BOULEZ'S SONATINE AND THE GENESIS OF HIS TWELVE-TONE PRACTICE DISSERTATION Presented to the Graduate Council of the University of North Texas in Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY By Sangtae Chang, B.M., M.M. Denton, Texas May, 1998 37? /V8U /#» V &
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BOULEZ'S SONATINE AND THE GENESIS OF
HIS TWELVE-TONE PRACTICE
DISSERTATION
Presented to the Graduate Council of the
University of North Texas in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Sangtae Chang, B.M., M.M.
Denton, Texas
May, 1998
37? /V8U
/#» V&
Chang, Sangtae, Boulez's Sonatine and the Genesis of His Twelve-Tone Practice.
Doctor of Philosophy (Musicology), May 1998,252 pp., 31 tables, 63 examples,
bibliography, 226 titles.
In a letter to John Cage (January 1950), Pierre Boulez proclaimed an end to his
'classical' period with the Livre pour quartuor (1948-49). Important biographical
events, personal correspondence, and published essays suggest that what Boulez
considered 'classical' frame his twelve-tone practice from 1945 to 1949, aiming to come
to terms with twelve-tone compositions by Schoenberg, Webern, and Berg. Despite such
a clear chronological frame, Boulez's twelve-tone practice appears paradoxical. While
modernist criticism overtly manifested itself against the predecessors and contemporaries
alike, a traditional organicist metaphor pervaded theoretical postulates that project the
conceptualization of musical structure.
This predicament of Boulez's twelve-tone practice becomes particularly
articulated in the Sonatine (1946/rev. 1949). The composer admitted that the Sonatine
systematically explored the twelve-tone row and rhythmic cells in an attempt to negate
his predecessors, while paradoxically modeling its structure upon Schoenberg's Chamber
Symphony Op. 9. This dissertation proposes that the Sonatine broadly unfolds a kinetic
structure that stems from the traditional tension-relief model and, consequently, its
dependence on tradition proves much deeper than Boulez would acknowledge. Chapter I
establishes the chronological frame of Boulez's twelve-tone practice and introduces
primary sources for twelve-tone compositions that predate the Sonatine, as well as those
for the Sonatine. Chapter II addresses an 'eclectic' approach to twelve-tone composition
in Douze notations. Chapters III, IV, and V address how twelve-tone exploration
determines the structural unfolding of the Sonatine. Finally, Chapter VI addresses
revisions of the Sonatine, taking into account the sketches, an early incomplete version of
which only the flute part survives, the final complete version, and the published score.
Examination of these primary sources indicates that revisions of the Sonatine enhance its
kinetic structure by amplifying subversion of row ordering and by deliberately expanding
motivic transformation throughout the composition.
BOULEZ'S SONATINE AND THE GENESIS OF
HIS TWELVE-TONE PRACTICE
DISSERTATION
Presented to the Graduate Council of the
University of North Texas in Partial
Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
By
Sangtae Chang, B.M., M.M.
Denton, Texas
May, 1998
37? /V8U
/#» V&
ACKNOWLEDGMENT
Research for the dissertation was supported in part by a grant from the Paul Sacher
Stiftung (Basel, Switzerland). I would particularly like to thank its resident scholar
Robert Piencikowski for his helpful comments and suggestions.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENT iii
Chapter
I. INTRODUCTION 1 Primary Sources
H. ECLECTICISM IN TWELVE-TONE PRACTICE: DOUZE NOTATIONS 24
Overview Exploring a Twelve-Tone Row (Group A) Exploring Row Properties (Group B) Exploring Aggregates (Group C)
III. PRINCIPLES OF KINETIC STRUCTURE: SONATINE . . . 83 Suggested Kinetic Structure in the Introduction Row Unfolding as a Foreground Determinant Aggregates as Foreground Determinants
IV. INTEGRATION OF ORDERED DURATION AND PITCH-CLASS SETS: TEMPO SCHERZANDO . . . . 1 3 1
Overall Structure Integrated Set in the Tempo Scherzando
V. RELAXATION OF KINETIC STRUCTURE . . , . 1 8 6 Twelve-Tone Usage Interplay between the Row and the Pitch-Class Set H
VI. CONCLUSIONS 225
BIBLIOGRAPHY 237
IV
CHAPTER I
INTRODUCTION
In a letter to John Cage, dated January of 1950, Pierre Boulez emphasized the
need for a new approach to composition while dismissing his early works as
insignificant:
Meeting you made me end a "classical" period with my quartet [Livre pour quatour], which is well beyond me now. Now we have to tackle real "delirium" in sound and experiment with sounds as Joyce does with words. Basically — as I am pleased to discover — I have explored nothing as yet and everything remains to be looked for in fields as varied as sound, rhythm; orchestra, voices; architecture. We have to achieve an "alchemy" in sound (see Rimbaud) to which all I have done so far is merely a prelude and which you have greatly clarified for me (Boulez 1993,45).
Although Boulez invoked here a historical perspective that implies periodization of his
early compositions, it remains unclear whether he actually suggested what may thread
together his early compositions, or merely categorized them with a pejorative term. By
taking into account important biographical events, personal correspondence, and
published essays, however, one may still construct a frame of reference for Boulez's
early compositional practice that responds to tenets of twelve-tone compositions by
Arnold Schoenberg, Anton Webern, and Alban Berg.
Boulez's interest in twelve-tone composition was stimulated by hearing for the
first time Schoenberg's Wind Quintet (Op. 26) directed by Rene Leibowitz in a concert at
2
the house of Claude Halphen in February of 1945.1 Immediately following the concert,
Boulez sought Leibowitz for informal instruction of twelve-tone composition.2 While his
association with Leibowitz continued well into 1946, Boulez completed at least five
twelve-tone compositions, including the Theme and Variations for the Left Hand, the
Quatuor for four Ondes Martenot, Douze notations, the Sonatine for flute and piano, and
the First Sonata for piano.3
Boulez's compositional output was momentarily interrupted by his trip to Brazil
to accompany the Compagnie Renaud-Barrault as music director from April to August of
1950. His urge to explore a new approach to composition, first expressed in his letter to
Cage (January 1950), took nearly a year to materialize. It was only at the end of 1950
that Boulez addressed specific compositional concerns that eventually led to the
development of a short-lived phenomenon called "total serialism." In a letter to Cage,
dated December 30,1950, Boulez suggested ways in which to organize rhythm,
transform a pitch-class organization by exploring quarter-tones, and theorize about an
'In the interview with Antoine Golea, Boulez generally dated the concert from 1945 (Golea 1958,27). In his biography of Boulez, however, Dominique Jameux more specifically dated the concert from February of 1945 (1991,15).
2For study purposes, Boulez appears to have copied out published scores of twelve-tone compositions that Leibowitz owned. Among these copies are Webern's Symphony (Op. 21), Concerto (Op. 24), String Quartet (Op. 28), First Cantata (Op. 29), and Variations for Orchestra (Op. 30). For a brief description, see the inventory of the Boulez manuscripts at the Paul Sacher Stiftung (Piencikowski 1988, 22-23).
3Boulez participated in a concert directed by Leibowitz at the Paris Conservatoire in December of 1945 (Golea 1958,27-28) and even dedicated the First Sonata for piano to Leibowitz. This dedication appears crossed out on a "scrap" paper used to cover the sketches for the first version of le Visage nuptial. I would like to thank Robert Piencikowski for clarifying this matter.
3
organization of the entire audible sound frequencies (Boulez 1993, 80-90).4
Indeed, the first encounter with a twelve-tone composition in 1945 and his
temporary inactivity in 1950 may well have framed Boulez's twelve-tone practice, which
can be distinguished from his subsequent "serial" practice. The distinction between the
twelve-tone and serial practices generally depends on where the principle of pre-ordering
applies. In Boulez's twelve-tone practice, this principle is applied only to the domain of
pitch-classes, while in his serial practice the application goes far beyond the domain of
pitch-classes and affects other musical domains incluidng duration, dynamics, and attack.
Such a general distinction may be further refined through Boulez's personal
correspondence and published essays, which document that, around the end of 1950, his
interest in serialism shifted from raising sheer criticism to launching actual composition.
In an essay completed as early as November of 1949, "Trajectories: Ravel,
Stravinsky, Schoenberg," Boulez criticized Schoenberg's atonal Pierrot lunaire and his
twelve-tone compositions in general on similar grounds:
It did not fall to Schoenberg to make the essential discovery, that is of necessity of deducing the structure of a work from its contrapuntal functions and from them alone. It was Webern who, in a far-reaching series of works ~ the Symphony for chamber orchestra, the saxophone Quartet, Op. 22, the Concerto for nine instruments, the Variations for Piano, the String Quartet — was to implant this audacious idea.. . . Schoenberg's contrapuntal constructions are formal rather than intrinsic and . . . the meaning of his language is not inseparable from its
4This discussion interestingly coincides with his changed opinion about the second version of le Soleil des eaux. In a previous letter to Cage, dated autumn of 1950, Boulez lauded the orchestration of le Soleil des eaux which, according to him, became inspirational in revising the orchestration of le Visage nuptial (Boulez 1993, 76). In a later letter to Cage, dated December of 1950, however, Boulez drastically shifted his opinion about le Soleil des earn, dismissing it as "a step in the wrong direction" (Boulez 1993, 86).
underlying structure: this seems to me the most serious charge one could level at Pierrot lunaire, the lack of profound coherence and a 'uterine' relation between its language and its architecture (Boulez 1949 in 1991,198).5
Schoenberg used the embryonic serial technique to subsume preclassical and classical forms into a world governed by functions antagonistic to these forms: since the architecture does not flow from the serial functions alone, a hiatus appears between the structure of the work and the natural tendency of its material (Boulez 1949 in 1991,199).
The crux of this criticism rests on two general assumptions: that the principal task of a
composer ought to be focused on discovering hidden potentials; and that structure ought
to be compatible and consistent with the premises of its means. From these assumptions,
Boulez concluded that Schoenberg had failed to unravel structural potentials hidden in
basic materials and had merely resorted to anachronistic structural means. Nonetheless,
Boulez remained short of explaining how to deduce "the structure of a work from its
contrapuntal functions" or how to make the "architecture" of a work flow from the "serial
functions."
When Boulez continued similar criticism of Schoenberg and Berg in "Bach's
Moment," an essay completed around May of 1950 and published in 1951, he
emphasized two important concepts, "serial principle" and "serial functions":
Schoenberg's work. . . goes in search of a new constitution of the sound world: an important discovery, if ever there was one, in the history of musical morphology. For it is perhaps not the fact of having worked out a rational organization of chromaticism by means of twelve-tone serialism that is the true measure of the Schoenberg phenomenon, but rather, it seems to me, the introduction of the serial principle itself: a principle which - 1 am inclined to think — could govern a sound world of far more intervals than just the semitone. For, just as modes and keys produced not only musical morphologies, but also,
5In a letter to Cage, dated November of 1949, Boulez spoke of completing this essay (Boulez 1993,33).
out of that, syntax and forms, so the serial principle conceals new morphologies as well as . . . a renewed syntax and new and specific forms. It must be said that one scarcely finds in Schoenberg so great an awareness of the serial principle as generator of serial FUNCTIONS as such except in an embryonic state: the use, for example, of the four possible variations of a series; the use of invariance between row-forms; the deployment of privileged regions within the series; in Berg, too, this kind of awareness is rare.. . . On the other hand, in Webern the MUSICAL EVIDENCE is achieved by generating structure from material. I mean that the architecture of the work derives from the workings of the series" (Boulez 1951 in 1991, 7-8).
By serial principle, Boulez meant the general notion of ordering itself, and by serial
functions he meant structural potential inherent in that ordering. While Boulez praised
Webern as the composer who understood and implemented the serial principle and its
functions, this conclusion still remained short of defining how to derive a complete
musical structure from a given material.
Boulez traced the origin of such an intrinsic structuring to J. S. Bach, particularly
exemplified in the Canonic Variations based on "Vom Himmel Hoch" (BWV 769) and
the organ chorale "Vor deinem Thron tret'ich hiermit" (BWV 668):
The progressive increase in the complexity of the canonic writing, and in the number of real parts, the increasing difficulty of the canons themselves, the process of augmentation — that is the rhythmic progression — and finally the changing of disposition of the canons with each variation and their arrangement in stretto: all this together defines the architecture of the chorale variations. We can thus see the rigour and logic with which the variations are linked, thanks solely to the contrapuntal technique and the superimposed structure, whose schema it is possible to abstract (Boulez 1951 in 1991,11-12).
The structure of the chorale melody generates the structure of the chorale itself. The chorale consists in effect of four sequences — developments which correspond to the four phrases of the melody. Notice that those four sequences respectively use as contrapuntal material only their own fragment of the figured chorale, and that we are therefore dealing here with a highly specialized developmental procedure, reinforced by the contrapuntal technique: a procedure which rejects all superfluous figures and makes use exclusively — through the multiple resources of counterpoint: imitation, inversion, augmentation — of the
phrase it is developing; all automatism is excluded. We may sum up by saying the 'theme' generates both the material of the development and its own architecture, and that the latter derives from the former (Boulez 1951 in 1991, 12).
Boulez stressed Bach's noble contrapuntal textures whose disposition constitutes a
flexible musical structure. Still, his account remained unclear about the relationship
between structure and its means; he did not elucidate how contrapuntal developments
themselves might inherently motivate their disposition.
Once Boulez suggested diverse potentials for serial organization to Cage at the
end of 1950 (Boulez 1993, 80-90), he mapped out on various occasions throughout 1951
ways in which to construct a musical structure based on the serial principle. In a letter to
Cage, dated between May 7 and May 21,1951, Boulez spoke of his experience in total
serialism in Structures, Book 1: "In this series of works [Structures, Book 1], I have
attempted to realize the serial organization at all levels: arrangement of the pitches, the
dynamics, the attacks, and the durations" (Boulez 1993, 90-91). In a later letter to Cage,
dated August of 1951, Boulez specifically laid out the basis for the Structures, Book 1,
detailing the ways in which to organize the pitches, dynamics, attacks, and durations
under a single principle (Boulez 1993,100-101).6 Moreover, in 1951, Boulez wrote his
most controversial manifesto, "Schoenberg is Dead" (1952b in 1991,209-214), and a
compendium of his serial practice, "Possibly..." (1952a in 1991,111-140), which not
only reintroduces the serial organization of Polyphonie X and Structures, Book 1, but
also demonstrates how to broaden the serial principle in compositions, such as le
6This letter, along with the letter dated December 30,1950, reappears as a single essay entitled "The System Exposed" in Orientations (Boulez 1986,129-142).
7
Marteau sans maitre and Etude de musique concrete.7
Indeed, the frame of Boulez's twelve-tone practice appears sharply articulated by
his encounter with a twelve-tone composition in 1945 and by his orientation toward
serialism that had gradually materialized since the completion of "Trajectories" in
November 1949. One may well speculate that, when Boulez spoke of his "classical
period," his reference would point toward his twelve-tone practice. The concluding
boundary of Boulez's twelve-tone practice roughly corresponds to the completion of the
Livre pour quatuor which, according to Boulez, marks the end to his "classical period."
Moreover, the initial boundary of Boulez's twelve-tone practice corresponds to what he
regarded as the beginning of his formative years:
When I was composing the Trios psalmodies, I did not know until then the existence of twelve-tone music, but I had a pretty good sense of the need for atonality. Meanwhile, I no longer want to acknowledge these Psalmodies [as my own] today; they have never been published and never will be, to say the least, with my permission (Golea 1958,20).8
By rejecting the Troispsaslmodies, Boulez strongly indicated that his formative years
began with compositions written after the Trois psalmodies, which come to articulate the
advent of his twelve-tone practice.
7"In a letter to Cage, dated between May 7 and May 21,1951, Boulez spoke of writing "Possibly . .a long with "Stravinsky Remains" (Boulez 1993, 91). In a later letter to Cage, dated December of 1951, Boulez spoke of completing "Schoenberg is Dead" and supplied its synopsis (Boulez 1993,117-118).
8Lorsqrue je composai les Trois Psalmodies, j'ignorais jusqu'a l'existence de la musique serielle, mais j'avais lesentimenttresnet de la necessite de l'atonalite. Cependant, ces psalmodies, je ne veux plus les reconnaitre aujourd'hui; elles n'ont jamais editees, et ne le seront jamais, tout au moins par ma volonte (Hereafter, translations will be mine unless otherwise indicated).
8
Although Boulez's twelve-tone practice can be chronologically framed, it is
premature to delineate the general outcome of such a practice.9 Still, what may have
conditioned Boulez's twelve-tone practice is addressed in two contemporaneous essays,
"The Current Impact of Berg" and "Proposals," published together in the second issue of
Polyphonie in 1948 (Boulez 1948a in 1991,183-187; Boulez 1948b in 1991,47-54).
"The Current Impact of Berg" projects Boulez's antagonism to his contemporaries,
especially Leibowitz who, according to Boulez, failed to recognize anachronism in
Berg's compositions. In contrast, "Proposals" primarily addresses technical concerns,
especially the ways in which to integrate rhythmic developments he inherited from his
predecessors, such as Stravinsky and Messiaen, with rigorous contrapuntal textures.
Despite such sharp contrast, these two essays share many characteristics, such as
the negation of tradition, a politically oriented rhetoric against the establishment, a
penchant for systematic presentation, and alienation from trends influenced by popular
and non-Western European cultures.10 In "The Current Impact of Berg," Boulez
Comprehensive scholarship is lacking in addressing either a single composition or a group of compositions that may elucidate Boulez's twelve-tone practice. There have been written synoptic accounts appended to the composer's biography (Hirsbrunner 1985; Jameux 1990), general overviews (Bennett 1986; Bradshaw 1986; Bradshaw and Bennett 1963; Griffiths 1978; Hirsbrunner 1987), anaylyses of instrumental compositions (Baron 1975; Grimm 1972; Hirsbrunner 1986; Jedrzjewski 1987; McCullum 1992; Mellot 1964; Trenkamp 1973), and textual accounts of vocal compositions (Stephan 1974; Worton 1991). These studies, however, often neglect many important primary sources, such as sketches, initial drafts, and revisions. Moreover, they tend to cite rather indiscriminately Boulez's later commentaries on serialism in an attempt to overcome intrinsic difficulties that these compositions may pose to analysts.
10These characteristics are included in the categories that Georgina Born suggests to delineate modernism: the negative aesthetic represented in a negation of the previous traditions, a concern and fascination with new media, technology, and science,
9
portrayed Berg as anachronistic and romantic, thereby conflicting with the prevailing
contemporary reception of him. The rhetoric Boulez adopted appears politically oriented
to attack his contemporaries, among whom Leibowitz became the most overt target. In
so doing, Boulez deliberately tried to prove his claim through specific examples,
systematizing his criticism. Moreover, when Boulez mocked popular elements that
surface in Berg's oeuvre, such as the Viennese waltz, the military march, and the polka,
he certainly professed a disdain for popular culture.
At the beginning of "Proposals," Boulez stated his aim to integrate a Franco-
Russian tradition, as exemplified in the innovative approaches to rhythm of Stravinsky
and Messiaen, with an Austro-German tradition, exemplified in the judicious
contrapuntal writing of Schoenberg, Webern, and Berg.11 Declaring a need for such an
integration, Boulez found Schoenberg and Berg completely indifferent to rhythmic
innovation, on the one hand, and Messiaen incapable of contrapuntal writing, on the
other.12 By adducing specific examples, Boulez systematically demonstrated ways in
which rhythm could be integrated with polyphony. Moreover, professing that music
ought to be "collective hysteria and magic, violently modern," Boulez negated some of
theoreticism, politics aiming to subvert and shock the academia and official art establishment as well as the bourgeois audience, oscillation between rationalism and irrationalism, and ambivalent relations with popular culture (1995,40-45).
"David Gable characterizes even Boulez's style in general as being rooted in the synthesis of Franco-Russian and Austro-German traditions (1990,426-456).
12Webern was spared in Boulez's double-edged criticism: "Only Webern ~ for all his attachment to rhythmic tradition — succeeded in breaking down the regularity of the bar by his extraordinary use of cross-rhythm, syncopation, accents on weak beats, counter-accents on strong beats, and other such devices designed to make us forget the regularity of metre" (Boulez 1948b in 1991,49).
10
contemporary trends that incorporate musical elements from non-Western European
traditions: he dismissed such an endeavor as "a simple ethnographic reconstruction in the
image of civilizations more or less remote from us" (1948b in 1991, 54).
Antagonism to tradition, which surfaces most prominently in the negation of the
predecessors and contemporaries who esteemed them, is a common thread between the
two essays. Nonetheless this antagonism appears paradoxically interwoven with
subscription to tradition, as manifested in the organicist metapor of theoretical accounts.13
The organicist metaphor is not suggested superficially in the mere use of terms like
"embryo" or "cell." Rather, it constitutes an intrinsic part of conceptualization.
Regarding the Subitement tempo rapide section of the third Tempo Scherzando part of the
Sonatine, Boulez wrote in "Proposals"as follows:
This is part of an athematic passage, where the development [of rhythmic cells] proceeds without the support of characteristic contrapuntal cells. We can see that the rhythmic cells are formed by a ternary rhythm in rational or irrational values . . . an embryonic rhythm suitable for multiple combinations. From different sequences of these cells, I produce three different rhythms.... Since these rhythms are not of equal length . . . their superpositions do not correspond exactly, and in this way we derive the maximum possible variation from the ternary pattern (Boulez 1948a in 1991, 52-53).
