Structural Analysis of Sofia Gubaidulina’s String Quartets.pdf

UNIVERSITY OF CINCINNATI Date: August 06, 2007

I, Joseph Williams

,

hereby submit this work as part of the requirements for the degree of: Master of Music in: CCM Division of Composition, Musicology and Theory (Theory) It is entitled: Discontinuous Continuity?: Structural Analysis of Sofia Gubaidulina’s String Quartets

This work and its defense approved by:

Chair: Dr. Catherine Losada, Ph.D. Dr. David Carson Berry, Ph.D. Dr. Robert Zierolf, Ph.D.

Discontinuous Continuity?: Structural Analysis of Sofia Gubaidulina’s String Quartets

A thesis submitted to the

Division of Graduate Studies and Research of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

MASTER OF MUSIC

in the Division of Composition, Musicology, and Theory at the College-Conservatory of Music

2007

by

Joseph Williams

B.M. Stephen F. Austin State University, 2004

Committee Chair: Dr. Catherine Losada, Ph.D.

ABSTRACT Since the early 1980s, the majority of studies on Russian composer Sofia Gubaidulina (b. 1931) have centered on her historical influence. These writings, although necessary and interesting, have neglected any detailed analytical approaches in favor of aesthetic descriptions of her experimental new sounds and manipulations of proportions in large scale form. In particular, these studies have placed considerable stress on the discontinuous nature of her compositions, characterized as having “episodic forms” and “lack of development.” This study demonstrates how Gubaidulina’s music is not as discontinuous as it first appears. Utilizing the composer’s four string quartets as specific examples, the author shows in detail how, although seemingly episodic on the surface, they each contain streams of continuity as a result of similarly conceived structural processes that organize the various musical elements, including pitch, rhythm, and large-scale form. Through an analysis of these quartets, the author illustrates the processes at work and demonstrates the specific manner in which each process aids in establishing continuity and coherence throughout. The first quartet illustrates the process of expansion, in which pitches (and sometimes rhythms) are organized in one of two ways. First, they may radiate outward in pitch space from some central point or, second, they may expand outward in pitch space using some interval as a measure of distance. The second quartet utilizes inversional symmetry as a means of organizing large collections of pitches about a single asserted axis. This quartet also demonstrates the process of gap-fill, in which an interval is established and gradually filled in chromatically. The third quartet presents an organizing principle based on the intervals inherent to the pitch collection. Specific intervals are assigned particular functions in order to structurally arrange the

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individual pitches both linearly and harmonically. In this case, ic5 and ic1 act as structural markers between sections and as a means of differentiating between individual voices. Finally, the fourth quartet illustrates an organizing process based on sound masses. These collections are shown to mutate into one another via transformations.

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CONTENTS

LIST OF FIGURES

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CHAPTER ONE – INTRODUCTION

1

The Religion of the Avant-Garde: “Classical” vs. Experimental

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Gubaidulina’s Place in the Avant-Garde

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Elements of Individual Style

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The Four String Quartets

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CHAPTER TWO – ANALYSIS OF STRING QUARTET NO. 1 (1971): EXPANSION PROCESSES

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Wedge Expansion

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Additive Expansion

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Conclusion

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CHAPTER THREE – ANALYSIS OF STRING QUARTET NO. 2 (1987): GAP-FILL PROCESSES AND INVERSIONAL SYMMETRY

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Section A

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Section B

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Conclusion

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CHAPTER FOUR – ANALYSIS OF STRING QUARTET NO. 3 (1987): INTERVALLIC DEFINITION

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Hierarchical and Formal Separation

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Motivic Development and Transformations

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Conclusion

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CHAPTER FIVE – ANALYSIS OF STRING QUARTET NO. 4 (1993): VERTICALIZED SOUND MASSES

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The Primary Pitch Collections

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Relating Individual Sound Masses

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Conclusion

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EPILOGUE – CONCLUDING REMARKS

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BIBLIOGRAPHY

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MUSICAL FIGURES

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LIST OF FIGURES Figure 1.1: Diagram of Avant-Garde dichotomy

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Figure 2.1: Abstract example of additive expansion Figure 2.2: Viola (mm. 1-2) Figure 2.3: Violin II (R1, mm. 1-4) Figure 2.4: Wedge expansion from F#6 (R8, mm. 3-10) Figure 2.5: Wedge expansion from C#4 in violin I, violin II, and cello (R6, mm. 4-7) Figure 2.6: Wedge expansion from E5 in violin II (R7, mm. 2-9) Figure 2.7: Wedge expansion from D4 (R21, mm. 1-15) Figure 2.8a: Violin II and cello (R25, mm. 1-11) Figure 2.8b: Reduction of cello (R25, mm. 1-11) Figure 2.9: Viola (R1, mm. 1-2) Figure 2.10: Violin II, viola, and cello (R4, mm. 1-4) Figure 2.11: Cello (R1, mm. 1-4) Figure 2.12: Additive expansion in rhythm and pitch (R48, mm. 4-15) Figure 2.13: Wedge expansion and additive expansion in violin I and cello (R42-43)

89 90 91 92 93 94 95 97 100 101 102 103 104 106

Figure 3.1: Emphasis on G4/G5 dyad in m. 1 Figure 3.2a: Violin I/cello motive (R5, m. 1) Figure 3.2b: Gap-fill process in violin I (R5-15) Figure 3.3: New violin I/cello motive (R15) Figure 3.4: Moment of gap-fill completion in violin I motive (R20) Figure 3.5: R22-25 Figure 3.6: R21-22 Figure 3.7: Reduction of R21-22 with inversional symmetry in voice leading Figure 3.8: Reduction of R25-27 with inversional symmetry in voice leading

107 108 108 109 110 111 112 113 114

Figure 4.1: Formal diagram of Third String Quartet Figure 4.2: Primary set verticalized as sonorities (R49) Figure 4.3: Melody and accompaniment in opening measures Figure 4.4: Violin II accompaniment with second melodic segment (R1-2) Figure 4.5: Alternation of primary-set melodic material and ic5 structural divisions (R13-14) Figure 4.6: Ic5 structural divider (circled) in viola and cello (R55-56) Figure 4.7: Structural opposition between ic5 (R10) and ic1 (R11) Figure 4.8: Structural opposition between ic5 and ic1 (R15-17) Figure 4.9: Simultaneous structural opposition between ic5 and ic1 (R50) Figure 4.10: Melodic/accompanimental emphasis of ic1 and [G#, A, Bb] (R23) Figure 4.11: Linear version of primary set replicating pitch-classes from first sonority at R49 (R52) Figure 4.12: Violin I melody replicating cello pitches from R52 (R53-56) Figure 4.13: Motivic development using <+4, –1> (R58-60) Figure 4.14: Motivic development of violin I melody using <+4, –1> (R66-68)

115 116 117 118 119 120 121 122 124 125 125

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126 127 128

Figure 4.15: Replication of primary-set cello pitch-classes from R52 (R71-73) Figure 4.16: Interaction between ic5 and ic1 (R37-38)

129 130

Figure 5.1: B-viola four-measure ostinato pattern Figure 5.2: Reduction of recorded ensemble pitches into verticalized sound masses Figure 5.3: First occurrence of “Live Collection” in each voice of live ensemble Figure 5.4: Pitch-space graph of violin I (R26) Figure 5.5: Pitch-space graph of violin I (R14) Figure 5.6: Pitch-space graph of live ensemble (R34-41) Figure 5.7: Reduction of lower duet illustrating ascending thirds (R36-41)

131 131 132 133 134 135 137

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Chapter 1 Introduction

Since the early 1980s, the majority of studies on Russian composer Sofia Gubaidulina (b. 1931) have centered on her historical influence. These writings, although necessary and interesting, have neglected any detailed analytical approaches in favor of aesthetic descriptions of her experimental new sounds and manipulations of proportions in large-scale form. The formal studies have been informed specifically by the composer’s intense beliefs in the religious significance and mystical nature of numbers along with their proportional relations to one another when applied to rhythmic and sectional durations. In particular, these studies have placed significant stress on the discontinuous nature of her compositions. Jonathan Kramer characterized discontinuous music as having “‘episodic forms’ and ‘lack of development.’”1 Gubaidulina’s musical style has often been described in this exact manner. The disjunct, micro-chromatic pitch structures and sudden shifts in harmony, rhythm, and texture create a musical environment that can be disturbing to an unwary listener. The goal of this study is to demonstrate that this music is not as discontinuous as it appears. Gubaidulina’s four string quartets, although episodic on the surface, contain streams of continuity as a result of similarly conceived structural processes that organize the various musical elements, including pitch, rhythm, and large-scale form. Through an analysis of these

1

Jonathan Kramer, “Beyond Unity: Toward an Understanding of Musical Postmodernism,” in Concert Music, Rock, and Jazz since 1945: Essays and Analytical Studies, ed. Elizabeth West Marvin and Richard Hermann (Rochester: University of Rochester Press, 1995), 23.

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quartets, I will illustrate the processes at work and demonstrate the specific manner in which each process aids in establishing continuity and coherence throughout. The following discussion will place the composer in the proper historical and stylistic context. It will first present the general attributes of the post-1945 avant-garde movement followed by an overview of Gubaidulina’s life as a Soviet citizen composing in this vein, including considerations of her particular aesthetic views and a survey of the basic characteristics of her individual style. Finally, it will address the problems of using conventional analytic tools in a technical explanation of Gubaidulina’s works along with my proposed solution.

The Religion of the Avant-Garde: “Classical” vs. “Experimental” One of the most salient features of twentieth-century musical aesthetics is the multiplicity of approaches that arose throughout the era. In earlier historical periods a common aesthetic approach was often embraced, but aesthetic approaches of the twentieth century seemed to fragment into countless possible methods of creating artworks. I will focus on the particular manner in which a dichotomy arose within the avant-garde “school” of composers in the years following World War II. Some composers were determined to find ways to control the musical elements in a strict fashion in order to create a coherent whole; other composers, however, set their sights on creating a phenomenological moment-by-moment experience of the work, thereby disregarding the importance traditionally placed on the artwork as a cohesive whole. Because the former were more traditional in their overall approach, I refer to them as “classical” avant-gardists while the

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latter group are the “experimental” avant-gardists. Although this dichotomy over-simplifies a period of music history that is characterized as one of the most multi-variant, these labels are meant to distinguish the basic artistic motivations of composers writing in the years following WWII until 1975. They were either focused on producing a holistically coherent work or they focused on details while unconcerned with the overall map of the piece. The following discussion will present a more detailed treatment of this dichotomy along with a general historical background.

Leonard Meyer, in his discussion of the aesthetic changes in the twentieth century, posits that a dichotomy arose in the artistic motivations of composers. He states: Changes in aesthetic attitudes are matters of emphasis, having to do with what is considered to be the main focus of appreciative attention—the locus of artistic significance and value. . . . Aesthetics and criticism were dualistic. On the one hand were the materials and form of a work of art, the means; on the other hand were subject matter, expression and meaning, the end or goal. The former were significant because they made the latter possible. If one asked what a work of art was “about,” the answer was almost always in terms of subject matter or whatever it was thought to express, symbolize, or signify.2 This duality of focus, either on the means or the goal of a work of art, characterized the radical changes that occurred in compositional practice. Specifically, in the first half of the century, the two stylistic trends could be distinguished in terms of their relation to past styles. Compositions organized primarily on tonal procedures were considered traditional while works utilizing novel strategies were considered “avant-garde.”

2

Leonard B. Meyer, Music, the Arts, and Idea (Chicago: University of Chicago Press, 1967), 210. It should be pointed out that Meyer’s duality is not universally accepted within scholarly circles. Ivo Supi…iƒ specifically characterizes the duality as between “matter” and “form” rather than between Meyer’s “material” and “content.” (“Matter and Form in Music,” International Review of Music Aesthetics and Sociology 1, no. 2 (December 1970): 157.)

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The primary characteristic of avant-garde art is the focus on novelty. According to Meyer: In the absence of viable critical criteria, the one thing that could be assessed was whether the artist’s method, style, or “idea” was new. Novelty, often mistakenly confused with “originality” from which it sprang, gradually became a criterion for judging works of art. And this gave further impetus to the search for the new—for novelty for its own sake. . . . The new is valuable because it enlarges the sensibilities and awareness of the art audience.3 This is further elaborated by Susan Sontag: “New art is valued for the novel state it induces in the spectator and for what it reveals to him about himself, the physical world, or simply his way of reacting to paintings.”4 The intense aesthetic focus on novelty linked composers across national borders creating what could be called an international style. In other words, the ability to be novel and intriguing initiated a commonality between artists—unity formed from disunity. In her book Musical Style and Genre, Marina Lobanova states that “in the twentieth century with the rejection of the canonical system of thinking in culture, artists acquired an exclusive right to experiment. . . . Art continued to develop under the banner of discovery.”5 In fact, this aesthetic carried a mystical and religious character: What had unified [the avant-garde composers] was a radicalism of aesthetic purpose. For the modernist tradition of the nineteenth and twentieth centuries, the great sin against the human spirit was that of boredom. It was a sin brilliantly and definitively epitomized by Baudelaire in his great poem, “Le Voyage.” 3

Ibid., 218.

4

Susan Sontag, Against Interpretation (New York: Noonday Press, 1966), 142. Further claims of “novelty” as the primary characteristic of avant-garde art can be found in Renato Poggioli, The Theory of the Avant-Garde (New York: Harper & Row, 1968), 214, and Frank X. Mauceri, “From Experimental Music to Musical Experiment,” Perspectives of New Music 35, no. 1 (1997): 191. 5

Marina Lobanova, Musical Style and Genre: History and Modernity, trans. Kate Cook (Amsterdam: Harwood Academic Publishers Association, 2000), 22.

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Repetitions and returns—forms that achieved closure by recapitulation, by submitting gratefully to the gravitational force of “home keys,” styles that knowingly acknowledged that desire through the manipulation of neo-baroque and neoclassical topoi—were not merely pardonable indulgences; they were denials of the quintessential potential of human life. They ministered—either resignedly or in calculation—to the regressive streak present in every human and massively encouraged by the popular culture of the years after the Second World War.6 A new dichotomy emerged in the latter half of the century. While neo-classical stylistic trends partially continued within certain circles of composers, the general school of “avantgarde” composers fragmented into two distinct groups—“classical” and “experimental” avantgardists.7 This duality was analogous to the split that occurred around the turn of the twentieth century between the traditionalists and the general avant-gardists. In other words, the classical avant-garde primarily focused on the end, while the experimental avant-garde leaned toward the individual means. Figure 1.1 presents a diagram summarizing the previous discussion. The classical avant-garde, in its earliest form, was associated with total serialism. In order to present an idea through musical means, complete composer-control over all musical elements was required: At the end of World War II the musical world was eager to pursue somewhat selectively the explorations of the first half of the century. . . . If tonal Neoclassicism, following a final flourish in the late 1940s, was finally laid to rest with Stravinsky’s The Rake’s Progress (1948–1951), the Serialism of Schoenberg and his followers, which had been pronounced a dead end by many a critic in the 1930s and 1940s, was now taken up with an individualizing touch by composers 6

David Osmond-Smith, “New Beginnings: The International Avant-Garde, 1945–62,” in The Cambridge History of Twentieth-Century Music, ed. Nicholas Cook and Anthony Pople (Cambridge: Cambridge University Press, 2004), 361. 7

Joaquim M. Benitez, “Avant-Garde or Experimental?: Classifying Contemporary Music,” International Review of the Aesthetics of Sociology of Music 9, no. 1 (June 1978): 63. The distinction between these two groups has sometimes been specifically referred to terms of “choice vs. chance.” (See Glenn Watkins, Soundings: Music in the Twentieth Century (New York: Schirmer Books, 1988) and Paul Griffiths, Modern Music: The Avant Garde since 1945 (London: J. M. Dent & Sons Ltd., 1981).)

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of various nationalities and stylistic inclinations. The message their new works carried was that any aesthetic is capable of being reinterpreted and vitalized in highly personal terms. The early recognition that serial organization need not imply a style was a revelation to many, and the consequent fervor of the subscription cannot help but suggest that in part it was a reflection of an almost compulsive search for order following a period of world-wide chaos.8 In fact, it is somewhat ironic that what originally began as a reaction against the traditional compositional strategies via focus on altering the musical materials (i.e., the means) eventually became aesthetically associated with the more content-oriented (i.e., less experimental) artistic perspective. The experimental avant-garde, on the other hand, continued to focus on new means of musical expression without any particular holistic goal. John Cage stated: “What is the nature of an experimental action? It is simply an action the outcome of which is not foreseen. It is therefore very useful if one had decided that sounds are to come into their own, rather than being exploited to express sentiments or ideas of order.”9 To composers of experimental music, the classical avant-garde tendencies were too stifling and narrow. They believed an artist should not be concerned with attempting to express some point in the final product; rather, the work of art should simply be purged of all composer-control. Following the 1950s, the classical avant-garde gradually began to incorporate more than just serial techniques. The “classical avant-garde,” in the specific way I use the term in this study, came to be associated with any non-traditional method of organizing and structuring sound. Watkins elaborated: “Following a mid-century crisis that seemingly demanded that a choice be made between Serialism and Neoclassicism, we have come to the point where we are 8

Watkins, 506.

9

John Cage, Silence (Cambridge, Mass.: M.I.T. Press, 1967), 69.

