contributes a partial ordering whose cardinality is seven, the remaining five rows each contribute one pitch class to the aggregate, or if one row contributes a partial ordering of cardinality two, then the remaining five rows also contribute partial orderings of cardinality two (see Example 1, the first subsection of the all-partition array from the piano piece Playing for Time).

As well as partitions involving the maximum number of rows in the array, Babbitt also uses all the partitions formed by some subset of the array's maximum number of rows. For an array containing a maximum of six rows, the subsets, of course, are partitions whose maximum number of rows are five, four, three, two, and one. The partition of one part contains an ordered segment whose cardinality is twelve, which means one partition in the array will always be the row itself. Unlike trichordal arrays, the row is relatively easy to uncover in all-partition arrays. Con- sequently, in all-partition arrays, it is much easier to relate the partial orderings to each other and find the single underlying abstract ordering of pitch intervals that determine the row. The maximum number of partitions in an all-partition array is determined by adding together the partitions for the maximum number of rows in the array and all the partitions formed by subsets of the array's maximum number of rows. Therefore, an all-partition array containing a maximum of six row forms will contain fifty- eight uniquely partitioned aggregates.41 The other all-partition arrays

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Babbitt uses have either a maximum of twelve rows generating seventy- seven uniquely partitioned aggregates or a maximum of four rows generating thirty-four uniquely partitioned aggregates.

Since each aggregate in an all-partition array has, in Mead's words, a unique partitional shape, the type of repetition that characterized the trichordal arrays no longer exists in the new array type.42 Mead does acknowledge the limited role repetition plays among the aggregates of an all-partition array. He informs us that Babbitt does not view repetition as undesirable; it is simply not part of the all-partition array's compositional landscape. Another type of repetition that is often not present in all-partition arrays is the reoccurrence of row forms. The six-lyne and some twelve-lyne all-partition arrays contain one appearance each of the forty- eight members of the row class. Since each member of the row class appears just once, Mead calls these arrays "hyperaggregates" or row- class aggregates. He cites the hyperaggregate as another example of the maximal diversity principle as a cohesive element at work in all-partition arrays, because aggregates are unique presentations of all twelve pitch classes, all-partition arrays are unique presentations of all the possible divisions of twelve elements into a specified number of parts, and hyperaggregates are unique presentations of the members of the row class. Once again, the structure of all-partition arrays reintroduces the row as the source of compositional material underlying the array.

All-partition arrays may lack any repetition of row-class members and partitions, but the array's aggregates can contain pitch-class repetitions. To be more specific, successive appearances of the array can contain pitch- class repetitions. Mead informs us that Babbitt's principle of representing all four serial transformations is expanded from the level of trichord within an array to the array itself. Arie da Capo, for example, contains five appearances of the six-lyne fifty-eight partition array: the To form and retrograded, inverted, retrograded and inverted, and T6 versions of the array. Of course, applying any of the four serial operations to the entire array preserves the integrity of the array's aggregates. That is, the transformed array will still contain all twelve pitch classes in each aggregate column. Applying a different inversion operator, however, to each of the array's lyne pairs that represent one of the row's harmonic areas produces pitch-class duplications in the partition columns of the array.43 Mead, following Babbitt's lead, calls partition columns with pitch-class duplications "weighted aggregates." Example 4 illustrates a portion of Arie da Capo's weighted array.

Since super-arrays are really an amalgam of all-partition arrays or all- partition arrays and trichordal arrays, the super-array components do not present any new technical developments with regard to array structure. However, like weighted aggregates, super-arrays will contain pitch-class repetitions but to a much higher degree. The number of pitch-class dupli-

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BbA A EDAbEb Eb C CBG F FF#CC

Bb F F# B C# GC C CEb E Ab Bb BbAD

BE EEb Eb FA A Bb B Db F#CDG FAb

C c C# Db Gb Ab D Bb Bb B B Eb F EA

G

EBC C Bb B Gb F F DA Eb Db Db Ab Ab G

Eb Ab GA A Db D F F Bb E F# B C

5322 42312 43213 5231 5314 3241 4231 3223

Example 4. Weighted sub-array from Section 2 of Arie da Capo Pitch-class duplications within aggregate columns are italicized

cations is determined, of course, by the number of simultaneous arrays unfolding at a given point in a work. Mead informs us that super-array pieces may contain from two to four array components, so each pitch class in any aggregate will have from two to four repetitions in a super- array. Although Mead does not offer any insights, at this point, into the "compositional responsibilities" of working with super-weighted aggregates, he does discuss one Babbittian strategy for regulating the flow of the polyphonic network of arrays. The polyphonic template for a super- array containing four component arrays is generated taking the fifteen possible subsets containing from one to four parts and arranging them in a sequence.44

VIII. Theoretical Fallout

Mead has presented pieces of the above-reconstructed analytic model outlined in part I of his book in other works as well.45 The present volume, however, both broadens the scope of the investigation and, as I stated earlier, attempts to generalize this approach over the entire corpus of Babbitt's works:

In a very real sense, Babbitt's whole body of work can be heard as a single gigantic composition, emerging intermittently and in different guises like an intricate shoreline seen through the mist. (p. 8)

Assuming that internal coherence is an essential feature of a composition, hearing all of Babbitt's works as a single gigantic composition im-

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plies all the compositions share common structural features, which in this case, will be revealed by the analytic model. Although Chapters 2, 3, and 4 are devoted to exploring the diverse ways the structure of trichordal, all-partition, and super-arrays manifest themselves in compositions, his underlying viewpoint, like the shoreline seen through the mist, centers on demonstrating that no matter how elaborate the surface or array structure becomes, all of Babbitt's works can be understood using the same basic set of twelve-tone listening/analytic strategies that has hearing aggregates as its first axiom. His approach is also somewhat teleological, since he views the all-partition array as an elaboration and continuation of earlier practices and the next logical step toward composition with super-arrays.46

Looking through Mead's communal filter, of course, focuses our attention on the features shared by all of Babbitt's compositions. For example, trichordal, all-partition, and super-arrays all represent different partition architectures, but they all have, as Mead informs us, a common underlying architecture consisting of hexachordally combinatorial row pairs representing each of a row class's harmonic areas. Another common feature that cuts across periods is using the sequence, derived from the Latin square, containing the fifteen possible subsets consisting of from one to four parts to control the progression of trichords in first period works and the progression of component arrays of super-arrays in third period works. The key feature leading to Mead's metaphor and perhaps his point of view is Babbitt's practice of using the same all-partition array without modifi- cation, transformed by one of the serial operations, with its components reordered, or a combination of the latter two procedures, for more than one composition. That is, a one-to-one relationship between array and composition does not exist for all arrays or compositions, depending on your perspective.

Obviously, the benefit of an analytic filter applied to a group of compositions is its ability to highlight common structural features that perhaps lead to generalizations about those structures, such as revealing a common source of coherence for the group of compositions. The limita- tion, of course, is that the general is highlighted at the expense of the particular or that the analytic method views coherence from only one perspective. A problem with an analytic method arises when the limitations start to outweigh the benefits.47 In my opinion, trying to demonstrate that all of Babbitt's works can be understood using one general set of twelve- tone listening/analytic strategies places limitations on our view of his music that do not outweigh the benefits of the approach. As was intimated earlier, adopting a position that stresses the general (i.e., one listening/ analytic approach to the precompositional structures shared by pieces from different compositional periods) over the particular (i.e., a context- sensitive approach to the precompositional structures where the context

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is the piece) with regard to Babbitt's music is what leads to inconsistencies in Mead's analytic theory. The strain of maintaining this singular point of view becomes more evident in the analyses as the book progresses from the early works, where the method is most successful, to the later works. I will also demonstrate that the singular view of coherence presented in this work-which essentially says "very fanciful mixes" of either pitch-class material or instrumental groupings on the surface only make sense when "we hear that they result from combinations of much more regular presentations of rows in individual instruments or registers," (p. 36)-overemphasizes the array's role as a source of coherence to the detriment of other sources of coherence, such as the surface set-class relationships generated from the union of the array's partial orderings.48 Fur- thermore, the model is essentially dependent on abstract array properties for structural divisions that may or may not be articulated in the pitch space of a composition. Finally, Mead's emphasis on the communal at minimum obscures and at worst limits how the compositional procedures of each period might radically alter our view of Babbitt's works.

IX. My Thoughts on an Alternative View of Some All-Partition Array Music

The change from trichordal to all-partition arrays, for instance, might also represent a radical shift away from the previous periods practices rather than just expanding and continuing them. I believe, even in its abstract pre-notation state, the structural implications of the all-partition array can have a profound effect on how we might view and listen to music of Babbitt's second period. His trichordal period works are, as Mead points out (p. 54), very much concerned with continuing and building upon Schoenberg's compositional foundation. Schoenberg, as many authors have noted, viewed the twelve-tone system as a new or alternative system with syntactic features as strong and as well defined as tonal syntax or perhaps just the next logical step in the development of the tonal system.49 In any event, Schoenberg found a creative world in the twelve-tone system that provided a framework for and facilitated his practice of creating structure and coherence through the development of motivic material, a technique important to all periods of his compositional output.

Mead acknowledges that Babbitt breaks from Schoenberg's vision of the twelve-tone syntax as a new version of tonality when he says, "Bab- bitt.. . saw that although twelve-tone syntax can support the dramatic strategies of tonal forms, it may also lead to entirely new compositional strategies vibrant and musically compelling in themselves" (p. 9). Mead's emphasis on an underlying structure in his model, however, suggests that although the compositional strategies may be different, they have the

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same goal as their tonal counterpart: coherence. I believe the works in Babbitt's all-partition period represent a radical break from the Schoen- bergian tradition, since I believe many works appear to abandon underlying coherence (i.e., a monist approach to coherence) in favor of multi- dimensional approach to continuity (i.e., a plurality of continuities). The all-partition works also abandon Schoenberg's generative motivic compositional process in favor of a holistic approach to motivic composition. Motivic composition can impose its own hierarchic structure on its materials without necessarily invoking the hierarchies among pitch classes that characterizes tonal syntax. I believe that the move from the trichordal arrays to the all-partition arrays represents a move from a generative and hierarchic model of motivic structure to a holistic and non-hierarchic model.50

Redundancy and certain forms of repetition are necessary systemic features of any compositional space that facilitates a generative approach to the development of motivic material. It is no surprise that in keeping with this tradition Babbitt's trichordal arrays are often highly redundant, containing forms of repetition that facilitate a generative and hierarchic approach to the development of motivic material. Trichordal arrays, for example, based on a single set class not only repeat the same set class, but many trichords in the array will be pitch-class identical as well.5' Consequently, trichordal arrays of this type are capable of generating a high degree of motivic unity.52 They also facilitate the generative motivic compositional process, which when viewed from the analytical side of the fence translates to either deriving all motivic material from a basic cell or viewing all subsequent instances of a motive as transformations that vary in their degree of relatedness to the basic cell. Both approaches imply that an underlying hierarchy and teleology exists.