When Boulez addressed the generation of large rhythmic structures by variously
13The organicist metaphor has been circulated so long since ancient Greek thinkers, such as Plato and Aristotle, that it may have become almost a cliche today (Orsini 1969). Organic form has been a major issue in investigating theoretical discourses in the nineteenth and the early twentieth centuries (Schmidt 1987; Thaler 1984). In particular, organicism in music has often been addressed in relation to Goethe's holistic epistemology (Don 1991; Spaethling 1992) and his influence on Heinrich Schenker (Neumann 1978; Pastille 1984,1985,1990), Webern (Essl 1989; Zuber 1995), and Schoenberg (Neff 1993). While organicist aesthetics has been claimed to be particularly useful for musical criticism (Solie 1980), its legitimacy as a compositional paradigm has been challenged (Hubbs 1990).
1 1
combining rhythmic cells, he invoked an organic growth metaphor, suggesting an
inherent relationship between the rhythmic cells and the large rhythmic structures
generated from them. The particular choice Boulez made to combine rhythmic cells is
referable neither to the organic growth potential inherent in the rhythmic cells, nor to an
inevitable choice that dictates the generation of extended rhythmic structures. Rather, it
articulates the way in which Boulez conceptualized the generation of large rhythmic
structures from an organicist perspective.
As Boulez's twelve-tone practice becomes concretely framed between 1945 and
1949, it unfolds in a paradoxical mingling of antagonism and adherence to tradition. This
dissertation addresses how such a twelve-tone practice is articulated in actual
composition, focussing on the Sonatine for flute and piano. Boulez acknowledged the
Sonatine, completed in Februaiy of 1946 and revised in April of 1949, as his earliest
definitive twelve-tone composition and acknowledged Schoenberg's Chamber
Symphony (Op. 9) as its formal model. The revision of the Sonatine, which spans almost
the entire chronological boundary of Boulez's twelve-tone practice, and its formal
modeling on a previous composition become compelling grounds for investigating the
articulation of Boulez's twelve-tone practice.
The remainder of this chapter will introduce primary sources for twelve-tone
compositions that predate the Sonatine, as well as those for the Sonatine. Chapter II will
address an "eclectic" approach to twelve-tone composition in Douze notations. Although
two compositions, the Theme and Variations for the Left Hand and the Quatuor for four
Ondes Martenot, predate Douze notations, they are excluded from this study since their
12
twelve-tone usage appears rather simplistic: the Theme and Variations successively
unfolds different row-forms at the outset and at the end as an enclosing device but the
row itself is hardly subjected to intricate variation: and the Quatuor explores only a row
succession as the subject of a strict canon. Chapters III, IV, and V will address how
twelve-tone exploration determines the structural unfolding of the Sonatine. Individual
parts of the Sonatine will be characterized not as independent entities but as
interdependent components that articulate the underlying structure. Finally, Chapter VI
will addresses revision of the Sonatine, taking into account sketches, the early incomplete
version of which only the flute part survives, the final complete version, and the
published score. In particular, the revision process will be shown to support the
structural unfolding characterized in the previous three chapters, which pointedly
articulate Boulez's twelve-tone practice.
13
Primary Sources
Boulez's twelve-tone compositions that precede the Sonatine comprise the Theme
and Variations, the first two movements of the Quatuor for four Ondes Martenot, the last
two movements of the Sonata for two pianos, and Douze notations. A chronology of
these compositions can be detailed (Table 1.1) since the fair copy of each composition is
always dated and signed at the end by the composer, except for Douze notations.
Table 1.1. Chronology of Boulez's twelve-tone compositions from the Theme and Variations to the Sonatine
Theme and Variations (for piano) June 1945
Quatour (for four Ondes Martenot; dedicated to Ginette and Maurice Martenot) I: Setember 1,1945 II: September 21,1945 III: March 8,1946
Douze notations (for piano; dedicated to Serge Nigg) December 1945/January 1946
Sonata (for two pianos) I: February 1948 II: September 1945 III: March 1946
rev. June 1948
Sonatine (for flute and piano) February 1946/rev. April 1949
In fact, the original score of Douze notations is lost and only its orchestral version
survives. According to Theo Hirsbrunner, the fair copy included among Boulez's
manuscripts at the Paul Sacher Stiftung is derived from his own copy of the original score
14
(1986).14 At the end of Hirsbrunner's copy, the composition is signed and dedicated to
Serge Nigg in the composer's hand, while its completion date, December 23,1945, is
entered in a different hand.15 Such a questionable means of dating may be somewhat
clarified by the orchestral version, which bears the completion date of December
1945/January 1946.
Primary sources for Boulez's early twelve-tone compositions, including those for
the Sonatine (Table 1.2), comprise principally a draft in pencil and a fair copy in ink.
The pencil draft normally corresponds to the fair copy. In particular, the pencil draft of
the Quatuor often complements the fair copy by clarifying ambiguous representation of
pitches or accidentals in the fair copy.16 Brief sketches also exist for the Quatuor and the
Sonatine. Since those sketches tend to correspond to the final version of the fair copy,
they hardly help us decipher the way in which the final version is reached. Nonetheless,
valuable aid to the analysis of the Quatuor may be found in the provision of the row table
for the second movement that represents all forty-eight row-forms and some potential
row segmentation, in the identification of row members in the score of the third
movement by their respective order numbers represented by integers from 1 to 12, and in
14The fair copy of Douze notation at the Paul Sacher Stiftung (Mappe A, Dossier 3d) is copied in blank staves of the music paper that contains two independent compositions for piano, dated in the composer's hand from July-August 1945 and November 1945 respectively. The Boulez manuscript catalogue of the Paul Sacher Stiftung, intended only for internal use, identifies these two compositions with the first and second versions of Psalmodie 3.
15I would like to thank Professor Hirsbrunner for providing me a copy of his own.
16In addition to the autograph fair copy, there is another fair copy in a different hand, but this fair copy appears unreliable because it often conflicts with both the pencil draft and the fair copy in the composer's hand.
15
what appears to be a permutation table that suggests a further manipulation of the twelve-
tone row in the third movement.17
The first version of the Sonatine, completed in February 1946, does not survive.
It may be represented only incompletely by the flute part Boulez sent to Jean-Pierre
Rampal in 1946. Still, one may wonder whether Boulez sent the flute part to Rampal for
consultation while he was completing the first version or after he had already completed
it. A reliable text of the Sonatine remains as yet unavailable because there are many
discrepancies between the fair copy sent to Amphion for publication and the published
score, and even between the flute and piano parts of the published score. The
discrepancies between the published flute and piano parts may be resolved by consulting
the fair copy. Since, whenever such discrepancies exist (Table 1.3a), one of the
published parts usually corresponds to the fair copy, the fair copy appears to be the most
reliable source. For example, when neither the flute nor the piano part correspond to the
fair copy at m. 41 and m. 184, one may claim an error in the published score, which must
be corrected according to the fair copy.
"Individual row members are identified on page 22 of the draft, and the permutation table is drawn on the left margin of page 23 of the draft (the composer's own pagination).
16
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19
Table 1.3a. Errata in the published score of the Sonatine
Measures Flute part Piano part 25
41
47
52
53
68
correct
r.h.: G clef
correct
r.h.: the second pitch as Ab5
r.h.: F clef
fl.: dotted quarter rest
fl.: the first pitch as C#6
r.h.: G clef
l.h.: the first grace note as Bb2
correct
correct
correct
106
142
146
154
160
166
correct
fl.: the first pitch as G4
correct
correct
correct
l.h.: F clef
l.h.: the uppermost pitch of the last chord as A3
correct
fl.: the first grace note as Bb4
r.h.: the last two repeated pitch as C#5
fl.: the first eighth note as C#5
correct
References to specific pitches follow the notation suggested by the Acoustical Society of America. The pitch-class is represented by an upper-case letter and its octave placement by a number following the letter. Hence cello C is C2, viola C is C3, middls C is C4, and so on.
20
Table 1.3a. Continued
Measures Flute part Piano part 184
200
210
211
232
247
269
330
337
393
399
442
r.h.: the first chord as G4 and D4
r.h.: F clef
the meassure number 210
r.h.: F clef
r.h.: grace note C5
the meter as 5/16
r.h.: the last duration as a 16th note
fl.: the middle two durations in triplets
fl.: the first two durations in triplets
the meter as 5/16
correct
correct
r.h.: the first chord as G4 and D4
correct
correct
correct
correct
correct
correct
correct
correct
correct
fl.: the 2nd pitch as B4
l.h.: the 5th pitch as Abl
References to specific pitches follow the notation suggested by the Acoustical Society of America. The pitch-class is represented by an upper-case letter and its octave placement by a number following the letter. Hence cello C is C2, viola C is C3, middls C is C4, and so on.
21
Table 1.3a. Continued
Measures Flute part Piano part
491 correct r.h.: the 2nd uppermost pitch B7 needs be added to the last chord
500 correct r.h.: Cb5 in parentheses as part of a trill
References to specific pitches follow the notation suggested by the Acoustical Society of America. The pitch-class is represented by an upper-case letter and its octave placement by a number following the letter. Hence cello C is C2, viola C is C3, middls C is C4, and so on.
22
The fair copy, however, does not remain free of errors. It may well contain clef
and pitch errors that conflict with idiosyncracies of the Sonatine, such as consistent
completion of an aggregate or a twelve-tone row, pervasive use of an ordered pitch-class
set whose members are smaller than those of a twelve-tone row, and sparse registral
disposition (Table 1.3b). Based on such idiosyncracies, one may suggest a clef change at
m. 12, m. 80, m. 162, m. 349, and m. 502.18 In contrast to the error in clef representation,
which can be easily remedied, the error in pitch representation appears much more
problematic to correct solely on the ground of consistent compositional premises, as they
could be intentional variants. For example, the suggested pitch corrections at m. 467 and
m. 476 follow the ordering of a twelve-tone row. Since the "error" takes place
consistently in the order number 7 of three different row-forms, however, the deviation
from the row ordering may have been deliberate. The remaining pitch corrections at m.
225 and m. 235 follow the pervasive use of an ordered pitch-class set. While the "errors"
do not reveal any consistent feature, the suggested corrections merely indicate one of
many potential solutions.
18A clef change is indicated in the right hand at m. 349 of the published piano part, and it is also implied in the left hand at m. 80 of the published piano part because the left hand of the subsequent system (mm. 81-86) is notated as in G clef.
23
Table 1.3b. Errata in the fair copy of the Sonatine
Measures Fair copy Flute part Piano part 12 l.h.: F clef l.h.: F clef l.h.: F clef
80 l.h.: G clef l.h.: G clef implied
162 r.h.: G clef before the 32nd-note flourish
r.h.: G clef before the 32nd-note flourish
r.h.: G clef before the 32nd-note flourish
225 uncertain r.h.: grace note as A4
r.h.: grace note as A4
235 fl.: the last two repeated pitches as Eb5/D#5
fl.: the last two repeated pitches as Eb5/D#5
fl.: the last two repeated pitches as Eb5/D#5
349 r.h.: G clef r.h.: G clef correct
467 fl.: the 3rd pitch as Eb6
fl.: the 3rd pitch as Eb6
fl.: the 3rd pitch as Eb6
476 fl.: the 3rd pitch as F#5; the last pitch as A4
fl.: the 3rd pitch as F#5; the last pitch as A4
fl.: the 3rd pitch as F#5; the last pitch as A4
502 l.h.: F clef l.h.: F clef l.h.: F clef
References to specific pitches follow the notation suggested by the Acoustical Society of America. The pitch-class is represented by an upper-case letter and its octave placement by a number following the letter. Hence cello C is C2, viola C is C3, middls C is C4, and so on.
CHAPTER II
ECLECTICISM IN TWELVE-TONE PRACTICE:
DOUZE NOTATIONS
Boulez's Douze notations represents his twelve-tone practice in what we may call
his formative years. This set of twelve short piano pieces was completed in December
1945/January 1946, when Boulez was still closely associated with Rene Leibowitz, the
mentor who guided him through Schoenberg's twelve-tone method and that of his
disciples.1 Douze notations was initially excluded from Boulez's definitive catalogue,
perhaps because of his aesthetic dissatisfaction with juvenilia.2 Over the years, however,
Boulez changed his views on this set.3 Douze notations was eventually published in 1985
'Theo Hirsbrunner suggests that Douze notations must have been completed shortly after Boulez left Leibowitz's informal analysis class (1986,2). Yet Boulez was closely associated with Leibowitz in December 1945 when he participated in a concert directed by Leibowitz at the Paris Conservatoire (Golea 1958, 27). Boulez even dedicated the First Sonata for piano to Leibowitz; the dedication appears crossed out on a "scrap" paper used to cover the sketches for the first version of le Visage nuptial.
2The unpublished juvenilia dating from 1945 include Nocturne for piano, Prelude, Toccata, et Scherzo for piano, Trois psalmodies for piano, Theme and variations for the Left Hand for piano, Douze notations for piano, and Quatour for four Ondes Martenot. For more detailed information see the inventory of the Boulez manuscripts at the Paul Sacher Foundation (Piencikowski 1988).
3In the conversation with Celestin Deliege, Boulez said that "what I composed in 1945 and 1946 I now consider as definitive for that period" (Boulez 1976, 35).
24
25
by Universal Edition, while the other juvenilia remain unpublished. Although Boulez
regarded the Sonatine for flute and piano (1946) as his earliest definitive twelve-tone
composition,4 the publication of Douze notations, though delayed by no less than four
decades, proves that the composer himself came to acknowledge its merits.
Two conflicting views on Douze notations are represented in current Boulez
scholarship. On the one hand, Theo Hirsbrunner emphasizes the unusual freedom with
which Boulez explored the twelve different pitch-classes. According to Hirsbrunner,
Boulez's "free" twelve-tone practice is indebted to Schoenberg, particularly to his
concept of developing variation (1986).5 On the other hand, Gerald Bennet emphasizes
several stylistic features, such as highly differentiated dynamics, minute phrase
articulation, chromatic pitch structures, and the brevity of phrases, themes, and motifs.
Bennett attributes these stylistic features to Boulez's simple and direct response to
Webern (1986).
Hirsbrunner's study remains the only one entirely devoted to Boulez's twelve-tone
practice as represented in Douze notations to date. Here Hirsbrunner informs us of the
transmission of the score whose original manuscript was lost; of the symbolic importance
4In the conversation with Antoine Golea, Boulez said that "this Sonatine represents my first phase in the path toward twelve-tone composition as I understand it" ("Cette sonatine est ma premiere etape sur le chemin de la composition serielle, telle que je l'entends") (Golea 1958, 38).
5Hirsbrunner describes Boulez's compositional concept around 1945 as one that "in succeeding the Schoenbergian school, subscribed to the permanent variation, which still continues to evolve" ("der sich in derNachfolge der Schule Schonbergs der permanenten Variation verschrieben hat, und diese noch steigert") (1986, 3).
26
of the number twelve6; of the stylistic features indebted to those who may have
influenced Boulez; of the features that foreshadow Boulez's later developments; and of
the twelve-tone technique that Boulez began to develop in his formative years. Most of
Hirsbranner's insightful account, however, seldom engages in detailed analysis. For
example, only one structural property of the twelve-tone row is singled out to
demonstrate a diatonic construct that distinguishes itself from a chromatic, symmetrical
construct favored by Webern and by Boulez in later compositions. Whenever the row
ordering cannot be exactly determined, such irregularities are merely attributed to a sort
of permutation that reflects Boulez's freer twelve-tone practice. Moreover, the
transformation of the twelve-tone row is hardly addressed in relation to the structural
unfolding of individual pieces and of Douze notation as a whole.
In examining Boulez's early compositions dating from 1942 to 1948, Bennett
finds a drastic stylistic change soon after Boulez came into contact with Webern's twelve-
tone compositions through Leibowitz in 1945. Bennett suggests that Webern remained
most influential through Boulez's early twelve-tone practice, though his response to
Webern continued to change: "Boulez took over many of the constructive principles he
found in Webern, extended them, enriched them, and gave them a breath and generality
they never had for Webern" (1986, 83). Except for a few descriptive remarks on stylistic
features, however, Bennett never penetrates how Boulez adopted, adapted, or transcended
Webern's constructive principles, not to mention those that may have crucially influenced
6This number symbolism stems from the fact that Douze notations comprises twelve pieces, each piece spans alike twelve measures, and each piece variously explores the twelve different pitch-classes.
27
Boulez.
In this chapter, I shall address the ways in which the exploration of the twelve
different pitch-classes defines the structure of each individual piece as well as that of
Douze notations as a whole. I shall focus on the formation of a twelve-tone row and/or
an aggregate as a principal means of structural articulation.7 By examining how the
ordered row and/or the unordered aggregate relates to the structural unfolding of each
piece, I shall suggest features that may best characterize Boulez's early twelve-tone
practice in late 1945.
Overview
The twelve pieces in Douze notations can be divided into three groups. Group A
(Nos. 1, 2, 3, 5, 6) is characterized by the use of the twelve-tone row as an unmistakable
linear theme. Group B (Nos. 7, 8,10) subverts row-ordering but explores important row
properties. And Group C (Nos. 4,9,11,12) exclusively explores aggregates.
7Aggregate is defined as an unordered set of the twelve different pitch-classes.
28
The twelve-tone row presented in No. 1 is consistently transformed to derive the
twelve-tone rows for the other pieces of Group A (Table 2.1).8
Table 2.1. Transformation of the twelve-tone row
No. 1: 8 ,10 ,3 ,2 ,9 ,4 ,0 ,5 ,1 ,7 ,6 ,11
j ROTFL,
No. 2: 10 ,3 ,2 ,9 ,4 ,0 ,5 ,1 ,7 ,6 ,11,8
No. 3: 3 ,2 ,9 ,4 ,0 ,5 ,1 ,7 ,6 ,11 ,8 ,10
No. 5: 2 ,3 ,10 ,8 ,11 ,6 ,7 ,1 ,5 ,0 ,4 ,9
No. 6: 4,0,5, 1 ,7,6,11,8,10,3,2,9
I ROTFL,
R(ROTFL4)
ROTFL,
One may derive the row for No. 2 by rotating the first pitch-class of the basic row to the
last order number; the row for No. 3 by rotating the first two pitch-classes of the basic
row to the last two order numbers; the row for No. 5 by rotating the first four pitch-
classes of the basic row to the last four order numbers and, then, retrograding the ordered
twelve pitch-classes; and the row for No. 6 by rotating the first five pitch-classes of the
basic row to the last five order numbers. This transformation can be represented by two
8Throughout the dissertation, the twelve different pitch-classes will be represented by integers from 0 to 11, so that the pitch-class C is represented by 0, the pitch-class C#/Db by 1, and so forth. Order numbers in the twelve-tone row are represented by underlined integers from 0 to 11.
29
different operations on order numbers, R (retrograde) and ROTFLn (rotation by which n
number of the first pitch-classes moves to n number of the last order numbers). Thus,
given the basic twelve-tone row for No. 1, the twelve-tone row for No. 2 is derived
through ROTFLj, the row for No. 3 through ROTFL2, and the row for No. 6 through
ROTFLj. Exceptionally, the row for No. 5 is derived through R(ROTFL4), since it
involves both and rotation retrograde.9
The transformational scheme unfolded so far leads us to predict that the twelve-
tone rows for the remaining seven pieces may systematically rotate the basic twelve-tone
row. Such potential, however, materializes only in No. 10, a piece from Group B. No.
10 does not complete a twelve-tone row but successively unfolds eight pitch-classes at
the outset, which correspond exactly to the order numbers 0-7 of the predictable row
derived through ROTFL9.
The basic twelve-tone row is all-combinatorial. Each row-form consists of two
hexachords of the same set type; for example, the two hexachords of TgP, <8,10,3,2,9,4>
and <0,5,1,7,6,11>, both belong to the [0,1,2,6,7,8] set type.10 As an unordered set, these
hexachords can be transformed into each other or into themselves through transposition
or inversion, fulfilling the condition for all-combinatoriality. Thus, all possible row-
9This transformation appears deliberately systematic, as the value n of ROTFLn corresponds exactly to the order number that each of the five pieces holds within this collection. Given the order number of No. 1 as 0, the order numbers of the remaining four pieces,! (No. 2), 2 (No. 3), 4 (No. 5), and 5 (No. 6), exactly corresponds to the value n of ROTFLn.
10Angles "< >" and the pitch-classes they enclose notate an ordered set. The set classification follows John Rahn's Tn/TnI-type (1980).
30
forms (TnP, InP, RT„P, and RI„P) can be grouped into three large families according to
the invariant pitch-class content of combinatorial hexachords (Table 2.2). It is in pieces
No. 7 and No. 8, both of which belong to Group B, that combinatorial hexachords (H,
and H 2) are explored. In these pieces, one hexachord constitutes a constant musical
element associated with an ostinato figure, while the other hexachord is deliberately
completed to articulate an aggregate boundary.