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comfortable with the wealth of our various alternatives, and an “anything goes” attitude prevails.”10 The primary aesthetic focus on novelty, however, also posed new compositional problems that required resolution. Experiments in new sounds and performance strategies forced composers to create novel means of organizing sound. On a local level, melodic and harmonic parameters required new systems of arrangement in order to sustain any sense of coherence. At the same time, large-scale properties such as rhythmic and metric proportions as well as formal dimensions also demanded some type of organized structure. After World War II, locating a means of unifying the various elements in a work instantly became the primary concern of classical avant-garde artists everywhere.11 Although it is impossible to formulate a specific stylistic trend,12 I will attempt to describe certain overarching concerns that arose in the compositional strategies in post-1945 classical avant-garde musical works. In structuring local-level elements (i.e., melody and harmony), an interesting pattern of stylistic influence arose among composers. Because of the nearly infinite possibilities of creating art, a similarly infinite array of potential organizational apparatuses could be conceived. The modernist movement saw a number of these enacted at some time; however, “memories” of past

10

Watkins, 688.

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The especially self-conscious nature of the classical avant-garde movement is made evident by the fact composers began writing more and more about their own music. Essentially, there seemed to be a particular need to both describe and justify individual organizational techniques. See Lucian Krukowski, “Hegel, ‘Progress,’ and the Avant-Garde,” The Journal of Aesthetics and Art Criticism 44, no. 3 (1986), 279 and Watkins, 520–23. 12

Mauceri notes that the avant-garde “is characterized as radically new but it is also posited as an historical category, a tradition in its own right. But as pointed out by art critic Harold Rosenberg, ‘The new cannot become a tradition without giving rise to unique contradictions, myths, absurdities.’ The irony of this historical category is the attempt to construct a genre out of work that by its own definition is radically different and highly individualistic.” The whole of this study follows from this sentiment in an effort to formulate a fairly complete picture of Gubaidulina’s individual style within the context of her avant-garde organizational procedures.

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structural means were often invoked. In other words, although composers were constantly seeking novel musical properties, they often reached, either consciously or unconsciously, to the past for inspiration. Lobanova calls this “mixed style”: The change of the musical paradigm . . . affected the “new style conception”: from “polystylistics” proper where one could detect fairly precise popular principles of quotation and allusion, composers moved to a different interpretation of style. The new stage was connected with the search for organics: tradition, which was offered “point-blank” in stuck-together collages and lost its meaning in them, was included more and more mediately and subtly in the style model of the composition. It was precisely on this difficult path of reflected transformation, accumulation of the traditional, that innovative treatments were found. The overcoming of distancing in the view and interpretation of the “other” style provided a vital link of the ages within an artistic work. Today the mixed style is not a means of polemic with attitudes of the preceding period, but a new system of artistic communication which admits of a multitude of individual treatments while preserving a connection with tradition.13 Thus, although the avant-garde aesthetic insisted on novelty and creative originality at the center of the role of art, it was ironically dependent on the art of the past. The property of intertextuality between the present and the past was originally posited by Harold Bloom in The Anxiety of Influence: “We need to stop thinking of any poet as an autonomous ego, however solipsistic the strongest of poets may be. Every poet is a being caught up in a dialectical relationship (transference, repetition, error, communication) with another poet or poets.”14 In other words, for any poet (or composer), the interpreter (or analyst) must always realize how the works from the past both influence and even define the works of the present. The “mixed style” and “anxiety of influence” can be illustrated with serialism. Although the pitches are arranged in such a way as to avoid any sense of pitch centricity or tonality,

13

Lobanova, 158 (italics original).

14

Harold Bloom, The Anxiety of Influence: A Theory of Poetry (London: OUP, 1973), 5.

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composers continued to limit themselves with their reliance upon the same pitches and rhythms used for centuries prior. In fact, the necessity of arranging the pitches as a linear “row” automatically retains a sense of hundreds of years of melodic organization. Ben Johnston goes so far as to present the twelve-tone pitch limit itself as unnatural: Over the whole of the historical period of instrumental music, Western music has based itself upon an acoustical lie. In our time this lie—that the normal musical ear hears twelve equal intervals within the span of an octave—has led to the impoverishment of pitch usage in our music. In our frustration at the complex means it takes to wrest yet a few more permutations from a closed system, we have attempted the abandonment of all systems, forgetting that we need never have closed our system. . . . In our laziness, when we changed over to the twelvetone system, we just took the pitches of the previous music as though we were moving into a furnished apartment and had no time to even take the pictures off the wall.15 Most avant-garde composers attempted to negate this “misreading” of the past, but present-day music theorists have demonstrated how modern art music is reliant upon the past traditions of tonality.16 Other composers, however, deliberately attempted to exploit the past in an attempt to create novelty. This could be done by simulating or modeling a past style, or even borrowing or quoting directly from a specific composition.17 Gubaidulina, in her creative efforts, has written works that avoid past influence as well as works that consciously utilize themes or styles from the past. For instance, The Seven Last Words of Jesus Christ (1982), Offertorium (1980), and Meditation on the Bach Chorale ‘Vor 15

Quoted in David Cope, New Directions in Music, sixth ed. (Dubuque, IA: Brown & Benchmark, 1993),

66. 16

See Joseph Straus, Remaking the Past: Musical Modernism and the Influence of the Tonal Tradition (Cambridge, Mass.: Harvard University Press, 1990); Kevin Korsyn, “Towards a New Poetics of Musical Influence,” Music Analysis 10, no. 1–2 (March–July 1991), 3–72; and Mark Evan Bonds, After Beethoven: Imperatives of Originality in the Symphony (Cambridge, Mass.: Harvard University Press, 1996). 17

For a discussion of the various “types” of utilizing past works, see Meyer, Music, the Arts, and Ideas,

193–208.

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deinen Thron tret ich hiermit’ (1993) all feature a quoted statement from J. S. Bach (1685–1750) or Heinrich Schütz (1585–1672). For Gubaidulina, these quotations are meant to symbolize some aspect of the work rather than to imitate a past style: “I used this theme [of King Frederick the Great of Prussia in Offertorium] not to refer to the style of Frederick the Great (or Bach, who developed the same theme in his Musical Offering), but to symbolize the idea of sacrifice.”18 As for the large-scale formal properties of avant-garde music, ingenuity abounded. Although “mixed” procedures were quite evident, rhythmic and formal attributes were subjected to numerous experimental attempts at novel organization. Lobanova states: The search for new musical syntax was connected with the idea of the fundamental unlimitedness of musical and objective time and the full or partial openness of musical form. This is an important symptom of the rejection of classical conceptions of time organisation [sic] and a single interpretation of artistic time as opposed to physical time. As a result there arose a plurality of individual treatments which broke completely or partially with the familiar causeand-effect connections of the classical type. . . . The consecutive application of simultaneous forms meant the interconvertibility of many musical parameters, the formation of new dimensions, and a new attitude to musical space and time structures.19 In other words, the musical elements on the local level were more limited and dependent on previous compositional thought. Formal proportions, being temporal in nature, were more easily transformed in the classical avant-garde because of recent philosophical trends that had reconsidered the nature of time itself. To summarize the discussion thus far, the main concept of the classical avant-garde aesthetic was based on creation and novelty; the means of achieving it, on the other hand, were

18

Sofia Gubaidulina and Vera Lukomsky, “Sofia Gubaidulina: My Desire is Always to Rebel, to Swim Against the Stream!” Perspectives of New Music 36, no. 1 (1998): 26. 19

Lobanova, 122–23 (italics mine). This lack of “cause-and-effect connections” adds to the notion that avant-garde music is fragmented and discontinuous, a position to which this study stands antithetically.

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not so straightforward. The musical elements basically necessitated recourse to the past because of the nature of music itself. Composers were forced to conceptualize new methods of presentation and organizational procedures. Formal manipulation was also necessary in order for classical avant-garde composers to produce novel works with little or no connection to previous thinking. Into this musical environment, Sofia Gubaidulina was born.

Gubaidulina’s Place in the Avant-Garde For a composer interested in the avant-garde aesthetic, Soviet Russia was a less than ideal environment. Paul Griffiths describes compositional practice in the years immediately following World War II as such: In the U. S. S. R., too, the symphony was being cultivated assiduously, though as a genre suited to the optimistic and aesthetically conservative doctrine of socialist realism . . . . The newly severe interpretation of socialist realism, announced to musicians by Andrey Zhdanov in January 1948, obliged composers not only in the U. S. S. R. but throughout the communist bloc to address their music to the people in the simplest and most direct terms. Thus musicians from eastern Europe were effectively prevented from participating in avant-garde movement for some years. . . . But equally and more widely significant was the tension set up between musical radicalism and commitment to political revolution.20 Further description is given by Lobanova: The concept of modern music, like that of world musical culture, hung around in the Soviet Union for a painfully long time: the rich contacts of the twenties were replaced by a long bout of “delirium.” After the Russian Association of Proletarian Musicians turned into the official censor in 1930, having established together with the NKVD (later the KGB) and the Communist Party control over musical culture, achieved the banning of “ideologically hostile” organisations (first and foremost, the Association of Contemporary Music), publishers, magazines and compositions, and repressed some of its political opponents as “enemies of the people,” while forcing others to renounce their convictions,

20

Griffiths, 15.

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connections with the West were broken, as were the threads linking the past, present and future in Russian culture. All “modernistic,” “formalistic,” “decadent,” “religious,” “bourgeois,” and “reactionary” works were banned from cultural usage. . . . East-West was interpreted by official ideology in accordance with the canons of mythological description as “them and us,” “left and right,” “bad and good.”21 Thus, experimental art of any type was strictly forbidden and censored from both publication and performance. Despite the stifling socio-political environment, the avant-garde aesthetic continued to thrive in some academic circles. Specifically, five graduate students came to be known as the primary leaders of avant-garde music in Moscow: Andrey Volkonsky, Alemdar Karamanov, Edison Denisov, Alfred Schnittke, and Sofia Gubaidulina.22 It was they who “began the intensive assimilation of the avant-gardist musical idioms. The ‘Sturm und Drang’ of the young composers allowed within a short span of time to overcome the local confinement of Soviet music and disrupt its actually existing isolation despite the world-wide recognition of Dmitry Shostakovich, Sergei Prokofiev, and Aram Khachaturyan.”23 The influence of Western compositional practice is apparent in the works of these composers. Nevertheless, the Soviet classical avant-garde composers, in general, acquired their own unique stylistic traits. Joel Sachs attributes this to “the fact that they received the new

21

Lobanova, 29.

22

The latter three are collectively referred to as “the great three.” However, both Volkonsky and Karamanov were essential in cultivating the experimental “school” of composers and must also be included in any list of influential avant-garde Soviet composers. See Levon Hakobian, Music of the Soviet Age: 1917–1987 (Stockholm, Sweden: Melos Music Literature, 1998), 256–61. 23

Svetlana Savenko, “Valentin Silvestrov’s Lyrical Universe,” in Underground Music from the Former U.S.S.R., ed. Valeria Tsenova, trans. Romela Kohanovskaya (Amsterdam: Harwood Academic Publishers, 1997), 67.

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Western techniques more or less in one burst, at a time when they, already experienced young composers, could pick and choose from them at will.”24 Of the five composers listed above, Gubaidulina is the only one who never held a university position. She was encouraged to pursue a career in composition, despite her affinity toward anti-communist artistic trends: And although we were accepted to the graduate school [in 1954], the Conservatory officials declared that, despite our giftedness and capacity for hard work, we had chosen the wrong way, or what they called “a false way” (the “right way,” of course, meant Socialist Realism). . . . [Shostakovich] told me personally: “Everybody thinks that you are moving in the wrong direction. But I wish you to continue on your ‘mistaken’ path.”25 Upon completing her compositional studies at the Moscow Conservatoire in 1963, she worked as a freelance composer developing film scores. Her “mistaken” career path allowed her numerous opportunities to experiment with unusual collections of instruments (both conventional and exotic) and unique methods of producing sounds. Eventually, the oppression of censorship became too much for the composer. Not only was it impossible to arrange a performance of her works, she was finding it increasingly difficult to be creative. As a result, her compositional output began to suffer.26 To relieve herself of 24

Joel Sachs, “Notes on the Soviet Avant-Garde,” in Russian and Soviet Music: Essays for Boris Schwarz, ed. Malcolm H. Brown, Russian Music Studies, vol. 11 (Ann Arbor: UMI Research Press, 1984), 307. He clarifies his claim further, stating: “They did not have strong preconceptions, because they did not participate in the agony associated with the evolution in the West of those techniques, and thus escaped the strongly partisan atmosphere so often transmitted in composition training. [They] seem, therefore, to have bypassed the aggressive modernity that can be found in many Western composers of their generation.” 25

Gubaidulina, Lukomsky, “My Desire is Always to Rebel,” 16.

26

Gubaidulina gives specific examples of blatant censorship of both Schnittke’s and her own music in Sofia Gubaidulina and Dorothea Redepenning, “Sofia Gubaidulina: An Interview with Dorothea Redepenning,” in Contemporary Composers on Contemporary Music, ed. Elliott Schwartz, Barry Childs, and James Fox (New York: Da Capo Press, 1998), 449–50 and idem, “Sofia Gubaidulina: Into the Labyrinth of the Soul,” in The Voice of Music: Conversations with Composers of Our Time, ed. Anders Beyer and Jean Christensen (Aldershot: Ashgate, 2000), 45–47.

13

compositional sterility, she traveled to Germany, visiting for long periods of time starting in 1985. In 1992, she permanently relocated to Hamburg. Aesthetically, for Gubaidulina, the liberty discovered in musical experimentation was religious in nature. In describing what caused her to become a composer, she resorts to mystical language, stating: A human being, even under the most difficult conditions, even in a murderous atmosphere, must have something to hold sacred. I remember extremely well my impressions at five years of age, when I began to have some understanding: all my life was gray and I felt good only when I entered the door of the music school. From that moment I felt myself in a sacred space. I heard the sounds coming from the halls, I felt the bond that united the students, and everything came together in that polytonal harmony of sounds, and I wanted to live in that world.27 Based on her experiences as a child, Gubaidulina turned to religion as a means of moral strength, allowing her beliefs to form her aesthetic attitude toward music, experimental music in particular. Because of this religious focus, she regards the act of composition as an ethical process; thus, much of her compositional output centers on a religious theme.28 Because of the intense socio-political situation, Gubaidulina’s artistic opportunities were limited. Her compositional ventures were tightly controlled by those who believed her music would not suit the common class, whether because of difficulty in comprehension or simply because of radical individualism. Either would contradict the socialist ideology. For these reasons, she considered her own compositions to be a sacred and holy object. To her, they were a

27

Quoted in Josiah Fisk, ed., Composer to Composer: Conversations About Music (St. Leonards, NSW, Australia: Allen & Unwin, 1993), 460. 28

Quotes concerning Gubaidulina’s thoughts on ethical composing are countless in number. Some examples can be found in: Fisk, 461; Gubaidulina, “Labyrinth of the Soul,” 50; Sofia Gubaidulina, Vera Lukomsky, “‘The Eucharist is My Fantasy’: Interview with Sofia Gubaidulina,” Tempo 206 (Sept. 1998): 31; Claire Polin, “The Composer as Seer, but Not Prophet,” Tempo 190 (Sept. 1994): 16–17.

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means by which to escape the mundane into a divine presence, requiring the sacrifice of the composer and giving a moral quality to the process of composing.29 In the end, Gubaidulina’s experimentation with musical style can be understood and appreciated only if her motivations are perceived. In the context of the above discussion, she can be placed firmly in the classical avant-garde: “We must worry about the incarnation of our idea, about meaning and formation. The contemporary artist is faced with an extremely important task: finding a correlation between intuition and intellectual work.”30 For Gubaidulina, the composer must find new musical materials and organizational means in order to create a work of art containing specific spiritual content. Based on her own admission, her compositions are formed with a particular end in mind, placing her squarely in the arena of the classical avantgarde aesthetic.

Elements of Individual Style Throughout her career, Gubaidulina has experimented with and incorporated various post-1945 avant-garde trends into a characteristically individual style. I will discuss these in four categories: melody, vertical dimension (harmony), timbre/texture, and rhythm/form.

29

Hakobian, 287.

30

Gubaidulina, Lukomsky, “My Desire is Always to Rebel,” 9.

15

Melody Gubaidulina’s melodies are especially characterized by their intense chromaticism. Any tonal implication is quickly subsumed by a shift toward extra-diatonic/atonal pitch collections. In Ruth Crawford Seeger’s music, Joseph Straus refers to this process as “dissonation”: “Melodies, . . . should be systematically ‘dissonated,’ that is, organized to suppress any traditional tonal or triadic implications. Since any consonant interval might be heard to suggest a major or minor triad, it will normally be followed by a dissonant interval, to undermine that implication.”31 Although Gubaidulina utilizes triads to some extent, the succeeding sonorities do not reference a tonal syntax. Another melodic characteristic involves the use of quarter-tones. The composer describes her view: I understand it as a unification of two spaces: the first is the twelve-semitonal space, and the second is another twelve-semitonal space a quarter tone higher. For me this is a metaphor of the image and its shadow, or a day and a night. From my point of view, in the twelve-tone compositions of the twentieth century, everything is as in the daytime; everything is enlightened and reationalized; there is no place for “night.” “Night” existed as a supplement of the diatonic system: the diatonic sphere was “day,” whereas the chromatic sphere was “night”: one could go there and return. That blessed situation gave us classical and romantic composers. In twelve-tone compositions we lost “night”: everything became “day.” But within the twenty-four tone scale, we may have not only “a day,” but also “a night.”32

31

Joseph N. Straus, The Music of Ruth Crawford Seeger (Cambridge, Mass.: Cambridge University Press,

1995), 18. 32

Gubaidulina, Lukomsky, “My Desire is Always to Rebel,” 11.