Mead's excellent analysis of the solo clarinet line from the opening of Composition for Four Instruments, for example, focuses on the generative function of the clarinet's first 3-3[014] trichord (pitch classes B, Eb, and C) and its relation to the overall structure of the solo and the composition (see pages 57-65). If we parse the line following the initial trichord through measure six by partitioning the line's nine notes into three groups containing three adjacent notes with no overlap between groups, the following trichord succession emerges: 3-9[027], 3-5[016], and 3-7[025]. The generative role of the 3-3 [014] trichord is hard to see, comparing the set classes of these trichords to the initial trichord. Mead takes us in another direction by directing our ear to the temporally non-adjacent but registrally adjacent pitches in the clarinet's lowest register, Ab3, E3, and G3. This line is, of course, a member of the opening trichord's set class, 3-3[014]. The trichords are ordered transformations of each other with ordered pitch-interval patterns <+4, -3> and <-4, +3>, respectively. Build- ing on this foundation, Mead untangles the upper-register dyads to reveal

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two more embedded transformations of the initial 3-3[014] trichord whose ordered pitch-interval patterns are <+3, -4> and <-3, +4>. Taken together, the four 3-3[014] trichords represent all four serial transformations. Another way to view this is that the initial trichord was the gener- ator for the other three.

Applying the same temporal adjacency partition to the second aggregate produces a succession of 3-2[013] trichords. Mead again looks past the pitch-class temporally adjacent trichords to reveal the registrally adjacent trichords. He shows that the first upper-register trichord is the initial trichord transposed up an octave. This is a very strong relation, since the two trichords not only are members of the same set class, they are pitch- class and ordered pitch-interval identities. The remaining embedded trichords of the second aggregate are also pitch-class and ordered pitch- interval identities (but not pitch identities) of the embedded 3-3[014] trichords of the first aggregate. In fact, the same four trichords appear in all the remaining aggregates of the solo, but in aggregates five through eight the pitch class order of each trichord is reversed. In this context, Mead's model is at its best, because it "begins to illuminate a way to parse the music's highly variegated surface into a more regular underlying pattern." (p. 58). The redundancy of the underlying pattern generates a high degree of motivic unity and facilitates the generative motivic compositional process. In the solo, the opening trichord generates the remaining embedded trichords by application of the serial operations to the first trichord. Similarly, each of the registrally distinct 3-3[014] trichords of the first aggregate generates the embedded registrally distinct trichords of aggregates three through four by application of the serial operations to each 3-3[014] trichord of the first aggregate. In this way, the first trichord generates the next three and the first aggregate generates the next three. In the end, the entire passage is generated by a single trichord, the opening 3-3[014], <B, Eb, C>.

Compared to its trichordal predecessor, moving to composition with the all-partition array's non-redundant partitional architecture seems like a break with tradition as well as an expansion of array technique. Mead acknowledges somewhat the effect the all-partition array will have on the compositional landscape when he says, "repetition [of partitions] would entail different kinds of compositional responsibility" (p. 35). He does not, however, fully explore how the compositional responsibilities might change the analytic responsibilities in moving from the highly repetitive trichordal array to the non-repetitive all-partition array. For example, since the repetition of partition shape, set classes, and pitch classes associated with set classes is easily achieved in the trichordal arrays, and it is the exception rather than the rule in all-partitions arrays, the generative motivic coherence present in the clarinet line from Composition for Four

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Instruments might also be the exception rather than the rule in all-partition pieces.

Two non-equivalent partitions can contain some of the same cardinality classes, but this does not guarantee that the partial orderings are from the same set class or contain the same pitch-class material. Each of the cardinality-equivalent partial orderings could be extracted from different parts of the row that contain non-equivalent set classes. In Example 1, for instance, all the partial orderings of cardinality five are member of different set classes, 5-30[01468], 5-7[01267], and 5-6[01256], respectively. Furthermore, even when partial orderings do contain the same set class, they may be in the same partition, adjacent partitions, or widely separated in the array, which is another limiting factor on making associations.53 Furthermore, whereas establishing an equivalence relation among partitions in a trichordal array is both possible and easy, establishing an equivalence relation among the partitions of an all-partition array is not possible. The only possibility is establishing a similarity relation among partitions.54 Even the trivial equivalence relationship of each partition containing all twelve pitch classes is not guaranteed. If the all-partition array contains weighted aggregates, its partitions contain fewer than twelve distinct pitch classes. All of these factors introduce an element of randomness to the all-partition array that was not present in the trichordal arrays. The consequence of the new partitional architecture, however, can only be fully appreciated when viewed in conjunction with another lack of repetition in the all-partition array.

A hyperaggregate, as Mead informed us, is a six- or twelve-part all- partition array that contains a single appearance of all forty-eight members of the row class (i.e., no member of the row class is repeated). Of course, Mead links this development to Babbitt's principle of maximal diversity, which certainly is the case, but it also seems to be an interesting and understated solution to a "problem" of composition that concerned early twelve-tone composers. Like a child trying to escape the influence of its parent, twelve-tone music sought to establish syntactic principles independent of tonality's syntax. Sometimes, however, rather than a complete break with the principles of the past, the new syntax rein- terprets them. Progression, and specifically chord progression, had a suf- ficiently strong influence on the early twelve-tone composers that they sought to incorporate the concept into twelve-tone music. The desire to emulate the concept gave rise to the following question: given any row, what criteria determine one's choice of the row that follows another row?

Schoenberg's solution, in some works, was to create the idea of the secondary set in which the second hexachord of one row forms an aggregate with the first hexachord of the row that follows. The aggregate principle that creates a polyphonic structure in twelve-tone music through

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hexachordal combinatoriality could also create progression from one row to the next by means of secondary aggregates.55 Of course, linking row forms by secondary aggregates only works with rows from the same harmonic area or region (i.e., with rows whose hexachords are pitch-class content invariant). Since the succession of two rows from different regions breaks the secondary aggregate progression of row forms, the break can demarcate a move to a new region. The introduction of a new region could be thought of as analogous to modulation, and the move to a new region, therefore, could imply a return to the original region. This model of progression in twelve-tone music characterizes Op. 33a, for instance, as a sonata form or at minimum an example of A-B-A form. Within a region, a hexachordally combinatorial row pair could function as the central node in a structure that relates other members of the region to the central row pair hierarchically. The hierarchic model of row-pair relatedness within a region could explain why the pair of rows stated at the beginning of Op. 33a also mark the return of the A section's harmonic region.56

The secondary aggregate procedure also appears in Babbitt's arrays and compositions. For example, it limits the succession of row forms in the first of Milton Babbitt's Three Compositions for Piano, and it limits the succession of rows within a lyne of the all-partition array of Arie da Capo.57 In the piano work, the technique has a function loosely corre- sponding to the technique's function in Schoenberg's in Op. 33a. Besides limiting the succession of row forms to those rows capable of creating secondary aggregates, the row pair that begins the work also marks the closing section giving the piece an overall A-B-A structure and perhaps establishing a hierarchy among the rows from the region.58 Unlike Op. 33a, however, the work only contains rows from a single harmonic region (i.e., the B section does not "modulate" to a different region). Outside of structuring the succession of row forms in the all-partition array of Arie da Capo, secondary aggregates do not perform functions similar to those found in Op. 33a or Three Compositions for Piano. In fact, the structure of the array appears to be designed to de-emphasize hierarchical structure and offer a different solution to row succession in serial music.

The all-partition array associated with Arie da Capo59 has six lynes that are filled by three pairs of hexachordally combinatorial rows. Fol- lowing the earlier model, all the lyne pairs should be from the same harmonic region. However, Mead informs us that each of the three lyne pairs is from one of the row's three harmonic regions so that all the row's regions are simultaneously represented in each of the array's divisions, or blocks, as Mead calls them. Consequently, harmonic regions can no longer function as modulatory goals, and they cannot be hierarchically distinguished from each other. Similarly, each lyne pair of the array uses all the rows from a region without repeating any row forms, so the type

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of hierarchical relationship among rows that depends on repetition to establish the hierarchy is not possible either.60

Presenting all the members of a row class in a work appears to be a holistic rather than a hierarchical approach to the question of row succession in serial music, since the function of a particular row and its parts in the holistic paradigm will be determined by the context of the "compositional language" as a whole. That is, having all the members of a row class represented in a work acknowledges and compositionally exploits the fact that every row can potentially follow every other row, and the parts of any row can form associations with the parts of all the rows in the row class.61 In this context, a particular row's function and the function of its parts will be determined by how all the other rows and their parts function in the composition. Since, in the holistic paradigm, any particular row form is structurally related to every row in the class by subset content (not just the complete pattern of intervals that identifies a row), a work must present all the members of the row class to fully max- imize the potential of an individual row and the potential of its subsets for making structural associations. In other words, the structural meaning of one row and its subsets from the row class in a composition depends on the structural meaning of every row and all their subsets from the row class in a composition.62 To twist an old expression about all roads leading to Rome, we can say that in a Babbitt hyperaggregate all-partition array composition that all rows lead to every other row. Furthermore, the path one decides to take between rows depends on what type of trip you want to have, since the only place one row is going to take you is to another row. Translating the idea to listening means we are focused on moving through the compositional space, not where that movement is taking us.