In Group C (Nos. 4,9,10,11), row ordering is abandoned, while aggregate
completion is ensured in a variety of ways. For example, spanning no less than five
measures, No. 4 deliberately projects an aggregate, the completion of which articulates
the first structural division. Although the remaining three pieces (Nos. 9,10,11)
continue to unfold aggregates, the completion of an aggregate no longer defines a point
of structural articulation. Rather, aggregates become structural components that can be
manipulated within larger structural divisions.
31
Table 2.2. Three large combinatorial families inherent in the basic twelve-tone row
Family I
T 8 P:
T 2 P:
I10P LP:
T „ P -
T 5 P:
IIP:
I7P:
H , H 2
8,10,3,2,9,4/0,5,1,7,6,11
2,4,9, 8,3,10/6,11,7,1,0,5
10, 8,3,4, 9,2/6,1,5,11,0,7
4,2,9,10,3,8/0,7,11,5,6, 1
H 2 H ,
11,1,6,5,0,7/3,8,4,10,9,2
5,7,0,11,6,1/9,2,10,4,3,8
I,11,6, 7,0,5/9,4, 8,2,3,10
7, 5,0,1,6,11/ 3,10,2, 8,9,4
Family II
H„ H„
T 9 P T3P I . . P
I5P:
T 0 P:
T 6 P:
I«P: I2P:
9,11,4,3,10,5/1,6,2, 8, 7,0
3,5,10,9,4,11/7,0, 8,2,1,6
11,9,4,5,10,3/7,2,6,0,1,8
5,3,10,11,4,9/1,8,0,6, 7,2
H B HA
0,2,7,6,1,8 / 4,9,5,11, 10,3
6,8,1,0,7,2 /10,3,11,5,4,9
8,6,1,2,7,0 / 4,11,3,9,10,5
2,0,7,8,1,6 /10,5, 9, 3,4,11
Family III
T10P: T 4 P:
I„P: I6P:
T,P:
T 7 P:
I9P: I3P:
HP HQ
10,0,5,4,11,6/2,7,3,9, 8, 1
4,6,11,10,5,0/8,1,9,3,2,7
0,10,5,6,11,4/8,3,7,1,2,9
6,4,11,0,5,10/2,9,1,7,8,3
Hq Hp
1,3,8,7,2,9 / 5,10,6,0,11,4
7,1,2,1,8,3 /ll,4,0,6,5,10
9,7,2,3,8,1 / 5, 0,4,10,11,6
3,1,8,9,2,7 /11,6,10,4,5,0
Exploring a Twelve-Tone Row (Group A)
The five pieces in Group A (Nos, 1,2,3, 5, 6) are structurally diverse,
encompassing bipartite, tripartite, ostinato, and canonic structures (Table 2.3).
The characteristics of individual structures affect the way in which structural divisions
are articulated. The first section of bipartite and tripartite structures corresponds to the
initial completion of a twelve-tone row. In non-sectional structures, based on an ostinato
or canonic imitation, however, a structural division is determined by means other than
row completion: in the main section, No. 2 combines three different row-forms as a linear
theme against a consistent dyadic ostinato; and No. 6 introduces a complete row-form as
a principal imitative subject that continues to be manipulated within a larger structural
division.
J J
Sectional structures, tripartite: No. 1
In No. 1, a tripartite structure is projected according to the way in which a single
ordered pentachord is reiterated and the way in which twelve different pitch-classes are
explored (Example 2.1). The five pitch-classes at m. 1 are treated as an ordered
pentachord that frames the piece, as its return at m. 12 maintains the pitch-class ordering
intact. In the second section, the same pentachord is transposed at m. 7 to constitute a
linear theme over constant chordal accompaniment; given the ordered pentachord at m. 1
and m. 12 as X, the transposed pentachord at m. 7 can be represented by T4X. The
tripartite structure is also projected according to the way in which the twelve pitch-
classes are organized. The first section unmistakably unfolds the twelve-tone row TgP as
a linear theme. The second section avoids deploying all twelve pitch-classes, either as a
row or as an aggregate. Finally, the last section successively unfolds aggregates in two-
part counterpoint.
34
Example 2.1. Pitch-class manipulation in No. 1 (Fantasque - Modere)
0,1,2,3,6,7]
0,1,4,5,7
j f subito
[0,1,2,3,6,7}
TiX-
^ I hJu H 7 = "J. > .
2 1
PP
7 i X _y ht
t > : K7 K7 kg
t)-? 5 ? * -p W f
aggregate [0,1,4,5,6,7,9]-
aggregate
[0,1,2,3,4,6]-
s EE
[0,1,2,4,6,8]
¥=$ w 5 ^ tat P-&-
soutenu
- ^ PP
i Pi # = t r e P^P 5*$
.[0,1,2,3,4,6] Fig [0,1,2,4,6,8]- 8"'-
35
The second section is distinguished from the outer sections because it does not
constitute either a twelve-tone row or an aggregate. Still the second section can be
associated with the first section through transpositional and complementary relations
(Example 2.2). Transpositional relation has already been shown between the ordered
pentachord X at m. 1 and the tranposed pentachord T4X at m. 7. Complementary relation
is suggested between the pentachord at m. 2 and its complement at m. 8; the
complementary heptachord is constituted rather subtly when Bb3 as a grace note replaces
the underlying B3. In fact, these two sets do not literally complement each other to
produce an aggregate. Rather, respectively classified under the [0,1,4,5,7] set type and
the [0,1,4,5,6,7,9] set type, these two sets project an abstract complementary relation.11
11 Complementary relation can be either literal or abstract. In literal complementary relation, two sets literally complement pitch-classes of each other, encompassing all the twelve pitch-classes. In abstract complementary relation, however, the complementation occurs between equivalent set types (Forte 1973, 75).
36
Example 2.2. Transpositional and complementary relations between the first and second sections of No. 1 (Fantasque - Modere)
1
JJ subito
[0,1,2,3,6,7}
1 1 - 1 •1
1 ^ •f i f r f l fflf 1 L*^ ' 1
\ an ' -j ban— ft
f - ~
_>> L , J -
7 v tf 0 . "#r •
pp
I a"w U _
ly-Z 1 I +
$ ? ? t
- y - i y 4
l ? ?
fti? —1° ....
[0,1,4,5,6,7,9]-
37
The outer sections are distinguished from each other in that the first section
conspicuously unfolds a twelve-tone row, while the third section successively unfolds
aggregates in two-part counterpoint. Nonetheless, there are many features that associate
these outer sections. In addition to the framing pentachord that has already been noted,
the third section is associated with the first section particularly in the way individual
contrapuntal voices are constructed and vertical dyads are unfolded in two-part
counterpoint (Example 2.3). The two contrapuntal voices of the third section (mm 9-12)
simultaneously unfold two different forms of the same hexachord, the [0,1,2,3,4,6]
hexachord at mm. 9-10, and the [0,1,2,4,6,8] hexachord at mm. 10-11, forming
successive aggregates.
58
Example 2.3. Commonalities between the first and third sections of No. 1 (Fantasque -Modere)
[0,1,2,3,6,7}
JJ subito
3 E
[0,1,2,3,6,7}
aggregate
[0,1,2,3,4,6]-[0,1,2,4,6,8]
& l|J i|JiI J 5
soutenu mf PP
l - i
9-m-[0,1,2,3,4,6]
[0,1,2,4,6,8]-
39
Such partition of the twelve pitch-classes into two forms of the same hexachord may
have already been suggested in the first section, where two forms of the [0,1,2,3,6,7]
hexachord are spliced. Moreover, the row ordering of the first section affects the way in
which a series of vertical dyads are unfolded in the third section (Table 2.4). The six
vertical dyads, which correspond to the six adjacent dyads of the twelve-tone row, appear
grouped into three harmonic units — A consisting of the first dyad, B consisting of the
second dyad, and C consisting of the remaining four dyads whose ordering remains
intact. The order in which these three harmonic units are unfolded enables one to relate
the successive aggregates in the third section to the twelve-tone row in the first section.
In the row of the first section these harmonic units are unfolded in the order of A, B, and
C. This ordering is retrograded in the first aggregate of the last section, where the
harmonic units are unfolded in the order of C, B, and A; and this ordering is rotated and
regrograded in the second aggregate of the last section, where the harmonic units are
unfolded in the order of A, C, and B.
4 0
J
fc
H
O
0 S
!<
.O
§
O
Z . 5
cn T 3
03
T 3
" 3 # o
t 5 <D >
O & <+H
o OX)
# g
t 3
§
c
D
( N 0>
s e i
EH
V O
r - "
i n
° 1
Q J O N
< N
» r n
o < s
£ c §
0 )
a 0
H
1 CD
j >
1 3
£
H
00
c T
C N
n 6
C q
P
i n
c T
u s S
n
CO - a
c d
Q
O
O N
1 3 o
6
€ > E <L>
>
< N
cn
^sO
< o
i n
u s S
°°„ cT
C/D . . T 3 r - s Cd r-H
> • * I— 1
Q o
1 3 * " " ]
V s £ B > w
co a ) GO <L>
+->
a <D
c3
a
# a
T 3 <D
t 3
O
* 3
. s
2 c d
CO
" i >%
- d
1 3 o
*£ <D
>
41
Sectional structures, tripartite: No. 3
No. 3 continuously unfolds different row-forms both as melody and as
accompaniment (Example 2.4). As principal melody, the twelve-tone row appears in the
right hand except for the last four measures, where it surfaces in the left hand. As
accompaniment, the twelve-tone row is often partitioned into dyads to form a harmonic
unit. In particular, the same means of presenting the row are simultaneously combined in
the outer sections to ensure a closed tripartite structure.
In both the first and third sections (mm. 1-8; mm. 9-12), T3P is simultaneously
combined with its retrograde, RT3P (Example 2.4a). The first section unfolds T3P as a
linear theme and RT3P as accompaniment, while the third section unfolds RT3P as a
linear theme and T3P as accompaniment. Each row-form is partitioned into two
hexachords at pitch-class 1 (m. 3 and m. 10). In the first section, pitch-class 1 begins the
second hexachord of T3P, while concluding the first hexachord of RT3P. In the third
section, the same pitch-class 1 demarcates the two hexachords of T3P and RT3P. Here,
the right-hand accompaniment does not complete T3P, omitting pitch-classes 9 and 4 (the
order numbers 2 and 3). But, as in the first section, the shared pitch-class 1 begins the
second hexachord of T3P, while concluding the first hexachord of RT3P. This
hexachordal pairing constitutes vertical aggregates in the outer sections, the similarity of
which projects a closed structure. Such a closed structure is further enhanced when the
harmonic progression of the first section (mm. 1-3) is exactly reversed in the third section
(mm. 9-12) by consistently verticalizing the order numbers 2-9 presented as a linear
theme (Example 2.4b).
42
Example 2.4a. Simultaneous combination of T3P and RT3P
mm. 1-3
I
m
p
-U1
i i" ~p p -
RT3P-aggregate
f b r ; • U
& $ aggregate
mm. 9-12
T3P
5
m m
RT3P incomplete aggregate
43
Example 2.4b. Harmonic correspondence between the first and third sections
Ipz: P T 9 "
H9- V
44
The middle section (mm. 4-8) distinguishes itself from the outer sections by
intensifying a contrapuntal texture (Example 2.5). Initially, a contrapuntal texture is
suggested at m. 4, where the order numbers 0-4 of T4P and IjP imitate each other through
retrograde inversion. The contrapuntal texture is conspicuously projected at mm. 6-8,
where I4P, the principal melody, is exceptionally accompanied by two different row-
forms, RT3P and I2P. Following the RT3 that forms a chordal accompaniment, I2P
constitutes a counter melody by linearly unfolding its twelve pitch-classes; notice that
pitch-class 0 becomes common to I4P and I2P.
45
00
a .2 o <D c/D
JJ
*3 T3
S3
o Z *n fs pSi 'E E ee H W
E;
IN I
mi
46
Such contrapuntal texture lingers at the outset of the third section, where the order
numbers 0-3 of I9P and I3P imitate each other through retrograde (Example 2.6). At m.
9, however, the remaining eight pitch-classes of I9P constitute a chordal accompaniment,
while I3P is interlocked with RT3P to unfold a principal melody for the third section.12
Thus, a closed tripartite structure is ensured by the return of the melody/accompaniment
texture that initially pervaded the first section.
l2It is possible to interlock these two forms because the first four pitch-classes of the two forms are invariant, regardless of their order numbers.
47
<N 00
G .2 V-i 0 <u 01
*5
<L> 43
s:
3 en 6 2
c4 -£l a 5 6 * w
v
<N| A
ou
48
Sectional structures, bipartite: No. 5
The bipartite structure of No. 5 can be attributed to distinct melodic and
accompanimental figurations, dynamics, and row unfolding (Example 2.7). Both
sections (mm. 1-6; mm. 7-12) begin with a sixteenth-note figure and conclude with a
figuration that emphasizes pitch-class 3 by a preceding grace note; both sections sustain a
single vertical hexachord as constant accompaniment13; dynamically, both sections
progress from p to sfz; finally, both sections unfold only one row-form as principal
melody, T2P (mm. 2-6) and T9P (mm. 9-12).
13The way this accompaniment distinguishes two different sections is rather subtle because the two accompanying hexachords share five pitch-classes in common -- 4,6,7, 8, and 10 ~ exhibiting a close similarity.
49
Co > O
&
o
Q
i n
o
2
t - '
< N
a s C3
w
Q - * n
i n .
i n
~ ~ H I
n : \
~ ^ l I
\ — + | I CO S=32S~
^ 4 ' - s
" 7 ^ 1
r p
& H - n *
* 5 -
! ( r
•*.
i » A C
if
Z - - d l l I I
P M
H *
- > l l
(
J J -
'S>i'
• f t • • «
• * *
| ¥ ^ S ! v am
j p .
Q _ j £ L _ Q & i S L -
'T± « 3 T
? s ~
taS
' H l ^ v
• A < i e
^ 5 * >
50
Between the two row-forms an aggregate is projected at mm. 6-9 to maintain a
degree of continuity (Example 2.8). The aggregate is temporally bounded by the same
underlying interval. At m. 6, interval-class 6 spans from pitch-class 9 to pitch-class 3
preceded by a grace note and, at m. 9, the same interval-class spans from pitch-class 3
preceded by a grace note to pitch-class 9. This pitch-class 9 remains in the same register,
demarcating the aggregate boundary, while the underlying interval-class 6 is combined
with the grace note to unfold two different forms of the same [0,1,6] trichord. Within this
boundary articulation, the remaining pitch-classes project two different forms of the same
[0,1,3,4,5,7] hexachord as melody and accompaniment and, thereby, extend an
introduction to T9P, which spans one more measure than the introduction to T2P.
51
Example 2.8. Aggregate projection in No. 5 (Doux et improvise), mm. 6-9
[0,1,3,4,5,7] .
[0,1,6]
[0,1,6] 6 ten.
£ 4 -v-
sfz
p
m
[0,1,3,4,5,7]-
52
Non-sectional structures, ostinato: No. 2
In non-sectional structures, several row-forms are combined to constitute a broad
section. In No. 2, the main section (mm. 4-11) successively unfolds three different row-
forms as a linear theme in the left hand, against a consistently dyadic ostinato in the right
hand (Example 2.9). The linear theme consists of three parts (mm. 4-6; mm. 6-8; mm. 9-
11), all of which are separated from one another by rests. It is only in the first part that
one row-form, T10P, is literally completed. When the second part unfolds T9P, it omits
pitch-classes 11 and 7 (the order numbers 5 and H) . The missing pitch-class 11 is
replaced by the pitch-class 10 in the middle of m. 7; this may be an anomaly attributable
to a scribal error. The other missing pitch-class 7 is concealed in the dyadic ostinato.
When the last part unfolds RT10P, it omits pitch-classes 7 and 8 (the order numbers 8 and
11). These two pitch-classes are supplied by the dyadic ostinato that has gradually
assumed significance in completing the twelve-tone row.
53
>2 £ (N o
Z C+H o c .2 o CD
C/3 . s *3 <L>
XI
Oti a i s <5 §
o Pi Q\ <N Jm TEL 3 ce x W
V
A.<
A.i
A •<
A .«
£ *:v
i i *
c i
CO
C U : ^
CO
** c r s
3
00
PH h "
t > - * OH
O
V - "
,v:i
v -
V
J • G k _
CO u t -
* L
c>*0®
itz\.
54
Despite the unfolding of three different row-forms, the linear theme tends to
emphasize pitch-class 10. As the order number 0 of T10P, pitch-class 10 temporally
frames the entire linear theme. It also concludes the middle part of the linear theme,
which conceals the pitch-class 7 of T9P (the order number 11) in the dyadic ostinato.
Perhaps, the pitch-class 10 in the middle of m. 7 deliberately substitutes for the pitch-
class 11 of T9P (the order number 5); paradoxically, the substitution itself forcefully
projects the pitch-class 10 by deviating from the strict row ordering.
55
Non-sectional structures, canon: No. 6
No. 6 introduces the twelve-tone row as the principal subject that continues to be
manipulated within a large canonic structure (Example 2.10). This canonic structure is
divided into a strict canon (mm. 1-7) and a canon by inversion (mm. 7-12). According to
the relationship between the two canonic voices, the canon by inversion is further
subdivided into one that relates its two voices through index number/sum of
complementation 8 (mm. 7-8) and the other that relates its two voices through index
number/sum of complementation 4 (mm. 8-12). In each canon, nine row-forms are
unfolded in the dux of the right hand (Table 5). Except for the first form of the strict
canon and the last two forms of the canon by inversion, all the remaining row-forms are
systematically truncated by one last note. This systematic truncation is intended to
overlap adjacent forms by one pitch, thereby creating a continuous succession of row-
forms. Moreover, the truncation prevents the commes of the canon by inversion from
literally duplicating that of the strict canon in reverse order. Thus, although truncation as
process is exactly reversed in the second section, the overall structure remains short of
becoming literally symmetrical.
56
Example 2.10. Canonic structure of No. 6 (Rapide)
Imitative Canon
r
Inverse Canon:
Index Number 8
Inverse Canon:
Index Number 4
57
Table 2.5. Systematic truncation of the twelve-tone row in No. 6 (Rapide), the dux
T4P: 4,0,5, 1,7,6, 11,8, 10,3,2,9
T9P: 9,5,10,6,0,11,4,1,3,8,7,(2]
T7P: 7,3,8,4, 10,9,2, 11, 1,6, [5,0]
T6P: 6,2,7,3,9,8, 1,10,0, [5,4, 11]
T0P: 0,8,1,9,3,2,7,4,[6,11,10,5]
T4P: 4,0,5, 1,7,6, 11, [8, 10,3,2,9]
T„P: H,7, 0, 8,2,1,(6,3,5,10, 9, 4]
TjP: 1,9,2,10,4,(3,8,5,7,0,11,6]
T4P: 4,0,5,1,(7,6,11,8,10,3,2,9]
9,4
T4P: 4,0,5,1,(7,6,11,8,10,3,2,9]
T,P: 1,9,2,10,4,(3,8,5,7,0,11,6]
T4P: 4,0,5,1,7,6,(11,8,10,3,2,9]
T6P: 6,2,7,3,9*8,1,(10,0,5,4,11]
T,P: 1,9,2,10,4,3,8,5,(7,0,11,6]
T5P: 5,1,6,2,8,7,0,9,11,(4,3,10]
T„P: 11,7,0,8,2,1,6,3,5,10,(9,4]
T!0P: 10, 6,11,7,1,0,5,2,4, 9,6, [3]
TjP: 8,4,9,5,11,10,3,0,2,7,6,1
T,P: 1,9,2, 10,4,3,8, 5,7,0, 11,6
Missing pitch-classes are indicated in square brackets. The asterisk (*) indicates the change from the index number/sum of complementation 8 to the index number/sum of complementation 4 in the commes of the left hand.
58
Exploring Row Properties (Group B)
The three pieces in Group B (Nos. 7, 8,10) subvert the row ordering while
revealing important properties of the basic row in the process of aggregate completion, as
presented in piece No. 1. No. 10 unfolds at the outset a partially ordered row, the first
eight pitch-classes of which correspond to the order numbers 0-7 of the row derived from
the basic row through ROTFL9. The remaining two pieces, Nos. 7 and 8, explore
combinatorial hexachords of the basic row, thereby associating themselves with multiple
row-forms.
No. 10 (Mecanique et tres sec)
In No. 10, a two measure-unit becomes the basis for a multisectional structure
(Example 2.11). This structural unit always begins with a group of five pitch-classes,
projecting a motto that retains its duration and dynamics whenever it reappears; this
motto is always realized in sixteenth-note quintuplets and its last note is always
articulated with Moreover, the two-measure structural unit coincides at the outset
with the completion of the partially ordered row, articulating the initial structural
division.
59
Example 2.11. Two-measure structural unit of No. 10 (.Mecanique et tres sec)
¥ h
i ZOE i
3
in
P 1,3
60
Following the partially ordered row, a number of aggregates are successively
unfolded to project structural continuity. Adjacent aggregates always share more than
one pitch, and these shared pitches are compositionally projected in a variety of ways
(Example 2.12). Adjacent aggregates may literally overlap (Example 2.12a); aggregate
boundaries may be articulated by a group of pitches that are discretely assigned to a
particular hand to constitute an independent voice (Example 2.12b); and when, adjacent
aggregates are interlocked, some of the aligned pitches are not necessarily part of either
aggregate (Example 2.12c).14
l4As the first aggregate is completed at m. 2, two pitches from separate aggregates, C#6 and B3, are aligned. While C#6 is included in the first aggregate, B3 is excluded from it. Instead, together with C3, B3 constitutes a dyad that introduces a new aggregate.