16

From her own words, it can be deduced that the traditional twelve tones make up the “tonal” system while those pitches micro-tonally related to the former are “chromatic.” This binary opposition is elsewhere referred to in terms of “consonant” and “dissonant.”33 Finally, glissandi are used fairly often, further adding to the ambiguous nature of her melodic material. Although upper and lower limit pitches may be notated, the glissandi serve to downplay the notated pitch’s importance by creating a continuous melodic line lacking precision in its placement in pitch space. In general, Gubaidulina avoids long melodic lines, preferring to link several short melodic segments together. In this way, cadential implications are circumvented, allowing for an open-ended quality. The segments are often sequenced through some systematic variation, yielding a dynamic melodic process as opposed to a static melodic object.

Vertical Dimension (Harmony) Because of the detailed stress placed on melodic structure, Gubaidulina’s harmonies almost always arise from some linear stratification. In fact, at times she has even described the individual lines and voices of her music as characters intersecting and crossing at specific points.34 For this reason, the term “vertical dimensions” seems best suited in describing the chordal character. One important aspect in Gubaidulina’s music must be stressed. As in the music of Edgard Varèse (1883–1965) and György Ligeti (1923–2006), octave equivalence is rarely granted a 33

See Dorothea Redepenning, “Staccato Existence: Russian Women Composers in Germany,” in Women Composers in Germany, ed. Roswitha Sperber (Bonn: Inter Nationes, 1996), 101. 34

Gubaidulina, Polin, 15–16.

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prominent place in Gubaidulina’s works. That is, notes an octave apart are denied equivalent status, and thus pitch-class space is generally irrelevant. Rather, the musical action occurs in pitch space. In this way, for example, the occurrence of C3 is a qualitatively different event than the occurrence of C4. Based on the organizational principle of pitch space, two other types of vertical dimensions are possible: sound masses and pitch clusters. Although similar, for this study I will distinguish between these two. I define sound mass as a group of pitches (melodically distributed or simultaneously occurring) irrespective of proximity to one another in pitch space. A pitch cluster, on the other hand, consists of pitches in extremely close pitch-space proximity performed simultaneously (i.e., as an actual sonority).

Texture/Timbre As implied above, Gubaidulina often organizes a composition into an intricate web of independently moving lines intersecting at various points. Because of the intensity of the counterpoint, thick pitch clusters and sound masses are inevitable; however, the density of these sections is almost always counterbalanced by sections containing a lighter texture and, often, solo instruments. Once more, it is apparent how much stress is placed on individual lines. Register plays an important structural role in the composer’s works. Registral extremes are often favored in order to produce some dramatic effect, such as melodic climax and heightened contrast between her self-described “characters.” Also, register is intimately linked with concerns of timbre. Gubaidulina often employs string harmonics in melodic structures. In her own words: “Sound can have an earthly, only too ‘human’ expressiveness. And yet if you

18

touch the same spot of the string in another way, if you change a bit your attitude, you are carried away from earth to heaven. The ordinary, the trivial becomes heavenly, if not sacred.”35 Thus, timbre is directly linked to the composer’s aesthetic values. Gubaidulina also experiments with unconventional performance methods. Performers are often instructed to manipulate the natural sound of their instruments in order to create a specific auditory effect. Along with experimental acoustic sounds, the composer has also worked with electronic sounds and devices.

Rhythm/Form According to Gubaidulina, temporal factors are the most important aspects of a musical work. In her view, numbers contain mystical properties that can be manifested within the music. Similar to Susanne Langer, Gubaidulina believes that music “makes time audible”36 and that it is the composer’s ethical duty to embody the spiritual within a musical form. This is achieved by means of accurate proportions and rhythmic durations. In her own words: “I form a certain profile of numbers. But in general, there is a beautiful picture of rhythmic calculation, proportionality, mathematic exactness in the large-scale formal organization—over the absolute freedom of all other musical elements: melody, harmony, rhythm.”37 More specifically, she is most concerned with the Fibonacci sequence, a series of numbers in which each member is formed by the addition of the previous two members. The

35

Quoted in Hakobian, 288.

36

Susanne Langer, Feeling and Form (New York: Scribner’s, 1953), 110.

37

Sofia Gubaidulina and Vera Lukomsky, “‘Hearing the Subconscious’: Interview with Sofia Gubaidulina,” Tempo 206 (July 1999): 30.

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sequence as a whole approximates a series of increasingly refined golden means since each member is related to its neighbor in a golden ratio. For Gubaidulina, this produces the most perfect and natural relation between two parts; thus, any musical form manifesting a goldenmean proportion is spiritually and physically balanced. Based on her aesthetic views, beauty is characterized by this perfect proportion.38 In a more practical sense, texture and timbre, more than any other musical element, are the most convenient means of articulating formal sections lacking a form of syntax. Sudden changes in texture or timbre generate an immediate sense of restarting (i.e., a new section). In fact, as will be illustrated shortly, she employed this method of formal division in three of her four string quartets.39 As for rhythmic properties in general, Gubaidulina places significant emphasis on the possibility of utilizing rhythm as a structural entity as opposed to mere local figuration: In the light of the very frequent speculation about the different ways in which modern composers try to control rhythm and proportion in a work, it is worth noticing one important and distinctive part of Gubaydulina’s [sic] attitude to the problem. In conversation she is most keen to stress that she cannot accept the idea (a frequent post-serial one) of rhythm or duration as the material of a piece. . . . To her, rhythm is nowadays a generating principle as, for instance, the cadence was to tonal composers of the Classical period; it therefore cannot be the surface material of a work. . . . [S]he expresses her impatience with Messiaen, whose use of rhythmic modes to generate local imagery, she feels, restricts the effectiveness of rhythm as an underlying formal level of the music.40 38

See Valeria Zenowa, Zahlenmystik in der Musik von Sofia Gubaidulina, trans. Hans-Joachim Grimm and Ernst Kuhn (Berlin: Verlag Ernst Kuhn, 2001). Although I admit the possibility of the formal and rhythmic inclusion of this property in the string quartets, I have failed to locate any specific example. Also lacking is any reference by the composer in her interviews to this formal aspect in the quartets. The discussion is included in this chapter primarily in an attempt to describe the composer’s stylistic tendencies as completely as possible. 39

See Swetlana Sarkisjan, “Die Streichquartette Sofie Gubaidulinas als Versuch der Erschleissung des Sonoristischen Raumes,” International Review of the Aesthetics and Sociology of Music 36, no. 2 (2005): 271–86. 40

Gerard McBurney, “Encountering Gubaidulina,” The Musical Times 129, no. 1741 (March 1988): 123.

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In other words, durations segment and organize time; thus, rhythm cannot be musical material on its own. It is a structural coordinator that arranges other musical parameters. In sum, Gubaidulina’s music is structured in terms of melodic properties that are organized by rhythmic proportions. Vertical dimensions arise out of melodic interaction while formal units similarly arise out of textural interactions. For her, the music is not a static object; rather, it is a living dynamic organism that grows and changes as it is performed. Melodies do not exist in themselves; they are formed through a process by which a small segment is expanded and developed. Formal divisions are not objectively present; rather, they arise out of the interaction of different textures and shifts in timbre. A formal scheme does not structure music; instead, the music produces a form as it grows in time.

The Four String Quartets The quartets were written over a period of thirty-two years. Following post-graduate studies in Moscow, Gubaidulina began working in the Moscow experimental studio for electronic music (1969–70). The first quartet, composed in 1971 (significantly earlier than the remaining three), was one of her earliest instrumental pieces not designed in this electronic studio. The remaining three were composed within a shorter time-span and betray influence from her involvement with the Astrea improvisational team (1975–80 and 1991 to present). The second and third were composed the same year (1987) and the fourth was composed six years later (1993). Also, during this later time in the composer’s life, her interests turned toward symphonic textures, giving more equality to the individual instruments than do her initial works. This shift

21

toward symphonic textures can be seen in Stimmen . . . Verstummen . . . (1986) and Pro et contra (1989). As in most of her orchestral works, all four quartets are designed to allow the maximum freedom for each member of the ensemble, but the later they were written, the more experimental they became. The composer not only found new possibilities for organizing sound, she also became more comfortable with her choices. Attempting to analyze a piece organized entirely on the basis of an individually created style that lacks any conventional syntactical system poses a problem. Understandably, endeavoring to employ traditional analytic tools on such idiosyncratic music proves fruitless. In the case of Gubaidulina’s four string quartets, conventional set-class analysis cannot explain the pitch organization. In general, her music does not depend on similar pitch groupings. Discrete passages often organize the same pitches in ways that are fundamentally different. This requires two different interpretations of the same pc-set class, severely limiting the amount and importance of the information a pc-set class analysis can convey. On the other hand, separate passages contain radically different pitch collections organized in a similar fashion. Once again, a pc set-class analysis would fail to yield valuable information concerning the comparative structure of these passages. The lack of available tools that might aid in a valid understanding of the work actually supports the notion that the composition is fragmented and lacks continuity. I argue that these quartets are coherent works and do retain some form of continuity throughout. I will demonstrate my point by exploring four different processes of organization unfolding within the quartets. The first quartet illustrates the process of expansion, in which pitches (and sometimes rhythms) are organized in one of two ways. First, they may radiate outward in pitch space from

22

some central point or, second, they may expand outward in pitch space using some interval as a measure of distance. The second quartet utilizes inversional symmetry as a means of organizing large collections of pitches about a single asserted axis. This quartet also demonstrates the process of gap-fill, in which an interval is established and gradually filled in chromatically. The third quartet presents an organizing principle based on the intervals inherent in the pitch collection. Specific intervals are assigned particular functions in order to structurally arrange the individual pitches both linearly and harmonically. In this case, ic5 and ic1 act as structural markers between sections and as a means of differentiating between individual voices. Finally, the fourth quartet illustrates an organizing process based on sound masses. These collections are shown to mutate into one another via transformations. The analyses that follow are not intended to be exhaustive, nor even complete. Instead, my primary purpose is to demonstrate continuity in these well-designed compositions by presenting specific processes that organize the musical elements throughout each individual work.

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Chapter 2 Analysis of String Quartet No. 1 (1971): Expansion Processes

Unlike the remaining three quartets composed by Gubaidulina, the first does not conveniently divide into large formal sections. Rather, the work seems to play out as a chain of small, often unrelated “ideas” formally separated only by rests or sudden shifts in texture. This “stream-of-consciousness” property only adds to the accepted notion that this composer’s works are discontinuous and unfold in a series of “moments.”1 However, I propose to demonstrate how one particular process of organization assists in uniting the various formal portions of the work and aids in establishing continuity. This process, which I will call expansion, coordinates the seemingly random pitches and rhythms into an organized and cohesive structure. The expansion process that occurs throughout the work can be divided into two basic categories. The first, which I call wedge expansion, begins with the assertion of a single pitch. From this starting point, the pitch space is gradually opened up by the statement of pitches on either side of the initial pitch. The size of the intervals between each of the “outer” pitches and the central pitch (and necessarily between the outer pitches themselves) gradually increases. This “wedge” can be achieved via contrary motion between two voices: as one voice states the various pitches below the initial pitch, an alternate voice balances the wedge with pitches above the initial pitch. The same process is also possible by means of a single voice. In this case, an

1

These properties are discussed at length in chapter 1.

24

initial pitch is followed by alternate statements of pitches of gradually increasing intervals in both directions from the original pitch. One caveat must be included with this discussion of wedge expansion. Although symmetrical structures do play an important part in the later quartets,2 the presence of exact inversional symmetry is interestingly lacking in this work. Here, the process of expanding or opening up pitch space rarely corresponds to exact interval sizes traversed in both directions, yet the wedge expansion does describe the specific manner in which the various pitches are organized in pitch space. Jonathan Bernard, in his discussion of Ligeti’s Kammerkonzert, presents a description that also applies to this quartet: “As the dimensions of occupied space continue to spread outward, the general idea of a symmetrical centre is effectively conveyed, but the actual, specific location of that centre is constantly in flux.”3 In other words, although exact inversional symmetry is rarely presented, the pitch organization mimics a symmetrical layout. In order to avoid implying an exact symmetrical center, however, I will refrain from using the term “axis” in reference to the initial pitch. In contrast to the wedge category, the second category of expansion, additive expansion, is more unidirectional. Instead of organizing pitches around a central tone, an interval distance is utilized as the starting point from which the rest of the process stems. This is best explained through demonstration. Figure 2.1 presents a chain of four pitches: E4, G4, C5, G5. The first

2

See especially chapter 3 for a discussion of Gubaidulina’s use of inversional symmetry.

3

Jonathan Bernard, “Inaudible Structures, Audible Music: Ligeti’s Problem, and His Solution,” Music Analysis 6, no. 3 (1987): 226.

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interval, E4–G4, is [3], that is, the pitches are separated by three semitones.4 This stated interval is then manipulated by the addition of another (unstated) interval. The resulting sum of the two intervals dictates the size of the interval immediately following the stated interval. In Figure 2.1, [2] is added to the stated interval [3] producing [5]. As can clearly be seen, the interval immediately following E4–G4 is G4–C5, which is [5]. This process is repeated, employing the same unstated interval, to produce the next interval. Here, [2] is added to [5]: (G4–C5) resulting in [7]: (C5–G5). This type of expansion is essentially a chain of transformations that extends each resulting interval by a constant.5 For the present example, the transformation could be named ADDTWO since it is defined as a procedure that adds two semitones to each ensuing interval. In the following analyses, I will present examples of each type of expansion as individual processes. Although the sections in the first quartet are texturally distinct and display relatively little surface-level commonality, I will demonstrate that, not only is it possible, but it is quite apparent that the expansion processes are indeed functioning as organizing principles for both pitch and rhythm.

4

Throughout this study, a number placed within square brackets will always refer to an exact interval size in semitones (e.g., [13] = 13 semitones). If the interval is more clearly represented as part of a “class” of intervals, this will be notated with the prefix “ic” (e.g., ic1 = 1 semitone, mod-12 and regardless of direction). Therefore, [13] = ic1, depending on the contextual emphasis. Finally, anytime it is necessary to define or clarify an interval size (exact or interval class) with exact pitches, the following shorthand will be utilized: [13]: (C4–C#5) or ic1: (C4–C#5). For both, the interval size before the colon is specifically defined by the pitches in the post-colonic parentheses. 5

The idea of “transformations” as performed on pitches and intervals was originally posited by David Lewin. See “Transformational Techniques in Atonal and Other Music Theories,” Perspectives of New Music 21, no. 1–2 (1982): 312–71 and Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987).

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Wedge Expansion Figure 2.2 displays the solo viola motive from the opening measure of the piece. The G#3 is first established as a dotted half-note, ornamented with a gradually strengthening vibrato, notated as a curved zig-zag line above the staff. Following a triplet A3-G#3-A3 figure, the G#3 descends through two quarter-tones to G3.6 A3 is similarly inflected by a quarter-tone descent, after which a two-quarter-tone ascent begins with G3. The return to G#3 is finally completed with the final quarter-tone descent from A-half-flat. Displayed beneath each interval in the score is the relative size in semitones (Figure 2.2).7 It can be seen that the unison [0] increases in size to a whole-tone [2] before collapsing back into a unison. This process is repeated in turn at the same tonal level (G#3) in each of the remaining voices in the ensemble. Thus, the opening motive establishes the model by which many of the processes operative throughout the work can be understood. In the first four measures of R1 (Figure 2.3),8 the second violin presents a small wedge expansion beginning with the center pitch C-half-#5. The first segment gradually expands outward in quarter-tone intervals until reaching D-half-#5 in the ascending direction and B4 in the descending direction. The second segment, separated from the first by a dotted eighth-rest,

6

Because of her interest in experimental sounds, Gubaidulina often composes works with microtonal elements. The first and fourth quartets employ these microtones and the notations require some explanation. Since the two quartets have different notational procedures, I will present each one in the context of that quartet (for the fourth quartet, see chapter 5). In this work, the half-sharp is notated as a sharp missing one vertical and one horizontal line while the one-and-a-half-sharp simply adds an extra vertical line to the half-sharp sign. The half-flat is notated with a flag (similar to an eighth-note flag) added to the flat sign and the one-and-a-half-flat simply includes a horizontal line through the regular flat sign. 7

It should be noted that the interval sizes involve fractions corresponding to the microtones. Also, [0] indicates a unison. 8

Throughout this study, rehearsal numbers are abbreviated with a capital “R” followed by the number.

27

discards the use of quarter steps and continues the asymmetrical expansion via semitones before ending with a final upward ascent (G#4-D#5-B-half-#5). It is interesting that not only is the initial expansion continued, but the expansion interval itself is affected by its own expansion (quarter-step expanded to half-step). A similar process occurs in mm. 3–10 of R8 (Figure 2.4) between three members of the quartet. The first violin discontinues its previous highly chromatic motion (not shown in Figure 2.4), focusing on two pitches: F#6–G6. The second violin maintains the F#6 as the first violin descends to F6. The procedure occurs once more by the second violin and viola while the first violin continues to expand in both directions to E6 and G#6. In m. 8, the expansion process reverses by collapsing back into F#6. However, rather than retracing the exact steps utilized in expanding, the contraction process employs quarter-tones. For instance, the first-violin E6 ascends to E-half-sharp6 while G#6 simultaneously descend a quarter step to G-half-#6. The process continues until all three voices intersect at a unison F#6. Thus, in a similar way to the second-violin line, shown in Figure 2.3, this passage exemplifies how multiple intervals can be utilized in the wedge expansion process. Figure 2.5 presents a passage in which a single wedge expansion is simultaneously applied to three different voices. Beginning in m. 4 of R6, the two violins and cello lines are canonic in their melodic structure while the rhythmic durations of individual pitches are contrasted throughout. They each commence by asserting C#4 and gradually expand in pitch space in both directions until all twelve chromatic pitches are stated.9

9

It is of interest to note how this passage invokes serial techniques since no pitch is repeated until the entire aggregate has been established. Gubaidulina, however, rarely made use of this technique as a means of structuring a composition.