Extending the holistic non-hierarchic account of Babbitt's practice of using all forty-eight row-forms to the motivic domain can have a profound effect on how we view Babbitt's music from this period. If the move from the trichordal arrays to the all-partitions arrays also represents a move from a generative and potentially hierarchic model of motivic structure to a holistic and non-hierarchic model of motivic structure, then the development of motivic material, like the succession of rows, is no longer a teleological compositional process. That is, a motive introduced at the beginning of a work does not necessarily set the stage for future developments, and coherence no longer depends on tracing the origins of every element of a work back to a single generative motive. In most of Babbitt's all-partition works, I believe that every motive introduced is a generative motive, and every grouping of pitch classes is a motive. Since, in this model, the temporal succession of motivic material does not imply progression, the move from one motivic group to another is more akin to a

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series of changing states. Again, our focus is on moving through the compositional space, not where that movement is taking us. If we replace the term "sound masses" with "motives," Varese's description of his own music captures the changing motivic states of Babbitt's music:

When these sound masses collide the phenomena of penetration or repul- sion will seem to occur. Certain transmutations taking place on certain planes will seem to be projected onto other planes, moving at different speeds and at different angles. There will no longer be the old conception of melody or interplay of melodies. The entire work will be a melodic totality. The entire work will flow as a river flows... In the moving masses you would be conscious of their transmutations when they pass over different layers, when they penetrate certain opacities, or are dilated in certain rarefactions.63

In the holistic model with all motives being equal and all material being motivic, the image of a composition could be reduced to a single plane. Furthermore, since everything is potentially a motive, it might seem as if the term motive has been reduced to a tautology. That is, if everything is a motive, then nothing is a motive; the plane is undifferentiated. Although an undifferentiated plane is the result of taking the holistic viewpoint to one extreme, a static plane is not the only form resulting from the holistic viewpoint. In the first chapter ("Introduction: Rhizome") of their book A Thousand Plateaus, Gilles Deleuze and F61ix Guattari describe plateaus rising from a plane to capture the idea of the individual and its simultaneous connection to everything else in the world.64 On the plane everything is connected, but periodically areas of coagulation form and rise up from the plane forming a plateau. The plateau exists for a time, then it is absorbed back into the plane and another coagulation forms and another plateau rises. Plateaus may be structurally unrelated, similar, or equivalent, but they all share the common connection of being connected to the plane. Motivic structure in some Babbitt all-partition array pieces and especially the super-array pieces are like the protrusions from the plane. Motives come into being, exist for awhile, and then recede back into the plane and new plateaus form with new sets of motivic material, which may be unrelated, similar, or equivalent to the material from other plateaus. If we add multiple planes and allow the protrusions from one plane to generate protrusions in another, then we have Varise's description of his music.

The holistic non-hierarchical approach to progression and the non- hierarchical non-generative approach to motivic development make Bab- bitt a post-modern composer. However, the non-hierarchical nature of the hyperaggregate all-partition pieces alone does not accomplish this. It is the fact that the protrusions or plateaus are dependent on the listener not the author. Babbitt has created the planes, and it is up to us, the reader, to

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"push our fingers into the plane" generating the plateaus. Babbitt's ac- complishment as a composer in this context has been to create an extremely rich and fertile plane for us to poke. In fact, Mead's compositional tutorial and the necessity of stressing the structural implications of every relation he finds in every piece analyzed is a consequence of the plane Babbitt has created. What Mead demonstrates by doing this is that the more we know or more importantly, what we know and what we look for to a great degree determines what we find. Mead's analyses, of course, take us through his reading of the plateaus he finds. What he looks for more often than anything else is the source of surface structures in the underlying structure. However, the model presented here of a plane with plateaus does not depend on an underlying structure. The plateaus are not the surface and the plane is not the underlying structure. The plateaus and plane are really the same thing with the focal point changed from the entire plane to one of its plateaus. By stressing a division between a more regular underlying structure and a possibly irregular surface structure and by necessitating the normalization of the irregular by the regular, we lose our focus on moving through the compositional space, and consequently we miss the trip.

X. Analytical Fallout from the Holistic Viewpoint

In fact, the context-sensitive component of Mead's model and the inconsistencies it gives rise to may be generated by his model coming into contact with the holistic non-hierarchic view of Babbitt's music. Like two planes colliding, the holistic model generates plateaus in Mead's model. We can see evidence of this in his analysis of Post-Partitions (pp. 178-88). The central focus of the analysis is the role the 26 mosaic (i.e., a partition of six unordered dyads [B,D], [A6, C], [E6, G], [Ct, Ft], [F-B6], [E-A] ) together with its particular content of pitch-class dyads plays in generating a hierarchical relationship among all the work's mosaics.65 Building on the hierarchic foundation, he produces an image of the work that is progressive, in which the process of the work unfolds in the first mosaic/ aggregate, while a cumulative function is attributed to the central and final aggregates in that they summarize the processes leading to their appearances:

... the progression of the piece is accumulative. All three of the initial mosaics are invoked at the central passage, and the final passage sweeps up material from the entire work. The accretive process is crystallized in the final aggregate, with its multiple references to different processes and places in the piece.66

Like the analysis of the clarinet solo, the initial mosaic/aggregate plays a generative role in determining the motivic material of the piece, six pairs

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of dyads as simultaneities. The dyads not only establish the interval material, which initially consists of three interval class 5s, two interval class 4s, and one interval class 3, they also fix the pitch-class content of the dyads. For example, interval class 3 is formed by the pitch classes B, and D. Consequently, the distinction between mosaics is based on interval- class content and the pitch classes assigned to each interval class. For example, only one interval class 3 appears in the mosaic determining the dyadic content for the fourth block of the array, and its pitch-class content is fixed to the pitch classes C# and E as opposed to the B and D of the first mosaic. In order to maintain the progressive and accretive image of the work, analytically it becomes necessary to demonstrate the work's coherence; that is, to demonstrate how all the material of the work leads back to its generative material.

The first aggregate's 26 mosaic partitioning with its particular interval collection and fixed pitch-class dyads persists through the first eight aggregates of the work. Although the reoccurrence of the six dyads in each of the first eight aggregates imposes the 26 mosaic structures the pitch- class groupings, the partitions of the array generating the partial orderings for each mosaic are unique. In order of appearance, they are 26, 3323, 642, 543, 424, 3313, and 322212. Only the initial mosaic of the first eight aggregates has an isomorphic structure with an array partition. As Mead takes us through the first eight aggregates demonstrating the cohesion the dyads generate, a few discontinuities begin to appear. The third aggregate (m. 3 and the first beat of m. 4) abandons the simultaneous temporal attack of the dyad's pitch classes in favor of presenting each note in its own temporal space. The new mode of attack generates new dyadic (as well as other) associations along with the original dyads. Although the fourth aggregate, according to Mead, restores the dyadic attack, one pair of pitch classes that form a dyadic motive cannot be attacked as a simultaneity. Since each member of the errant dyad is contained in non-adjacent order positions of a partial ordering, pitch classes that occupy the order positions between the members of the errant motivic dyad prevent the temporally non-adjacent pitch classes from forming the motivic dyad. We are informed, however, that experience of dyadic content allows us to group these pitch classes in spite of surface interference from the pitch classes in the partial ordering.

In other words, the generative and hierarchic status of the initial aggregate's six dyads allows us to determine that it is more important to over- ride the surface groupings in favor of maintaining the coherence of the model. If we examine the array partitions associated with aggregates three and four, we find that the restrictions a 642 and a 543 partition imposes on forming the generative dyads accounts for deviations from the model. The new material is deviant or lesser in status, because the model dictates a greater role for the generative dyads. In the holistic model, it may just

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be that the plateau supporting the dyads is receding and a new plateau with new material equal in status is beginning to emerge. A hint of this multiplicity is evident in Mead's discussion of the transition from the opening mosaic's dyadic material to the mosaic that determines dyadic content for the next group of aggregates. If we take one pitch class, pc E, from the first new dyad (B, E) and pitch class A from the second new dyad (A, G#), their union, of course, produces the dyad (A, E), which was one of the dyads from the first mosaic (see Mead's Example 3.41, p. 180). The multitude of dyadic memberships a particular pitch class can participate in leads Mead to make an interesting observation about the first aggregate of the second block:

As may be seen, the aggregate can be interpreted as the new set of dyads, but it can also be heard in terms of the dyads of the first block. This is typ- ical of Babbitt's music: virtually every aggregate participates in a multitude of trajectories over different time spans. (p. 180)

Another crack in the coherence/hierarchic model appears when Mead discusses the continuation of blocks two and three:

The second and third blocks are composed to project the new mosaic as much as possible. As aggregates grow longer-that is, contain more elements ordered in a single lyne-it becomes more difficult to project desired groupings. If two notes of a dyad are nonadjacent members of the same lyne, Babbitt must resort to filling in the dyad gradually with the intervening elements and their appropriate pairs, gradually building up larger collections. Frequently these larger collections can be alternatively parsed into dyads including some from the earlier section, keeping them alive in the surface of the composition. (pp. 180-81)

The partial orderings of the array are imposing a restriction on generating dyadic simultaneities to either form new mosaics or dyads from the first mosaic. As Mead implies, new processes, such as the appearance of B-type hexachords, are manifesting themselves in these sections. Al- though maintaining the transitive coherence of the dyadic structures becomes more difficult, it must be maintained and supported to preserve the model of coherence for the work. Forcing the dyadic model over the entire work normalizes the irregular by the regular. In my opinion, Mead's view on the multiple trajectories is in conflict with the view of the work as progressive. While I agree with his multiplicity idea, since it is in es- sence the holistic non-hierarchic view I stated earlier, I do not see a way one can privilege one trajectory over another. Mead accomplishes this by creating a hierarchic role for mosaics: the process of the work unfolds in the first mosaic/aggregate, while a cumulative function is attributed to the central and final aggregates in that they summarize the dyads leading to their appearances. The cumulative effect and nexus role Mead attributes

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to the central aggregate that begins the fifth block and outlines the dyads of the new mosaic has as its foundation the fact that this aggregate contains dyads from all the other mosaics.