61
Example 2.12. Projection of shared pitches between adjacent aggegates in No. 10 (Mecanique et tres sec)
» -
• f
S - — 1
•
3 0 f/ ™
k = j — W d "
t
. b
' 7 ' ^ 4 —
-r H
H - 1
" f - T
I "
m
B
62
Aggregate interlocking is considerably extended in two instances (Example 2.13).
Beginning with the last four notes of m. 6, an aggregate is broadly unfolded across mm.
6-8 (Example 2.13a). Before this aggregate is completed by Al, the lowest pitch at the
first beat of m. 8, however, it is temporally interrupted at the end of m. 7, where another
aggregate begins to emerge. While the second aggregate is unfolded continuously, the
first aggregate is interrupted by the left-hand dyad (Eb2 and C2), which constitutes part
of the second aggregate alone. Aggregate interlocking becomes more complex between
mm. 9-10 (Example 2.13b). The initial aggregate encompasses all pitches of m. 9 and the
left-hand dyad (B3 and D#4) at the outset of m. 10. Before this aggregate is completed
by G2 — the lowest pitch at the end of m. 10 ~ it is temporally interrupted by another
aggregate that begins with the right-hand dyad (C#5 and C6) at the end of m. 9,
encompassing all the pitches of m. 10.
63
Example 2.13. Extended aggregate interlocking in No. 10 (Mecanique et tres sec)
a.
i 'fK fA^4f=%==T %= "91
7
5
. ' r , —
/ 1 V ^
*i i */ ^ •/
1 —3— 1 Qvb_ -J- j
1 1
b.
F 4 i — » —
n
~t|j 1 i p * *4
Y y itr = m
k •/ hj
7 1 ^ = R - V —
it
7
w
t --
Y y itr = m
k •/ hj 1 y Ijf
" I ^
7
w
t ---4 1 i — ? • - 1
1 " 3 1
64
No. 7 (Hieratique) and No. 8 (Modere jusqu'a tres vif)
No. 7 and No. 8 share several features. Both pieces are sectionally structured:
they explore combinatorial hexachords of the basic row, which limit their content to
twelve specific pitches (not pitch-classes) with one exception; and, in both pieces, one of
the combinatorial hexachords constitutes a constant musical element associated with an
ostinato figure, while the other hexachord is deliberately completed to articulate an
aggregate boundary that corresponds to a structural division.
Despite such similarities, the two pieces differ from each other considerably in
the treatment of the two combinatorial hexachords, H, and H2, that form the basic row.
In No. 7, H, constitutes a constant musical element, while H2 is deliberately completed to
articulate a structural division (Example 2.14) Hexachord H, combines the left-hand
ostinato with the right-hand dyad (C#5 and G4) that always appears in the same register
and with the same attack ">." Against this constant musical element, hexachord H2 is
deliberately completed ~ frequently interrupted by the C#5-G4 dyad ~ to articulate
individual structural divisions (mm. 1-4; mm. 5-8; mm.9-12). At mm. 1-4, H2 is
completed without pitch-class repetition. Before H2 appears in its entirety at m. 8 and at
m. 11 respectively, some of its members have already been combined with the constant
C#5-G4 dyad to unfold a broad linear event. Indeed, such deliberate completion of H2
indicates the precise moment when the two combinatorial hexachords complement each
other to constitute an aggregate, articulating each structural division!
65
Example 2.14. Combinatorial hexachords in No. 7 (Hieratique)
Ha
Hi
I - i f f l j T i J l v f '
/
CL 1 K h
p •If r D "p 7 f
I * j -—| 1
b w — ! — 1 — |
^ I
— h 1
^ $ $ 'Jt' y - y ^ \*i ^ mf sempre
ii
¥ * / > < / />
i d ' : i ' i > i d ;
o I M J XT
66
Similarly in No. 8 (Example 2.15), the combinatorial hexachords complement
each other, projecting a bipartite structure in which the first section is telescoped in the
second section. The functions that the combinatorial hexachords perform, however,
cannot be distinguished from each other as clearly as in No. 7. As a constant musical
element, two pitch-classes of H2 constitute the right-hand ostinato throughout. H2
remains incomplete until the outset of m. 7, however, where the remaining four pitch-
classes appear simultaneously. Although H, remains incomplete until m. 6, its five pitch-
classes have already been repeated as a vertical pentachord from m. 3 onwards. Thus, the
two hexachords perform a dual function: they provide a constant musical element, while
their completion deliberately projects an aggregate that articulates a structural division.
67
Example 2.15. Combinatorial hexachords in No. 8 (Modere jusqu'a tres vif)
1 r 3 ^ 3 , n simile
b
^ P J p J " p * V w 1 P- s ' P
•f .V h - - -
-mj
-pa is
Lift? . : / ,
| 3 | simil
n j
e
i 7 P - J
"1 | 3 | | 3 | simile
^ B * ^ i ' • ' ^ p p p r
m k. U •
«7P*
K- '
v 7 T p p \> r p r
>
K n n < y ] | l 8 .
i f - * M *1 °
H 2
I 3 | simile
/ J *4
nile
%
— ^
K-1 Uln
1
1 * i ho
W
V p p p y ^ p M M
< (fl) 4W— j f « -
L-yj.
-
Hi
68
Exploring Aggregates (Group C)
The four pieces in Group C (Nos. 4, 9, 11,12) explore aggregates in two ways. In
No. 4, an aggregate is deliberately completed to articulate the first structural division. In
the remaining three pieces (Nos. 9,11,12), aggregate completion no longer performs a
structure-defining function. Rather, aggregates constitute structural components that can
be manipulated within a large structural design.
No. 4 reveals three ordered hexachords (indicated as OHJ, one of which is
repeated throughout as an ostinato in the left hand (OHJ, while the other two (OH2 and
OH3), which share the same pitch-class content, are deliberately unfolded in the right
hand (Example 2.16). Once pitch-classes 2 and 0 initiate the right-hand hexachords at m.
1, a new pitch-class is consistently added in each of the subsequent four measures to
"compose out" an ordered hexachord (OH2), <2,0,1,11,10,3>. At the same time, another
ordered hexachord (OH3) is locally unfolded at m. 5 in the right hand, <11,10,3,2,0,1>.
69
O Z .5 00 •s O cd X <u & CS *S <L> 6 jd H. s o L) so ?"H c4 QJ
5 6 X W
a: o
70
Table 2.7. Relation between the left-hand ostinato and the right-hand hexachords in No. 4
The Lefit-Hand Ostinato
OH,: <9,4,5,7,6,8>
The Right-Hand Hexachords
OH2: <2,0,1,11,10,3>
OH3: <11,10,3,2,0,1>
RT,
ROTFL
ROTFL3RT6
These two ordered hexachords are related to the ostinato closely as shown in Table 2.7.
The ostinato OH, is transformed into OH2 through transposition and retrograde (RT6).
Since OH2 is transformed into OH3 through rotation(ROTFL3), the ostinato OH, is
transformed into OH3 through a multiple operation that combines transposition,
retrograde, and rotation (ROTFL3RT6). The two right-hand hexachords (OH2 and OH3)
globally and locally complement the ostinato, deliberately projecting an aggregate that
corresponds to the initial structural division. Moreover, the two right-hand hexachords
are highlighted independently of each other in the subsequent two sections, where the
first three pitch-classes of OH2 and OH3 respectively determine sectional boundaries
(Example 2.17): the first three pitch-classes of OH3 - 1 1 , 1 0 , and 3 - temporally frame
the second section (mm. 6-8), while the first three pitch-classes of OH2 - 2, 0, and 1 -
temporally frame the last section (mm.8-12).
71
£ o £ .s
S o T3 § <N
a o X> G .2
3 o
cd a .2 %-> o <u CO
n QJ
"E S 08 *
w
72
The remaining three pieces in Group C (Nos. 9,11,12) manipulate aggregates as
structural components in a variety of ways. No. 9 introduces diverse aggregate partitions
that respectively characterize individual sections. No. 11 layers aggregates that are
articulated by pervasive musical features. Finally, No. 12 partitions an aggregate
exclusively into two forms of the same hexachord.
No. 9 (Lointain-Calme)
No. 9 characterizes three sections through distinct aggregate partition (Table 2.8).
In the first section (mm. 1-3), an aggregate is partitioned into three unordered sets of
unequal size — A (5,11,0,7,10), B (8,3,4), and C (9,2,6,1) ~ that continue to be
systematically rotated to unfold four aggregates (Table 8a). In the second section (mm.
4-7), an aggregate is partitioned into two unordered sets of unequal size — M
(7,11,6,8,10,3,2,9) and N (0,4,5,1) — that are systematically rotated to unfold three
aggregates (Table 8b). In the last section (mm. 8-10), however, there is no systematic
aggregate partition manifested (Table 8c). In contrast to the first two sections,
aggregates are not intersected but interlocked.
73
Table 2.8. Aggregate partition in No. 9 (Lointain-Calme)
The complete music is abstracted in integer notation in the first row. Beneath the first row aggregate completion is indicated in layers. Vertical sonorities (or chords) are indicated in parentheses,"()." Individual measures are demarcated by a slash,"/."
74
The three sections of No. 9 are also enhanced by textural correspondence
(Example 2.18). In the first section (mm. 1-3), three textural segments (I, II, III) are
consecutively unfolded without interruption. Although all of the three segments appear
in the second section (mm. 4-7), the segments I and II are interrupted at m. 5 by a
chromatic tone-cluster. In the third section (mm. 8-12), the segments I and II are also
interrupted by the same tone-cluster, while the textural unfolding is incomplete by the
omission of segment III.
Example 2.18. Textural correspondence in No. 9 (Lointain -Calme)
75
Lointain - Calme
1 I
g y i PP / sourd
%
X 1 ii iii
m m ILIOL
9E=
3 D PPP PP
m ft
II ^•Lj
vJLT S i
M a p fat
ppp PP ppp
. | s * I s * | » J f j P ^ !
ppp
m
76
No. 11 (Scintillant)
No. 11 unfolds a symmetrical structure of which the first half is exactly
retrograded by the second half in both pitch-class content and rhythmic organization
(Example 2.19). Within this transparent structure, aggregates are articulated by a
progression from one principal pitch-class to another. In the first section, this
progression is filled in by a thirty-second-note ornament. Such an ornamented
progression appears insignificant in articulating the second aggregate that encompasses
twelve consecutive pitch classes, beginning with the last pitch-class of m. 1. It is in the
completion of the first and third aggregates, however, that the ornamented progression
comes to be articulated. In the completion of the first aggregate, all ten pitch-classes of
m. 1 are combined with the two principal pitch-classes of m. 2; the ornament at m. 2
proves to be auxiliary. In the completion of the third aggregate, which also begins with
the last pitch-class of m. 1, all five pitch-classes of m. 3 are combined with the principal
pitch-classses of m. 2 and m. 4. The ornaments at m. 2 and m. 4, which begin alike with
pitch-classes 8 and 9, however, prove to be more than auxiliary, since their initial two
pitch-classes complete the third aggregate.
J3
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77
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78
No. 12 (Lent - Puissant et apre)
No. 12 unfolds an uneven bipartite structure in which the first section is
telescoped in the second section (Example 2.20). A parallelism between the two sections
appears unmistakably in both pitch-class content and textural design. The pitch-class
parallelism, however, does not correspond to the textural parallelism. As indicated on
Example 2.20, the two sections begin with the same five vertical hexachords, except for
pitch-class 10 omitted at m. 3, while the corresponding hexachords do not share the same
texture. In contrast, the final measures of the two sections share only the same texture. It
is at the penultimate measure of the two sections (mm. 6-7; m. 11) that the pitch-class
parallelism corresponds to the textural parallelism, except for pitch-class 9, omitted at m.
1 1 .
Throughout the first section (mm. 1-8), an aggregate is consistently partitioned
into two forms of the [0,1,2,3,6,7] hexachord that appear adjacently. Yet this procedure
is not followed in two instances. At mm. 3-4, an incomplete aggregate is partitioned into
a vertical pentachord and a vertical hexachord.15 When an aggregate is completed at mm.
6-8, it repeats pitch-class 1. Since the repeated pitch-class is sustained as the same pitch
C#5 across the three measures, one may interpret the aggregate partitioned into two
forms of the [0,1,2,6,7,8] hexachord.
"Given the missing pitch-class 8, the aggregate would have been partitioned not into two forms of the same hexachord but into a [0,1,4,5,6,9] hexachord and a [0,1,2,5,6,9] hexachord.
79
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80
Despite the pitch-class and textural parallelism between the two sections, the
second section completes an aggregate only at the outset by unfolding two forms of the
[0,1,2,3,6,7] hexachord. Thereafter, adjacent hexachords nearly complete an aggregate,
always omitting one pitch-class. Earlier at mm. 3-4, as mentioned above, all the pitch-
classes of the pentachord/hexachord pair nearly complete an aggregate, omitting pitch-
class 8. At the corresponding measures in the second section (mm. 9-10), an unexpected
pitch-class 10 is added, instead of the expected pitch-class 8, to prevent aggregate
completion. Likewise the subsequent hexachord pair (mm. 11-12) fails to complete an
aggregate, omitting pitch-class 9.
In Douze notations, the twelve pitch-classes are explored in three ways. (1) As a
twelve-tone row, the twelve pitch-classes are temporally ordered to unfold a linear theme.
In sectional structures, a single row-form constitutes a linear theme and its completion
coincides with the first structural division. In non-sectional structures, however, many
row-forms are combined to constitute a broad linear theme within a large structural
division. (2) The twelve pitch-classes are explored to reveal properties of the twelve-
tone row, though the row ordering is not strictly maintained. In No. 10, a partially
ordered row is unfolded at the outset to articulate the initial structural division. In No. 7
and No. 8, the combinatorial hexachords of the basic twelve-tone row deliberately
complement each other, completing an aggregate that coincides with the initial structural
division. (3) As aggregates that entirely abandon the strict row ordering, the twelve
pitch-classes are explored in two distinct ways. In No. 4, an aggregate is deliberately
completed to articulate an initial structural division. In Nos. 9, 10, and 12, aggregate
81
completion no longer performs a structure-defining function. Rather, aggregates are
explored as structural components that can be manipulated within a large structural
division.
Such a diverse exploration of the twelve pitch-classes suggests preexisting models
upon which Boulez's twelve-tone practice may have been based. One may speculate
about a Schoenbergian model both in the treatment of the twelve-tone row as a linear
theme and in the partitioning of the twelve-tone row (or an aggregate) into two forms of
the same hexachord. One may also speculate about a Bergian model in the generation of
diverse twelve-tone rows by systematically rotating the basic row. Moreover, all three
composers ~ Schoenberg, Webern, and Berg - could serve as models in strict canons and
symmetrical structures.
These associations, however, can easily be refuted. The partition of an aggregate
into two forms of the same hexachord is a condition built into hexachordal
combinatoriality ~ one of the features that characterize Schoenberg's mature style
(Haimo 1990, 8). Yet combinatorial row-forms are never explored to produce structures
as complex as those in Schoenberg's twelve-tone compositions (Mead 1984; Peles 1983).
The systematic rotation (ROTFLn) that transforms the basic twelve-tone row points
towards the famous canons by rotation and transposition in the Allegro misteriosos (III)
of Berg's Lyric Suite (mm. 45-67).16 This is, however, the same movement that Boulez
adduced to characterize Berg's anachronism (Boulez 1948a in 1991, 185). Finally, it is
,6Before Berg completed the Lyric Suite, he explained the systematic rotation in a letter to Schoenberg (Harris and De Voto 1971).
82
debatable to associate Boulez's contrapuntal writing — strict canons or free counterpoint
-- with any preexisting models, since he had shown a keen interest in counterpoint by
continuing studies with Andree Vaurabourg long after concluding his formal education
(Heyworth 1986, 5). Indeed, it appears doubtful for us to pinpoint the models upon
which Boulez based Douze notations. Rather, the diverse exploration of twelve pitch-
classes in Douze notations genrally suggests Boulez's early compositional strategy,
which can best be characterized as being eclectic, in an attempt to come to terms with the
twelve-tone method of the three Viennese composers.
CHAPTER III
PRINCIPLES OF KINETIC STRUCTURE: SONATINE
In characterizing the Sonatine for flute and piano, Boulez acknowledged its
formal indebtedness to Schoenberg's Chamber Symphony (Op. 9) that, according to
Boulez, fuses four parts of a sonata-allegro form ~ first theme, second theme,
development, and recapitulation - with the four movements of a sonata ~ a sonata-
allegro movement, a slow movement, a scherzo, and a finale. He further complemented
this specific formal attribution by emphasizing the transformation of a single theme,
through which to ensure a unity and generate kinesis that propels the structural unfolding
of the Sonatine.
This account amalgamates Boulez's remarks about the Sonatine, which are often
inconsistent with one another. In 1948, Boulez detailed the way in which to explore
rhythmic cells at the end of the third Tempo Scherzando part (Boulez 1948b in 1991,50-
51). Ten years later in 1958, general commentaries appeared in the interview with
Antoine Golea where he acknowledged the Sonatine as his earliest representative twelve-
tone composition, adduced Schoenberg's Chamber Symphony as its formal model, and
characterized its structural unfolding in terms of contrast between thematic and athematic
83
84
developments (Golea 1958,38-39).1 In a lesser known letter of May 31,1963,
responding to the questionnaire from George K. Mellot, Boulez generally commented on
revision of the Sonatine, its structural derivation from a principal theme, and placement
of subsidiary developments in various points (Mellot 1964, Appendix H).2 Later in the
conversation with Celestin Deliege, Boulez consciously distanced himself from
Schoenberg on stylistic grounds. Still, the Sonatine remained affiliated to Schoenberg's
Chamber Symphony in its form and, particularly, in its transformation of a single theme
as a basis for a unified whole (Boulez 1976,27-28).3
Boulez's concern for kinetic structure, first manifested in the interview with
Deliege, was not specifically directed to the Sonatine. Rather, it articulated an important
general compositional process called "proliferation of basic materials":
This means that in general I start off with relatively simple materials; my basic ideas, and even the overall plan of my works, are fairly simple, but of course within the plan there will be very highly developed textures. When I have in front of me a musical idea or a kind of musical expression to be given to a particular text of my own invention, I discover in the text. . . more and more possible ways of varying it, transforming it, augmenting it and making it proliferate. For me a musical idea is like a seed which you plant in compost, and suddenly it begins to proliferate like a weed. Then you have to thin it out. . . . So in many cases I have found it necessary to reduce, to thin out the possibilities, or else to put one after
'A similar remark appeared one year later in 1959 in liner notes that accompanied the first recording of the Sonatine performed by Severino Gazzelloni (flute) and David Tudor (piano). This recording of the Domaine musical concerts for the 1957 season was produced by Vega (C30 A139). I would like to thank Nancy Lorimer at Stanford University Music Library for providing me a copy of the liner notes.
2These accounts, however, appear less reliable. Boulez, who at the time taught at Harvard University, depended only on his memory and did not have a score, as acknowledged in the correspondence.
3Boulez even claimed that the Sonatine was more unified than the Chamber Symphony.
85
the other so as to create an evolution in time and not a superposition that would have been too compact (emphasis mine) (Boulez 1976,15).
This compositional process facilitates a structural concept that rests on constant
transformation of a given material. The structural outcome, therefore, depends on the
transformational potential of a given material; it would be unthinkable to impose any pre-
existing formal model.
As Boulez traced the model for such a concept back to the chorale variations and
chorale preludes by J. S. Bach, he nearly reiterated his earlier postulates introduced in
"Bach's Moment," an essay published in 1951. The earlier postulates were originally
intended to support the relevance of a short-lived phenomenon called "total serialism."
They also allowed Boulez to criticize Schoenberg harshly for using preexisting forms in
his twelve-tone compositions and, at the same time, to praise Webern for integrating the
structural concept based on transformation with twelve-tone composition:
It is perhaps not the fact of having worked out a rational organization of chromaticism by means of twelve-tone serialism that is the true measure of the Schoenberg phenomenon, but rather, it seems to me, the introduction of the serial principle itself: a principle which — I am inclined to think — could govern a sound world of far more intervals than just the semitone. For, just as modes and keys produced not only musical morphologies, but also, out of that, syntax and forms, so the serial principle conceals new morphologies as well a s . . . a renewed syntax and new and specific forms. It must be said that one scarcely finds in Schoenberg so great an awareness of the serial principle as generator of serial FUNCTIONS as such except in an embryonic state: the use, for example, of the four possible variations of a series; the use of invariance between row-forms; the deployment of privileged regions within the series; in Berg, too, this kind of awareness is rare . . . . On the other hand, in Webern the MUSICAL EVIDENCE is achieved by generating structure from material. I mean that the architecture of the work derives from the workings of the series (Boulez 1951 in 1991, 7-8).
Nonetheless, Boulez's afterthought manifested in the interview with Deliege
considerably reoriented the earlier postulates. To be sure, he maintained the crux of the
86
earlier postulates that hinge on the structural derivation based on a given material.