28

The second violin in mm. 4–9 of R7 (Figure 2.6) employs the identical method of wedge expansion demonstrated in the semi-canonic passage described above. Beginning with an asserted E5, the melodic material expands outward in pitch space until the entire twelve-tone aggregate is established, ending with A#5 in m. 6. This passage is differentiated from the one previously discussed by the inclusion of repeated pitches. In Figure 2.6 the circled pitches represent those involved in the wedge expansion, while those not circled are repetitions of some previously established pitch, always retaining the exact pitch level. In this way the expansion process continues to function as the primary means of organization. The remainder of the passage following the aggregate completion continues to be structured by the expansion begun in m. 4. Following the A#5, C#6 is asserted as a break in the ascent, which, until now, has expanded by semitones. The descending portion follows suit by stepping a whole-tone from the previous B4 to A4 and continuing to descend chromatically. However, the Bb4 is immediately stated, re-establishing the possibility of a continued semitone expansion. Following the Eb5-A5-Ab4 segment (all repeated pitches), the line ends with another semitonal descent from F4 to B3. Although the descending portion of the wedge receives heavier emphasis, it can be viewed as an acutely unbalanced continuation of the initially balanced wedge expansion process. A particularly salient example of wedge expansion occurs at R21 (Figure 2.7). Beginning on D4, the first violin gradually ascends by semitones (circled in Figure 2.7), alternating each new pitch with a restatement of the initial D4. The remaining voices in the ensemble support the expansion through the reiteration of the newly established pitches. For example, in m. 2 the firstviolin line expands from D4 to E4, passing through Eb4. Once the first violin reaches E4

29

(doubled by the viola), the second-violin and cello lines repeat the Eb4. Thus, as the first-violin line ascends, the remaining quartet members chromatically saturate the created interval with repetition. As the interval becomes larger, the literal repetition of every asserted pitch must cease for a practical reason (i.e., there are too few instruments). Interestingly, the first instance of “losing” a previously stated pitch (m. 7), in which the pitch segment lacks Eb4, occurs just prior to the commencement of the descending component of the wedge in the cello and viola lines (m. 9). Throughout this passage, the voices retain the D4 as a continuous drone that functions as both a stable point of repose and a catalyst for an expansion— simultaneous motivation for continuity and change.10 A complex example of wedge expansion can be found at R25 (Figure 2.8a). The cello and second violin provide an accompaniment to the improvisational first-violin melody (omitted from Figure 2.8a). These accompanying voices are structurally based on the wedge-expansion process outlined above. D5 is asserted in the cello for four beats before “sliding” down through Bb4 and returning upward to D-half-b5, one quarter-tone below the initial pitch. Another “slide” to Bb4 is followed by another ascent to Db5. This process continues, always ascending to a pitch one quarter-tone below the previous ascent, until C5 is attained. At this point the “slide” is followed by an ascent to a half-note Eb5, and the process recommences from D5. As is demonstrated in the analytic voice-leading graph of the cello line (Figure 2.8b), each descent from D5 terminates on a pitch expanded downward by one quarter-tone from the previous

10

This passage also exemplifies how the wedge expansion principle is not bound by temporal placement. Either component may occur first or they may occur simultaneously. Just as the passage in Figure 2.6 was imbalanced by the emphasis on the descending portion of the wedge, the passage in Figure 2.7 is heavily imbalanced toward the ascending component. This is partially based on the fact that the descending component begins over halfway through the passage.

30

termination. Each of these pitches is also immediately followed by an ascent to a pitch expanded upward by one semitone in relation to the previous ascent. In other words, each descent and ascent expands away from D5 in both directions; however, the pitch space below D5 is organized by quarter-tone expansions while the space above D5 is expanded by semitones. The process culminates on the final pitch B-half-b4, one quarter-tone from the “slide” pitch Bb4. The D5 for the cello also acts as the center for a wedge expansion in the second-violin line, although the method of expansion is less consistent than for the cello. As a counter-balance to the cello drone and the rhythmically fluctuating melodic material for the first violin, the second-violin line is decisively rhythmic. The initial interval [4]: (C#5–F5), defined by the pitches in the first measure, has already expanded the pitch space in both directions around D5. Gb5 and G5 are asserted in mm. 6–7, creating an unbalanced ascent compared to the descending component, which has not exceeded C#5. Balance is achieved in mm. 10–11 when the wedge expands downward to Bb4, the lower boundary of the cello line. Thus, the wedge expansion process not only structures each of the individual voices of the duet, it also organizes the pitches between the voices through the emphasis on D5 and B4. The emphasis on descent by the cello is balanced by the emphasis on ascent in the second violin, even though the voices as individual lines are imbalanced. In sum, wedge expansion organizes the pitches within a passage of music around a previously asserted central pitch. Not only can this process occur among multiple voices, it can also structure a single line. Most importantly, wedge expansion does not consistently organize the pitches symmetrically or even in any balanced manner. However, because of the nature of the wedge expansion, some form of symmetry does often arise.

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Additive Expansion Whereas wedge expansion is specifically bound to a particular pitch in pitch space, the second type of expansion process, additive expansion, uses a constant to create a chain of events. Each event is a transformation of the preceding event rather than a transformation based on a single pitch. In this way, the expansion is bound to a temporal process as well as a spatial organization. For example, the viola line in mm. 2–3 of R1 (Figure 2.9) consists of three ascending intervals, each starting on a different pitch. The first interval [9]: (Eb4–C5) is expanded by [1] in the next interval [10]: (A4–G5). This process is repeated as the next interval is expanded once more ([11]: (D5–C#6)). Thus, a single operative transformation can be defined as “add [1] to the previous interval.” Throughout the remainder of this chapter, I will refer to this particular transformation as ADDONE. Figure 2.10 displays mm. 1–4 of R4, in which the second violin presents a descending melody based on the ADDONE additive transformation. The initial interval [1]: (F#7–F7) is expanded by [1], resulting in [2]: (F7–Eb7). As can be seen by the labels above the score in Figure 2.10, each interval in this descent is the result of ADDONE applied to the immediately preceding interval. However, the line produced in this passage is fundamentally different from that in the previous discussion. In Figure 2.9, the first interval [9]: (Eb4–C5) is separated from [10]: (A4–G5) because they do not share any pitches. In the present example, each new interval formed as a result of ADDONE is built on the terminating pitch of the previous interval. Thus, [2]: (F7–Eb7) shares its first pitch with the previous interval [1]: (F#7–F7). This transformational expansion continues until the pitches are out of the second-violin range, at which time they are

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continued by the cello line. The pattern terminates only after the cello has reached its registral nadir. The possibility of rhythmic organization utilizing a wedge formation is difficult to imagine since rhythms are limited to temporal space. Not only must temporal aspects occur in sequence whereas pitches can occur simultaneously, pitches have an added benefit of being perceived on “vertical” levels in space corresponding to a listener’s impression of high and low. Thus, it is possible for pitches to “move” outward and expand in pitch space in two different directions. Sequential events are incapable of spatial movement and are cognized as occurring in chains, which necessarily do not allow for bidirectional perception. Based on this understanding, wedge expansion, because of its dependence on multiple directionality, is only capable of structuring pitch material. However, the additive expansion process is perfectly able to formally structure rhythmic material. Additive expansion applied to rhythmic space follows the same basic procedure as it does when applied to pitch space. Rhythmic values are expanded in duration by some constant rhythmic value. As with intervals, the transformation acts upon each resulting duration in order to produce the next duration.11 For example, the cello melody in mm. 2–4 of R1 (Figure 2.11) is made up of four (012) melodic trichords: , , , and . Each trichord is rhythmically articulated as a short-short-long segment; however,

11

This idea was inspired by the notions of “projection” and “deferral” as presented in Christopher Hasty, Meter and Rhythm (Oxford: Oxford University Press, 1997) and by the work of David Lewin: “On Formal Intervals Between Time-Spans,” Music Perception 1, no. 4 (1984): 414–23 and Generalized Musical Intervals, 60–81. The concept of constantly-undermined expectations of rhythmic continuity is especially prominent in Gubaidulina’s experimentation with the “stream-of-consciousness” character of time perception. By avoiding any constancy of metric division, countless manners of temporal perception within a composition become possible. In the particular context of this analysis, an additive character lends a unique quality to the piece as a whole, in which time seems to be constantly expanding.

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though the “short” is consistently presented as a sixteenth-note, the “long” is systematically lengthened in each consecutive statement. The first statement employs a “long” duration of three sixteenth-notes, and the second statement includes an expanded “long” duration of four sixteenth-notes. The third and fourth statements continue the process of expanding the previous “long” duration by one more sixteenth-note. This additive expansion could be seen as the rhythmic analogue of expanding an interval by [1]. Therefore, I will define this particular transformation as “add one sixteenth-note duration to the previous duration.” For the remainder of this chapter, I will refer to this defined transformation as ADD16th.12 Returning to Figure 2.10, the identical additive process ADD16th structures the rhythmic properties of the descending melody. Although the pitch statements retain a constant sixteenthnote duration, the duration of the rests between these pitches fluctuates. Between the first two (F#7–F7) exists a “zero” duration of silence. The transformation ADD16th is applied to this “zero” duration resulting in a one-sixteenth-rest between the second and third pitches (F7–Eb7). Applying ADD16th once more results in a two-sixteenth-rest between the third and fourth pitches (Eb7–C7). This additive expansion continues unhindered, even as the melody passes into the cello line, resulting in a duration of nine sixteenth-rests between the final two pitches (B2–C2). The process is graphically demonstrated with labels below the score in Figure 2.10. Thus, the additive expansion process effectively organizes both the pitch material and rhythmic durations of this passage. Another example of this interaction of pitch and rhythm via additive expansion is demonstrated in mm. 4–15 of R48 (Figure 2.12). B4 is asserted by the first violin, viola, and 12

To clarify, although the example under discussion involves transformations acting upon durations of pitches, the ADD16th function can also (and will) apply to durations of rests as well.

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cello in unison. As the viola sustains B4, the first violin ascends by a quarter-tone, and the cello descends the same distance. This apparent wedge expansion is immediately subsumed by an additive expansion. Following an eighth-rest, the first violin begins once more from B4 and expands upward two quarter-tones to C4 while the viola ascends one quarter-tone and the cello sustains B4. The interval in each voice has been expanded upward by one quarter-tone. In this second iteration, the viola repeats the first violin’s first statement, and the cello repeats the viola’s first statement. This process continues throughout, so the first violin can be thought of as acting as a dux in a pitch canon in which the pitches are organized as quarter-tone additive expansions, always starting from B4. The viola, acting as a comes, is delayed by one quartertone expansion while the cello lags behind by two quarter-tone expansions. In other words, following every rest, each voice returns to B4 and presents a quarter-tone expansion of its own previous interval. The additive process also organizes the rhythmic structure of both the B4 repetitions and the intervening rests. The initial B4 duration, four sixteenth-notes, is expanded via ADD16th to a five-sixteenth-note duration when B4 is repeated. The duration of the rest between the initial first-violin interval (B4–B-half-#4) and the second interval (B4–C4) is two sixteenth-rests. Parallel to the pitch rhythms, the next silence is expanded by one sixteenth-rest. As is graphically demonstrated in Figure 2.12, the ADD16th process continues in the rhythms of both pitch (labeled above the first-violin line) and the intervening rests (labeled below the first-violin line) until m. 9. At this point (circled in Figure 2.12), a single eighth-rest is inserted, successfully discontinuing the pattern and, in this passage, signaling a second attempt at the process. The rhythm of the B4 reinitiates the process beginning with a two-sixteenth-note duration while the

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intervening rest durations resume the process beginning with a one-sixteenth-rest duration. The newly begun process continues uninhibited through the remainder of the passage. Figure 2.13 (R42–R43) presents a similar additive expansion functioning on both pitch and rhythmic levels, creating an interesting interaction. The ascending first-violin line in R42 establishes a consistent pattern, albeit not as simple as ADDONE. The pattern alternates between expansions via [2] and [3]. By using a [3]-expansion twice, the pattern is broken with the final pitch. The rhythmic additive expansion pattern is also much less transparent than the previous ADD16th. As Figure 2.13 demonstrates, the second and third pitches are expanded by one sixteenth-note compared to their previous durations. The fourth and fifth pitches, on the other hand, double the additive constant (i.e., sixteenth-note), making each new duration two sixteenth-notes longer than the previous one. Finally, the sixth and seventh pitches in the melodic chain are each expanded by a four-sixteenth-note duration, once again doubling the twosixteenth-note additive constant. In this way the melodic line is expanded exponentially in both pitch space and temporal space. The cello follows the first-violin pattern in a similar manner while inverting the overall direction of the line. At R43 the alternation of additive [2] and [3] begins as expected; however, once E4 is reached the pattern is completely broken. By traversing a descending interval of [8] to A3, the pattern, which should have added [3], subtracts [1]. Instead of [8] becoming [11], the chain follows [8] with [7]! This is compounded by the remarkably large [18]: (A3–Eb2) immediately following.

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The rhythmic pattern performs a stabilizing role in the cello line. The exponential quality of the durational expansions carries over from the first-violin line; however, instead of starting the chain with sixteenth-note durations, the descending cello line is structurally based on eighthnote durations. In other words, although the ratios between the individual durational transformations is identical to ratios in the first-violin transformations, the constant is doubled (sixteenth-note becomes eighth-note). The cello pattern deviates by allowing one extra eighthnote addition before resuming the expected rhythmic transformations. Aside from this, the rhythmic pattern retains its structural stability at a point when the pattern of transformations acting on the pitch structure is discontinued.

Conclusion The two categories of expansion processes presented in this chapter aid in giving the quartet some sense of continuity and growth. Although the individual sections may seem to be connected in only the most superficial manners, the ubiquitous property of expansion in pitch space and rhythmic space assists in establishing a method or process by which these individual sections are connected. One final note concerning performance will reinforce these various musical aspects. According to the composer’s instructions, the performers should begin the work sitting in a traditional string quartet arrangement. At various moments throughout the performance, each of the members is individually directed to pick up his or her chair and move away from the stillplaying group toward a preselected corner of the stage. The “expansion” of the ensemble occurs both temporally, over the course of the entire work, and spatially, since only one member moves

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by a small amount each time. In other words, the expansion processes are not limited to the notes on the score. It seems appropriate in the context of this discussion to note the similarity between the aural and visual aspects of the performance.

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Chapter 3 Analysis of String Quartet No. 2 (1987): Gap-Fill Processes and Inversional Symmetry

Gubaidulina’s second quartet can be divided into two distinct sections separated by a slight pause just before R21. The two sections exhibit two drastically different textures, creating a distinct mood for each section. The first section (A) lacks any metric notation and, with one exception, avoids strict rhythmic divisions. The music is notated to allow for a maximum of performance freedom. The first section consists of a non-stop drone and several tremolo pitches moving at a quick pace, though lack of meter is often interpreted as a method of achieving temporal stasis. In this case the overall effect is one of frustrated restlessness and agitation rather than motionless serenity. The second section (B), on the other hand, although governed by temporal divisions of metric notation, consists primarily of long durations, sometimes lasting several measures. Often, one voice changes pitch while the remaining voices continue holding their respective tones in the sonority. The voices also feature intensely chromatic lines, requiring small melodic steps, and employ several harmonics. The lack of simultaneous motion, along with the acute focus on chromaticism and harmonics, yields a quality of ethereal dreaminess. Thus, at first glance the piece actually consists of two dissimilar and unconnected sections rather than a single cohesive work; however, the processes at work within the each section assist in creating an overarching structure and establish a means of achieving continuity.