He claims that the first three mosaics do not share any dyads. Here is where the abstract properties of the mosaics come into conflict with the pitch realization of that material. In his Example 3.41, he demonstrates that the aggregate that begins the second block can be interpreted as a new set of dyads or as the previous set that had been in effect. The multiplicity of dyadic interpretations present at the beginning of the second block, in my opinion, weakens the cumulative effect of the central aggregate, because they are only cumulative in the abstract. In fact, one can find references to other mosaic dyads in other parts of the work. Mead implies this himself in discussing the new material that will be important later in the piece resulting from the individual attacks of dyadic pitch classes in aggregate three. Perhaps what really grants the first, central, and final aggregates their hierarchical status is they are the only mosaic partitions that are either isomorphic to or closely represent the partition of the array that produces their partial orderings (see Mead's Example 3.43, p. 183).

By linking the array to the aggregates that play a central role in determining the form and structure of the piece, Mead solidifies the array's role in determining that structure. The array would play a much more indirect role, and his model of array-based underlying structure essential to our comprehension of the work would not be a prominent feature of his analysis without the link. However, if we remove the hierarchic and progressive constraints, we will be free to follow the work's multiple trajectories. When the dyadic material recedes in the blocks based on the second and third mosaic, rather than mourning what was lost, we can revel in what now is replacing it. In the holistic model, we move from one plateau to another, and in each plateau we can partially see where we have been and we can see a multitude of places we might go.

XI. Towards a New View of the Array

I understand the desire to find in the all-partition pieces the types of coherence that may have structured the earlier trichord pieces and, the types of coherence that generate much of the structure of tonal music. However, by abandoning the desire for a singular approach to coherence, we may produce models of these works that are more informative and closer to the experience of these works. My own analysis of Playing for Time produced an analytic model very close to Mead's. In this work, the partial orderings of its all-partition array are frequently realized in pitch to produce a form of the row that is used to construct the array (see Exam- ple 5). These partial order-generated rows appear in suggestive locations

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RT4I(P)

4.5=

p

Partition 33

Bb A F# C# F G CB

Eb E Ab D

8-22

Example 5. Partial orderings of an array aggregate column realized in pitch as a transformation of the row (Measure 33 of Playing for Time)

of the movement through the array, such as the first and last partitions and at least once in each array block.67 I tried to relate the music both before and after these appearances by means of transformations to the partial order-generated rows. In other words, I tried to establish a generative function for the partial order-generated row forms. I was unhappy with the results, because they did not produce a hierarchy. After working through Mead's book, I realize that taking the holistic approach to Bab- bitt's all-partition and super-array music may be more rewarding. In this model, the partial order generated rows are not nodes higher in a tree; they are plateaus on a plane. Analyzing pieces such as Arie da Capo, that do not even exhibit the motivic regularity of Playing for Time and Post- Partitions, with a non-hierarchic, non-generative, and holistic approach to motivic structure, may yield richer models of these works that may capture some elements of the experience of these works. The job now is to construct these models using the seminal materials Mead provides.

Of course, Mead looks to the array as the source of unity. What may be irregular on the surface can be understood in terms of the underlying regularity provided by the array. If access to the array were transparent, then perhaps it could perform this function. A key feature of gaining access to the array is the ability to hear aggregates. For me the question is moot. I think we can hear aggregates. However, a much more relevant

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question in this case is, can we hear aggregates in every context (i.e., every composition or in every part of every composition)? That is, how general is the model with regards to Babbitt's entire corpus of works? We can generalize the question to include the rules for parsing the pitch material of a piece, because another key feature of his analytic model is the necessity of untangling the pitches of a line to reveal the lynes (i.e., the partial orderings that represent row segments and the underlying regularity). Per- haps the biggest problem I have with Mead's model is its underlying assumptions for parsing or segmenting the pitches of a composition.

In the reconstruction of the analytic model, we uncovered the structural features that form the foundation for determining the efficacy of the model as an analytical tool for exploring the space between a structure's compositional realization in pitch (its musical realities) and its abstract properties determined by the pitch-class array. That is, the structures that play a crucial role in articulating the divisions in the model of underlying structure. Mead outlined several properties, such as repetition of pitch class, invariance, and excluded interval, that would aid in parsing aggregates, which leads to discovering the underlying source of surface combinations. Applying the model to the first four measures of the first of Three Compositions for Piano produces good results (see Example 6).

The opening measure contains complementary pitch hexachords in each hand that, of course, form the work's first aggregate. Measure two continues the pattern of mutually exclusive hexachords in each hand. A little careful listening reveals that the last pitch class heard in the left hand of the first measure is the first pitch heard in the right hand of measure two. The same holds true for the other pair of hexachords. A little more careful listening reveals that not only are the first and last pitch classes of each group identical, the hexachords are collectionally invariant. Listen- ing to the hexachords within each register rather than across registers reveals that the interval class separating the hexachords is ic 6. Again, the same holds true for the upper register hexachords. Aiding our perception of this detail is the fact that the instance of interval class 6 contains the same pitch classes, Db and G, in each appearance. Interval class 6 is the excluded interval of the B-hexachordal type that produces the rows and aggregates in the work.

There are other factors, not discussed by Mead, aiding our perception of the structure of measures one and two. Each aggregate is separated by a slight rest, and the temporal unfolding of aggregates is constant, one aggregate per measure. Contour, the most basic of listening tools, also aids our perception of structure. The contour of the collectionally invariant hexachords is virtually identical.68 Extending our observations to measures three and four, we find the pattern established in measures one and two repeats. One abstract property discussed by Mead does not play a crucial role in our listening to the structure of this passage. The interval

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(• = 108)

mp •

mf M P.

TI

Mf poo

)-,I

OOF , I I I- ' " L- Iil

.i i i

p = -1?6t

Example 6. Measures 1-6 of the first of Three Compositions for Piano

between row intervals generated at each order number when a pair of combinatorial rows are placed in a 1-1 species counterpoint setting will

produce a pattern of intervals for the first pair of hexachords in the combinatorial pair that will be repeated, but perhaps not in the same order, in the second pair of hexachords in the combinatorial pair. This potential structure does not manifest itself in this piece. Consequently, it plays no role in our parsing of the pitch material into aggregates. From the preceding discussion, we can conclude that the composition of the pitch material can either enhance or de-emphasize the properties derived from the abstract structure of the array that are necessary for parsing aggregates.

In fact, some of the properties may not even be part of the abstract

array. To solidify the point, we can compare the opening measures of Duet (Example 6a) with the previous example. The array for Duet is very similar to the previous array, except it is based on the 6-32[024579], C

327



( = 120)

p1 3

(una corda)

o 2

-pp mf p mf

o 3

f mp p mf

" __

" "r L ..-.I,,

b

. i. -

Example 6a. Measures 1-11 of Duet

type hexachord. Both the B and C type hexachords share the same excluded interval, interval class 6. In keeping with the lyric character of the work, the sharp edges that demarcated the aggregate divisions in the first of Three Compositions for Piano are softened, if not eliminated, from Duet. The division between the first and second aggregates of the treble staff line, for example, comes between the C#5 and D5 of measure 6. The D5 in measure 5 is the first repeated pitch class to occur in the upper staff, and more importantly, it is a repeated pitch. The D5 occur- ring in the same register could aid hearing this as an aggregate division. However, the context of the D5 leads me not to hear it as a division, but as a continuation from the previous aggregate.

The first two phrases reach their conclusion in measure 4 with a gesture that is similar to the opening, and the dynamic decrescendo to pianis- simo. The gesture in measure 5 recalls how the opening rhythmically iso- lates pitch class D in the left hand. The "cadence" in measure 4 and the recollection of measure 1 in measure 5 leads me to hear a stronger asso-

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ciation between the instances of pitches D5 and D4 than I hear between the D5s in measures 1 and 6. Why should I not hear D4 as an aggregate division? It is a pitch-class repetition. For pitch-class repetition to function as a demarcation of aggregate division, we must perhaps add another rule, such as "only pitch-class repetitions within the same octave qualify as aggregate demarcations." Does the addition of the register rule now make the repeated Gb5 across measures 8 and 9 an aggregate division? In this case it is not; the division comes between the Db5 and G5 in measure 10. We could add another rule to disambiguate this situation as well: only pitch-class repetitions separated by additional pitch classes qualify as demarcations. This will not work either, because there are many compositions in which a repeated pitch with no intervening pitch classes is a member of two different aggregates. This is an extremely commonplace occurrence in all-partition arrays (in fact, they cannot be constructed without pitch-class repetitions). In this case, we might have to add more rules pertaining to duration of the repetition to keep the demarcation function alive.

In the case of weighted aggregates and super-arrays, even more rules will be necessary. For example, in the sections of Arie da Capo based on weighted aggregates it is very difficult to tell which group of twelve notes forms an aggregate unless you use a non-weighted aggregate as a model for parsing the composition's pitches into aggregate bundles. Mead informs us, however, that in these sections we can follow the midlevel aggregates formed by the hexachordal combinatorial lyne pairs within the array:

When listening to music composed with weighted aggregates we can still use twelve-tone listening strategies to good effect, however. In such passages, classical aggregates unfold over longer spans, by hexachordally combinatorial lyne pair and by individual row. The octaves and unisons found within weighted aggregates themselves serve a dual function, articulating collections in the surface, while indicating the contrapuntal independence of their disparate lyne sources.69

By focusing on the lyne pair aggregates, Mead is essentially admitting that pitch-class repetition can no longer articulate the divisions of the larger all-partition array. Pitch-class repetition now functions, according to Mead, as an articulator of the contrapuntal independence of the hexachordally combinatorial lyne pairs. Of course, the same thing can be said of the super-array pieces, but extended to the array.70 I believe the weight of the extra rules and the difficulty of determining the context so one knows which rules to apply outweighs the benefit of pitch-class repetition rules for parsing or segmenting aggregates.

The model finds a way to keep aggregates active, since one of its basic premises is that we hear this music in terms of aggregates. In forcing the

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model on the music, we lose a potential opportunity to follow the music's trajectory in a new direction. If, in the weighted aggregate sections, the hexachordally combinatorial lyne pairs are contrapuntally independent, as Mead implies, then the streaming element of the music is similar to the streaming element in Ives's music.71 Rather than independent marching bands, the music has independent contrapuntal elements coming in and out of focus. Extending these observations to the super-array pieces again suggests a link between Varese's interacting planes and Babbitt's multiple arrays colliding to produce the phenomena of penetrations or repul- sions and similar music moving at different speeds and different angles projected from one plane to another.