Nonetheless, the association with total serialism disappeared completely, as it became an
irrelevant phenomenon. By adding a new structural dimension, moreover, Boulez
claimed a relief from the influence of his predecessors, especially that of Webern:
From Webern... I had taken a particular conception, not of silence (in which I was not especially interested—it seemed to me that silence, in Webern, was something peculiarly his own which would be difficult to use without becoming imitative) but of a certain texture of intervals. Beyond this, the dynamic character of my early works is really quite different from the agogics found in Webern. I had no inclination for vague objects or states of mind, but preferred extremely dynamic works whose texture would therefore be sufficiently dense to propel this dynamism further and further forward (Boulez 1976,14).4
Indeed, the discrepancy between Boulez's earlier and later accounts raises some
doubts whether any of his accounts represents an important aspect of the Sonatine. In the
interview with Golea, for example, Boulez's account of the thematic/athematic contrast
became anything but clear:
One finds [in the Sonatine]... developments based on characteristic motifs, deduced evidently from the twelve-tone row but maintaining still a thematic character; these developments are opposed to other developments that result from the encounter between the twelve-tone row and rhythmic cells, to put it differently, the beginning of «athematicism». It is in this Sonatine, if you will, that I have tried for the first time to impose independent rhythmic structures, the possibilities of which Messiaen had shown to me, upon classical twelve-tone structures (Golea 1958,38-39).5
4Boulez's rejection of silence, however, appears to contradict his earlier remarks in "Poss ib ly . . an essay published in 1952, where he validated silence and proclaimed Webern as its first proponent (Boulez 1952a in 1991,124-125).
5On y trouve . . . des developpements construits sur des motifs caracterises, deduits de la serie, evidement, mais conservant encore un caractere thematique; ces developpements s'opposent a d'autres developpements, resultant de la rencontre de serie et de cellules rythmiques, autrement dit, un commencement d1 «athematisme». Si vous voulez, c'est dans cette sonatine que j'ai essaye pour la premiere fois d'articuler des
87
The description of athematic development appears particularly puzzling, since very little
can be made of the "encounter between the twelve-tone row and rhythmic cells."
Moreover, while thematic development is associated with the twelve-tone row, it remains
still unclear how constituent motifs for such a development maintain a "thematic
character."
The discrepancy and ambiguity surrounding Boulez's accounts of the Sonatine do
not necessarily undermine their relevance. Rather, they may very well suggest a
structural predicament of the Sonatine whose salient features are selectively and
variously emphasized to come to terms with different circumstances. By the time the
interview with Golea was released, Boulez already abandoned the notion of total
serialism and, in the commentaries on the Sonatine, he deliberately avoided even
suggesting any compositional features that may pertain, however remotely, to the dead
phenomenon. In 1972, when the interview with Deliege was released, Boulez no longer
needed to concern himself with a specific compositional agenda, as he assumed the
directorship of IRC AM, which symbolized the culmination of his multifaceted career as a
composer, teacher, conductor and administrator. More keenly aware of his position in
history, Boulez appears to have shifted the focus of his earlier postulates from a specific
compositional agenda to a general structural principle that, according to him, affects a
wide range of his oeuvre.
structures rythmiques independantes, dont Messiaen m'avait revele les possibility, sur des structures serielles classiques.
88
Since Boulez's emphasis on athematic development represents a phenomenon
localized only in the third part of the Sonatine (Tempo Scherzando), the role of
thematic/athematic contrast considerably diminishes as a general structural principle. In
contrast, kinetic structure, which stems from a traditional tension-relief model, comes to
assert itself as a legitimate principle through deliberate row subversion and motivic
transformation (Table 3.1). Row subversion stimulates one level of kinetic structure by
opposing the ordered row and unordered aggregate whose roles constantly oscillate
between a foreground and a background structural determinant throughout the Sonatine.
Motivic transformation stimulates another level of kinetic structure by deliberately
obscuring and, then, clarifying the association between the row and a motif; this
association surfaces in the introduction, develops seemingly independently of the row in
the third part, and finally confirms its affiliation with row in the fourth part. This chapter
will address the genesis and development of kinetic structure in the Sonatine. It will
demonstrate the way in which kinetic structure surfaces in the introduction, suggesting
the dichotomy between the ordered row and unordered aggregate. It will further examine
how kinesis increases its momentum throughout the first two parts by shifting the roles of
The boundary dyads that project interval-class 1 (order numbers 0-1 and 10-11)
are strategically important for the selection of row-forms for the introduction, as they
become invariant in their ordering and pitch-class content through RT6. This unique
property assumes that, given a certain inversionally equivalent pair of row-forms, the
pitch-class/order invariance occurs between the beginning dyad of one form and the
concluding dyad of the other, or the pitch-class invariance alone occurs between the
opening dyads and between he concluding dyads of the two inversionally equivalent
forms; and, given a certain transpositionally equivalent pair of row-forms, the pitch-class
variance alone occurs between the beginning dyad of one form and the concluding dyad
of the other. The unfolding of four row-forms at the outset of the introduction (mm. 1-9)
suggests one way in which to take advantage of such a row property (Table 3.2). The
pitch-class/order invariance allows adjacent, inversionally equivalent row-forms to be
linked by an overlapping dyad that concludes the preceding row-form and, at the same
time, begins the subsequent row-form.
Table 3.2. Invariance in the boundari dyads
I„P: 0, 11, 7, 1, 8, 4, 3, 9, 2, 10, 5, 6
TSP: 5, 6, 10, 4, 9, 1, 2, 8, 3, 7, 0, 11
T„P: 11, 0, 4, 10, 3, 7, 8, 2, 9, 1, 6, 5
I6P 6, 5, 1, 7, 2, 10, 9, 3, 8, 4, 11, 0
93
The pitch-class invariance allows all four row-forms to become commutative, thereby
establishing a ground for kinetic structure through "deliberation" and "contradiction": by
deliberation, I mean a row unfolding in which the identity of a row-form appears
ambiguous at the onset but gradually becomes clear; and by contradiction, I mean a row
unfolding whose anticipated identity becomes contradicted at the end.6
Several row-forms are variously suggested at the outset of the introduction (mm.
1 -22) through characteristic pitch-class ordering, partitioning scheme, and invariance
(Example 3.2). The pitch-class ordering is not yet established as a principal factor in
dictating chronological events. The invariance property remains only secondary in
importance, as it normally serves to link adjacent row-forms through a common dyad. In
contrast, the characteristic partitioning of the row into two pentachords and a dyad
surfaces prominently and, in particular, its deliberate projection (mm. 1-9) and
subversion (mm. 10-22) primarily contributes to a kinetic unfolding of the twelve-tone
row.
The pitch-class ordering remains incomplete in a chronological event at mm. 1 -2,
since the first five pitch-classes of I0P appear as a vertical pentachord whose ordering has
to be deciphered from bottom to top. At the same time, all twelve members of I0P are
distinctly configured — a vertical pentachord, a linear pentachord in quintuplets
6Wilma Anne Trenkamp considers the four row-forms in the introduction as a single group called "region" (1973, 24). Contrary to her claim, however, this twelve-tone region does not encompass all row-forms in the introduction, nor does it play any significant role in the subsequent structural unfolding of the Sonatine, except in the middle section of the first part (mm. 52-79). Her notion of twelve-tone region, therefore, appears to address a particular compositional potential inherent in the row, which is never extensively explored.
94
articulated alike with staccato, and an ornamented flutter-tonguing note — to bring out the
partitioning of the row into two pentachords and a dyad. Such row partitioning appears
compositely conceived in the unfolding of subsequent two row-forms, TSP and TUP.
The ordering of these two row forms is considerably scrambled.7 Distinct configuration
of these row-forms, however, compositely projects a partitioning of the row into a dyad
and two pentachords. This row partitioning is initiated by order-numbers 0-6 of T5P.
which are distinctly configured as an ornamented flutter-tonguing note and a linear
pentachord articulated with staccato. While order-numbers 0-5 of T n P, as well as the
remainder of TSP, become amorphous in a vertical hexachord, order numbers 7-jJ, of
T n P, articulated with staccato, are distinguished from order-number 6, a flutter-tonguing
note, to complement the row partitioning initiated by TSP. The pitch-class ordering
almost becomes a moot factor in dictating chronological events when row segments are
simultaneously juxtaposed in two-part counterpoint (mm. 7-9). In contrast, the
characteristic row partitioning becomes most unmistakable, since the two contrapuntal
voices respectively unfold order-numbers 2-6 of RI6P in the piano and order numbers 7-
jJL in the flute, while the remaining members of RI6P, order-numbers 0-J_, are configured
and articulated differently from the preceding pentachords.
7The progression from order-number 6 to order-number 7 of TSP is interrupted by a vertical hexachord in which the remaining four members of TSP (order-numbers 8-11) are embedded as the lowest four pitches. The ordering of T n P becomes further subverted, as order-numbers 0-5 constitute a vertical hexachord and order-numbers 7-8 swap their positions.
95
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96
Once the pitch-class ordering has become a lesser determinant of row unfolding at
mm. 1 -9, its status remains unchanged at mm. 10-22 and, therefore, unable as yet to
stimulate any degree of structural kinesis. In contrast, the characteristic partitioning,
which has already asserted itself as a prominent determinant of row unfolding at mm. 1-
9, comes to be deliberately subverted at mm. 10-22 to suggest a kinetic structural change.
The subversion of the characteristic partitioning begins to surface when a single row-
form, RI6P, is extended at mm. 10-13. The retrograded ordering is partially represented
by the flute passage at mm. 10-12 that unfolds order-numbers 6-11 in reverse order.
Nonetheless, it merely surfaces in the succession of three vertical dyads of the piano at
mm. 12-13, which demonstrates that adjacent pitch-classes of the remaining members of
RI6P (order-numbers 0-5) are paired as a single unit. Coupled with such an ambiguous
pitch-class ordering, the partitioning of RI6P completely deviates from what has already
been established at mm. 1-9.
Both the ambiguous pitch-class ordering and inconsistent partitioning persist
throughout the rest of the row unfolding (mm. 12-22). The linear representation of the
row becomes completely dissipated in the initial two row-forms, RTnP and T3P, whose
identity can be construed only by the way in which some of their members are vertically
configured. For example, the identity of RT,,P rests on the vertical configuration of
order-numbers 10-11, which constitute an initial dyad, and of order-numbers 0-4, which
are further partitioned into two dyads (assigned respectively to the two hands of the
piano) and a flutter-tonguing note; the remaining members of RT,,P (order-numbers 5-9)
are completely scrambled in the flute passage at m. 13. Similarly, the identity of T3P
97
rests on the vertical configuration of all twelve members whose partitioning exhibits a
pattern of alternating a dyad and a trichord: <(3,4), (8,2, 7), (11, 0), (6,1, 5), (10, 9)>.8
The last two incomplete forms, I0P and I7P, exceptionally adhere to the established pitch-
class ordering but, by virtue of incompleteness, they remain far short of asserting the
unfolding of the twelve-tone row.
The subversion of the row partitioning certainly stimulates a kinetic structural
change. In contrast, the invariance property inherent in the twelve-tone row remains only
secondary in importance, as it normally serves to link adjacent row-forms through a
common dyad. Still, the invariance stimulates a degree of kinesis, especially between
TnP and RI6P through contradiction. These two row-forms are linked by a secondary
one (indicated with an asterisk on Example 3.2) that shares its initial dyad with TUP and
its concluding dyad with RI6P. As for pitch-class ordering, the boundary dyads of the
secondary row-form contradict each other: the initial dyad, <6, 5>, corresponds to order-
numbers 11-10 of RI0P or to order numbers (M of or I6P; and the concluding dyad,
coupled with the preceding flutter-tonguing note, project a linear trichord, <7, 0,11>,
that corresponds to order-numbers 9-11 of T5P.
The kinetic structural change, stimulated by the subversion of the characteristic
partitioning, culminates at mm. 23-31 where even a glimpse of the row appears to
dissipate (Example 3.3). Complete subversion of the twelve-tone row may well be
represented by an aggregate completion in the flute at mm. 23-28. In particular, the
derivation of a single motif, which articulates the boundaries of the introduction, appears
8Vertical configurations are indicated in parentheses.
98
further to enhance row subversion. This motif is initially suggested at mm. 1 -4 (the
second system on Example 3.3) to comprise a vertical trichord (indicated in a circle) and
a broad progression in the lowermost voice (indicated in square brackets). It is
associated with the twelve-tone row ambiguously, since the progression of the lowermost
voice encompasses three different row-forms. When this suggested motif appears
crystallized as an independent entity to conclude the introduction at mm. 26-31, its
previous frail association with the twelve-tone row becomes completely subverted.
99
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Row Unfolding as a Foreground Determinant
The overall structure of the first part (Rapide, mm. 32-96) is determined by the
way in which different row-forms and aggregates are explored to construct a pervasively
contrapuntal texture. The first part comprises three sections that explore both the row
unfolding sustained in a single voice and the row unfolding distributed in different
voices. In the first section (mm. 32-52), the twelve-tone row conspicuously emerges as a
linear passage in the flute, to which successive aggregates provide a counterpoint in the
piano. The second section (mm. 53-79) combines the row unfolding distributed among
different voices with the row unfolding sustained in a single voice whose surrounding
counterpoint often utilizes various row segments, suggesting simultaneous manifestation
of multiple row-forms. In the last sections (mm. 80-96), however, each row-form is
always distributed between individual voices of a two-part counterpoint.9
The first section clearly demonstrates a linear row unfolding in the flute, which is
accompanied by the succession of aggregates distributed between the two hands of the
piano (Example 3.4). The row unfolding and aggregate succession appear to complement
each other for musical continuity. The initial row-form. TSP, is separated at m. 40 from
the remaining two row-forms, T7P and I2P, which are connected by two common pitch-
classes.
9To this norm there is only one exception at mm. 82-85 where the linear row unfolding in the flute momentarily interrupts the ongoing two-part counterpoint in the piano.
101
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103
Aggregates continue to succeed one another except at mm. 44-47 where T7P is still being
unfolded. Thus, the aggregate succession compensates the separated row unfolding,
while the continuous row unfolding compensates the interrupted aggregate succession.
Aggregates sometimes reflect the row ordering. For example, the aggregate at
mm. 35-37 partially represents the linear ordering of T4P, which is rounded off by the
retrograded order-numbers 0-2 (indicated by a square bracket on Example 3.4). The first
two pitch classes, <4, 5>, and last three pitch-classes, <6,11,10>, respectively
correspond to the order numbers 0-1 and 9-lj.. The remaining pitch-classes of the left
hand whose simultaneous unfolding appear to obscure the row ordering suggest
nonetheless a dyadic partitioning of the row. The initial vertical dyad at m. 35
corresponds to the order-numbers 2-3. Of the ornamented vertical dyads at m. 36, the
upper voice corresponds to the retrograded order-numbers 5-6, and the lower voice to the
retrograded order-numbers 7-8. Such a dyadic portioning becomes apparent in the
aggregate that begins with the last two pitch-classes of TSP in the flute (mm. 38-41). The
ordering of IQP is suggested by the vertical dyads in the piano. Although consistent
dyadic partitioning is subtly offset by the missing order-number 3 (pitch-class 1), the
sequence in which these dyads appear unmistakably corresponds to the row ordering.
Similarly in the aggregate that begins with the order-numbers 5-6 of I2P (mm. 49-50),
consistent dyadic partitioning is manifested, as the initial linear dyad in the flute is
followed by three vertical dyads that migrate from the right hand to the left hand of the
piano. The sequence in which these four dyads appear corresponds to the ordering of
I0P, while the remaining four pitch-classes deviate from the row ordering.
104
The partitioning of the twelve-tone row aside, dyads are further explored to
expand an aggregate at mm. 42-44. Once an aggregate is completed at the outset of m.
43 (demarcated by a dotted line on Example 3.4), it begins to expand by converting
vertical dyads into linear dyads and vice versa. The initial two vertical dyads in the left
hand at m. 42 turn into respectively the uppermost linear dyad in the right hand at m. 43.
<5, 1>, and the grace notes at m. 44, <4, 8>. Of the quintuplet figure in the right hand at
m. 42, the last four pitch-classes are treated as if they were two consecutive linear dyads,
<10, 9> and <3, 2>, which later turn into respectively two vertical dyads in the left hand
at m. 43.
As the second section explores both the row unfolding sustained in a single voice
and the row unfolding distributed among different voices, it takes advantage of the
invariant boundary dyads of certain T- and I-forms to set large row-families, not row-
forms, against one another. Given the invariant property, all possible row-forms can be
grouped into six families (Table 3.3). Of these six row-families, however, only row-
families III and IV are extensively opposed to each other for the main row unfolding of
the second section (mm. 56-76). It is at the end of the second section (mm. 75-79) that
new row-families, I and II, glimpse against the prominent row-family IV, disrupting the
prevalent opposition between row-families III and IV. Such disruption not only
demarcates the end of the section but expands the kinetic structure built on the opposition
of two different row-families.
105
Table 3.3. Six row families
Family I T0P: <0,1,5, 11,4,8,9,3,10,2,7,6> T6P: <6, 7,11,5,10,2,3,9,4, 8,1, 0> IJP: <1,0,8,2,9,5,4,10,3,11,6, 7> I7P: <7,6,2,8,3,11, 10,4,9,5,0,1>
The opposition between row-families III and IV is suggested at the outset (mm.
52-57) where two different row-forms are simultaneously manifested (Example 3.5). The
unfolding of the two row-forms, T2P and I4P, which respectively project row-families III
and IV, appears anything but conspicuous. T2P is only partially represented by the
repetition of its first five pitch-classes (Example 3.5b); when the five pitch-classes are
repeated, their ordering becomes completely scrambled. In contrast, I4P is fully
represented by all its members whose ordering, however, somewhat deviates from the
row ordering (Example 3.5c). The row identity is contrapuntally projected, as the upper
voice at mm. 53-54 linearly unfolds order numbers 0-4 as principal notes to which order
numbers 10-11 constitute a counterpoint at m. 54. In fact, order number 5 precedes order
number 3 but, as part of an ornamental figure, it hardly affects the pervasive row
ordering. Thus, actual deviation from the row ordering takes place as order numbers 6-7
swap their positions at mm. 55-56 and as order numbers 10-11 constitute the lower
contrapuntal voice at m. 54. Despite such deviation the close juxtaposition of order
numbers 0-1 and order numbers 10-11, in particular, brings out the boundary dyads that
enables one to identify a particular row-form and, through their invariant property, to
group different row-forms into a single row-family.
107
Example 3.5. Row families III and IV represented by simultaneous row unfolding
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108
The main row unfolding of the second section (mm. 56-76) juxtaposes both
successively and simultaneously different row-forms that respectively represent row-
families III and IV (Example 3.6). At the outset (mm. 56-64), I9P, which represents row-
family III, is followed by two successive row-forms, T3P and I10P, both of which
represent row-family IV, to project a linear opposition between two different row-
families. As members of each row-form are distributed between different contrapuntal
voices, the row ordering does not always manifest itself contiguously. Nonetheless,
while some inner row members scramble their positions, boundary row members always
strictly follow the row ordering, projecting a particular identity of each row-form. In the
rest of the main row unfolding (mm. 64-76), row-family IV appears to assert itself
prominently: following the migrating T3P (mm. 64-67), I10P as a single row-form
considerably expands (mm. 67-75).
109
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110
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Example 3.6b. Assertion of row family III
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112
The prominence of row-family IV becomes further enhanced toward the end of the main
row unfolding (mm. 71-76); here, an incomplete RT9P, bounded by order-numbers 0-3,
constitutes a principal counterpoint to the ongoing I10P. These three row-forms,
however, do not strictly adhere to the row ordering. For example, the migrating T3P.
which follows most of the time the row ordering through a notationally explicit dotted
line in the score, cannot be completed unless its missing order numbers 7-9, <6.1. 5>. are
supplied from the surrounding counterpoint. When I10P expands from m. 67 to m. 75, its
complete form emerges only in the middle, beginning with order numbers 0-1 in the left
hand at m. 71-72; at the outset, only order numbers 0-4 strictly follow the row ordering
(mm. 67-69), roughly suggesting the identity of I10P. Finally, when the incomplete RT9P
begins with four pitch-classes, <9, 10,2, 8>, these four pitch-classes correspond to order
numbers 0-3 only in terms of pitch-class content, and they completely deviate from the
row ordering.
As a counterpoint to row-family IV, row-family III appears weakly represented
by a row segment at m. 73 in the flute and by an aggregate at mm. 67-71 in the left hand
that, by reiterating pitch-classes 2 and 3 (indicated in square brackets on Example 3.6a),
emphasizes one of the boundary dyads of row-family III. It is only at the outset of the
main row unfolding (mm. 64-66) that row-family III asserts itself a bit more forcefully,
as RI9P constitutes a counterpoint to T3P (Example 3.6b). The identity of RI9P,
however, appears only partially represented by order numbers 0-3 at the end. While the
upper two voices at the outset (mm. 64-65) emphasize the pitch-class content of the
boundary dyads common to all for forms of row-family III, their deviation from the row
113
ordering fails to point to any specific row-form.