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Two primary structural processes are at work in the second quartet: gap-fill and inversional symmetry. In the first, gap-fill, the voices establish a particular pitch space within which the immediately following musical material unfolds.1 The interval gap is established through the assertion of two pitches of significant distance in pitch space. This can be attained in two ways: a single voice may leap from an asserted pitch creating a gap between the two pitches; or, two voices may state two different pitches, simultaneously or in melodic sequence, at different registral levels. Based on Leonard Meyer’s concepts of “affect” and “meaning,” the produced gap creates an expectation for pattern completion, which is achieved by means of chromatic saturation.2 Because the music avoids any reference to conventional functional tonality, the only recourse is to saturate chromatically the established gaps. Carrying Meyer’s reference to “affect” and “meaning” one step further, the systematic blocking of gap-fill completions becomes necessary for establishing long-range formal sections in the second quartet. In this way, a section of music organizes the pitches contained therein by opening up a space and eventually filling the gap with every missing chromatic pitch. The second process, inversional symmetry, involves the arrangement of pitches, both melodic and harmonic, around some pitch or dyad. This central pitch acts as an axis that may

1

Much work has been done with the concept of gap-fill and chromatic saturation within established gaps. See Leonard B. Meyer, Emotion and Meaning in Music (Chicago: University of Chicago Press, 1956); Joseph N. Straus, The Music of Ruth Crawford Seeger (Cambridge: Cambridge University Press, 1995); and Catherine Losada, “A Theoretical Model for the Analysis of Collage in Music Derived from Selected Works by Berio, Zimmerman, and Rochberg,” (Ph.D. diss., City University of New York, 2004). 2

Meyer’s discussions of gap-fill in a tonal context is applicable to the present study. He presents these needs for pattern completion and closure as “products of a more general psychological need for structural completeness—for the elimination of structural gaps not only in the melodic line of the individual piece but also in the tonal system itself. . . . A structural gap, then, creates a tendency toward “filling in.” And if this tendency is delayed, if the completion of the pattern is blocked, affect or the objectification of meaning will probably follow.” (Meyer, 134)

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either be literally asserted in the music or merely implied based on the symmetrical organization of those pitches within its gravitational pull.3 In most cases the central axis has some functional connection to the previous and following musical material, thus creating a connective link between the various segments of music.4

Section A5 Gap-fill, as a process that organizes the musical elements of this work, primarily accomplishes two significant goals: melodic coherence and sectional articulation. The melodic coherence occurs on a purely local level, connecting musical elements within an immediate temporal vicinity. The methods of melodic organization are similar to Ruth Crawford Seeger’s melodic structures with regards to chromatic completion; however, in this quartet there is an almost exclusive use of gap-fill in pitch space with the resultant avoidance of abstract pitch-class

3

It should be noted that the word “axis” was avoided in chapter 2 in order to deflect any connection to exact inversional symmetry and focus only on expansion outward in pitch space, regardless of distance. In this chapter, however, the concept of exact symmetrical distances from a central point is quite important; thus, the term “axis” will be employed as needed. 4

A massive amount of work has been produced with inversional symmetry as the theme. For instance, see Milton Babbitt, “Twelve-Tone Rhythmic Structure and the Electronic Medium,” Perspectives of New Music 1, no. 1 (1962): 49–79; David Lewin, “Inversional Balance as an Organizing Force in Schoenberg’s Music and Thought,” Perspectives of New Music 6, no. 2 (1968): 1–21; idem, “A Label-Free Development for 12-Pitch-Class Systems,” Journal of Music Theory 21, no. 1 (1977): 29–48; George Perle, “Berg’s Master Array of the Interval Cycles,” Musical Quarterly 63, no. 1 (1977): 1–30; Elliott Antokoletz, The Music of Béla Bartók: A Study of Tonality and Progression in Twentieth-Century Music (Berkeley: University of California Press, 1984); Jonathan Bernard, “Space and Symmetry in Bartók,” Journal of Music Theory 30, no. 2 (1986): 185–201; and Philip Stoecker, “Studies in Post-Tonal Symmetry: a Transformational Approach” (Ph.D. diss., City University of New York, 2003). 5

As will be illustrated in the following discussion, the two organizational processes cooperate on many levels. Because this interaction is so pertinent to the understanding of their particular organizational functions, the following analysis will be structurally divided based on the musical section being discussed rather than the individual processes.

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relations.6 The goal of sectional articulation, on the other hand, is achieved via chaining several of these musical events into larger formal segments. The following discussion will clarify the different manners in which the gap-fill process achieves these ends. In the opening passage of the work (Figure 3.1), the second violin asserts a harmonic G5, which is sustained at a soft dynamic level throughout the entire A section. The cello initially supports the harmonic G5 before inserting articulations of non-harmonic G4. This same alternation of G4–G5 occurs in different rhythmic forms in each of the two remaining voices (i.e., first violin and viola). Thus, the only two pitches asserted in the first four rehearsal numbers are G4 and G5. The sheer repetition of this octave establishes a gap, successfully opening up a pitch space in which the melodic material of A section unfolds. As can be seen in Figure 3.2a, the first-violin line at R5 comprises thirteen pitches beamed together and followed by a brief pause. This group of pitches initiates the process of gap-fill that will occupy the remainder of section A. Into the originally established G4–G5 gap, G#4, A4, and D5 are inserted while, outside the established space, F#4 is added.7 This grouping of quickly-stated pitches followed by a short rest takes on a motivic character in section A. The first-violin line invariably incorporates similar formations of pitches and pauses, creating a chain of independent pitch collections. The motivic aspect is further strengthened by the particular pitch contour (initial ascent followed by a descent that returns to the opening pitches) and

6

“Once an initial note has been asserted, a second note can be added to open a musical space that then becomes the locus of the ensuing action. . . . A previously opened space (either pitch of pitch-class space) is often filled in chromatically. This strategy is closely related to the drive of Crawford’s melodies toward chromatic completion.” (Straus, Ruth Crawford Seeger, 41–42) 7

The assertion of pitches outside the boundaries of an established space is closely related to the expansion processes presented in chapter 2. The importance of F#4 will be elaborated in further discussion; however, it must be ignored for the time being.

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dynamic markings (gradual crescendo followed by decrescendo) established with the first group (Figure 3.2a). Each new declaration of this motivic collection integrates one or more previously unstated pitches. In this way, through several statements of the “motive,” the G4–G5 gap is eventually saturated with every chromatic pitch within its lower and upper boundaries (i.e., G4 and G5, respectively). However, as the reduction in Figure 3.2b demonstrates, the order of additional pitches is not random. Beginning from the D5 in the first motive statement (Figure 3.2a), each succeeding motive includes a pitch a semitone higher than the previous one. In other words, the next motive statement includes Eb5 and the following contains E5. In similar fashion, except for C5, the pitches below D5 are also added in semitonal increments. This process continues until the pitches have extended to the upper boundary, G5.8 The chain of motive statements is terminated at R15 with the introduction of a contrasting idea in the first violin (Figure 3.3). The new first-violin statement is distinguished from the previous motive in many respects. First, the line begins from its highest point and descends, establishing a new contour. Second, the initial dynamic level is forte, discontinuing the preceding trend of gradually increasing in dynamic from an initial piano. Third, this motive also represents the only deviation from a “free” rhythm because the first five pitches are notated in specific rhythmic durations. Finally, this is the moment at which the pitches above D5 have ascended to the upper boundary pitch G5. Thus, the various musical elements combine to

8

Once more, it should be noted how this particular use of gap-fill corresponds to the process of expansion (ch. 1). The only difference lies in the assertion of a goal pitch. The expansion process simply asserts an initial pitch from which the remaining pitches radiate. On the other hand, the specific gap-fill process utilized in Gubaidulina’s string quartets involves the assertion of both an initial pitch and a goal pitch. Thus, in a sense, the process of expansion is still a major organizing force in this passage.

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constitute a formal boundary with respect to the A section (hereafter, divided into A1 and A2). Based on the intense markedness of this new motive,9 one would assume that the gap-fill process is correspondingly completed; however, C5 remains unstated. The remainder of section A (specifically, A2) follows a process similar to A1. The motive-chain, now based on the newly established line (R15), is resumed by the first violin. However, as foreshadowed by the assertion of a pitch beyond the established boundary at R5 (F#4), the newly established pitches in each statement continue the semitone expansion, ascending beyond G5. This, combined with the gradually reduced durations of the intermittent pauses, generates a significantly enhanced momentum in the final moments of section A. Figure 3.4 demonstrates how the original catalyst for this continued expedition, C5 (circled in Figure 3.4), is finally attained in the final statement. This statement also marks the simultaneous cessation of the ascending expansion with an upper boundary of F#6.

Inversional symmetry, the second process at work in this piece, is closely related to the gap-fill procedure presented above. Not only does inversional symmetry tie local motives and relate different voices to one another, it also aids in establishing large-scale connections across the work as a whole. I will discuss these aspects of inversional symmetry as well as its association with the formal boundaries previously outlined.

9

“Markedness, a concept traceable to Roman Jakobson and the Prague School of linguistics, can be defined as the valuation given to difference.” Robert Hatten, Musical Meaning in Beethoven: Markedness, Correlation, and Interpretation (Bloomington, Indiana: Indiana University Press, 1994), 34.

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At R5 (Figure 3.2) the cello line, neglected in the previous discussion, features an exact inversion about G4 of the first-violin statement. Based on its structural function and its salient presentation in the preceding music, G4 is the most logical axis point. Symmetrical inversion becomes apparent upon closer examination of the pitches involved. Beginning with an asserted G4, the intervallic contour of the first-violin statement is <+1, +1, –2, +1, +6, –6, –2, +1, +1, +1, –2, +1> with each sign/number pair equivalent to a directed interval. In other words, the first violin changes pitches from starting pitch G4 up [1], then another move up [1], followed by [2] down, etc. The contour of the cello line, beginning with the same G4, is <–1, –1, +2, –1, –6, +6, +2, –1, –1, –1, +2, –1>. The interval distances remain constant when compared to the first violin, but the direction of each individual motion is inverted, creating what Philip Stoecker refers to as a “successive mirror.”10 Thus, not only is the expansion process attempting to fill the gap between G4 and G5 in the first violin, an implied gap is simultaneously being filled via inversional symmetry. The cello line mirrors the first-violin line and, by implication, will eventually fill the gap between G4 and G3.11 Every time the first violin adds an ascending pitch, the cello adds its inversionally related mate. As is demonstrated in Figure 3.4, in the same passage in which C5 successfully completes the first-violin G4–G5 gap, D4 appears in the cello line, completely saturating the G4–G3 gap with its own chromatic pitches. In sum, the opening passage establishes one level of activity (the alternation of G4 and G5), and the material beginning at R5 constitutes a second level of activity. Throughout section

10

Stoecker, “Studies in Post-Tonal Symmetry,” 5. It is “successive” because the cello line does not occur simultaneously with the first-violin line (the lack of exact temporal correspondence is apparent in Figures 3.1–3.3). 11

To convey the inversional property, I present the established gaps as extending from the axis. Thus, G4–G3, although “backward” from the conventional notation of intervals (e.g., G3–G4) directs attention to the fact that it actually is “backward” since it is based on an inversion.

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A, the second violin and viola preserve the initial activity of the opening passage with a focus on G4 and G5. The two levels interact as the first-level activity maintains the initial octave gap as a structural basis, and the second-level activity (first violin and cello) saturates the space (along with the inversionally related space) with every possible chromatic pitch. Once this saturation is essentially satisfied (excepting one pitch), the second-level voices are forced to exceed the boundaries of the primary octaves. As for large-scale articulation, section A can be conveniently divided into two subsections (A1 and A2). The first is governed by the process of gap-fill within the boundary pitches G4–G5 and, through a symmetrical process, G4–G3. The formal boundary is established at the moment G5 is attained, articulated in the melody by a sudden change in various musical parameters. Section A2 consists of an expansion beyond G5, which is terminated with the assertion of C5, the only remaining unstated pitch within the G4–G5 gap. Throughout section A, the cello line mirrors the inversional partner of every pitch asserted by the first violin.

Section B The character of the second quartet’s second large section (B) is antithetic to section A. Whereas section A is rhythmically and metrically vague, section B maintains a consistent meter and rhythmic structure throughout. The melodic material of section A is aggressively asserted in short bursts alternating with pauses. Section B, on the other hand, comprises consistent and slow-moving melodic lines. The accompanimental material in section B focuses on the sustained [3] and [4] at varied pitch levels, whereas section A is limited to the articulated repetitions of [12] on an exclusive pitch level (i.e., octave G4–G5).

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Notwithstanding the noticeable differences in character, section B, beginning at R21, is fundamentally structured in a fashion similar to section A. In the opening measure of R21, the first-violin line contains a harmonic F#6 immediately followed by F#4. Tantamount to the focus on G4–G5 in the opening of section A, these two pitches similarly establish the gap that is eventually filled in section B. The structural importance of F#6 is supported by the fact that section A halted the ascending expansion on this exact pitch. Thus, the two sections are interrelated based on the structural importance of the final pitch of section A and the initial pitch of section B. Ascension by [1] is the primary means of gap-fill in section B, again constituting another structural property in common with section A. At R23 (Figure 3.5), [1]: (A4–Bb4) is stated melodically by the viola before being sustained as a dyad for five beats. This [1] figure initiates a three-voice canon, answered by the first violin (m. 2) followed by the second violin (m.1 of R24).12 The viola’s A4 is invariably sustained while the upper note of the dyad ascends by increments of [1]. For instance, at R24 the A4–B4 dyad, expanded from the originally established A4–Bb4, continues to expand. The same process occurs in the remaining two canonic voices, terminating once the viola reaches the [8]:(A4–F5) dyad. F#5 is stated by the unaccompanied first violin at R25, effectively ceasing the [1] expansion. Although the first octave seems completely saturated, the initial pitch in the viola line is A4. Therefore, two pitches remain unaccounted for, G4 and G#4; however, they can be located in the cello and viola parts at R28. In this way the first octave of the F#4–F#6 gap-fill is achieved

12

The canon here is based on the viola’s pitch sequence alone. Rhythms are varied and essentially unsystematic. Nonetheless, generally, in any given measure, one voice sustains a dyad for a significantly long duration while the other two continue their stretch upward.

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primarily by means of an ascending [1] expansion, paralleling a similar procedure in section A. Two significant aspects can be seen here: first, the F#4–F#5 octave is completely filled before the upper voices resume the [1] expansion at R30 (discussed below). Thus, the first half of the gap is successfully saturated before any attempt is made at the second octave (F#5–F#6). Second, the structural importance of G4 in section A is apparent, and the fact that it is one of the final pitches attained in the gap-fill process of section B seems significant. In a sense, it exposes section B’s structural reliance on material previously presented in section A. Following a lengthy interlude reminiscent of the opening of section B, a second [1] expansion process begins at R30 by the violins. Beginning with F#5, the first-violin line resumes the [1] ascent as the second violin accompanies a semitone below. The sustained dyads parallel the A4–Bb4 dyad that initiates the canon at R23. Although some pitches, immediately after their initial statement, are repeated as harmonics two octaves higher, the process at R30 is essentially unhindered until R31. The expansion temporarily ceases once C6 is reached by the first violin. In other words, the gap-fill process once again fails to successfully saturate the originally established space. It is interesting to note the correlation between this renewed process and the first expansion occurring at R21–R25. The first expansion spans A4–F#5 as part of the gap-fill of the lower half of the originally established F#4–F#6. In other words, F#5, the terminating pitch, bisects the initial space into two [12], and the first expansion process, along with the statement of G4 and G#4, only fills the lower [12], leaving the upper [12] empty of its chromatic saturation. At R31 the second process is terminated at C6, once again bisecting the remaining empty space

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into two equivalent intervals (i.e., [6]). Similar to the first expansion, this second process only fills the lower portion of the remaining gap. In m. 7 of R31, the C6 is reasserted in the first violin while the second violin adopts the familiar accompaniment a semitone below. In this way the second expansion begun at R30 is resumed from this point. The gap-fill is successfully completed at R33 with the assertion of F#6 (notated as Gb6) by the first violin. However, in similar fashion to section A, the expansion process continues by both violins beyond the functional boundary pitch, finally ceasing at A7, more than an octave above F#6. Taking stock of the many parallels between section A and section B, we are invited to find formal correlations as well. Section A is interpreted as consisting of two individual subsections differentiated by individual processes contained therein. Sub-section A1 is governed by the gap-fill process of G4–G5, and sub-section A2 is governed by an expansion beyond the boundary pitch, G5. In section B, the same criteria reveal a similar formal structure. Sub-section B1 (R21–R29) contains the first octave gap-fill, and sub-section B2 (R30 through m. 2 of R33) is characterized by the second octave gap-fill. The individual sections are also contrasting in their individual textures. B1 comprises a canonic melodic structure, whereas the homorhythmic violins carry the melody in B2. The final sub-section, B3 (beginning in m.3 of R33), consists of long sustained sonorities, each voice rhythmically synchronized, in similar fashion to a chorale. However, in pitch structure the violins continue the [1] ascension. Therefore, it can be seen that section B consists of three subsections, the first two representing an octave gap-fill (parallel to sub-section A1), and the third contains an expansion beyond the boundary pitch F#6 (parallel to sub-section A2).

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In section B, inversionally symmetrical relationships create connections between adjacent sonorities. Throughout the passages presented in the ensuing discussion, a chain of inversional transformations manipulates each individual set of pitches in order to produce the resulting set. Each resulting set is then exploited as a temporary inversional axis in order to produce more pitches. For example, at R21 (Figure 3.6) the dyad containing pitch classes D and F# is the prominent sonority for three measures.13 Following the F#4–F4 statement by the first violin, pitch class A is asserted via inversion about F/F# in pitch-class space. Thus, A is the inversional counterpart of D, a relationship further strengthened by their simultaneous statement in m. 4.14 This is graphically portrayed in Figure 3.7.15

Similarly, at R22 the opening D5–F#6 dyad

is inversionally mapped onto a harmonic A4–F6 dyad about the F/F# axis (in pitch-class space). Thus, the addition of pitch class A is achieved by means of symmetrical inversion. In m. 2 of R22 (Figure 3.6), the semitone A4–Bb4 in the first-violin line is symmetrically balanced by an inverted statement in the cello line (A3–G#3). This simultaneous statement establishes the newly asserted pitch class A as the new axis, usurping the structural title previously held by the F/F# axis. The second-violin assertion of [7]: (A4–E5) in m. 3 of R22 is

13

Inversional symmetry can structurally act on pitch classes without recourse to actual pitch space. In order to portray this in the analysis, the lack of a number after a pitch name indicates that octave equivalence is enacted. In other words, F#4 refers to a specific pitch located [6] above “middle C” while F# (no number) refers to any F# in pitch space. 14

This is an instance of what Straus terms PIVOT: “As a melody progresses, single notes or pairs of notes are often established as momentary pivot point or centers of inversion around which other notes balance.” (Straus, Ruth Crawford Seeger, 42) It is important to note that a temporary PIVOT can affect a single pitch or several of the pitches that immediately follow. 15

The graphic representation of the voice-leading used in this and following examples is originally taken from Henry Klumpenhouwer, “A Generalized Model of Voice-Leading for Atonal Music,” (PhD diss., Harvard University, 1991) and expanded by Joseph N. Straus, “Voice-Leading in Atonal Music,” in Music Theory in Concpt and Practice, ed. James Baker (Rochester: University of Rochester Press, 1997), 237–274 and “Uniformity, Balance and Smoothness in Atonal Voice-Leading,” Music Theory Spectrum 25/2 (2003), 305–352.