Returningto measure 5, there is a way one I can hear an aggregate division, but the perception of the division happens in retrospect. The right hand of the new phrase in measure 5 consists of two rising whole step dyads, (B, C#) and (D, E). A disjunctive leap and return to the piano dynamic that began the two previous phrases focuses attention on the quarter notes G, F, and C in measures 6 and 7. The pitch realization of these notes not only continues the whole step dyad motive, but they repeat the opening gesture's ordered pitch interval pattern, in the right hand of measures 1 and 2. Since the intervals create a connection between the gestures, with more careful listening we might become aware that both gestures share the dyad (G, C). Making this association may help make the association between the (C#, B) dyad accompanying the right hand in measure 6 and its retrograde and contour inversion counterpart in measure 5. All these factors taken together may help one realize an aggregate boundary has been crossed. The important thing to realize is that the perception of an aggregate boundary came about from the cumulative effect of motivic, rhythmic, contour, and pitch-class associations. These are all factors that can either emphasize or de-emphasize, depending on the compositional context, the abstract division between aggregates. For every clear aggregate division present in a Babbitt composition, there is an obscured division.72 Consequently, the first axiom of Mead's analytical model, parsing the surface into discrete bundles of aggregates, is not a secure foundation upon which to build an analytic theory that covers all three periods of Babbitt's career. I would assume that what is most important to Mead is making connections between subsets, such as the 3-9[027] trichords in m. 1 (D, C, G) and mm. 6-7 (G, F, C). Ironically, strongly demarcated divisions between aggregates in a composition are not necessary to make this type of association.

Another key feature of his analytic model is the necessity of untangling the pitches of a line to reveal the lynes (i.e., the partial orderings that represent row segments and the underlying regularity). Again, whether or not the lines of a composition can be parsed or segmented into lynes depends on the composition. Mead's analysis of Composition for Four

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p i •d'<S

P2 >_ •P

Partition 1 Middle Register B C

EAA6

High Register F

GD Eb Db Low Register B b 43213

Example 7. First aggregate in the flute line of Groupwise

Instruments demonstrates that this can be done, and parsing the music this way will reveal some interesting structural properties. In fact, it is even easier to do in some all-partition array pieces, such as Arie da Capo, because all the instruments except for the designated soloist only have one lyne. Practically anything you hear within an instrumental line is isomorphic to an array's lyne. In other pieces, however, a single instrument often unfolds an entire six-lyne array itself. Example 7 is the opening flute line from Groupwise. The flute's pitch-class material is generated by six- lyne all-partition array. In other words, the flute alone has as much pitch- class material as the entire ensemble of Arie da Capo. The example also contains a temporal reduction and the source aggregate. According to the array, the B5 and C5 belong to the same array lyne. In the music however, they appear to belong to different registral lines: the B5 leads to Gb6 and the Bb4 leads to C5. The tangles produced by the registral overlaps and the temporal disjunction make it very difficult if not impossible to untangle the lynes to reveal the partial orderings. For example, the next note in the G46's lyne, C06, does not appear until measure 6.

Here is where we hit a brick wall erected by Mead's analytic model. If we continue to try and hear this music in terms of the model, then our job is to peel away its layers, because our comprehension depends on our understanding of its underlying structure, which is the array. The only way we will be able untangle the web is to have a priori knowledge of its structure, which seems to defeat the goal of the model. In fact, my biggest problem with this approach is it seems to assume the very thing it is trying to uncover. Almost all the segmentation of a composition's pitches done in the book seem to be made with prior knowledge of which groups to extract, and the pitches extracted are just those groups that happen to

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be partial orderings found in the array. Often these relationships are highlighted, while other important relationships are overlooked.

In his discussion of the first Interlude of Du, Mead calls our attention to trichords based on their mode of attack, simultaneities, and articulation, slurring (see Mead's Example 2.57, p. 112).73 The mode of attack and articulation groups pitch classes across large registral spans and highlights each group, possibly setting them into relief. In other words, our immediate attention is focused on across register groups rather than the lines in each register. The significance of these trichords for Mead is that in the first part of the Interlude, each trichord unfolds linearly within one of the registral lines. Rather than hearing them as a succession of chords, the trichords are heard as contrapuntal lines in the first part of the Inter- lude. Of course, the contrapuntal lines are also partial orderings from the work's array, so the trichord succession leads to hearing the contrapuntal lines and therefore it leads to hearing a segment of the array or vice versa.

While the association between the partial orderings from the array and trichords formed from grouping individual pitch classes from partial orderings across registers is important, it is not the complete picture. For example, two trichords do not fit into this puzzle (the 3-2[013] (G#, A, B) and the 3-4[015] (D, Bb, Eb) trichords), because all three of their pitch classes are neither slurred nor simultaneities. Are these just leftover pitches from the array that because of order restrictions could produce one of the contrapuntal array trichords? This might be one way of accounting for these groups, if the focus was centered on only uncovering those groups of pitch classes related to array segments. However, the different mode of attack and articulation performs a different type of differentiation.

In calling our attention to the modes of attack and articulation, Mead has inadvertently provided us with a tool for segmenting the passages and possibly accounting for the two errant trichords. Although three trichords are slurred, two share the same rising contour, (Eb, Ab, E) and (Ab, E, F), and the central trichord, (G, F#, E), first rises then falls. We hear the (Eb, Ab, E) and (Ab, E, F) trichords as more similar to each other than they are to the (G, F#, E) trichord, because their contours are similar. Each of the contour-similar trichords is followed by three trichord simultaneities, while the central slurred trichord is followed by the two errant trichords and a trichord simultaneity. Digging a little deeper, the central trichord together with the trichords that follow are rhythmically continuous, while rests group the other trichords into pairs of hexachords. Examining the set classes of the trichords reveals that four trichords in the 3/4 measure, 3-3[014], 3-2[013], and two 3-4[015]s, form a retrograde progression of the first four trichords in the 5/4 measure, which of course, are already linked by articulation and contour relations. (Incidentally, the dynamics are also retrogrades of each other.) Digging even deeper reveals that the hexachords formed from the unions of the rest-articulated trichords in

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each progression are identical with regards to pitch class content. Each trichord of each progression also shares two pitch classes. If the hexachords are identical with regard to pitch-class content, then they are all members of the same hexachordal set class, which is 6-14[013458]. The central slurred trichord and its following trichords can also be grouped into two contrasting hexachords, 6-Z24[013468] and 6-Z46[012469], which contrast with the 6-14[013458] hexachords surrounding them. Overall, the passage has an A-B-A form in which the B section contrasts with the A section both in hexachordal set classes, rhythm, and articulation. The two errant trichords in terms of articulation now seem to be an integral part of the passage's structure. An additional feature of the passage is that each section of the A-B-A form is also an aggregate.

The interesting point about the preceding analysis is that it started with nothing more than noticing a difference of articulation, a difference that is immediately perceivable. It was not necessary to invoke the array to find the passage's deeper structure. Perhaps extending the preceding observations to the first part of the Interlude would lead to uncovering the connection between the slurred and simultaneous trichords and their contrapuntal counterparts. Even making this connection does not necessitate invoking the array. It would only be by extending these observations over the entire piece that the array might emerge. To be sure, this section of the interlude does have its genesis in the array, but that does not have to be either the starting point or the point of arrival before comprehension of the music occurs.

I sympathize with the effort of trying to preserve the array. After all, it is a fact of a composition. It can be reconstructed from the composition. Furthermore, making the array vital to our hearing of the piece is one way to answer criticisms about its use being an "intellectual conceit." However, in the effort to make it essential, we cannot lose sight of all the other ways that Babbitt's music is vital. Of course, we could always over- look this and attribute the book's focus on the array in keeping with its introductory nature, but even introductions to subjects should point to ways that the journey should continue.

In a very direct way, Mead's work has performed this function for me, since it forced me to think of other possible models for the array. In those pieces where Mead's model would be most problematic, I view the array as the machine language portion of a computer. When we work in a word- processing program, all we really care about are the letters on the screen. All of our typing actions are being interpreted by a computer program that translates them into programming language commands in a language like C11. The programming language commands are themselves translated by a compiler into machine code and perhaps further translated into assembly language, the language the computer's CPU can understand. All of these translations are opaque to the user, except when the computer

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crashes. We then become aware that some action we performed may have had dire consequences further down the chain. Although we are for the most part unaware and do not need to be aware of the parts of the chain to perform actions, they do exert an influence over what we do. For example, if we can not find a way to use a program to execute a machine code the CPU understands, we will not be able to perform that action. In a very real way, the limitations of the machine code place limitations on what we can expect the CPU to accomplish.

In this model, we do not have to have access to the array to feel its influence. Much of Mead's work does take steps in this direction: his discussion of how dynamics might function in Babbitt's music is one example. Nevertheless, much more work needs to be done with just those ravishing surfaces. As theorists, if we want to put those criticisms of "intellectual conceit" to rest, we should first revel in those surfaces and then develop compilers to reveal their connection to the other parts of the chain.

I would like to make one final note on the appropriate reader for this book. Mead says it is for anyone interested in Babbitt's music, which implies that the book is self-contained in that it contains all the information one needs to grasp the subject. I cannot recommend the book for people who do not have some background in atonal serial theory, since many of the implications and idiosyncratic presentations of the pitch- class structures Mead discusses will be lost on readers lacking a firm foundation in this area.

In closing, I would like to applaud Mead's effort in trying to conceptualize the web of relations that is Babbitt's world of composition. What at first seems overwhelming, Mead has made more manageable, informative, and enjoyable. The theoretical community should appreciate this work, since it is an important contribution to the field of Babbitt scholarship.

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NOTES

1. Lawrence Fritts, for example, cites this omission as a fault with Mead's work. See Fritts's review in Music Theory Spectrum 19, no. 1 (1997): 93-103.

2. For example, see Fritts, ibid., and David Lewin, "Generalized Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio," Music Theory Spectrum 17, no. 1 (1995): 81-118.