The opposition between row-families III and IV is disrupted at the end of the
second section that interweaves row-family IV with two new row-families, I and II
(Example 3.7). Each row-family is represented by a specific row segment or by a group
of pitch-classes whose identity, however, becomes ambiguous. Row-family I is
suggested only by three pitch-classes that correspond to order-numbers 0-2 of I7P
(Example 3.7b). Row-family II is initially represented by contiguous row segments in
the flute (mm. 75-77), which correspond to order numbers 2-4 and order numbers 7-H.
of T,P (Example 3.7b). The association with a specific row-form, however, immediately
becomes subverted, as the boundary dyads of I2P (mm. 77-78) complement the preceding
segments of TjP. Similarly, row-family IV also reveals an ambiguous row identity.
When a broken chord at m. 78 (right hand) is partitioned into a vertical dyad and a
vertical trichord at m. 79 (left hand), the resulting vertical sonority of the two hands
suggests contiguous segments of T9P; that is, the two vertical pentachords that combine
the two hands comprise respectively order numbers 2-6 and order numbers 7-JJL- Such an
emergent row identity, however, conflicts with the initial dyad at m. 75, which
corresponds to order numbers 0-i of I10P instead.
Example 3.7. Interwoven row families I, II, and IV
114
cresc
i< • E
cresc.
115
When the last section (mm. 80-96) successively unfolds several row-forms,
individual row members are distributed in different contrapuntal voices to subvert
deliberately the row ordering (Example 3.8). Initially at mm. 80-87, members of two
successive row-forms, InP and T4P, constitute a two-part counterpoint, and they
consistently project a dyadic partitioning, as adjacent dyads of the twelve-tone row,
marked by slurs, alternate between the two hands of the piano. While InP expands
considerably by repeating itself in part (indicated in a dotted box on Example 3.8) and in
full from the right hand to the flute (mm. 82-84), the dyadic row partitioning remains
intact; even the single flute passage that unfolds order numbers 2-11 articulates the
dyadic partitioning by slurs and by the same kinetic patterns as those applied in the initial
InP. Once the last two pitch-classes of IUP is configured as a grace note and a principal
note, however, the consistent partitioning scheme begins to dissipate, since the
ornamented note is connected with another note to unfold order numbers 0,1, and 3 of
T4P at the end of the flute passage. While order numbers 2 and 3 of T4P are aligned to
follow the row ordering, order numbers 2 and 4 are slurred as a linear dyad for the right
hand (m. 85). The partitioning into adjacent dyads, therefore, applies only to order
numbers 5-10, allowing order number H. to be excluded.
116
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117
Although row members are distributed in different contrapuntal voices to subvert
the row ordering, one may yet trace the row ordering inasmuch as partitioned dyads are
linearly configured. When partitioned dyads are vertically configured and distributed in
different contrapuntal voices (mm. 88-92), however, the row ordering becomes harder to
recognize. The subversion of the row ordering progresses from the linear configuration
to the vertical configuration of partitioned dyads. This progression appears deliberate,
not abrupt, through a transitional unfolding of I3P at mm. 86-88. At the outset of I3P.
linear dyads are opposed to each other to constitute a two-part counterpoint. Once order
number 3 (left hand) is aligned with order number 4 (right hand) at the end of m. 97,
however, vertical dyads begin to prevail. Throughout such a gradual progression from
the linear to vertical configuration, the row ordering becomes considerably obscured;
order numbers 2 and 3 swap their positions in the two initial linear dyads, and so do order
numbers 7 and 8 at the end of m. 88 to create a vertical dyad that consists of
discontinuous order numbers 7 and 9. Nonetheless, while the unfolding of I3P remains
short of establishing a norm for vertical configuration, its intersection with the following
I,P substantiates the transitional function of I3P; order number 7 and 9 aside, order
numbers K) and U. of I3P are embedded in the lowermost voice at m. 88 as order
numbers 2 and 4 of I,P.
The row ordering becomes further obscured as segments of different row-forms
are combined to complete an aggregate. Such an aggregate completion is initially
suggested by the contrapuntal passage that connects T6P with I10P at mm. 91-92. There
are three different row segments manifested to complete an aggregate, which include
118
order numbers 8-11 of T6P, order numbers (M and 4-5 of I3P, and order numbers 0-J. and
5-6 of I6P (indicated in dotted circles on Example 3.8). Except for the segment of T6P.
the row segments unfold discontinuous dyads, but each dyad itself comprises adjacent
members of the row, still maintaining the principle of dyadic row partitioning.
It is only at the end (mm. 94-96) that the hitherto established norm of dyadic
partitioning becomes completely subverted. There are two aggregates manifested around
a core row segment that comprises order numbers 7-11 of I2P in the right hand (mm. 95-
96). This core row segment is complemented in the flute by a continuous segment that
comprises order numbers 0-6 of RTSP with a minor adjustment; that is, the pitch-class for
order number 3 is substituted (indicated by a circle on Example 3.8) to avoid doubling
pitch-class 4, which appears as order number 8 of I2P. In contrast, the core row segment
is complemented in the left hand by a discontinuous segment that comprises order
numbers 3-4 and order numbers 7-11 of TnP.
119
Aggregates as Foreground Determinants
The subversion of the row ordering, which has deliberately been set up in the first
part, begins to surface prominently in the second part (Tres modere, presque lent, mm.
97-150). The twelve-tone row retreats to the background, framing the overall structure of
the second part. Within the twelve-tone frame, aggregates, freed from the strict row
ordering, come to claim a foreground structural determinant.
The twelve-tone row manifests itself in a series of trills (Example 3.9). The
semitone trills simultaneously unfold two different row-forms, T7P and TgP, bringing out
interval-class 1 projected in some of the adjacent dyads of the row, order numbers 0-1, 5-
6, and 10-11. The disposition of these trills, however, becomes anything but
proportionate. The single trill that simultaneously represents order number 0 of the two
row-forms expands so much as to constitute the first section (mm. 97 115). The
remaining trills that represent order numbers 1 - H quickly succeed one another to
constitute the second section (mm. 116-140). In contrast, the twelve-tone row suspends
its broad structural articulation in the last section (mm. 141-150) where a semitone trill
locally introduces a row segment that gradually emerges as a source from which to derive
an ordered pitch-class set for the subsequent third part (Tempo Scherzando).
Example 3.9. Row unfolding as a structural frame
120
mm. 97-115
e -
mm. 116-118 mm. 119-121 mm. 122-125 mm. 126-127
( I , . ) | j t ° IE ZIE
mm. 128-130
m
m. 130 Qva
O p ) mm. 131-133
m: p o -
m. 134
fcjxi. ( ^ ) m. 135
t ]o (!?•)
mm. 135-137
tjxi ( t ± ) mm. 138-139
• f a
121
Aggregate completion appears forcefully projected in the first section that
deliberately unfolds two simultaneous aggregates, one that is associated with the
prominent trill and the other with the flute solo passage. While the same trill recurs
throughout, it is either followed by a thirty-second-note flourish or accompanied by a
vertical chord to suggest aggregate completion (Example 3.10). The thirty-second-note
flourish projects an aggregate ~ though never completing it ~ by adding pitch-classes to
its initial form whenever it recurs, except at m. 114 (added pitch-classes indicated by a
circle on Example 3.10). Momentarily interrupted by vertical chords, the progression
from one flourish to another becomes deliberate, especially in the last two flourishes that
share the same pitch-classes; while the initial two pitch-classes remain intact to allude to
the trill, the remaining pitch-classes are rotated to amplify how closely these flourishes
are related.
Similarly, the vertical chords that accompany the trill deliberately suggest
aggregate completion. The progression from one vertical chord to another, momentarily
interrupted by the thirty-second-note flourish, becomes deliberate from the outset. While
the same vertical chord is repeated, two new pitch-classes, pitch-classes 5 and 6, are
added to suggest an emergent aggregate (Example 3.10b). Later, however, the vertical
chord either changes its content by substituting two new pitch-classes (pitch-classes 0
and 1, Example 3.10c) or completely disappears at the end (Example 3.1 Od). Still, the
two substitute pitch-classes enhance a deliberate aggregate completion. Moreover, pitch-
classes 5 and 6 continue to accompany the trill, maintaining a close relation between the
accompanying figures.
122
Example 3.10. Suggestion of aggregates a. mm. 97-100
n# 1 F —r—TTmTM- IY71
/ H s U , f i m ) UJ'>m
vs. *
^ ' i ' 1 —
k&-V 5 :
b. mm. 101-106
tr
i
mm. 107-113
4 d. mm. 114-115
^ It
123
In the flute passage, a simple, linear aggregate is surrounded by another that
emerges gradually (Example 3.11). The linear aggregate (mm 102-106), repeating pitch-
class 11, comprises three segments, the last of which corresponds to the ordered pitch-
class set explored later in the third, Tempo Scherzando part (indicated by an asterisk on
Example 3.11a).10 The surrounding aggregate is suggested at the outset of the flute
passage (mm. 98-100) in two unequal segments that, clearly separated by a rest, remain
short of completing an aggregate (Example 3.11b). These segments become transformed
into two new segments and complemented as an aggregate by the piano at mm. 108-113
(Example 3.11c). The new aggregate segments are clearly articulated and separated from
each other by a comma and a rest that enclose a new, intervening pitch-class 4. While the
ordering of all the pitch-classes of the initial segments remain intact, the new segments
share pitch-classes 2 and 1, and swap their positions that stem from the initial pitch-class
ordering. There is a minor adjustment to the ornamental figure by adding pitch-class 11
to pitch-class 2 at the outset of m. 108. This addition, however, not only enables the two
new segments to correspond to each other by beginning with the same pitch-class but
also outlines the juxtaposition of the initial segments.
10This correspondence was already noted by Mellot (1964,235) and Trenkamp (1973, 69). But the interpretation of this segment merely as a motif (Mellot) or as permutation of a row segment (Trenkamp) fails to take into account how deliberately a row segment (order numbers 5-U.) is contrapuntally set against an emerging pitch-class set, which will become definitive only at the outset of the Tempo Scherzando part, to suggest a close association between the twelve-tone row and the ordered pitch-class set explored in the third part.
124
Example 3.11. Gradual aggregate completion
a.
ft H ' f f l
6 it I T = 5 r t
- 3—1 —'3' *»/ poco sfz
:/^=- pz==-pp
< r > n f
jo subito poco piu f
i . ) J"1
108 mf 3
125
In the second section (mm. 116-140), a quick succession of different trills
globally articulates a structural frame based on the row unfolding. Often followed by a
thirty-second-note flourish, the semitone trills also locally underlines aggregate
completion at surface. While these trills are usually accompanied by row segments, the
deployment of row segments in particular is intended not to privilege the twelve-tone row
but to suggest deliberately the way in which a row segment serves as a basis for deriving
a pitch-class set explored in the third, Tempo Scherzzando part.
The deployment of row segments becomes systematic and deliberate at mm. 116-
134. The initial three trills are consistently accompanied by order numbers 0-4 of I6P
(mm. 116-118, right hand), RI8P (mm. 119-121, flute), and I4P (mm. 122-124, left hand).
While the row segment that accompanies the trill at mm. 126-127 momentarily becomes
reduced to including only order numbers 0-2 of I9P, the systematic use of order numbers
0-4 persists and even extends at mm. 131-134. Once order numbers 0-4 of I6P, which
start off by accompanying the semitone trill at mm. 116-118, are reintroduced at mm.
131-133 (left hand), they continue to expand by repeating order numbers 2-4 at m. 134
(left hand), rounding off the systematic accompaniment to the semitone trill.
126
Example 3.12. Transformation of a row segment
T5P: 5-11 •fjp w w w w w v
138 ^ >
vS-J C i ' ) - J 1 fi
1 ?
Prototypical pitch-class set explored in the third part
127
Such a consistent, systematic exploration of order numbers 0-4 deliberately
prepares for the projection of a complementary row segment that comprises order
numbers 5-U.. This projection does not aim at privileging the twelve-tone row. Rather,
it paradoxically opposes the previous row segment. When order numbers 5-H of TSP are
introduced at mm. 148-150 (flute), they complement the previous row segment only in
abstract (Example 3.12); that is, complementation applies not to the actual pitch-class
content but to order numbers alone. Instead of privileging the twelve-tone row, the
projection of order numbers 5-1.1 hints at how a row segment can transform into a pitch-
class set that will be explored in the third, Tempo Scherzando part. The complementary
row segment includes as its first six members all the pitch-classes that constitute an
prototypical pitch-class set explored in the third part. Moreover, the leaping figure of
interval-class 3 in the left hand (indicated in a circle on Example 3.12), which exactly
corresponds to the "motto" characteristic to the prototypical pitch-class set, may
foreshadow the transformation of a row segment into a differently ordered pitch-class set.
The association between the complementary row segment and the prototypical
pitch-class set gradually emerges in the last section of the second part through an intimate
counterpoint (Example 3.13). A complementary row segment, order numbers 5-11 of
TgP, begins with a semitone trill in the flute, the same way as previously at mm. 138-140,
but it overlaps with another row segment that comprises order numbers 0-6 of T3P.
128
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129
These overlapping row segments are preceded by what appears to approximate the
prototypical pitch-class set; its pitch-class ordering hardly conforms to that of the
prototypical pitch-class set, but its durational pattern adheres to that of the prototypical
pitch-class set. As a counterpoint to the overlapping row segments, this approximate set
(a on Example 3.13) continues to transform into another (b, b', c, and c'). While such a
transformation deliberately leads toward forming the prototypical pitch-class set, it
constitutes an intimate counterpoint to the overlapping row segments, suggesting how
closely the prototypical pitch-class set can be associated with order numbers 5-H. of the
row.
Stimuli for kinetic structure surface at the outset of the Sonatine — including the
introduction, first part, and second part ~ at two different levels that stem respectively
from the subversion of the row ordering and deliberate motivic transformation. An
immediate stimulus arises from the introduction (Tres liibrement - Lent, mm. 1-31) where
the bare suggestion of a twelve-tone row is abruptly subverted to create the dichotomy
between the ordered row and unordered aggregate. This dichotomy appears further
broadened in the first part (Rapide, mm. 32-96) and the second part (Tres modere,
presque lent, mm. 97-150). In the first part, the structural unfolding rests on the
succession of different row-forms, which is normally accompanied by aggregates. In the
second part, however, the twelve-tone row emerges as a background structural frame
within which aggregate completion surfaces prominently.
130
While the dichotomy between the twelve-tone row and aggregate generates an
immediate kinesis, constant motivic transformation suggests another dimension of
kinesis, which is conceived more broadly. Derivation of a principal motif is suggested
in the introduction whose boundaries project a single pentachord configured and disposed
in register alike. This motif associates itself ambiguously with the twelve-tone row at the
outset of the introduction and at the end of the second part. Its ambiguous association
with the twelve-tone row culminates in the third part of the Sonatine (Tempo Scherzando)
where it appears developed independently of the row. The emergence of such an
independent entity amplifies the subversion of the row ordering, which has deliberately
been initiated in the first two parts by gradually projecting aggregates as a foreground
musical event. It is only in the last part of the Sonatine (Tempo rapide, mm. 342-510)
that such a kinesis subsides, as the ordered pitch-class set, seemingly developed
independently of the row in the third part, becomes unambiguously related to the twelve-
tone row.
CHAPTER IV
INTEGRATION OF ORDERED DURATION AND PITCH-CLASS SETS:
TEMPO SCHERZANDO
Boulez's commentaries on the Tempo Scherzando are relatively extensive. He
describes its overall structure, suggests underlying principles of structural unfolding, and
makes some claim to its historical importance. As the Sonatine is structurally likened to
the four-movement scheme of a sonata that assimilates four parts of a sonata-allegro
form,1 the Tempo Scherzando is suggested to reveal a developmental, tripartite structure
that integrates the third movement of a sonata ~ a scherzo as the tempo marking indicates
-- with the development of a sonata-allegro form (Golea 1958, 38; Boulez 1976,27).
While Boulez stresses a contrast between free and strict rhythms and between
thematic and athematic developments as an important structural principle, his discussion
of the strict rhythm and athematic development points toward the Tempo Scherzando. As
opposed to Messiaen's eclectic assimilation of non-Western rhythm, Boulez expresses a
need to devise a new approach to rhythm, one that he applies in the Sonatine:
We have to invent our own rhythmic vocabulary in accordance with our own norms. In this sense, even in my earliest works, there is what one might call a contrast between free forms . . . and on the other hand extremely strict sections. This is something I still practice; it is one of my main ideas. Even in my first published composition, the Sonatina for flute and piano, there are certain passages
'Boulez considers the sonata-allegro form as consisting of four parts, which correspond to the first theme, the second theme, development, and recapitulation.
131
132
made up of elaborate, highly-developed rhythmic structures; these are still worked out in a simple way, being based on straightforward schemes and written in a classical style, but even so they are elaborated to the limit of their potential (Deliege 1975,13).
Such rhythmic exploration may well contribute to what Boulez dubs athematicism:
One finds [in the Sonatine]... developments based on characteristic motifs, deduced evidently from the twelve-tone row but maintaining still a thematic character; these developments are opposed to other developments that result from the encounter between the twelve-tone row and rhythmic cells, to put it differently, the beginning of athematicism (Golea 1958, 38-39).2
As an example of the athematic development, Boulez selects the last forty-three measures
of the Tempo Scherzando, marked as Subitement tempo rapide, in which rhythmic cells
are developed to generate a rhythmic canon that abandons the contrapuntal exploration of
pitch materials:
This is part of an athematic passage, where the development [of rhythmic cells] proceeds without the support of characteristic contrapuntal cells. We can see that the rhythmic cells are formed by a ternary rhythm in rational or irrational values . . . an embryonic rhythm suitable for multiple combinations (Boulez 1991, 52).
This athematic development, Boulez further maintains, captures the historical importance
of the Sonatine by articulating independent rhythmic structures upon twelve-tone
structures:
If you will, it is in this Sonatine that I have tried for the first time to articulate independent rhythmic structures, the possibilities of which Messiaen had shown to me, upon classical twelve-tone structures (Golea 1958, 39).3
2On y trouve . . . des developpements construits sur des motifs caracterises, deduits de la serie, evidement, mais conservant encore un caractere thematique; ces developpements s'opposent a d'autres developpements, resultant de la rencontre de serie et de cellules rythmiques, autrement dit, un commencement d'«athematisme».
3Si vous voulez, c'est dans cette sonatine que j'ai essaye pour la premiere fois d'articuler des structures rythmiques independantes, dont Messiaen m'avait revele les
inn 1JJ
Boulez scholars often recognize thematic/athematic contrast as a general
structural principle, but their accounts of this principle, as it applies to the Tempo
Scherzando, vary from one to another. George K. Mellot, for example, locates the
thematic/athematic contrast exclusively in the Subitement tempo rapide where row
segments interrupt a systematic deployment of rhythmic cells (1964,222). In contrast,
Wilma Anne Trenkamp argues that, while the first Tempo Scherzando explores thematic
and rhythmic variations, its return and the Subitement tempo rapide exemplify an
athematic development that assimilates free pitch and rigorous rhythmic organizations
(1973, 75). Furthermore, Gerald Bennett, who likens the tripartite structure of the Tempo
Scherzando to a three-strophe poem, suggests that all three sections of the Tempo
Scherzando reveal an extended rhythmic organization based on rhythmic cells (1986, 60-i
61). |
These different readings of the Tempo Scherzando result from diverse j
interpretations of Boulez's own account. Mellot attempts to elaborate on Boulez's
suggestion of the thematic/athematic contrast fcjy defining what "thematic" and
"athematic" may mean:
The "thematic" sections are those in which themes or motives are used in a more or less conventional maanner. These sections occupy the larger portion of the work. For the most part, thematic ideas are derived from the row, but there are exceptions . . . . "Athematic" parts are those where themes are absent (1964, 222). !
Based on this distinction, Mellot suggests that, except for the Subitement tempo rapide
and the intervening middle section, the Tempo Scherzando is entirely based on one pitch-
possibilites, sur des structures serielles classiques.
134
class motif that, according to him, is not derived from the row; the Subitement tempo
rapide, occasionally interrupted by ''thematic" row segments, continues to unfold a three-
part rhythmic canon that voids any explicit theme or motif (1964,235-242). Mellot's
account, however, remains merely descriptive. It is not entirely clear what to make of the
thematic/motivic usage "in a more or less conventional manner," which falls short of
explanation. His identification of a single motif in the Tempo Scherzando addresses, at
best, surface musical events from one measure to another; it remains as yet
unsubstantiated how such motivic usage contributes to the structural unfolding of the
Tempo Scherzando. Moreover, Mellot neglects to acknowledge that his discussion of the
rhythmic canon hinges on Boulez's own account, which was published as early as in
1948.4
In her dissertation (1973), Trenkamp also recognizes a single motivic usage but
her account considerably differs from that of Mellot. The predominant motif in the
Tempo Scherzando, Trenkamp argues, is derived from the permutation of a row segment,
and its usage is limited to the first Tempo Scherzando, while the return of the Tempo
Scherzando, as well as the Subitement tempo rapide, explores free pitch and rigid
rhythmic organizations (1973, 69, 75). In fact, Tenkamp observes a large-scale rhythmic
organization in the return of the Tempo Scherzando that she likens to a fourteenth-
4Boulez's account was initially included in the essay, "Propositions," which was originally published in the second issue of Polyphonie in 1948. Later, it was included in the collected essays entitled Releves d 'apprenti (Boulez 1966). These collected essays have been translated into English twice, by Herbert Weinstock (Boulez 1968) and by Stephen Walsh (Boulez 1991). Given the date of Mellot's dissertation (1964) and difficult access to the journal Polyphonie, Mellot may not have consulted Boulez's own account.