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the exact counterpart about the newly established A axis of the first-violin [7]: (D4–A4) in m. 5 of R21. Finally, the following [11]: (A4–G#5) in the second-violin line (mm. 4–5), an exact repetition of the first-violin statement in mm. 6–7 of R21, is also related by inversion about A to the [1]: (A4–Bb4) in pitch-class space. Thus, almost every pitch throughout R21–R22, though superficially unassociated, can be related by means of inversional symmetry and shift of axes (either in literal pitch space or in pitch-class space). As graphically presented in Figure 3.8, a similar procedure involving the same axes can be located in the interlude material following the completion of the gap-fill process on the first F#4–F#5 octave (R25–R30). In m. 3 of R25, the D/F# dyad is once again asserted by the first violin and cello, closely followed by a statement of F#3–F3 in the cello line, paralleling the F#4–F4 in the passage presented above. Once again, a statement of pitch classes D and A (inversionally related about the F#/F axis) is immediately followed with a statement of pitch classes A and E. As before, A becomes the structural axis inversionally mapping pitch classes D and E onto one another. The violins immediately follow with two more open fifths (C4–G4 and B5–F#6), which are related by inversion about D/E, the pitches attained in the previous sonority via inversion about A. Following this, the viola pitch changes from A5 to Bb3 while the cello line shifts to Db4 from D4. The pitch classes A and D are inversionally mapped on to Bb and Db via the axis B/C, the exact pitch classes in the violins’s open fifths. Finally, the viola asserts [1]: (F5–E5) followed immediately by a cello statement of [1]: (F#4–G4). Pitch class F will map onto F# and pitch class E will map onto G when inverted about Bb/Db, the pitches previously attained by the cello and viola.

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Inversional symmetry functionally organizes the individual pitches within these passages into a constellation of intervals relating each pitch to at least one other pitch. In other words, although the individual pitches may appear unrelated, random, and disjunct, a single process of inversional symmetry is functioning as an organizing force, establishing continuity throughout.

Conclusion Far from being discontinuous, the second string quartet displays a cohesive design. The motives and themes are developed by the same means by which large-scale unity is maintained. The gap-fill process paves the way for chromatic saturation in order to establish specific pitches as more structural than others based on their inclusion in (or lack thereof) in a specific group. Inversional symmetry allows certain pitches to assert their structural and formal priority based on their function as a central axis about which other pitches are organized. The two compositional processes are keys to the cohesive nature and continuity of the work.

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Chapter 4 Analysis of String Quartet No. 3 (1987): Intervallic Definition

In the third quartet, another process is at work to aid in establishing continuity is at work. It involves the assignment of a particular function to specific interval classes contained within the primary pitch-class set employed. The assigned function assists in delineating the formal structure. This is specifically achieved in two ways. First, ic5 and ic1 each act as structural markers between sections and as a means of differentiating between voices. Second, when a primary set is disposed as a series, the intervals within the series are used to vary motives via repetition of intervals and additive chains. This corresponds to Christopher Hasty’s notion of intervallic definition.1 Figure 4.1 graphically presents the formal outline of the composition. Formal divisions are articulated primarily through shifts in textures.2 The opening through R18 (section A) is

1

Christopher Hasty, “An Intervallic Definition of Set Class,” Journal of Music Theory 31, no. 2 (1987): 183–204. Hasty’s notion consists of utilizing the interval vector of a pitch-class set as a means of defining said pc set. In other words, rather than dependence on the numbered pitches, Hasty proposes defining a pc set by its potential interval usage. For my own purpose in this analysis, I have particularized and focused this “intervallic definition” to demonstrate how the contained intervals themselves may also be functionally defined, creating the potential for formal delineation. Not only is the collection of possible intervals important for defining the pitch collection, each individual interval class involved is specifically limited in its potential use for organizing pitches. In essence, it is simply a further delineation of levels. Not only is a specific pc set structurally employed within the various sections of the piece in question (via its intervallic definition), the individual intervals also function on an abstract “micro-level” to establish the manner in which the pc set itself is utilized. 2

Claire Polin specifically discusses the aesthetic consequences of Gubaidulina’s focus on changes in texture and timbre as formal articulation within this work: “Note that Gubaidulina appears to be most interested in and draws her material from the manner in which sound is produced. Of her Third String Quartet she says ‘as a composer I am most interested in the distinctions such as . . . whether the sound is produced by means of touch or by plucking . . . such small changes can create the climax of the piece, when it changes from pizzicato to arco [R49].’” (“The Composer as Seer, but not Prophet,” Tempo 190 (Sept. 1994): 16)

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characterized by its emphasis on a single melodic line accompanied by ostinato repetitions of arpeggiated open fifths in other voices. The melody exhibits aspects of Webernian Klangfarbenmelodie as two or more instruments perform successive pitches in the melodic line, endowing individual pitches with a unique timbre. Section B (R19–R48) is more contrapuntal, characterized by its emphasis on semitonal relationships, both within and between each instrumental voice. Although a single meter temporally organizes the four instrumental lines, the rhythmic divisions exhibited by the individual voices are completely independent of one another. Each line is organized into an idiosyncratic rhythm lacking any connection to the remaining voices. Section C (R49–R51) ensues following a five-second pause. Beginning with six chords, each held in long notes and consisting of seven individual pitches, the section continues with material similar to the accompanimental arpeggiated fifths from the A section. Section D (R52–R76) contains qualities from each of the previous sections. The texture consists primarily of imitative counterpoint between the main voices, but the rhythmic meandering of section B is conspicuously absent. Melodically, the passage resumes melodic material similar to that initially utilized in section A, creating an overarching connection to the single-line melody/accompaniment texture of the opening. The final section (R77–R78) functions as a small coda that echoes musical material from section C. The primary pitch material for the work as a whole is epitomized in the chords at R49 (Figure 4.2). These chords, each with seven different pitches,3 are arranged so that the outer pitches (the highest in the first-violin voice and the lowest in the cello voice) move in contrary

3

Interestingly, this quartet does not depend on pitch-space organization as much as the other quartets in this study. Much of this analysis relies on the means through which pitch-class collections are organized without the intense recourse to exact pitches so ubiquitous in the remaining three analyses.

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motion toward each other. This wedge-like motion, reminiscent of the process utilized in the first quartet, culminates in the final septachord, which consists of stacked semitones [A, Bb, B, C, C#, D, Eb]. The first five sonorities belong to a single set-class—namely, 7-27 (0124579). The first sonority comprises the pcs [G, A, B, C, D, Eb, E], and the second comprises the pcs [Db, D, Eb, F, Gb, Ab, Bb]; each of the remaining three chords consists of the exact pcs contained in the first sonority. The relation between the first two chords is the inversion that maps Eb onto D (i.e., I5 transformation). This set class, verticalized into a single sonority in this passage, constitutes the primary melodic and harmonic material for the entire work. Because the exact pitch collection contained in the first chord is so ubiquitous in this work, I will refer to it as the Source Chord. In some cases the actual pitch-class content of this chord organizes the pitch structure. In other sections the intervallic definition inherent to this pitch collection is the primary compositional construct.

Hierarchical and Formal Separation The pitches asserted in the third quartet are sometimes arranged by assigning specific functions to an interval class, bestowing upon those interval classes the potential of structurally organizing the musical material. By analyzing the intervallic structure of the primary set class 727 (0124579), it can be seen that the set is an all-interval septachord with an interval vector [344451]; thus, any interval is a possible compositional choice. Specific to this work, ic5 functions as a method of separating material, both between sectional units and between simultaneously sounding voices. This individual intervallic focus is further supported by the fact

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that ic5 is the most pervasive interval in the vector. I will focus on each issue separately in the following discussion. At the opening of the work (Figure 4.3), the second violin performs a series of arpeggiated open fifths as harmonics on open strings: . This continues as an ostinato beneath the first violin’s prominent melodic statement. Punctuated by fermatas, the individual units of the melodic line are completely void of ic5; however, the connection between the two voices is made apparent by the melody’s emphasis of the figures D5–E6 and A5–D#6, which, except for D#6, assert the exact pitches that the second violin articulated into units of ic2 and ic6. In this way the redundant “ic5 line” is relegated to the background while articulations of the remaining ics are imbued with a melodic structural status. This textural separation by means of intervallic distribution continues throughout the A section as the voices carrying the melodic material avoid utilizing ic5. For instance, Figure 4.4 presents the second-violin accompaniment pattern following the first violin’s initial melodic statement. As can be seen, another fifth is included in the accompaniment collection: . This addition is made even more salient since the first melodic statement terminates on the exact same pitch (B6). In other words, when combined into a single collection the melodic and accompanimental lines contain the exact pitch-classes from the Source Chord, yet the individual lines emphasize different ics. The Eb5–G4 asserted by the first violin upon reentry at R2 (Figure 4.4) marks the point at which the 7-27 set class is literally formed. Thus, the pitch-class content of the Source Chord (R49) structures the opening of the work. Another important observation concerns the consistent manner in which the melodic material temporarily ceases each time a new pitch is added to the accompaniment collection.

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Because the ic5-saturated accompanimental material clearly articulates melodic segments, by implication, ic5 assumes a formal role, functioning as a structural divider between distinct phrase units. In this way, an interval inherent to the primary pc set is endowed with a functional status. The structural role of ic5 is operative in other sections of the work. For example, at R13 (Figure 4.5) each pitch of the melodic line is stated by a different instrument. Throughout this passage, two distinct motives alternate. The first is characterized by relatively shorter rhythmic value, louder dynamic level, and the inclusion of a glissando “tag” attached to each note.4 The second consists of a group of harmonics and is characterized by the longer rhythmic value, softer dynamic level, and the lack of a glissando “tag.” If the glissandi pitches are compared, it can be seen that they consist of a series of several intervals; however, ic5 never appears between any consecutive pitches. The harmonic pitches, on the other hand, make up the same accompanimental pattern from the opening, transposed to begin on C4: . At R14 the melodic material embodied by the glissandi pitches is structurally articulated into two distinct statements by interpolations of accompanimental material. Another example can be found in section D (Figure 4.6). Throughout this section the primary melodic material for the first violin is based solely on the previously defined set-class 727. The remaining voices imitate various short motives following the initial statement by the primary melodic voice. Also, each voice imitates the pitches in a unique rhythmic pattern. In a sense, the first-violin line functions as a dux voice of a quasi-canonic texture; however, the comes voices do not imitate the line in toto, but simply focus on short distinct motives while

4

In the process of analyzing this passage, I do not consider the pitches to which the main notes “gliss to.” In performance, since the tempo is relatively quick in this passage, the glissando itself seems to be the primary effect and the termination of the glissando is not actually perceived. Therefore, I will focus on the organization of the pitches that are readily apparent.

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altering the rhythms. Each time an especially long duration appears in the primary melodic voice, a series of ic5s is asserted in one or more voices. The ic5 statements function as structural separators, segmenting the melodic material into phrases. In Figure 4.6, each of the comes voices imitate the first-violin motive in different rhythmic sequences. In fact, the second violin omits the first pitch of the motive altogether, focusing instead on the final two pitches. Three measures after R56, the viola and cello perform the first ic5 in the D section. Because it occurs just prior to the presentation of new motivic material in the first-violin line (m. 4 of R56), this ic5 series functions as a structural divider between the first-violin phrases. The same process of organization continues to structure the entire D section, recreating the contrast between melodic material and the ic5 accompanimental material that is characteristic of section A. Ic5 is not the only interval whose function is to establish structural and formal boundaries; ic1 is also imbued in some sections of the work with a similar function. In the opening section, for instance, the viola joins the first violin’s melodic line just prior to R1 (see Figure 4.3). Along with the first-violin statement of an Eb harmonic, the viola asserts a harmonic E, emphasizing ic1. In fact, throughout R1–R9 every pitch in the viola line sounds a semitone from the first violin’s simultaneously sounding pitch. Because ic1 is so saliently exposed in the melodic material, a special functional contrast is established with the accompanimental material. In other words, because the ic5 accompanimental material was shown to be functionally distinct from the melodic material and ic1 was shown to have special status among the melodic intervals based on its particular salience, it follows that the ic5-ic1 antagonism has the potential to be

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exploited as a means of formal structure. Thus, just as ic5 is utilized as a structural divider, the specific contrast between ic5 and ic1 also establishes formal contrast. The most striking example, found at R10–R11, is presented in Figure 4.7. In section A the ic5 material can be found in all instruments simultaneously only twice, and, because of the status of ic5, each time it is a structurally strong formal marker. At R10 all four instruments utilize the ic5 material in various rhythmic durations for approximately three seconds before each, in turn, is silenced. Immediately following, at R11, is a section in which each voice is saturated with ic1 material. The violins consist of two repeated patterns containing the pitches [G3, Ab3, A3] while the viola and cello focus on [F#3, G3, Ab3]. In essence, this section is quoting the first-violin/viola simultaneities from the preceding section directly in the viola while using the same material in a linear fashion in the violins. In this way the salient focus on ic1 in the melodic material of R1–R9 is maximized here in both the vertical and horizontal dimensions, saturating the present section with varied ic1 presentations. The dynamic level and distinct manner of pitch articulation support a melodic interpretation of the viola line. The remaining three lines are softer and redundant, forming a static pitch structure.5 Overall, the sole use of ic5 in R10 followed immediately and suddenly by singular employment of ic1 in R11 creates a tension that is conveniently exploited for its structural and formal value. The same idea can be found in R15–R17, displayed in Figure 4.8. R15 corresponds to R10 as each individual voice consists entirely of repeated ic5 chains. The following material is

5

This terminology points to the fact that the music is perpetually moving with no goal orientation. In other words, the material in this section of the quartet has a distinct sense of rhythmic and melodic motion, but the syntax required for some goal is absent, creating a mere perpetual motion machine.

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an interpolated echo of the melodic material presented in R12; however, the primary structural contrast appears in R17, which functions as the formal analogue of R11. The upper three voices contain a persistent repetition (occurring for about eighteen seconds) of the pitch collection [A#3, B3, C4, C#4]. The cello remains tacit except for a single statement of the Source Chord [G, A, B, C, D, Eb, E], again echoing the melodic material from the previous sections (especially R14). Beginning at R18 the contrapuntal lines completely avoid ic5 between any two consecutive linear pitches. Because this has been the main criteria for locating hierarchical difference between simultaneously sounding voices, the lack of any ic5 creates a stronger sense of true melodic and contrapuntal equality among the individual lines. Thus, not only does the texture immediately shift at R18, becoming more overtly contrapuntal, it also becomes less clear which voice is carrying the principal melody. Based on this sudden change in texture and melodic emphasis, the contrasting ic1 material at R17 functions as a large-scale formal articulation between the larger A and B sections. Another example occurs at R50 (Figure 4.9), which follows the section in which the outer voices complete the wedge motion through the six chords (see Figure 4.2) and eventually come to rest on a cluster of ic1s: [A, Bb, B, C, C#, D, Eb]. Immediately following this, as the ic1 cluster is softly sustained, each instrument performs a pizzicato version of the accompanimental material (ic5). The subordinate placement of ic5 is unmistakable since the ic1 drone continues to sustain past the pizzicato texture. Thus, not only are the two separated as melody and accompanimental material, they also articulate a formal boundary between sections C and D. One final example should be noted, as it demonstrates the manner in which ic1 is capable of functioning alone as a structural entity without relying on the ic5 contrast. At various points

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throughout the B section, the contrapuntal material focuses on three pitches (different at each distinct point), always arranged as two ic1s, thus giving ic1 the function of a structural marker. For instance, the first occurrence of this phenomenon is at R23 (Figure 4.10). Prior to this point the individual voices have focused on their individual and distinct pitch collections linearly distributed with wide ranges and various intervals. At R23 all four lines coalesce into a common pitch collection—[G#3, A3, Bb3]. While the upper three voices assert these pitches in various rhythmic patterns, the cello echos the viola strategy from R11 (Figure 4.7). Just as the viola followed a simultaneous ic1 assertion with a glissando to a pitch a semitone away, the cello at R23 states a pizzicato Bb-A simultaneity and slides to G#. Following this short period of repose, the previous contrapuntal activity resumes. Similar events occur several times throughout the B section, always with a different pitch collection.

Motivic Development and Transformations The previous discussion focuses on one manner in which intervals are given a structural function. A second method of using intervals consists of endowing them with a developmental function. Typically, a motive consisting of pitches from the primary set is expanded and developed by an interval inherent to the set itself. Essentially, this can be achieved in two ways: the interval or intervals in question can simply repeat at a local melodic level, or the motive can be restated in toto at another transpositional level based on the functional interval. It can be argued that, since every motive is developed or transposed by an inherent interval, an all-interval set will, by default, always contain this structural property. The nature of my discussion, however, will demonstrate how the first manner of applying the process

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systematically avoids utilizing ic5 in expanding and developing motives. The second manner in which this process occurs, on the other hand, applies only ic5 in transpositional transformations of motivic material. I begin the discussion with the first method. Figure 4.11 presents R52, in which the cello utilizes each pitch from the first chord of R49 (i.e., the primary set), even repeating the Eb that is doubled in the verticality. However, the melody continues once the pitch source has been exhausted by repeating the same intervallic series used for the opening pitches of the melody, illustrated in Figure 4.11. Beginning with G3, the directed interval series is <–10, +6, +9, +11, –9> and immediately following with the pitch D3 is <+2, –6, +9, +11, –9>. Although the first interval <+4> is bypassed in the repetition, the remainder of the series is intact.6 It should come as no surprise that ic5 is avoided completely while every remaining interval class is present. Although the order has been altered, the first-violin melody in the following section (Figure 4.12) retains the exact pitch classes from the cello melody and, by direct relation, the primary set. It is apparent, however, that the interval series has been altered in order to emphasize ic1. The remaining voices imitate the main line’s pitches with varied rhythmic patterns until the viola initiates a structural division with an ic5 chain just prior to R56. Beginning at R56 the first-violin melody utilizes the pitch-classes from the second chord of R49 (related to the first by I5), again repeating the doubled D from the verticalization. Nevertheless, the ic5 structural division in the cello and viola parts occurs before the pitch-class source is exhausted (m. 3), thus establishing a new melodic phrase beginning with a descending [6]: (D6–Ab5). The imitative counterpoint resumes until the ic5 structural division is stated once 6

Because of octave equivalence in pitch-class space, any directed interval and its inversion in the opposite direction are considered identical moves. For instance, +10 is commensurate with –2.