3. Babbitt, Words about Music, 167. 4. For example, footnote 40 in Chapter 1 repeats the same pattern. The text on page

31 implies that Babbitt's partitioning of four distinct elements into two or fewer parts in a trichordal array is the result of a non-trivial operation: " ... the example actually contains two representations of the list of partitions of four, in terms both of the lynes of the array and of the array's trichordal elements. This is not a trivial consequence, as either representation could be maintained without the other, simply by swapping the positions of the (3+1) partitions." Rather than leading the reader to specific information demonstrating why the consequence is non-trivial and why this fact is important to know for the discussion, the footnote simply refers the reader to the book Words about Music and the article "Since Schoen- berg" with the comment by the author that "Babbitt discusses the dispositions of trichords in arrays" in both works.

5. For example, I, for one, had no idea that Babbitt wrote a little article titled "The Synthesis, Perception, and Specification of Musical Time," for the Journal of the International Folk Music Council in 1964.

6. I call these documents, because unlike previous periods in music history, our period has a record of the performance practice, for better or worse, surrounding a body of work. It will be interesting in and of itself to study the changes in that performance practice.

7. A very informal definition of a Babbittian all-partition array is that it is an abstract structure of intersecting rows and columns. The rows of the array are the members of a 12-tone row's row class and the columns contain row segments (partial orderings). Like the rows of the array, each column contains the complete chromatic formed by the 12-tone row segments generated by the intersection of the array's rows and columns, and no column contains the same distribution of partial orderings that another column has, and all the possible ways of dividing up 12 into parts that add up to 12 are represented by the array columns. That is what makes the array all-partition. (See Example 1.)

8. For instance, the array from which Example 1 is extracted is also the source of pitch-class material for the solo violin composition Melismata. In this case, the array is not transformed under any of the TTOs, except the trivial To, and the order of the array blocks, as Mead calls them (sections of the array in which all the linear twelve-tone rows that form the lynes of the array complete the aggregate), is the same in both compositions. The same array transformed by TsM is the source of the pitch-class material for the composition Arie da Capo.

9. Babbitt was 31 at the beginning of the first "official" musical period. 10. Whereas the array columns in the trichordal arrays contain the same distributions

of partial orderings (three pitch classes from each of the four linear row forms), the columns from an all-partition array each contain a unique distribution of partial orderings. In other words, while the total number of pitch classes in each col-

335



umn remains twelve, the distribution of partial orderings that sum to twelve in each column is unique.

11. Nicholas Cook, for example, in his afterword to the volume Concert Music, Rock, and Jazz (Rochester, New York, University of Rochester Press, 1995), 422-39, makes this point about a contribution to the volume by Mead titled Twelve-Tone Composition and the Music of Elliott Carter (pp. 67-102). He says "Mead adopts the same explanatory strategy as Schenker did in Derfreie Satz: he sets out abstract structures and then shows how they are "realized" or "manifested" in the fin- ished composition" (p. 423). He goes on to characterize this approach as Platonic: "Nevertheless, this distinctly Platonic concept of composition imposes limits upon the composer's authority that are hardly less severe than Schenker's" (p. 424). Apparently, Cook views the similarity of approach, moving from an abstraction to its realization in a composition, between Schenker and Mead, to signify an appeal to Platonism by Mead. Schenker's approach could certainly be considered Pla- tonic, because it could be argued that his background structures and the transformations that lead from the surface to the background represent musical universals. Mead, however, stakes no claim of musical universal status for the array. Further- more, since the array is created by the composer, and since it can assume practically any form the composer wishes, the limits the array might or might not impose on the composer's authority are self-imposed, and creating the array could be considered an act of composition. The latter point of view is a fundamental premise guiding the creation of arrays in Morris's Composition with Pitch-Classes.

12. Unity, in this sense, is not the same as a composition exhibiting the property of coherence. Unity just distinguishes one object from another by establishing the boundaries between the object and its environment.

13. Hasan Ozbekhan, "Planning and Human Action," Hierarchically Organized Sys- tems in Theory and Practice, ed. Paul A. Weiss (New York: Hafner Publishing Co., 1971), 180-81.

14. The author, however, is not implying that he would not want to transfer every aspect of the literary system to music. In the music system, for instance, it may not be necessary that each subsystem have a hierarchical organization.

15. A trio grouping, for example, would be produced by assigning three instruments of the quintet two lynes each from the six-lyne array.

16. Milton Babbitt, Words about Music, ed., Stephen Dembski and Joseph Straus (Madison, Wisconsin: The University of Wisconsin Press, 1987), 167.

17. Lawrence Kramer, Classical Music and Postmodern Knowledge (Berkeley, Cali-

fornia: University of California Press, 1995), xii. 18. While I was a graduate student at the University of Washington, I studied and per-

formed East Javanese Gamelan music. 19. To explore an alternative technical path for modeling structure in Babbitt's music,

see Dora Hanninen's analysis of Tableaux. She explores contour relations and their role in creating associations among gestures. This study of Tableaux consists of five analyses. The first two involve individual passages; the last three, pairs of passages. The five analyses progress in complexity from the first, which concerns only contour relations composed in realization, to the last in which contour relations rooted in the array interact with those composed independently and with other kinds of Csc and Ci,, criteria. Together, they investigate how "Scseg associations composed solely in musical realization (1) produce chains of association among

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gestures; (2) relate to musical contours determined by the lyne--octave realization rule and partitions of the aggregate; and (3) involve and interact with other musical dimensions such as pitch-class and set-class to produce associative sets and associative subsets." Dora Hanninen, "A General Theory for Context-Sensi- tive Music Analysis: Applications to Four Works for Piano by Contemporary American Composers," (Ph.D. diss., University of Rochester, Eastman School of Music, 1996, p. 154).

20. I share Mead's opinion that the expressive power of a musical work is tied to its syntax. However, I also believe that a compositional syntax is a necessary but not sufficient component of a musical work's expressive power.

21. Milton Babbitt, "Twelve-Tone Rhythmic Structure and the Electronic Medium," Perspectives of New Music 1, no. 1 (1962): 49-79.

22. Robert Morris appears to share this sentiment when he says "few composers of electronic music in the 1960's took Marshall McLuhan's slogan 'the medium is the message' literally." Robert Morris, "Listening to Milton Babbitt's Electronic Music: The Medium and the Message," Perspectives of New Music 35, no. 2

(1997): 85-99. 23. We will attempt answers to these questions and examine the analysis in detail later

in the review. 24. In his article, "Cognitive Constraints on Compositional Systems" [Generative

Processes in Music, edited by John A. Sloboda (Oxford: Clarendon Press, 1988), 231-59], Fred Lerdahl's attack on serial composing theories takes the step from the inability of a listener to hear rows as entities and the transformations relating those entities in Boulez's Le marteau sans maFtre to generalize that listeners have a cognitive inability to hear rows and their transformations. When I suggest rows as candidates for the underlying structure in an analytic listening model, I am not implying that one should only hear rows and their transformations as entities according to the principles outlined by Lerdahl. To be more precise, one should say, "to hear rows as the underlying structure in an analytic listening model, one should hear the effect of rows or hear rows as entities." Context determines which strategy to employ. To hear the effect of rows means that one is aware of segments that reoccur in a composition that share ordered interval patterns, and consequently a transformational relationship. Given the correct context, we can hear these segments (the partial orderings produced by an array) as manifestations of the row's abstract ordered pitch-class interval pattern, a point Mead makes. Again given the correct context, one should be able to ascertain the ordered pitch-class interval pattern that is the basis of "the row" and its transformations. This is especially easy to do in an all-partition array composition, because the complete row is always presented in a single register, by a single instrument (in a multi-instrument work), and unaccompanied at some point in the work. The multiplicative operations employed by Boulez to "the row" of Le marteau sans maitre create an environment in which neither hearing rows as entities nor hearing the effects of rows is facilitated, because the ordered interval patterns that would identify segments with the abstract row are not maintained. Therefore, in many ways, Le marteau sans maitre is not a serial composition, in my opinion. Consequently, it should be no surprise that Lerdahl would conclude that listeners exhibit cognitive opacity with regards to rows and their transformations in Le marteau sans maitre. For a more complete criticism of Lerdahl's argument, see the first chapter of my

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dissertation: Ciro G. Scotto, "Can Non-Tonal Systems Support Music as Richly as the Tonal System?" (D.M.A. diss., University of Washington, 1995).

25. For example, the first variation of the second movement of Webern's Symphonie, op. 21, polyphonically combines four row forms without any apparent fixed columnar divisions.

26. Babbitt's arrays contain even numbers of row forms (12, 6, or 4) as lynes, therefore each pair of rows in the array can form one hexachordally combinatorial pair. The harmonic regions of a row are all the transformations of the row that maintain the pitch-class content of the row's hexachords. In Example 1, although all the rows are members of the same row class, the pair of rows at the top of the array are in a different harmonic region from the pair of rows at the bottom of the array, because the pitch-class content of the hexachords in each pair of combinatorial rows is different. All the rows capable of forming a hexachordally combinatorial pair with a given row will be in the same harmonic region. Therefore, as the number of combinatorial pairs a given row can form with other members of the row class increases, the number of harmonic regions for that row class will decrease. For additional information on harmonic regions and their relation to the all-combinatorial hexachords see Babbitt, Words about Music, 52-57.

27. If we assume that the array underlying sketch A consists of three row forms in which each row form contributes four pitch classes to form three columnar aggregates (a 3 X 4 array), then the only structure determined by the array is the order of the pitch classes in each row segment. Any structures produced by the interaction of row segments, such as the trichord simultaneities in sketch A, result from post-array construction compositional decisions. Any number of configurations that maintain the order of the row segments and generate new discrete bundles of pitch classes from among the segments are possible. Of course, one of the configurations, suggested by the array's own structure, is a note-against-note counterpoint of temporally discrete trichords.

28. The following statement is one of many that supports this claim: "As most aggregates in his all-partition pieces are built of segments from more than one row, his music can contain very fanciful mixes of instruments and registers-but over the long span we hear that they result from combinations of much more regular presentations of rows in individual instruments or registers." Mead, An Introduction to the Music of Milton Babbitt, 36.