135
century isorhythmic motet. Since Trenkamp suggests that the first Tempo Scherzando
and its return are neither motivically nor rhythmically related, however, one must
conclude that what compells her to characterize the last Tempo Scherzando as "the return
to the scherzando" is limited to the identical tempo marking of the two sections.
Moreover, her account of the Subitement tempo rapide remains anything but substantive;
it illuminates neither the Subitement tempo rapide nor the Tempo scherzando to quote
simply Boulez's account that was known since 1948.
In his overview of Boulez's early oeuvre (1986), Bennett also recognizes a large-
scale rhythmic organization in the return of the Tempo Scherzando .5 Motivated by
Boulez's account of 1948 that describes a way to generate rhythms by combining
different rhythmic cells, Bennett suggests a hierarchical rhythmic organization. As the
basic element, Bennett argues, rhythmic cells are initially combined to generate rhythmic
motifs. Those rhythmic motifs are, then, combined to generate different rhythms.
Finally, those rhythms are combined to generate a large-scale rhythmic organization that
pervades the return of the Tempo Scherzando (1986,60-62). Such an insightful account,
however, remains anything but complete. As Bennett notes only three sections of the
Tempo Scherzando — the first Scherzando (strophe), its return (anti-strophe), and the
Subitement tempo rapide (epode) — he completely dismisses the intervening section that
separates the first Scherzando and its return. Although Bennett acknowledges variants of
the large-scale rhythmic organization in the return of the Tempo Scherzando, his account
5In his discussion of extended rhythmic structure, Bennett misrepresents the return of the Tempo Scherzando as the first Tempo Scherzando. His reference to the first strophe, in fact, address the return of the Tempo Scherzando.
136
is represented only by a diagram whose simplicity renders it unverifiable. Moreover, he
likens the first scherzando to its return by merely noting a similarity in texture and
rhythmic structure; one may wonder to what extent and in what way a similar rhythmic
structure is manifested in the first Tempo Scherzando and its return. Finally,
characterizing the Subitement tempo rapide as an epode, Bennett simply acknowledges
Boulez's account of 1948 as the authoritative one that requires no further explanation.
In this chapter, I shall examine ways in which pitch-class and rhythmic motifs
project themselves respectively as an ordered pitch-class set and as an ordered duration
set. In doing so, I shall clarify the relationship between the ordered pitch-class and
duration sets, which significantly contributes to the structural unfolding of the Tempo
Scherzando. In fact, the ordered pitch-class set fully realizes the potential motif, which
was ambiguously related to the row in the introduction. The absence of the row in the
Tempo Scherzando, however, suggests as if the ordered pitch class set were motivically
developed indepently of the row and, thereby, articulates the culmination of row
subversion that has gradually been enhanced in the previous parts of the Sonatine,
including the introduction, the first part, and the second part.
137
Overall Structure
The sectional structure of the Tempo Scherzando is represented by different
tempo markings and by the recall of earlier parts (Table 4.1).
Table 4.1. Sectional Structure of the Tempo Scherzando
Measures Tempo marking/Comments
151-184 Tempo Scherzando
185-221 Intervening section Recall of preceding parts
222-295 Tempo Scherzando
296-341 Subitement tempo rapide
The first Tempo Scherzando is separated from its return by an intervening section that
recalls many features of the preceding parts of the Sonatine, including the introduction
(Lent), the first part (Rapide), and the second part (Tres modere, presque lent). Despite
the absence of a specific tempo marking, a three-voice counterpoint (mm. 185-191),
which constantly alternates short and long durational values, projects a texture similar to
that of the first part (mm. 80-93). In contrast to such a straightforward recall, the
remaining section (mm. 192-221) combines distinct features of individual parts both
simultaneously and consecutively. At mm. 192-194, for example, the tempo marking of
the second part — Tres modere, presque lent — and its characteristic trill are
simultaneously combined with a contrapuntal texture from the first part; at mm. 195-200,
the characteristic trill of the second part is simultaneously combined with a tempo
138
marking reminiscent of the third part ~ Subito Tempo Scherzando — and with a
characteristic motif of the third part; at mm. 201-211, under the tempo marking of the
second part, a characteristic tremolo passage of the second part alternates with a
charateristic motif of the third part and with a row unfolding in a thirty-second-note
flourish reminiscent of the first part; finally, at mm. 212-221, a pervasively slow tempo
and flutter tonguing, both characteristic to the introduction, are combined with a
row/aggregate unfolding from the first part.
The Subitement tempo rapide projects a rhythmic canon in which, according to
Boulez, "the development [of rhythmic cells] proceeds without the support of
characteristic contrapuntal cells" (1991, 52). As a basis for this rhythmic canon, three
extended rhythms are generated by combining four different rhythmic cells (Example
4.1). Occasionally interrupted (mm. 314-317; mm 327-328), the three rhythms (indicated
as 1,2, and 3 on Example 4.1) and their retrogrades are combined to unfold a nearly
symmetrical structure (Table 4.2); this rhythmic canon is also interrupted subtly at the
outset (mm. 304-305), where the flute continues to unfold the patterned rhythm, while the
piano separates the ending of one rhythm from the beginning of the other by inserting
additional rhythmic cells, i (the right hand, mm. 304-305) and ii (the left hand, m. 304).
139
Example 4.1. Rhythmic cells and their combinations in the Subitement tempo rapide (mm. 296-339)
Four rhythmic cells
J — : / n / J J ) / * J i I 3 1
J • / 3 7 J 3 1'
j J 3 ? J 3 1"
J* J R ~ } I J 3
- 3 — — j
r~3 J I ) J 3 r~3
2 '
r~3 J J 3 J 3
2 " I 3 ~
r~i J J 3 n 3 I 3 1 r — 3 — I
J 3 J I J> J
3' I 3 1 I 3 i
J 3 J ? J
140
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141
Initially, this structure is suggested by the exact correspondence between the first
and the last, and between the second and the penultimate superpositions of three different
rhythms. But the remaining rhythmic superpositions are prevented from becoming
completely symmetrical. The contents of the matched rhytmic superpositions (indicated
by dotted lines on Table 4.2a) do not exactly correspond to each other because some of
the component rhythmic cells are retrograded and/or substituted; as shown in Example
4.1, the substitution of a rhythmic cell can be found in two variants of Rhythm 1 (1' and
1"), two variants of Rhythm 2 (2' and 2"), and one variant of Rhythm 3 (3'). In the
matched rhythmic superpositions at m. 301 and m. 329, for example, the flute rhythm
remains the same, while the left-hand rhythm is retrograded and the right-hand rhythm is
altered by substituting one of its component rhythmic cells. In the rhythmic
superpositions between m. 310 and m. 318, however, the component rhythmic cells are
neither retrograded nor substituted. Still, the simultaneous unfolding Rhythm 3 and its
retrograde (flute, m. 310) prevents a one-to-one correspondence.
Despite such variants, the content of each rhythmic superposition appears
virtually intact. In fact, the retrograde does not affect the intrinsic ordering of rhythmic
cells in each rhythm. Moreover, the substitution of one rhythmic cell for another proves
to be an insignificant factor because the substitute rhythmic cell is simply a retrograde of
the original rhythmic cell. Underneath these minor alterations there lies a background
symmetrical structure (Table 4.2b) in which not only the first half of the Subitement
tempo rapide is mirrored in the second half, but each half also projects its own
142
symmetry.6 Although the unfolding of the symmetrical structure is interrupted at mm.
314-317 and mm. 327-328, these interruptions pointedly articulate the symmetrical
structure; the first interruption at mm. 314-317 separates the first half of the symmetrical
structure from the second, while the second interruption at mm. 327-328 suggests an axis
around which the second half projects its own symmetry.
Integrated Set in the Tempo Scherzando
In the first Tempo Scherzando (mm. 151-184) and its return (mm. 222-293), a
particular sequence of durational values is correlated with a particular sequence of
interval-classes to project an integrated duration/pitch-class set upon which the
structural unfolding of the Tempo Scherzando is based.7 In this duration/pitch-class set
(Example 4.2) that integrates an ordered duration set,«, with an ordered pitch-class set,
H, the sequence of two sixteenth notes is always correlated with interval-class 3 and with
a repeated pitch-class; the sequence of an eighth note and a sixteenth note, on the other
hand, is correlated with interval-class 5.
6When Bennett characterizes the Subitement tempo rapide as tripartite, he seems to suggest the generation of organic structure: "As the Scherzo has three sections, so the epode is clearly tripartite So Boulez moves from the smallest motivic element to an entire movement, nesting smaller structures within ever large ones to form a unified whole" (1986, 61). Apparently he overlooks the symmetrical structure that underlies the Subitement tempo rapide.
7Since a particular durational value is not assigned to a particular pitch-class, the ordered duration set can be correlated with any forms of the orderd pitch-class set, which are equivalent under Tn or In.
143
Example 4.2. Integrated duration/pitch-class set in the Tempo scherzando
a!H
i t a / R H
144
The placement of a grace note, however, slightly alters the underlying principle. When
the duration/pitch-class set is retrograded (Example 4.2b), the grace note is attached,
not to the pitch-class 1 that it initially ornaments, but to the pitch-class 9 that initially
precedes the grace note. Thus, the ornamented durational sequence is correlated either
with interval-class 6, or with interval-class 2. Moreover, interval-class 6 may
additionally be correlated with three sixteenth notes, the last two of which repeat the
same pitch-class.
As shown in Example 4.3, many different subsets of the duration set a and its
retrograde are manifested throughout the Tempo Scherzando; this collection of duration
subsets does not exhaust the whole segmental possibility but empirically maps out the
duration subsets that pervade the Tempo Scherzando. Subsets b through h are first
introduced in the first Scherzando, while subsets / and j are first introduced in the return
of the Tempo Scherzando. Subsets d,f, and h cannot be properly retrograded because the
grace note ought to precede, not follow, the principal note. Moreover, subsets g and i
cannot be retrograded because they are symmetrical.
145
Example 4.3. Subsets of the duration set a and its retrograde
a
r~i . - j j
rrn
8
d
h
!<
j
z
rr~}
Ra
J
Rb
Re
f.n
Rc
Kj
n
f
*
146
According to the duration/pitch-class integration, pitch-class subsets can be
similalrly represented (Example 4.4). Since duration subsets from g to j are correlated
either with a repeated single pitch-class or with an interval-class, pitch-class subsets
considered here comprise an ordered pentachord, P', an ordered tetrachord, T\ an ordered
trichord, Tr, and their retrogrades; this collection also empirically maps out the pitch-
class subsets that pervade the Tempo Scherzando. While the subset pentachord and
tetrachord are correlated with their corresponding duration subsets, the subset trichord is
correlated with two different duration subsets, d and e. Nonetheless, one may find
exceptional the co-relationship between Trand Re, and between Rtr and e because the
principal pitch-class subsets are normally correlated with the principal duration subsets,
and the retrograde pitch-class subsets with the retrograde duration subsets.8
The norms of the integrated duration/pitch-class set are suggested as a reference
at the outset of the first Tempo Scherzando. In the rest of the first Tempo Scherzando,
however, the duration set a and its subsets are explored independently of the pitch-class
set H and its subsets. One may find an initial systematic use of the duration/pitch-class
set in the intervening middle section where the subset pentachord Pl serves as an
imitative subject for a three-part counterpoint. It is in the return of the Tempo
Scherzando that the duration/pitch-class set, as well as its subsets, is fully explored to
yield a broadly balanced duration/pitch-class structure.
8The subset trichords correlated respectively to d and Re are inversionally equivalent through index number (or sum of complementation) 9.
147
Example 4.4. Subsets of the pitch-class set H and its retrograde
v . it
f
148
The potential for exploring the duration/pitch-class set is deliberately suggested
in the first Tempo Scherzando (mm. 151-184). The first four measures (mm. 151 -154). as
shown in Example 4.5, unmistakably demonstrate the pitch-clas/duration set and its
retrograde in the flute and piano. In particular, the pitch-class ordering is uniquely
characterized at the outset by a 'motto' interval, interval-class 3, that can be formed only
once out of all fifteen possibilities combining any two pitch-classes of the ordered pitch-
class set.9 As an ordered set, the pitch-class set H is transformed through transposition,
inversion, and their retrogrades, while the duration set a is transformed only through
retrograde. For example, the pitch-class set T6H (the right hand at m. 151) is both
inverted and retrograded to unfold the subsequent RIUH, while the first duration set is
simply retrograded to unfold the second duration set.10
9This can be easily proven by the interval vector of this ordered pitch-class set, 421242. This interval vector represents, out of all fifteen possibilities, four possibilities producing interval-class 1 and 5, two possibilities producing interval-class 2, 4, and 6, but only one possibility producing interval-class 3. For a formal discussion about interval vector, see Forte (1973,15-21).
l0The value n of Tn represents the level of transposition, and the value n of In
represents the index number (or sum of complementation).
149
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150
Subsets of the duration/pitch-class set are also explored, but the pitch-class
sequence does not always correspond to the duration sequence as suggested by the norms
of the duration/pitch-class set. For example, when two subsets precede a complete set
in the flute (RT9H, mm. 151-153), the sequence of two sixteenth notes is only
exceptionally correlated to interval-class 4 (D4-F#4 at the outset of m. 152). In the left
hand of the piano, moreover, a group of sixteenth notes is frequently unfolded (m. 151,
m. 152, and m. 154) to undermine the suggested duration set. To be sure, the norms of
the integrated duration/pitch-class set remain merely suggested at the outset of the first
Tempo Scherzando.
The deviation from the norms of the duration/pitch-class set culminates in a
thirty-second-note flourish that frequently interrupts the unfolding of a complete
duration/pitch-class set and its subsets. Such interruptions in the flute and piano
articulate subdivisions of the first Tempo Scherzando (Table 4.3).
Table 4.3. Subdivision of the First Tempo Scherzando
Measures Remarks
151-162 Duration/pitch-class set & its subsets
162-163 Interrupted by a thirty-second-note flourish
164-176 Duration/pitch-class set & its subsets
177-178 Interrupted by a thirty-second-note flourish and a trill
178-184 Duration/pitch-class set & its subsets
The norms of the duration/pitch-class set, initially suggested as a reference at mm. 151-
154, are subverted in most of the first Tempo Scherzando. The ordered pitch-class set is
151
not always correlated with the ordered duration set. Rather, these two sets and their
subsets are often explored independently from each other.
The independent exploration of duration subsets may well be demonstrated when
the ordered pitch-class set H is delineated either by a non-corresponding duration set or
by the combination of different duration subsets (Example 4.6). In the flute at mm. 165-
166 (the first staff of Example 4.6), two diferent duration subsets, Re and/, are combined
to delineate a retrograde of the ordered pitch-class set, RT,H. When the duration set a is
completed in the flute at mm. 173-174 (the second staff of Example 4.6), it delineates,
not the intended pitch-classs set, but a retrograde thereof, RI2H. The exploration of
duration subsets becomes a little more complicated in the flute at mm. 155-156 (the
fourth staff of Example 4.6). Here, two duration subsets,/and d, are combined to unfold
six pitch classes, <10, 0,9, 5,11,4>, the ordering of which rotates the first three notes of
an ordered pitch-class set (RT9H, the third staff of Example 4.6) to the last three
positions. Moreover, when the same six pitch-classes are unfolded in the right hand of
the piano at mm. 169 (the fifth staff of Example 4.6), its ordering changes further by
retrograding the last three pitch-classes of the previous flute passage at mm. 155-156,
while maintaining the ordering of the first three pitch-classes intact. Finally, the norms
of the ordered pitch-class and duration sets are completely subverted in the flute at m.
171 (the last staff of Example 4.6) where any trace of the durational subset vanishes and
the original pitch-class ordering is scrambled. One may associate this scrambled six
pitch-classes with the ordered pitch-class set only as an unordered set, since both of them
belong to the same [0,1,2,3,7,8] set type.
152
Example 4.6. Independent exploration of duration subsets in the first Tempo Scherzando
RTiH fl.: mm. 165-166
te
Rc f
RI2H fl.: mm. 173-174 u * * -Jr h i *1:
RT9H fl.: mm. 153-154
£
ROTFL3RT9H fl.: mm. 155-156
rh.: m. 169
H
fl.: m. 171
153
When the ordered pitch-class set is partially represented by the subset pentachord
P'. the pitch-class ordering remains intact, but the corresponding duration subset b
becomes considerably varied (Example 4.7). The subset pentachord P4 is initially
suggested by separating the last pitch-class of the complete set T0H from its first five
pitch-classes correlated with the duration subset b (the first staff of Example 4.7). Soon
the suggestion of P4 is confirmed in the right hand of the piano at mm. 161-162 (the
second staff of Example 4.7), since the same five pitch-classes recur as an independent
entity, T0P4.
The duration subset b is a composite in which the duration subsets g and d are
embedded. In turn, the duration subset d is also a composite in which the duration subset
h is embeded. Thus, variants of the duration subset b can be produced by simply altering
any of these component subsets. For example, when two forms of the subset pentachord,
I7P' and I5P4, are overlapped (mm. 183-185, the third staff of Example 4.7), the first form
is correlated with the corresponding duration subset b, while the second form is
correlated with its variant, b\ In fact, the variant b' results from a chain reaction: the
modification initially takes place when a duration subset variant, h ' is created by simply
repeating the duration subset h\ the variant h', in turn, changes the duration subset d to a
variant, d'; and, finally, the variant d' changes the duration subset b to a variant, b'.
Other variants of the duration subset b result either from substituting the duration subset/
for the inherent subset d (b" in the fourth staff of Example 4.7) or from modifying the
inherent subset d by augmenting the principal durational values (b'" in the fifth staff of
Example 4.7).
Example 4.7. Exploration of the subset pentachord P'
154
ToP*
fl.: mm. 153-154
& 5
T o P 1
rh.: mm. 161-162
'J 7 ] >
hP' fl.: mm. 183-185
T 2 P ' rh.: m. 156
f UP* lh.: mm. 174-175
/ R O T L F i I u P ' ft hv. J W " ^ : — 4 * - ^ T7^
R O T L F i T n P ' rh. & l h . : m m . 171-172
/ "
m — . *
P 4 *
155
The pitch-class ordering appears dissipated in the piano at mm. 168-169 and mm.
171-172 (the last two staves of Example 4.7) where five pitch-classes are spread over
two-part counterpoint. This two-part counterpoint, however, partitions the subset
pentachord into an ordered trichord and an ordered dyad as its two independent voices.
Respectively delineated by the duration subsets/ and h', the ordered trichord and dyad
can be derived, not directly from the subset pentachord, but from a rotation thereof. At
mm. 168-169 (the sixth staff of Example 4.7), the subset pentachord (I,,P),
<11,8,10,4,9>, is transformed into its rotation (ROTLF,InP), <9,11,8,10,4>, by moving
the last pitch-class to the first position. Of this rotation, the first three pitch-classes
constitute the <9,11,8> trichord in the right hand, and the last two pitch-classes constitute
the <10,4> dyad in the left hand.
The same trichord/dyad partition recurs in two-part counterpoint at mm. 171-172
(the last staff of Example 4.7). The ordered trichord in the right hand, <1,11,2>, and the
ordered dyad in the left hand, <0,6>, are respectively delineated by the same duration
subsets/and h'. Inversionally related to their counterparts at mm. 168-169 ~ the
inversional relation being represented by the index number (or sum of complementation)
10 — the <1,11,2> trichord and the <0,6> dyad are also derived from a rotation of the
subset pentachord (ROTLF,TnP), <1,11,2,0,6>.
156
The partial representation of the ordered pitch-class set H becomes more
transparent when it involves an ordered subset tetrachord (Tl) as shown in Example 4.8.
It unfolds the first four pitch-classes of H, accompanied by the corresponding duration
subset d. Only a minor modification occurs to d in the left hand of the piano at mm. 179-
181 where the ornamented note is repeated either with the grace note to project d' or
without the grace note to project b (the third staff of Example 4.8).
In contrast, the partial representation of H becomes ambiguous when it involves
an ordered subset trichord (T1). At first, the <3,9,4> trichord (T3Tr) in the flute at mm.
152-153 (the fourth staff of Example 4.8) is derived from the order numbers 2-4 of T2H,
<2,5,3,9,4,10>, along with the corresponding duration subset e. When this trichord is
retrograded (the last staff of Example 4.8), the duration subset remains unchanged,
because the grace note cannot follow the principal note.
157
Example 4.8. Exploration of the subset tetrachord T' and trichord Tr
TNT1
fl.: m. 169
T0T' rh.: m. 175
T2T' lh.: mm. 179-181
_ -eg LJ?