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again at R58.7 Although ic6 has been uninvolved in the structural articulation of phrases, the placement of the structurally-dividing ic5 in m. 3 places enough emphasis on ic6 to endow it with structural importance, further strengthened by its repetition in the following passages. Thus, ic6 is specifically utilized as the initial interval of each individual first-violin melodic phrase throughout section D. In m. 2 of R58 (Figure 4.13), another statement of this first-violin melody occurs, beginning with descending [6]: (C6–Gb5). However, the statement is not an exact repetition; rather, the material is expanded by the interpolation of a chain of ic4-ic1 units. Important here is the formation of a motive from the only two intervals uninvolved in the intervallic pattern repetition in the cello line. In other words, the initial ic4 and the final ic1 are estranged from the <–10, +6, +9, +11, –9> pattern, clearly marked in Figure 4.11. Combined into a single <+4, –1> melodic fragment, these two intervals become the primary means of thematic expansion and development in the following statements of the primary set throughout the entire D section. Beginning with another descending [6]: (C6–Gb5) in mm. 2–3 of R66, presented in Figure 4.14, another first-violin melodic segment is initiated, extending only as far as R67 before a statement of five consecutive <+4, –1> melodic fragments develops. This expansion forms the dramatic climax of the entire section by carrying the melody to an extreme register. The climax is further supported by the quicker rhythmic motion and a focus on ic1: [F#5, G5, Ab5, A5] in the accompanimental viola and cello voices. In R70–R73 (Figure 4.15) the melodic material is

7

Though tedious, I have consciously chosen to walk through the previous passage chronologically in order to demonstrate and justify my division of the melody into motivic segments based on the structural divisions of ic5 material. These segments are based entirely on intervals drawn from the primary set and are varied in the remainder of the sections.

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successfully brought to a close with an exact restatement of the first six pitch classes from the cello’s opening melody at R52 (circled in Figure 4.15). The second manner of structurally utilizing intervals contained within the primary pc-set class 7-27 is more abstract and operates on a deeper level. Rather than using a motive or ic chain per se, the operations transform the set as a whole entity. However, it will be shown that these transpositional relations are strong enough to establish a structural and formal function. The first example occurs at R14 (Figure 4.5). As was previously pointed out, this section consists of two statements of the primary set separated by the ic5 structural divider. Focusing on the primary-set statements, it is apparent that the first set is [F#, G#, Bb, B, C#, D, Eb] while the second set contains the pitch-classes [Db, Eb, F, F#, G#, A, Bb]. (The first set appears to have an asserted G as its final pitch, but this can be seen as an anticipation of the first pitch in the immediately following ic5 structural divider, which is also G.) A comparison of these sets reveals that the second is a transposition of the first down by five semitones (in pitch-class space). Thus, although ic5 is utilized on the surface as a means of separating statements of the primary set, at a deeper structural level it is the interval used to relate them. Another example can be found at R38 (Figure 4.16), in which ic5 is shown to relate the motivic material. Three different transpositions of set class 3-1 (012) are present in this passage: [C#, D, Eb], [G#, A, Bb], and [D#, E, F].8 When compared, an interesting relationship is revealed. The [G#, A, Bb] collection is first presented at R37 and remains in effect for five measures before the remaining two collections are presented. The [C#, D, Eb] collection is

8

The primary set 7-27 is absent in this example because this passage occurs in section B. Since this section is characterized by its structural dependence on ic1, it is only logical that the primary set 7-27 would be subsumed by a pitch collection consisting of stacked [1]s.

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related to the first by T5, or “transposition up by ic5,” while the [D#, E, F] collection is related to the first by T–5, or “transposition down by ic5.” As the surface motives are emphasizing ic1, the underlying relationship between the motives is controlled by ic5. The final example occurs in the cello motive at R52 (Figure 4.11). In the discussion above it was shown how the cello pitches are organized by means of a repeated interval series, continuing to add pitches after those of the primary set (i.e., the first sonority at R49) are depleted. By basing the melody on the same interval series, intervallic definition dictates that the same primary set will be formed again (at some transpositional level). Thus, the first eight pitch classes of the Source Chord, [G, A, B, C, D, Eb, E], are directly related to the final eight pitch classes, [D, E, F#, G, A, Bb, B]. When compared, it is apparent that the second collection is related to the first by T–5. The connection is further strengthened by the fact that the only repeated pitch class in the first eight cello pitches (Eb) directly maps onto the only repeated pitch class in the final eight pitches (Bb). Once again, even though ic5 is conspicuously absent from the melodic material itself, it is structurally present, governing the overall relationship between the pitch collections.

Conclusion In sum, the pitch organization in the third quartet is structured by means of assigning particular functions to two specific interval classes found in the primary set 7-27. The distinct roles of ic5 and ic1 allow the pitches to be hierarchically separated into melodic and accompanimental levels. The same functional assignments form a contrast between ic5 and the various remaining interval classes (especially ic1), endowing ic5 with the potential to articulate

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formal boundaries. Repetition of the remaining intervals of the set as a series (excepting ic5) is also utilized as a means by which motives are developed and expanded. Finally, ic5, although avoided melodically on the surface level, is the primary means by which distinct statements of the primary set 7-27 are structurally related.

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Chapter 5 Analysis of String Quartet No. 4 (1993): Verticalized Sound Masses

Composed considerably later than the other three, the fourth quartet best represents Gubaidulina’s mature style. In many ways it seems to exaggerate various aspects of the avantgarde experimental textures, making it distinct in comparison to the other three quartets. For instance, it is the only quartet utilizing three separate ensembles. Two of the ensembles are recorded onto a tape while the third performs live along with the recording. This work also illustrates the composer’s interest in experimenting with the natural timbre of the instruments. The performers are instructed to attach a ping-pong ball to the instruments with an elastic band. When the strings vibrate, the ball bounces on the sounding board creating a “clicking” noise. This effect is maintained throughout most of the work.1 The quartet can be divided into two large sections. The first is characterized by two distinct layers of activity that occur simultaneously. The two recorded ensembles are grouped together based on the canonic organization of their musical elements, especially rhythm, while the live ensemble is clearly differentiated by means of its idiosyncrasies in pitch, rhythm, and texture. However, as I will demonstrate in the following analysis, although the two layers of

1

Another interesting experiment concerns the use of colored lights in the performance of this work. Because I have chosen to limit my analytical focus to purely musical parameters, any discussion of this aspect and its consequences on the structural properties of the piece, though interesting, are beyond the scope of this study. Gubaidulina herself discusses the “musical” use of colors in Sofia Gubaidulina and Vera Lukomsky, “Sofia Gubaidulina: ‘My Desire Is Always to Rebel, to Swim against the Stream!’” Perspectives of New Music 36, no. 1 (1998): 28–31.

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activity organize and structure their relative elements in different ways, the musical material being organized is actually comparatively similar based on intervallic structure. The recorded ensembles discontinue beginning at R30. This second large section contains a series of gradually changing sonorities performed by the live ensemble. Although the most immediate distinction between these sections lies in the inclusion or lack of the recorded ensembles, the live ensemble itself is also fundamentally different within each section. The first section maintains an anxious character as the voices of the live ensemble each feature their own individual rhythmic structures. In other words, each individual voice in the ensemble carries its own rhythmic patterns, completely unconnected to the other three voices. The superimposed rhythms occurring throughout the first section contrast sharply with the lengthy durations of the individual sonorities in the second section. Because of the avant-garde or experimental quality of the various musical elements, the fourth quartet defies analytical explanation. Although they are constantly in motion with tremolos, the voices are fairly static in pitch space.2 Regardless, I will illustrate how these fairly static segments of pitches can often be “verticalized” into a collection, demonstrating how they relate to one another in pitch space.3

2

Compare this to the third quartet, in which the voices sometimes maintain a superficial rhythmic motion but are actually confined to semitonal motion within two boundary pitches outlining an especially small interval. 3

Many of the analytical tools utilized in this chapter were first presented in recent studies of Ligeti and Varèse: Jonathan W. Bernard, “Pitch/Register in the Music of Edgard Varèse,” Music Theory Spectrum 3 (1981): 1–25; idem, “Inaudible Structures, Audible Music: Ligeti’s Problem, and His Solution,” Music Analysis 6, no. 3 (1987): 207–36; idem, The Music of Edgard Varèse (New Haven: Yale University Press, 1987); idem, “Voice Leading as a Spatial Function in the Music of Ligeti,” Music Analysis 13, no. 2–3 (1994): 227–53; Jane Piper Clendenning, “The Pattern-Meccanico Compositions of György Ligeti,” Perspectives of New Music 31, no. 1 (1993): 192–234; Michael Hicks, “Interval and Form in Ligeti’s Continuum and Coulée,” Perspectives of New Music 31, no. 1 (1993): 172–90; Miguel Roig-Francolí, “Harmonic and Formal Processes in Ligeti’s Net-Structure Compositions,” Music Theory Spectrum 17, no. 2 (1995): 242–67.

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For example, Figure 5.1 displays the opening pitches in the B-viola voice.4 Since this line is repeated in toto numerous times in the ensuing measures, these four measures conveniently group into a whole with the pitches organized into a four-note ostinato. Because the line relentlessly continues to return to its initial pitch (via repetition), the line represents a static sound mass comprised of restless energy but lacking actual motion in pitch space. The sound mass, therefore, can be conveniently verticalized into a single pitch collection [F4, G4, A4, Bb4] and related to other similarly formulated sound masses. In other words, a short melody can be treated as an unordered pitch collection and conceived of as a vertical sonority.5 In this chapter, I will illustrate the various manners in which “static” melodies, verticalized into sound masses, can be related to one another based on their intervallic structures.

The Primary Pitch Collections The two distinct layers in this work, although differentiated by means of their idiosyncratic organizational and structural design, are both based on a characteristic collection of intervals. The verticalized sound masses contained within each layer of musical activity establish

4

Because the score contains two recorded quartets and one live quartet, I label the members of each quartet differently. The two recorded ensembles are labeled A and B, and the live ensemble is conventionally labeled (i.e., without a letter). The individual instrument labels include the ensemble letter (if needed) followed by a hyphen and the conventional instrument name. For example, “B-viola” is the viola line in the second recorded ensemble and “first violin” refers to the first violin in the live ensemble. 5

Bernard describes this procedure as “melody interpreted as harmony, . . . in classic Ligetian fashion.” (“Ligeti’s Restoration of Interval,” 10) Interestingly, Gubaidulina herself seems to compose in a similar manner: “In the first phase the sound comes to me, often in a form I have trouble breaking up into its component parts. It seems to be built up vertically, so I begin to guess what this sound could eventually mean. . . . I rearrange the sound from a vertical to a horizontal shape, so to speak, and with that the form comes into being.” (“Sofia Gubaidulina: Into the Labyrinth of the Soul,” in The Voice of Music: Conversations with Composers of Our Time, ed. Anders Beyer and Jean Christensen (Aldershot: Ashgate, 2000), 50. Italics mine.)

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a primary intervallic collection and utilize these same organized intervals in registral space in order to define specific sonorities. This will become clear in the following discussion. Each member of the recorded ensembles contains a four-measure ostinato rhythmic pattern, presented in Figure 5.1. Furthermore, the voices in each individual ensemble each consist of the exact same pitches as their counterpart in the alternate ensemble. In other words, the A-viola states the same < Bb4, A4, G4, F4, G4, Bb4> segment as its counterpart, B-viola (Figure 5.1). However, even though the rhythmic and pitch patterns remain consistent between voice pairs, the metric placement is varied for each individual instrument. The B-viola voice begins on beat one while the entrance of the A-viola occurs on beat four, creating a canonic structure. Since the rhythmic durations are consistent for all eight members of the two ensembles, the canon is not limited to pairs of voices. The two recorded ensembles are organized together into a single level of rhythmic canonic activity while each instrument, along with its counterpart, asserts its own particular ordered pitch collection. Thus, the canon is completely based on rhythmic patterns instead of pitch patterns. Figure 5.2 presents an abstract reduction of the various pitch collections in order of their presentation in the music.6 Because of the consistent preservation of exact pitches, instrument pairs are displayed together as a single collection. The numbers above the segments roughly correspond to the placement in the score; however, since the canonic structure of the individual lines rhythmically displaces the related pair, the individual “bars” in the figure should not be read as directly coinciding with the metric structure of the actual music. The bar-lines instead delineate four-measure segments (rhythmically normalized in this abstract example). 6

In the abstract reduction, the pitches contained in each individual collection are presented in ascending order, regardless of their order in the music.

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The A-/B-viola and A-/B-second violin maintain the exact same pitches throughout section one, creating an actual ostinato. The A-/B-first violin and A-/B-cello, while consistently organized with the same rhythmic pattern, contain different pitches in each successive four-bar pattern. Together, these two instrument pairs create a wedge-like pattern over the course of the first section, the first-violin pair moving in contrary motion to the cello pair. When compared, the first four verticalized sound masses delineated in the abstract reduction display an interesting consistency. The individual intervallic designs are also notated beneath the pitch collections in Figure 5.2.7 The intervallic structure in registral space defining the initial pitch collection of all four instrument pairs is exactly the same, [2 2 1], further supporting the posited notion of canonic structure in this layer. The pc set defined by this intervallic structure, 4-11 (0135), will henceforth be referred to as the Ostinato Collection. Beyond the first four collections, the Ostinato Collection is never again presented in the recorded ensembles; however, the importance of this collection will become apparent in the ensuing discussion. Figure 5.3 displays the live ensemble at R5. The initial quintuplet in the cello line contains the ordered pitches with an intervallic structure [1 1 3]. Each successive statement transposes this quintuplet by ic1. The exact same sequence of events occurs at R8 followed by yet another sequence at R9. Its sheer repetition endows this quintuplet with motivic and structural significance. Henceforth, I will refer to this pc set, 4-4 (0125), as the Live Collection.

7

In order to avoid confusion, I have basically adopted Roig-Francolí’s shorthand model for displaying intervallic subdivisions. (“Harmonic and Formal Processes,” 246, n. 13) However, I will include one difference: rather than omitting the brackets altogether, interval chains defined by a single pitch collection will be contained within a single bracket. For instance, [3] can be subdivided as [1 2], [2 1], or [1 1 1].

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It should be clear from the preceding discussion how the Ostinato Collection and the Live Collection are each firmly established as structural entities within their relative layer of activity. Both are presented relatively early in the ensemble with which they are associated, asserting an apparent independence in relation to the other. However, the remainder of this chapter will reveal this apparent independence to be superficial by demonstrating how the interaction between these two collections is a primary means of achieving structural coherence and relating different sound masses.

Relating Individual Sound Masses Adjacent melodic segments are directly related via the intervals inherent to each; however, these intervals are not always evident until the melodic segments are verticalized into sound masses. Different transformations reveal how one sound mass can morph into another, illustrating the interaction between adjacent sound masses that cannot be related via traditional operations of transposition and inversion. In other words, the intervals inherent to a collection of pitches can be transformed by means of specific operations in order to bind the group to an adjacent collection, establishing a continuity between sets that are normally distinct and dissociated. A melody is presented by the first violin beginning in m. 3 of R26. Although seemingly continuous, the line conveniently divides into distinct segments, punctuated by a consistent rhythmic “cadence.” Without exception, each individual segment of this melody terminates on a pitch with a rhythmic duration of three eighth-notes. This duration is reserved solely for the

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function of cadential articulation, making it particularly salient when it occurs.8 Further supporting its functional role, the terminating pitch is always followed by a [2] descent. In other words, each individual segment of this melody begins [2] below the terminating pitch of the previous segment. Figure 5.4 graphically displays these segments as verticalized sound masses. The vertical axis maps the individual pitches in pitch space, each unit representing the interval of a semitone. The horizontal axis plots time points; in this case, however, the horizontal units are not equal divisions of time (e.g., seconds or measures). The points on the horizontal axis (labeled with letters in Figure 5.4) are relative time points simply mapping the individual verticalized sound masses based on the order in which they occur in the music. Thus, point a and point b are unequal in their temporal durations, but they occur in succession. In relating the different verticalized sound masses, I have relied heavily on Jonathan Bernard’s transformational operations utilized in his studies of the music of Ligeti and Varèse. The primary difference between these operations and those conventionally employed transformations is the preoccupation of the former with pitch space as opposed to pitch-class space. These operations are designed to exhibit the specific manner in which pitch collections mutate in pitch space. I will focus on three: projection, GLIDE, and FLIP. “Projection” is defined by Bernard as a “transference of structure to a new pitch/registral level.”9 Essentially, the operation simply transposes some intervallic structure up or down, but

8

As is discussed in chapter one, the possibility of utilizing rhythm as a means of formal articulation is entirely in line with the composer’s expressed views: “I am preoccupied with the issues of rhythm and rhythmic proportionality in musical form, which I consider the main experiment in my life.” (Vera Lukomsky, “‘Hearing the Subconscious’: Interview with Sofia Gubaidulina,” Tempo 209 (1999), 27.) 9

Bernard, “Pitch/Register in Varèse,” 9.