29. Mead does acknowledge the important role pitch space plays in making comparisons among aggregates. In a later part of the chapter, he says, "in a trichordal array generated from a single trichord type, such as that found in Mead's Exam- ple 1.10, p. 28, every aggregate may consist of a simple redistribution of four ordered collections. Except for the reassignment of the four collections to what- ever articulative means are used to distinguish lynes of the array, the structure of the aggregates provides no mode of telling aggregates apart. Distinction among such aggregates is created by the composition of their details, the generation of significant collections and intervallic patterns from the counterpoint of lynes." (Mead, An Introduction to the Music of Milton Babbitt, 28.) There are two important differences between Mead's invocation of pitch space and mine. First, he appears to limit the necessity of invoking pitch space criteria to arrays with a very uniform structure. I believe pitch-space criteria are necessary to create distinctions among aggregates even in arrays in which the distribution of partial orderings and

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set types creates a unique profile for each aggregate. This leads directly to the second point that underscores the main discussion. The relationship between the properties exhibited by the array and its pitch realization is not always one-to-one (a point I will explore later in this review).

30. Although registral criteria determine the discernibility of the partial orderings in sketch D as discrete segments, this should not be taken to indicate that the only method of individuating the segments is by register. The point is the pitch compositional context plays a crucial role in determining the discernibility of an array's partial orderings. Furthermore, the preceding discussion on the discernibility of aggregate parts in pitch space should dispel any notion that using arrays is a mechanistic method of composition.

31. Of course, the benefit of approaching analytical listening from the direction of what is available to hear is it can speed up the process of directing the ear to and focusing the ear on hearing specific musical structures.

32. For example, Tinctoris's or Morris's treatises on composition will undoubtedly inform any analytic theory formulated about either composer's works. Similarly, Lewin's analytical work on transformational networks will undoubtedly inform some composer's theory for the generation of structure.

33. See note 30. 34. Tn, TI, RTn, or RTnI are equivalent to the more traditional nomenclature for row

forms, P, I, R, and RI. 35. Mead's criterion of repetition of pitch classes as a signal of aggregate boundaries,

for example, might be particularly easy to perceive using combinatorial rows from the same harmonic area.

36. See page 25 for the full discussion of excluded intervals and the all-combinatorial hexachords.

37. Mead traces this development in Babbitt's compositional technique to Webern as exemplified in the Concerto for Nine Instruments, Op. 24 (p. 26).

38. Mead justifies the multiple row classes in this type of trichordal array differently: "However, Babbitt, employing the all combinatorial hexachords, also wished to be able to signal the hexachordal boundary in his lynes with the hexachord's excluded interval, and this was not possible if all four lynes were members of the same row class." Since it is possible construct a trichordal array that preserves the excluded interval as an indicator of the hexachordal boundary using the all-combinatorial hexachords and a single row class, this does not appear to be the determining factor for using rows from more than one row class. In Example 2 for instance, the rows To(P) and T7I(P) can be combined with T6(P) and RT7I(P) to form a trichordal array that preserves the excluded interval as an indicator of the hexachordal boundary. Although the rows in the array are now all members of the same row class, the bottom two quadrants of the array will no longer contain all four serial transformations of the generating trichord. The only way to maintain all four serial transformations of the generating trichord in the lower two quadrants is simply to reverse the order of the trichords from the upper quadrants in the lower quadrants. While the reversal ensures that all four quadrants will contain all four serial transformations of a single trichord, it also generates a pair of rows that are hexachordally combinatorial, but not members of the same row class as the other two rows in the array.

39. Of course, the first trichord of To(P) can also combine the first trichord of To(Q)

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and T71(Q) as well as the first trichord of T7I(P), since the system contains no restrictions on how the partial orderings in individual aggregates can combine to form larger collections. The former combination produces another instance of the A hexachord, while the latter combination produces a member of the hexachordal set class 6-30[01679], which is not one of the all combinatorial hexachords. Al- though these types of combinations do arise in the music, Mead does not address their significance, choosing instead to focus on the all-combinatorial family. I will address the significance of this omission later in the review.

40. Although adding another lyne pair to a hexachordally combinatorial lyne pair and dividing by two produces a trichordal array (i.e., dividing hexachords by two produces trichords), you obviously cannot add a third lyne pair to the array and divide by two again. Dividing trichords by two would, of course, produce partial orderings containing 1.5 pitch classes. However, an array generated by tetrachords has three lynes and divides the original row into segments whose cardinality is four. A tetrachordal array requires three lynes to form aggregates. Adding three more lynes to the array and dividing the tetrachords by two produces an array with six lynes and partial orderings of cardinality two. Adding six more lynes to the dyadic array and dividing by two again produces an array that contains no partial orderings, since a single pitch class would occupy each array position. The final step in this progression would, of course, simply reproduce the original row matrix.

41. The six row all-partition array will contain eleven partitions for six rows, thirteen partitions for five rows, fifteen partitions for four rows, twelve partitions for three rows, six partitions for two rows, and one partition that just contains the row.

42. Although abstractly any similar partitions can determine subsets of equal cardinality, such as 623 and 632 which both contain segments of size six, the partitions only guarantee repetition with regards to cardinality. Since the segments of size six could be drawn from numerous consecutive segments within a row, there is no guarantee that the segments will be members of the same set class.

43. Besides producing pitch-class duplications, Mead says the procedure maps a hexachordally combinatorial lyne pair representing a harmonic area onto a different hexachordally combinatorial lyne pair from the same harmonic area. Applying a single inversion operator to the entire array maps a hexachordally combinatorial lyne pair representing one harmonic area onto a different hexachordally combinatorial lyne pair from a different harmonic area. In the former procedure, the lyne pairs representing a harmonic area maintain their location in the array and its transformation. In the latter procedure, lyne pairs representing a harmonic area shift their location in the transformed array. When lyne pairs representing a harmonic area maintain their location in the original array and its transformation, a single instrument assigned to the lyne pair will continuously cycle through the same hexachordal pitch-class collections. When lyne pairs representing a harmonic area change their location in the array's transformation, a single instrument assigned to the lyne pair will not continuously cycle through the same hexachordal pitch-class collections. The hexachordal pitch-class collections the instrument plays will change with the move to the transformed array.

44. As Mead points out, this is the same procedure Babbitt used to control the polyphonic unfolding of the components in a trichord array. Example 1.12 on p. 30 illustrates one possible sequence of the fifteen subset components.

45. Specifically, in Mead, "Detail and the Array in Milton Babbitt's My Complements

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to Roger," Music Theory Spectrum 5 (1983): 89-109; idem, "Recent Developments in the Music of Milton Babbitt," Musical Quarterly 70, no. 3 (1968): 310-31.

46. The Chapter titles reflect Mead's teleological view of the technical developments associated with Babbitt's compositions: Chapter 2, Mapping Trichordal Pathways (1947-1960); Chapter 3, Expansion and Consolidation (1961-1980); and Chap- ter 4, The Grand Synthesis (1981-).

47. Richard Cohn's evaluation of Schenkerian theory with regard to its treatment of motives is one example of how the limitations of a theory might outweigh its benefits. Richard Cohn, "The Autonomy of Motives in Schenkerian Accounts of Tonal Music," Music Theory Spectrum 14, no. 2 (1992): 150-70.

48. A similar point is made by Joseph Dubiel in the first of his three essays on Milton Babbitt: "And in an appropriately extended sense a listener who follows them might be given credit for catching on to the array-since conceivably an awareness of the twelve-tone structure embodied in the array would help motivate, and thereby facilitate, identification of these "surface" hexachords, while, recipro- cally, the perception of these hexachords might find extension and reinforcement in detection of the twelve-tone structure embodied in the array. Yet it would be a misconception to consider this a case of the readily apparent working to the good of the ultimately regular [italics mine], inasmuch as there is no reason to think 'the twelve-tone structure embodied in the array' any more the twelve-tone structure than these ad hoc but pointed expressions of it." Joseph Dubiel, "Three Essays on Milton Babbitt," Perspectives of New Music 28, no. 2 (1990): 246. Dubiel has discussed some of the issues raised in this review from another perspective in his three essays. The present work will attempt to use these points of coincidence to generate a third view of Babbit's music that lies somewhere between the very general and the extremely particular.

49. Mead, for example, expresses this point of view in the section titled "The Twelve- Tone System": "That twelve-tone compositions could effectively be made to be- have in ways similar to tonal works was a reflection of Schoenberg's yearning for the syntactical power of Mozart's and Brahms's language in a music that would employ the sorts of materials he had explored in his contextual works." An Intro- duction to the Music of Milton Babbitt, 9.

50. It should be noted that Robert Morris has investigated similar issues of coherence and continuity in Schoenberg's music. Robert Morris, "Modes of Coherence and Continuity in Schoenberg's Piano Pieces, Opus, 23, No. 1, "Theory and Practice 17 (1992): 5-34.

51. In fact, the trichordal array in Example 2 only contains four pitch-class distinct trichords.

52. The pitch realization of the trichordal array's partial orderings could obscure the trichordal array's redundant structure. At this point, however, we are just considering array structure abstractly from its pitch realization.

53. Consequently, the type of motivic development discussed by Jack Boss, in which there is a progression from a motive form to motive forms considered remote, is not facilitated by the all-partition array. Jack Boss, "Schoenberg's Op. 22 Radio Talk and Developing Variation in Atonal Music," Music Theory Spectrum 14, no. 2 (1992): 125-49. Of course, Schoenberg's motivic technique expanded in his twelve-tone works, to include new types of associations. This idea is discussed by Stephen Peles: "... statements ... made by Schoenberg ... would seem to sug-

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gest that he considered a 12-tone set as (among other things) a sort of 'super motive': not necessarily as a motive in itself but as least as a common denomina- tor relating and hierarchically organizing its sub-global thematic interpretations into a comprehensive global structure encompassing and saturating the piece." Stephen Peles, "Interpretations of Sets in Multiple Dimensions: Notes on the Sec- ond Movement of Arnold Schoenberg's String Quartet #3," Perspectives of New Music 22, nos. I and 2 (1983-84): 303. Babbitt's motivic development in the trichordal period is much closer to Peles's view of motivic function in Schoenberg. One could view the all-partition as taking the idea of "saturating the piece" with the set's influence to another level. The shift to the new level, however, is exactly the agent that produces the break with the previous practice.

54. At this point, we are only considering the abstract array. That is, we are considering the array, not the array and its realization in pitch, which can layer another partition in pitch over the array's pitch-class partition. For example, Robert Morris and Brian Alegant have investigated generating equivalence classes among partitions based on the mosaics produced by even partitions of the aggregate in trichordal arrays. At the end of the article, they suggest how the equivalence classes generated by mosaics could be extended to all-partition arrays by comparing the partitions of the first three aggregates of Babbitt's Post-Partitions, which are all members of the same partition class, 26, or Q: { { 2B} { 08 } {37} { 16 {A5} { 94 } } (in fact, the first seven aggregates use the partition Q). Parsing the succession of pitch dyads into 26 partitions, however, generates the equivalence of dyadic partitions across the first seven aggregates of Post-Partitions. Only the first pitch aggregate's partition is isomorphic to the first aggregate's pitch-class partition in the array. The other six pitch-class partitions, in order, are 3322, 642, 543, 424, 3313, 322212. Of course, the equivalence generated by the pitch partitions of the composition does not exist among the array's pitch-class partitions. The pitch motivic structure of Post-Partitions will be discussed later. Robert D. Morris and Brian Alegant, "The Even Partitions in Twelve-tone Music," Music Theory Spectrum 10 (1988): 74-101.

55. Schoenberg's piano piece Op. 33a is the classic example of this procedure. 56. Of course, in Op. 33a the coincidence of the opening pair of rows marking the

return of the opening harmonic region is almost determined, since Schoenberg's row produces twelve harmonic regions and each region contains only four row forms or two hexachordally combinatorial row pairs. The return of the opening row pair coinciding with the return of the opening harmonic region, however, is not completely determined, since the final A section could have begun with the retrograde forms of the row pair that begins the work (i.e., the other members of the harmonic region). Consequently, the row pair that marks the return to the A section's harmonic region appears to occupy a higher node in the hierarchy than the other members of the region. In fact, of the two rows,

Tt(P) and T3I(P) (C=0),

Tt(P) probably occupies a higher node than T3I(P) in a hierarchy that relates rows within a hexachordally combinatorial pair. T3I(P) is presented in its retrograded form, RT3I(P), at the work's opening, which could be interpreted as the work's second row being from the other row pair in the harmonic region. The presentation of Tt(P) appears to determine whether T3I(P) is in its retrograded or non-retrograded forms. In the final A section beginning at measure 32, Tt(P) is still presented as a succession of tetrachords, but the tetrachords unfold linearly rather

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than as simultaneities as in the work's opening. Since the unfolding of Tt(P)

is accompanied by a contrapuntal companion, it must be T3I(P) not RT3I(P) to maintain the hexachordal combinatorial structure. When RT3I(P) follows the tetrachordal simultaneities of Tt(P) at the works opening, hexachordal combinatoriality is not an issue, so T3I(P) can appear in either its retrograded or non-retrograded form. By following Tt(P) with RT3I(P), the succession of tetrachords forms a retrograde symmetrical structure that anticipates the work's overall A-B-A form. Since in each case the structure Tt(P) emphasizes appears to determine T3I(P)'s presentation, Tt(P) would occupy a higher node in a hierarchy relating the two row forms. Furthermore, since Tt(P) is hierarchically stronger it is sufficient to distinguish Tt(P)/T3I(P) from RTt(P)/RT3I(P). Therefore, the presence of RT3I(P) following Tt(P) at the work's opening does not signal a move to the other member of the harmonic region.

57. Since Babbitt's row in the first of the Three Compositions for Piano is generated from the B hexachord (6-8[023457]), it generates six harmonic regions each containing eight row forms or four combinatorial lyne pairs. Consequently, the choice of what row follows another is limited to the rows from the region but not determined, since there are several choices from the region that will generate a secondary set with a given row form.

58. Repetition of row forms, in this case, could be said to facilitate the repetition of motivic material.

59. Since arrays do not have names that identify them, they can be identified by the name of the piece with which they form an association. However, the name of the piece should also function as a class name, since arrays are often associated with more than one piece. The Arie da Capo array, for example, is also associated with Melismata, Playing for Time, and Groupwise, among others.

60. It should be noted that no algorithm exists that will generate an all-partition array. Even computer programs that can generate all-partition arrays use lemmas and proceed on a trial-and-error basis. From my own attempts at creating all-partition arrays, I found it much easier to generate them using Babbitt's scheme of starting with lyne pairs from each harmonic area. This guarantees a good distribution of all twelve pitch classes in the various order positions of the rows, so aggregates have a better chance of being generated when "slicing" up the rows into partial orderings. A counterexample will clarify the point. The row for the Arie da Capo array has three harmonic regions and each region contains sixteen row forms or eight hexachordally combinatorial row pairs. If I attempt use six rows from one region whose first hexachord was invariant as the building blocks for the array, then it will not be possible to generate aggregates. Since the ordering of the row's pitch classes must be maintained, order positions 0 through 5 and order positions 6 through 11 of the six rows will each contain the same six pitch classes. There- fore, the six-row polyphonic structure would not generate aggregates. I could use three hexachordally combinatorial lyne pairs from the same harmonic region, which would allow the generation of aggregates. However, since the rows are all from the same harmonic region, and depending on the internal structure of the row (i.e., invariance within the hexachords), highly fractured partitions, such as 21334, may still be hard to generate. If, however, each lyne pair in the six-lyne array is from a different harmonic region, then the distribution of the twelve pitch classes across all twelve order positions will be more even. This arrangement somewhat over-

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rides the invariances of the combinatorial row pairs and the invariances within the hexachords and facilitates the generation of a variety of partitions. In this light, the simultaneous and exhaustive presentation of a row's harmonic regions appears to be a facet of array construction rather than a compositional consideration like the secondary aggregate structure in the first of Three Compositions for Piano. This does not preclude, however, that this structural property of the array can manifest itself compositionally.

61. Mead makes a similar but subtly different point: "... the use of each member of the array's row class exactly once. We have named these hyperaggregate or row class aggregate arrays to reflect their analogy with pitch class aggregates. This aspect of array structure has two immediate consequences. First, it guarantees each instance of the array's underlying interval pattern a specific distribution of the twelve pitch classes and provides any row invocation arising from some aggregate's partial ordering a specific point of reference in the underlying array" (p. 133). The important fact for Mead appears to be that when the surface combination of an aggregate's partial orderings produce a row or a row segment that is not one of the rows contributing the partial orderings to the aggregate, there will be a row somewhere in the array to which the surface combination is making a reference. I view this take on the hyperaggregate as supporting Mead's view that the array is a composition's underlying structure, and it is the array that makes sense out of the surface combinations of pitch classes. In this case, the surface row invocation simply provides another clue in the listener's effort to uncover the array.

62. If one thinks of the holistic paradigm in terms of group structure, then having all the members of the row class present in a work provides the group property of clo- sure. The holistic view of Babbitt's hyperaggregates was derived from Donald Davidson's theory of meaning in language. Donald Davidson, "Truth and Mean- ing," Synthese 17, no. 3 (1967): 304-23; reprinted in The Philosophy of Language, ed. A. P. Martinich (New York: Oxford University Press, 1985), 72-83. In his attempt to formulate a theory of meaning, Davidson overcomes the limitations of syntax while still fulfilling the criterion that the parts of the sentence contribute to the meaning of the whole by formulating a theory of meaning based on truth conditions. Given two sentences (assumed to be translations of each other), one in an object language and the other in a metalanguage (or when the object and meta- languages are the same, one sentence is a structural description of the other sentence), put them into a biconditional relationship using the connective "is true if and only if," rather than the connective "means that" (e.g. "Snow is white" if and only if snow is white). Each side of the biconditional can now be related by means of its truth value rather than a semantically undefined intentional connector (i.e., "means that"). The truth value of the biconditional is derived from the known truth conditions of the sentences and their parts. "The work of the theory is in relating the known truth conditions of each sentence to those aspects ("words") of the sentence that recur in other sentences, and can be assigned identical roles in other sentences." Consequently, when the truth value of the biconditional is true, according to Davidson, one knows the meaning of the sentence. However, Davidson is not implying that only knowing the truth conditions of sentences as stated by the theorems of his theory is sufficient for interpreting those sentences. His bicondition- als interpret a sentence in the context of the language as a whole. That is, he believes that the meaning of any sentence or word can only be given by giving the

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meaning of every sentence and word in the language. "Frege said that only in the context of a sentence does a word have meaning; in the same vein he might have added that only in the context of the language does a sentence (and therefore a word) have meaning" (p. 75).

63. Edgard Varbse, "The Liberation of Sound," Perspectives of New Music 5, no. 1 (1966): 11-12. Reprinted in Boretz and Cone, eds., Perspectives on American Composers, 25-33.

64. Gilles Deleuze, F61ix Guattari, A Thousand Plateaus. Translation and Forward by Brian Masumi (Minneapolis: University of Minnesota Press, 1987), 3-25.

65. " ... it is worth noting that in addition to the hierarchical relationship established by dyadic content among the mosaics, a distinction is made in the piece with regard to the sources in the array." Mead, An Introduction to the Music of Milton Babbitt, 188.

66. Mead, An Introduction to the Music of Milton Babbitt, 187. 67. The work uses a single 6-lyne all-partition array consisting of 58 partitions. 68. The contour changes between in each hexachord between the interval connecting

the first four sixteenths to the last two sixteenths of the hexachords. 69. Mead, An Introduction to the Music of Milton Babbitt, 35. 70. For example, when I was working on Groupwise, I first found the array in the

piano. Once I worked it out, I used it as a model to uncover the other four arrays. 71. As much as I would like to take credit for this wonderful insight, Robert Morris

suggested the link between Ives's music and the later works of Babbitt to me in private conversation.

72. This observation can be extended to the hexachordal divisions within a lyne as well. Perhaps in keeping with the lyric character of the piece, the division between hexachords is not strongly demarcated. For example, the excluded interval does not mark this structural juncture. It is not that the excluded interval is compositionally de-emphasized; it is not a structural feature of Duet's array.

73. There are several errors in Example 2.57. First, all the slurs are missing. The groups of pitches that need slurs are, in the order of their appearance, (Eb, Ab, E), (G, F#, E), (D, Eb), and (Ab, E, F). Also, the last two trichords in the passage are mislabeled as 3-2[013] types, when they should both be 3-4[015] type trichords.

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The Conflict between Particularism and Generalism in Andrew Mead's "Introduction to the Music of Milton Babbitt"

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