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the term projection is preferred over “transposition” in this context because of its visual/spatial reference to projecting something onto various planes (comparable to pitch levels here). The property of “partial projection” is also possible. Basically, although some of the structure is transferred, something is also changed during the projection. This operation is less straightforward than full projection, but it does require at least an apparent resemblance between the operand and the result.10 The remaining two operations allow for transformations between sets with different cardinalities. The first, GLIDE, occurs when “two or more pitches, preserving their pitch-interval structure, are transposed to a new location without canceling their presence at the original location. This operation leaves a ‘trace’ and is therefore irreversible.”11 Essentially, GLIDE functions as a projection without moving the original pitches. For instance, if GLIDE (–1) is performed on at point m, the result would be . The second operation, FLIP, occurs when a “pitch moves about a pivot to a new position, giving up its old one and ending up the same distance, measured by unordered pitch interval, above the pivot as it was originally below, or vice versa.”12 In other words, if FLIP (D4) is performed on Eb4, then Eb4 would “flip” below D4 to C#4 (i.e., [1] above becomes [1] below).

10

Bernard’s concept of partial projection is similar to Straus’s “fuzzy” transposition. See Joseph N. Straus, “Voice-Leading in Atonal Music,” in Music Theory in Concept and Practice, ed. James Baker (Rochester: University of Rochester Press, 1997), 237–74 and “Uniformity, Balance and Smoothness in Atonal Voice-Leading,” Music Theory Spectrum 25, no. 2 (2003): 305–52. 11

Bernard, “Ligeti’s Restoration of Interval,” 15 (original italics). Bernard compares GLIDE to Richard Cohn’s transpositional combination “with the obvious qualification the glide functions strictly according to the exigencies of pitch space.” (15, n. 26) 12

Ibid. Bernard notes that this operation essentially combines his unfolding and infolding and even has an etymological similarity to David Lewin’s versions: FLIPEND and FLIPSTART. See Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987), 189.

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Finally, it is important to point out that, contrary to conventional transformational operations, none of these operations is required to affect the entire set. They are free to transform any portion of the set so long as the operation is fulfilled completely. The relation between pitch collections is also not limited to a single operation; rather, it can be achieved by two or more operations occurring in sequence. Returning to Figure 5.4, following the [1] dyad (point a), a collection of five pitches with an intervallic structure [1 2 1 1] is stated (point b). This grouping can be interpreted as an interlocked Ostinato Collection [E6, F6, G6, A6] = [1 2 2] and Live Collection [E6, G6, G#6, A6] = [3 1 1]. Because I defined these collections in terms of pc sets, inversions of the original statements are plausible and likely. However, since intervallic structures depend on order (listing bottom to top), different intervallic dispositions can not be considered equivalent. Therefore, inversion of pc sets will necessarily entail a reordering (retrograde) of the subdivision labels in the intervallic structure. In the present discussion, the Ostinato Collection [2 2 1] becomes [1 2 2]; likewise, the Live Collection [1 1 3] become [3 1 1]. Because the intervallic structure is reordered when the collection is inverted, Bernard utilizes the term “rotation” in lieu of “inversion” in order to emphasize the intervallic relations in pitch space (as opposed to pitchclass space).13 As is graphically demonstrated in Figure 5.4, the same collection occurs at points c and d, each successively projected down by [1]. Point e exhibits an expanded collection of pitches, interpreted as two groups of interlocked collections with a common pitch (C#6). As can be seen from the graph, the upper group [C#6, D6, Eb6, F6, F#6] continues the succession of descending [1] projections (points

13

Bernard, “Pitch/Register in Varèse,” 10.

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b–d). The lower group [G#5, A5, B5, C6, C#6] is a rotation of the projected collection about C#6. This expanded collection of a set and its rotation also occurs at point g. Thus, point e is projected downward by [7] to point g. An interesting anomaly takes place at points i and j. A partial [7] projection of the same expanded collection occurs between points g and i, omitting the lowest pitch (F#4). A new collection is presented at point j with the intervallic structure [1 1 1 1 1]. Essentially, this set, if reinterpreted, contains the pitches of the interlocked-collection [E4, F4, G4, G#4, A4] with an added pitch (F#4). It is interesting that this is the exact pitch omitted from the collection at point i. In a sense, it is as though F#4 is tardy, sounding at an inappropriate moment. By performing the GLIDE and FLIP operations in succession, the pitch collection at point m can be directly related to the modified set at point n. First, GLIDE (–1) is performed on the partial collection [C#4, Eb4], resulting in the complete collection [B3, C4, C#4, D4, Eb4]. FLIP (D4) and FLIP (C4) are simultaneously performed on Eb4 and B3, respectively. This results in the sacrifice of B4 and Eb4, yielding the collection [C4, C#4, D4]. Thus, the set at point n can be directly related to those at point m. A similar process takes place at R14 (Figure 5.5), in which the first-violin line contains a lengthy melodic line accompanied by the remaining voices. The accompanimental texture of the second violin and viola supports the melody with arpeggiations of (037) trichords and singlepitch repetitions against the continuous glissandi in the cello voice. Because of the static nature of the repetitions in the accompaniment, the analysis of this passage will focus exclusively on the melodic sound masses.

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The first-violin line is segmented based on the exact criteria employed above. Basically, the terminating pitch with a rhythmic duration of three eighth-notes followed by a [2] descent exclusively articulates the individual segments. Points a and b contain the interlocked Ostinato Collection and Live Collection with an intervallic structure [1 2 1 1]. The collection at point c is attained via FLIP (D#6) and FLIP (A#5) respectively applied to D6 and B5 followed by a GLIDE (–1) performed on [C#6, E6] (shown in Figure 5.5). Similarly, FLIP (C#6) and FLIP (A5) performed on C6 and A#5 result in the pitch collection at point d. The collection at point e results when FLIP (D6) is applied to Eb6 along with the [3] descending projection of [G#5, A5]. Figure 5.5 graphically demonstrates how the remaining pitch collections are derived. The final example will focus on the second large formal section of the quartet. As stated above, the recorded ensembles are absent from this section, and the musical material composed for the live ensemble is characterized by long durations and gradually transforming sonorities. The voices are structurally divided into two duets, each consisting of its own organizational procedures. In a sense this echoes the two layers of activity embodied in the different ensembles in the first section of the piece. Lacking an alternate ensemble, the live ensemble is divided into its own two activity layers. The cello and viola are structurally related not only by registral similitude but also on the basis of their intervallic interaction. Both voices utilize [3] and [6] as organizational properties within their individual lines while simultaneously employing these intervals as structural entities between their lines. The upper-voice duet is more overtly linked, maintaining a strict canon at the imitative distance of a quarter-tone. Although this canon does not preserve a rigorous rhythmic pattern, the deviations are slight.

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Rather than pointing out the transformations connecting the individual sound masses, the analysis of this passage will focus instead on the intervals that have structural significance within and between sound masses. Specifically, [3] and its multiples are essential to the organization of this passage. The sound masses in R34–R41 (Figure 5.6) correspond to each individual sonority, regardless of duration. The individual time points, therefore, are not temporally consistent; rather, as in the previous graphs, they are relative and simply ordered according to their appearance in the music. The initial four-measure cello solo is verticalized at point a, revealing the same ubiquitous interlocked Ostinato/Live Collection. The cello expansion to an upper boundary of E4 establishes [6] above its lower boundary (Bb3). The viola Bb4 over the cello E4 asserts another structural [6], perfectly bisecting the opened pitch space [12]: (Bb3–Bb4). The [6]-projection of [E4, Bb4] onto [C#4, G4] further divides the two already bisected [6] spaces into four [3] spaces (point b). The structural [6]: (Gb4–C5) also occurs at point c, opening a new pitch space that is immediately bisected by the viola A4. This point also marks the establishment of a new boundary pitch (Gb4) below which the cello voice does not descend throughout the remainder of the section. Finally, the only other structural [6] between the viola and cello occurs after R39 at point d (C5–F#5), relating point c and d. This corresponds to the original [6] established by the viola’s initial pitch, Bb4. In other words, just as the viola and cello bisect the [12]: (Bb3–Bb4), point c and d bisect the [12]: (Gb4–F#5) with [6]. Along with its role in further bisecting [6], [3] also functions as a structural interval. At point c, the viola contains two interlocked 3-1 (012) melodic trichords. [Bb4, B4, C5] corresponds to [G4, Ab4, A4], although the statements overlap. This correlation is demonstrated

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with an arrow in Figure 5.6. Throughout this section, the viola and cello lines essentially traverse two levels of pitch space primarily separated by [3]. Figure 5.7 illustrates this characteristic in a reduction. The interval [9] also bears structural weight in this passage. The first literal presentation of [9] occurs at the entrance of the upper duet (point e), between viola and second-violin pitches (B4–G#5). This successfully establishes [9] as a structural boundary between the upper and lower duets. [9] also delineates the space within which the upper voices themselves unfold. The lowest stated pitch in the upper duet is F5 while the uppermost boundary is the final first-violin pitch, D6. Thus, the same interval initially established as a structural boundary between the two duets gradually delineates and structures the space in which the upper duet itself unfolds.

Conclusion In sum, the fourth quartet resists analytical explanation when recourse is made to tools similar to those exemplified in the previous chapters. Instead, the quartet requires evaluation from the perspective of verticalized sound masses. By reducing the seemingly unrelated pitches contained in melodic segments into a single sound mass, specific relations become apparent. Intervallic structures assist in locating similarities between successive sound masses. Transformations, such as GLIDE and FLIP, reveal how one sound mass can morph into a following one, assuring the potential for continuity between melodic segments. The passages discussed in this chapter illustrate how the linear organization of certain voices is based on one of the two primary intervallic collections, which is apparent only when the line is verticalizated into a sound mass. Finally, the specific intervals inherent to these individual verticalized sound

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masses establish unique connections and ties to adjacent pitch collections that are otherwise incompatible.

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Epilogue Concluding Remarks

With this study, my primary goal has been to illustrate through analysis the cohesive nature of Gubaidulina’s four string quartets. Much of her music has been accused of discontinuity and fragmentation; however, in the analyses above, I have demonstrated that these works maintain structural continuity at the deepest levels of organization. I do not submit the previous chapters to the reader as complete analyses of these works. The purpose has been to demonstrate possible ways of accounting for the organization of those works that thus far have defied analytical explanation. In this way, it is hoped that a better understanding and, by implication, appreciation of the compositional craft is achieved. The first quartet was shown to utilize the process of expansion. Pitches are organized in pitch space through wedge expansion or additive expansion. The second illustrates the processes of gap-fill and inversional symmetry. The third utilizes ic5 and ic1, two intervals inherent to the primary pitch collection, as structural and organizational markers. The intervals are functional in the ways in which they structure various musical elements. Finally, the fourth quartet demonstrates how transformations in pitch space can specifically relate adjacent sound masses to one another and reveal intervallic connections within the different sections. Despite strong contrasts in musical language and style, the works discussed in this study are only superficially different in structure. The distinct processes presented in these analyses reveal a deep similarity among the individual works. For instance, in all four quartets the melodic and harmonic materials are invariably organized in such a way as to defy octave 81

equivalence as a guiding principle. In other words, the musical elements specifically take place in pitch space rather than pitch-class space. Each of the processes presented above illustrate the importance of pitch space in each of these works. Also, because every pitch is an individual entity in a spatial arrangement, the organization of pitches in pitch space places an intense focus on intervallic design. The processes utilized as structural forces in these four quartets rely considerably on interval-distance between pitches in pitch space, even endowing them with organizational function. The structural and formal properties of these works betray a fairly consistent compositional style. Many similar features can be found in works of Varèse, Ligeti, and Crawford Seeger. These composers have resisted placement in any particular school or sustained alignment with any particular method of composition. Thus, their stylistic similarities can be seen as striking in the context of stylistic trends in twentieth-century music. The styles of Varèse and Ligeti are comparable in their individual and idiosyncratic approaches to the organizing of pitches in pitch space and the emphasis on intervallic design, whereas Crawford Seeger’s style is similar to Gubaidulina’s in its intense chromaticism and use of transformational chains in order to shift inversional axes about which the pitches are arranged. Finally, it is important to understand that each of these processes is not limited to the one quartet that exemplified it in this study. Each of these processes can be traced in all four quartets; however, I specifically chose to present one process per quartet with the hope of illustrating the extent to which each of these analytical tools is capable of explicating the musical structure. This does not imply, for instance, that wedge expansion can only be found in the first quartet.

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In conclusion, I hope this study has stirred interest in the musical works of a wellrespected composer. I close with a quote by Hakobian: Gubaydulina [sic] is an absolutely unique, unclassifiable, indefinable phenomenon. It seems that one might—mutatis mutandis—apply to Gubaydulina the words once said about Iosif Brodsky [by Dovlatov], one of rare figures of more or less the same stature on the cultural scene of contemporary Russia: “The notion of soul was the central, decisive one in both his literary and everyday existence. He perceived the humdrum life of our State as the dying of a body abandoned by the soul. Or as the apathy of a sleepy world where only the poetry is awake . . . Beside Brodsky, the rest of . . . his nonconformist friends seemed to have an another métier.”1

1

Levon Hakobian, Music of the Soviet Age: 1917–1987 (Stockholm, Sweden: Melos Music Literature, 1998), 294.

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Musical Figures

Figure 1.1 Diagram of Avant-Garde dichotomy

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Figure 2.1 Abstract example of additive expansion

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Figure 2.2 Viola (mm. 1-2)

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Figure 2.3 Violin II (R1, mm. 1-4)

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Figure 2.4 Wedge expansion from F#6 (R8, mm. 3-10)

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Figure 2.5 Wedge expansion from C#4 in violin I, violin II, and cello (R6, mm. 4-7)

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Figure 2.6 Wedge expansion from E5 in violin II (R7, mm. 2-9)

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Figure 2.7 Wedge expansion from D4 (R21, mm. 1-15)

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Figure 2.7 (cont.)

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Figure2.8a Violin II and cello (R25, mm. 1-11)

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Figure 2.8a (cont.)

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Figure 2.8a (cont.)

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Figure 2.8b Reduction of cello line (R25, mm. 1-11)

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Figure 2.9 Viola (R1, mm. 1-2)

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Figure 2.10 Violin II, viola, and cello (R4, mm. 1-4)

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Figure 2.11 Cello (R1, mm. 1-4)

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Figure 2.12 Additive expansion in rhythm and pitch (R48, mm. 4-15)

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Figure 2.12 (cont.)

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Figure 2.13 Wedge expansion and additive expansion in violin I and cello (R42-43)

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Figure 3.1 Emphasis on G4/G5 dyad in m. 1

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Figure 3.2a Violin I/cello motive (R5, m. 1)

Figure 3.2b Gap-fill process in violin I (R5-15)

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Figure 3.3 New violin I/cello motive (R15)

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Figure 3.4 Moment of gap-fill completion in violin I motive (R20)

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Figure 3.5 R22-25

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Figure 3.6 R21-22

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Figure 3.7 Reduction of R21-22 with inversional symmetry in voice leading

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Axes:

D-------------------A

D------E

C–F# G–B

A–Db D–Bb

F—E F#–G

F/F#

A

D/E

C/B

Bb/Db

Figure 3.8 Reduction of R25-27 with inversional symmetry in voice leading

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Section: Rehearsals: Texture:

A

B

C

D (+ Coda)

R1-18

R19-48

R49-51

R52-78

Melody and Accompaniment

Contrapuntal/ Free

Homophonic

Contrapuntal/ Imitative

Figure 4.1 Formal Diagram of Third String Quartet

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Figure 4.2 Primary set verticalized as sonorities (R49)

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Figure 4.3 Melody and accompaniment in opening measures

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Figure 4.4 Violin II accompaniment with second melodic segment (R1-2)

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Figure 4.5 Alternation of primary-set melodic material and ic5 structural divisions (R13-14)

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Figure 4.6 Ic5 structural divider (circled) in viola and cello (R55-56)

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Figure 4.7 Structural opposition between ic5 (R10) and ic1 (R11)

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Figure 4.8 Structural opposition between ic5 and ic1 (R15-17)

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Figure 4.8 (cont.)

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Figure 4.9 Simultaneous structural opposition between ic5 and ic1 (R50)

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Figure 4.10 Melodic/accompanimental emphasis on ic1 and [G#, A, Bb] (R23)

Figure 4.11 Linear version of primary set replicating pitch-classes from first sonority at R49 (R52)

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Figure 4.12 Violin I melody replicating cello pitches from R52 (R53-56)

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Figure 4.13 Motivic development using <+4, –1> (R58-60)

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Figure 4.14 Motivic development of violin I melody using <+4, –1> (R66-68)

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Figure 4.15 Replication of primary-set cello pitch-classes from R52 (R71-73)

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Figure 4.16 Interaction between ic5 and ic1 (R37-38)

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Figure 5.1 B-viola four-measure ostinato pattern

Figure 5.2 Reduction of recorded ensemble pitches into verticalized sound masses

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Figure 5.3 First occurrence of “Live Collection” in each voice of live ensemble

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Figure 5.4 Pitch-space graph of violin I (R26) 133

Figure 5.5 Pitch-space graph of violin I (R14) 134

Figure 5.6 Pitch-space graph of live ensemble (R34-41) [N.B. + indicates half-sharp and * indicates sharp-and-a-half]

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Figure 5.6 (cont.)

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Figure 5.7 Reduction of lower duet illustrating ascending thirds (R36-41)

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