N ^ ^ T3T
R
fl.: m. 152-153 i
EJtf
!>3
*
rh.: m. 158-159
E # E fir JF
RT6TR RT4T
R-
158
An ambiguity may arise, however, as the same ordered trichord can be derived from the
last three pitch-classes of I,H, <11,8,10,4,9,3>, though the three pitch-classes, <4,9,3>,
should be retrograded. Indeed, such ambiguity stems from pitch-class invariance in the
last four pitch-classes of certain forms of H (Table 4.4).11
Table 4. Pithc-Class Invariance in Order Numbers 2-5 of H
The independent exploration of duration subsets may be further exemplified in a
'rhythmic counterpoint' where a duration subset does not necessarily delineate a pitch-
class subset (Example 4.9). At mm. 155-162, the pitch-class set and its subsets are
sporadically projected 1) to enclose a passage in the flute (ROTFL3RT9H at mm. 155-
156 and T0P' at mm. 161-162), 2) to initiate a passage in the left-hand of the piano (T,T'
at m. 155), or 3) to unfold a quasi sequence in the right hand of the piano (RT6Tr at m.
158, RT4Tr at m. 159, and IsT
r at m. 162). In contrast, a limited number of duration
subsets, as well as the duration set a itself, are constantly deployed ~ regardless of
whether they delineate pitch-class subsets or not — to yield a broad durational pattern in
each voice. Although the duration set a is varied at the outset (a', mm 155-156) and
temporarily interrupted by a thirty-second-note flourish at m. 157, it is most constantly
"As an unordered set, these four pitch-classes can be mapped into themselves under certain transposition and inversion because they belong to a symmetrical set type [0,1,6,7].
159
manifested in the left-hand of the piano to constitute a broad durational pattern.
Likewise, although the duration subset b is exceptionally extended by repeating its first
two members at m 159, it still delineates the durational pattern of the flute at mm. 157-
162. In contrast, the durational pattern of the right hand, which breaks up at m. 157 and
at mm. 160-161, is constituted, not by a single durations subset, but by two different
duration subsets, b and d. For the second half of the durational pattern, on the one hand,
the duration subset d is repeated at mm. 158-159. For the first half of the durational
pattern, on the other hand, the repetition of the duration subset b is subtly disguised: at
the ouset of m. 155, two of the duration set b are overlapped; and at the end of m. 156, all
four principal durational values of b are equalized to produce the variant b
160
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The entries of individual voices are subtly offset to project this three-part
durational counterpoint, one that coheres with the hierarchical relationship between the
ordered duration set a and the principal duration subsets utilized in this counterpoint, b,
d, and g. Such hierarchy is exemplified in the left hand of the piano at mm. 156-158
where dotted slurs unmistakably distinguish individual duration segments. At the lowest
level, three duration subsets ~ g, d, and g ~ are combined to constitute the duration set a.
At the middle level where two duration subsets — g and d — are combined to constitute b,
the duration set a is manifested by the combination of b and g. Although the broad
durational pattern differs from one voice to another, the close interchange of the duration
subsets among the three voices projects a durational counterpoint analogous to an
imitative pitch-class counterpoint.
162
Fully integrated duration/pitch-class set
Before the return of the Tempo Scherzando, the integrated duration/pitch-class
set has already been systematically explored, though briefly, in the intervening section
that separates the first Tempo Scherzando from its return. At mm. 195-200 (Example
4.10), the pitch-class subset P', accompanied by its corresponding duration subset b.
serves as an imitative subject for a three-part counterpoint. When the initial right-hand
subject, T0P', returns to the flute at m. 199-200, its entry is deliberately articulated by a
preceding pentachord. Although this pentachord is disguised as a pitch-class set type
different from P', it not only shares all pitch classes with the subsequent T0P', except for
one, but it is also delineated by the same duration subset b.n The ready transformation of
this pentachord into T0P\ which requires only a single pitch-class substitution and a
minor permutation, deliberately articulates the return of T0P' as a closure of the three-part
counterpoint.13
12While P' belongs to the [0,1,2,3,7] set type, the pentachord at m. 198 belongs to the [0,1,2,4,8] set type.
l3This transformation may have already been suggested earlier in the flute at mm. 159-162 (Example 4.8). When the same ordered pentachord, <11,3,2,1,7>, is unfolded at mm. 159-160, the initial two pitch-classes are repeated — marked by separate dotted slurs — to suggest a further partition of the pentachord into a dyad, <11,3> and a trichord, <2,1,7>. By substituting pitch-class 0 for pitch-class 11 in the initial dyad and rotating the firs pitch-class to the last position (ROTFLj)in the subsequent trichord, the ordered pentachord, <11,3,2,1,7>, can be transformed into T0P
l, <0,3,1,7,2> at mm. 198-200.
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The ordered pitch-class subset Tr appears more consistently explored at mm. 284-
292. Accompanied by the duration subset e, this ordered trichord is constantly paired
with its retrograde to serve as an imitative subject for a three-part counterpoint (Example
4.11). In this counterpoint, a true imitation is manifested, not by the corresponding
entries of the three voices, but by the isomorphic transformation of Tr. The three voices
do not literally imitate one another. The flute begins with the retrogrades of the pitch-
class and duration subsets, RT0Tr and Re, while the two hands of the piano begin with
the principal subset trichords ~ T2Tr in the right hand and T3T
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accompanied by the retrograde of the duration subset Re. Nonetheless, an isomorphism
emerges as various forms of Tr are consistently transformed into one another in all three
voices (Table 4.5).
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According to the transformational scheme, the three-part counterpoint can be
divided into two subsections (separated by a dotted line on Table 4.5). In the first half,
two forms of Tr in the initial pair are consistently transformed into those in the
subsequent pair through T5. At the same time, in each pair, the initial form of Tr is
consistently transformed into the subsequent form through RT2. In the second half of the
counterpoint, however, the transformation scheme is subtly altered. While the
transformation of the initial pair into the subsequent one can be represented by RT4,
individual forms of Tr are matched differently. That is, under the same operation RT4,
the first member of the initial pair is transformed into the second member of the
subsequent pair, and the second member of the initial pair into the first member of the
subsequent pair. Such isomorphic transformations break down at the end of the left hand
where an unexpected form of Tr, <0,7,1>, is substituted for the expected <0,5,11>. Since
this substitution takes place at the end of the three-part imitative counterpoint, it may
well suggest a beginning of a transition to the subsequent Subitement tempo rapide.
It is in the return of the Tempo Scherzando proper (mm. 222-284) that the
integrated duration/pitch-class set and its subsets are combined to yield a broadly
balanced duration/pitch-class structure. Momentarily interrupted by a flutter tonguing
flute passage with chordal piano accompaniment at mm. 252-259, the return of the
Tempo Scherzando can be divided into two sections (mm. 222-251; mm. 259-284). Both
sections hierarchically structure their broad rhythmic organization by repeating a few
number of rhythmic units that combine in various ways the ordered duration set with its
subsets.
168
The repetition of rhythmic units becomes most evident in the first section
(Exampe 4.12) where the repetitive pattern articulates a further structural division. As
shown in Example 12a, the first half of the flute passage is delineated by the repetition of
a rhythmic unit, x, and its retrograde, Rx. Rhythmic units are considerably lengthened in
the second half, however, where two rhythmic units, y and Ry, are unfolded only once in
turn. Ry, in particular, is further lengthened by repetition toward the end.14 In the right
hand of the piano (Example 4.12b), the repetition of r and its retrograde Rr takes place in
the second half, while two contrasting rhythmic units,/? and q are combined with their
retrogrades to delineate the first half of the right-hand passage. Although p, q, and their
retrogrades are also combined to delineate the first half of the left-hand passage
(Example 4.12c), the positions o f p and q, and of Rp and Rq, are swapped. Nonetheless,
such broadly balanced rhythmic organization appears to dissipate in the second half
where the initial rhythmic unit q is considerably extended.
14A durational value is sometimes replaced by an equivalent rest and vice versa. For example, the ornamented dotted eighth note at m. 225 is replaced by an ornamented sixteenth note and an eighth-note rest at m 230-231; and the sixteenth note and sixteenth-note rest at m. 243 is replaced by an eighth note at m. 246.
169
Example 4.12a. Repetition of rhythmic units in the flute, mm. 222-252
222
i f j ^ j b Nj. s j .!
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230
J73 \i rnnnnrfifii Rx Rx —
238 ^ * n
245
n ?\rm inrr^.ra Ry
170
Example 4.12b. Repetition of rhythmic units in the right hand of the piano, mm. 223-252
223
i j t ? I \ I s u n n\JFTI p , q : ,
231
t T T J I 3 M I * H I J 3 3 3 J 3 I J -
Rp Rq
239
n i rt? j i /ij *rz ijm j^jumnn
245
n n U U 3 J 3 I rm \jzn^n ? ft ? n i j ^ j 3 ? Rr Rr
171
Example 4.12c. Repetition of rhythmic units in the left hand of the piano, mm. 223-252
223
q , p
231
IJ3333J3IJ 3 ? ? / 3 ? t f T r r ~ r a / l ? I J 3 ? J 3 ? l Rq ' Rp —•
239
/ f l J T O l l / T 7 J I J 3 S 7 f i / S / 9 . 0 1 3
247
j i j } j j f n J 3 1 / 1 ? -T31? r z ? I ? / 3 ? / 3 Rq
172
The rhythmic organization tends to be more broadly conceived in the second
section of the return of the Tempo Scherzando (mm. 259-284). Its structure can be
further subdivided around a midpoint according to the way in which rhythmic units are
unfolded. In the flute (Example 4.13a), the outer boundary is delineated by / and its
retrograde Rl, suggesting a symmetrical structure, although Rl becomes short of
completely retrograding /; the first three pitch-classes of / are absent in Rl. In contrast,
the inner boundary is delineated by two contrasting rhythmic units, the ending of the first
half by m, and the beginning of the second half by n. The subdivision of the second
section becomes most apparent in the right hand of the piano (Example 4.13b) where a
single rhythmic unit, s, is repeated only once; the first unfolding of s corresponds to the
first half, and the second unfolding of s to the second half of the right-hand passage.
Although, as in the flute, four rhythmic units --/, Rl, m, and n —are deployed in the left
hand of the piano (Example 4.13c), a symmetrical structure is suggested, not in the outer
boundary, but in the inner boundary; the ending of the first half is delineated by Rl, and
the beginning of the second half by /.
173
Example 4.13a. Repetition of rhythmic units in the flute, mm. 259-284
259
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Example 4.13c. Repetition of rhythmic units in the left hand of the piano, mm. 259-284
259
\ n * ffi n nrn urn n
266
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176
Indeed, the two sections of the return of the Tempo Scherzando are articulated by
the way in which the same rhythmic units are explored in the two hands of the piano in
the first section and in the flute and the left hand of the piano in the second section. As
shown in Table 4.6, which represents the hierarchical rhythmic organization in the return
of the Tempo Scherzando, the exploration of the same rhythmic units projects two
contrasting voice exchanges. A local voice exchange is repeated in the first section to
delineate its first half. In the second section, however, a voice exchange is conceived
more globally across its subdivision to project a balanced overal structure.
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Such a broad rhythmic organization corresponds to a broad pitch-class
organization, enhancing the integration of the ordered duration set with the ordered pitch-
class set. As shown in Table 4.7, identical rhythmic units are always coupled with pitch-
class unfoldings that are equivalent through transposition, inversion, or their
retrogrades.15 Even when the rhythmic unit Ry considerably extends the rhythmic unit y
in the flute at mm. 238-252 (indicated by an asterisk in the top row of the second columm
on Table 4.7), their corresponding pitch-class unfoldings, T,,Py and RT,Py, come to
articulate deliberately the integration of the rhythmic units with their corresponding
pitch-class unfoldings. As shown in Example 4.14, the extension of Ry involves a
repetition of the duration subset b and transformation of the ordered pentachord I9P' into
I4P' through T7. By virtue of repeating the same duration subset and transforming one
form of an ordered ptich-class subset into another, the final duration subset b and its
corresponding pentachord I4P' (indicated in square brackets on Example 4.14), which do
not belong to the J?jVRT,P5 proper, become auxiliary, forcefully articulating pitch-class
1, the last member of RT,Py.
15Pitch-class unfoldings, which combine the ordered pitch-class set and its subsets, are represented by capital letter "P." Their association with rhythmic units are indicated by superscript letters following "P." Since the pitch-class unfoldings for the rhythmic units / and R/ are not equivalent through retrograde, however, they are distinguished respectively as P1 and P'.
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The balanced structure of the return of the Tempo Scherzando is further projected
by the way in which one pitch-class unfolding is transformed into another. The two
sections of the return of the Tempo Scherzando distinguish themselves from each other
by the extent to which an isomorphic transformation is applied to the pitch-class
unfoldings. In the first section (Table 4.8a), an isomorphic transformation locally takes
place between the two hands of the piano in its first half. Given the T8Pp/T,,Pq pair and
the RT,,Pp/RT10Pq pair, the first member of the initial pair is always transformed into the
first member of the subsequent pair through RT3, and the second member of the initial
pair into the second member of the subsequent pair through RTn. In the second half of
the first section, however, an isomorphic transformation, represented by RT2, takes place
among all three voices; even the pair of Pr forms are transformed alike into the pair of
RPr forms through RT2.
182
Table 4.8a. Isomorphic transformation of pitch-class unfoldings in the first section of the return of the Tempo Scherzando (mm. 222-251)
First half Second half
Fl.
T5F T,PX \
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R T |
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r RT,PY
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p
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183
Although an isomorphic transformation is also manifested in the second section of
the return of the Tempo Scherzando (Table 4.8b), the transfomation, involving inversion
alone, is globally conceived across the subdivision of the second section. As the
rhythmic unit Rl does not complete / in reverse order, missing the first three members of
/, the pitch-class unfoldings that correspond to Rl and / become inequivalent; for
example, T8P' at the beginning of the flute cannot be transformed into RT7P''. Thus, the
isomorphic transformation, represented by I5, comes to articulate the global voice
exchange between the flute and the left hand of the piano, projecting a balanced structure
in the second section of the return of the Tempo Scherzando.
Table 4.8b. Isomorphic transformation of pitch-class unfoldings in the second section of the return of the Tempo Scherzando (mm.259-284)
First half Second half
Fl.
Rh.
Lh.
T.P1 T6Pn TgP
n RT7P'
T7Ps
/ V I,0PS
I9pn RI10P' I9P' I„pm
h
184
The exploration of the integrated duration/pitch-class set distinguishes the two
Tempo Scherzando sections from the middle intervening section that recalls earlier parts.
It also distinguishes the two Tempo Scherzando sections from the concluding Subitement
tempo rapide section, which exclusively develops an extended 'cellular' rhythmic
structure. Such juxtaposition of contrasting elements becomes idiosyncratic even within
individual sections. In the Tempo Scherzando sections, for example, the unfolding of a
complete duration/pitch-class set and its subsets is interrupted by a thirty-second-note
flourish (the first Tempo Scherzando), or by a flutter tonguing flute passage with chordal
piano accompaniment (the return of the Tempo Scherzando) to articulate a structural
division. In the Subitement tempo rapide section, a similar structural articulation takes
place as the underlying 'cellular' rhythmic structure is interrupted to project an intrinsic
symmetry. In the middle section, however, the juxtaposition of contrasting elements goes
beyond structural articulation. It becomes a principal means of microstructural unfolding
that simultaneously combines distinct features of the earlier parts with one another;
structural articulation takes place when one combination of distinct features is shifted to
another.
Despite such idiosyncracies, the contrasting sections of the third part of the
Sonatine appear threaded together. The middle section explores the integrated
duration/pitch-class set in imitative counterpoint, foreshadowing its systematic usage in
the return of the Tempo Scherzando. Moreover, when the extended 'cellular' rhythm is
developed in the Subitment tempo rapide, it comes to amplify one of potential strategies
of the first Tempo Scherzando that explores an ordered pitch-class set and an ordered
185
duration set independently of each other. Indeed, when the middle section combines
distinct features of the earlier parts with those of the third part, it seems to suggest a unity
that may prevail across individual parts of the Sonatine. Nonetheless, it remains as yet
unclear how to associate the exploration of the integrated duration/pitch-class set in the
third part with the unfolding of the twelve-tone row that pervades the first part as a
principal structural component and the second part as a broad structural scaffold.
CHAPTER V
RELAXATION OF KINETIC STRUCTURE
The fourth part of the Sonatine (Tempo rapide) threads together all previous parts.
It recapitulates the first part by consecutively unfolding twelve-tone row-forms, which
correspond inversionally to those of the beginning of first part. At the conclusion of the
Sonatine, the fourth part also recalls distinct features of the introduction, the first part,
and the second part: textural characteristics of the introduction and the second part are
interwoven with the row unfolding of the first part. Most importantly, the fourth part
explains the relationship between the twelve-tone row and the pitch-class set H explored
in the third part, which is initially developed independently of the row.
The fourth part can be divided into three sections according to the way in which a
previous part is reintroduced (Table 5.1). In particular, each section distinctly explores
the twelve-tone row. As a recapitulation of the first part, the first section consecutively
unfolds different row-forms in the lower voice of the piano, while members of each row-
form are simultaneously combined with the upper voice of the piano to constitute an
aggregate. The second section also consecutively unfolds different row-forms, which
are, nonetheless, systematically truncated.
186
187
Table 5.1. Overall structure of the fourth part (Tempo Rapide)
Measures Comments
I. 342-361 Tempo rapide Recapitulation of the first part
362-378 Ralentir. . . encore plus large Row segment
II. 379-429 Tres rapide Interplay between row segments and
the ordered pitch-class set H from the third part
430-473 Tres progressivement de plus en plus rapide et tourbillonnant jusqu'a la mesure 474
Systematic truncation of the row
474-490 Extremement rapide
491-495 Precipite .. . Elagir peu a peu
III. 496-510 Tres modere, presque lent Recall of the introduction, the first part, and the second part
In contrast, the last section, a brief appendage, deploys only a single row-form partitioned
between the two hands of the piano until it is rounded off by the row-form that starts the
first part; while the recurrence of the partitioned row serves as an ostinato figure, the
ordering of the row segment in each hand is constantly retrograded.
Structural continuity is maintained across different sections. Toward the end of
the first section, for example, five adjacent pitch-classes of the twelve-tone row become
an independent entity that underlies a principal linear event. This row segment serves a
dual function here. On the one hand, its pitch-class ordering partially represents the
twelve-tone row, thereby maintaining a close link to the previous row unfolding. On the
188
other hand, its independent development paves the way for a deliberate transformation
into the pitch-class set H explored in the third part. As this transformation takes place at
the beginning of the second section, it prefigures an interplay between the twelve-tone
row and the pitch-class set H throughout the second section.
Twelve-Tone Usage
The row unfolding of the first section (mm. 342-361) exactly corresponds to the
row unfolding of the first section of the first part (mm. 33-52). As shown in Emample
5.1, three consecutive row-forms in the lower voice ~ I6P, I4P, and T9P ~ are
inversionally related to three consecutive row-forms at mm. 33-52 - -TSP, T7P, and I2P ~
through index number (or sum of complementation) 11. Such inversional relation is not
limited to the row unfolding, as it is further manifested in all the other pitch-classes,
except for one of the two grace notes at m. 357. Indeed, the parallelism becomes
unmistakable, since both sections share the same metric change, the same rhythm, and
the same placement of attacks and kinetic markings.
The first section of the fourth part, however, does not completely replicate the
first section of the first part. When three row-forms are consecutively unfolded in the
flute of the first section of the first part, some pitch-classes of the row are doubled in the
piano. Except for a grace note at m. 350, however, such row doubling is avoided in the
first section of the fourth part where the row unfolding takes place in the piano.
189
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As members of the twelve-tone row are combined with its accompanying pitch-classes to
constitute an aggregate in both sections, the way in which an aggregate is completed,
therefore, varies from one section to the other. In the first section of the first part, the
row unfolding in the flute is usually separated from the aggregate completion in the piano
because of the pitch-classes doubling in the piano. In the first section of the fourth part,
however, the row unfolding becomes an integral part of aggregate completion (indicated
in square boxes on Example 5.1) as members of the twelve-tone row are normally
combined with their accompanying pitch-classes to complete an aggregate.
In the second section of the fourth part, consecutive row unfolding constitutes an
entity that alternates either with a non-twelve-tone passage or with a simultaneous
multiple row unfolding. As shown in Emample 5.2, the consecutive row unfolding
comprises seven row-forms. Except for the first row-form, however, the subsequent row-
forms are not completed but systematically truncated. This truncation operates under two
principles: first, when adjacent row-forms are overlapped by one pitch-class, the number
of pitch-classes in the subsequent row-form is to be reduced by one; and, second, when
adjacent row-forms are overlapped by two pitch-classes that project interval-class 1, the
number of pitch-classes in the subsequent row-form is reduced by two.
191
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Throughout the second section, this systematically truncated row unfolding takes
place alternately in the flute and the piano, except for the incomplete, last unfolding at
mm. 480-487 (Table 5.2a). Since truncation as a process is reversed in the second and
fourth row unfoldings, a local symmetry emerges between the first and the second row
unfoldings, and between the third and fourth row unfoldings. At the same time, the
pairing of the first row unfolding with the second, and the third with the fourth, furnishes
a global symmetry whose axis falls at midpoint (m. 440). The local symmetries are set
up to contrast with each other through distinct transpositional relations (Table 5.2b).
While in each local symmetry all members of the first row unfolding are transformed into
those of the second row unfolding in reverse order, the transformation is performed
differently, through RT2 in the first local symmetry and through RT, in the second.
Despite such contrast, a global symmetry comes to be projected through inversional
relations between the two local symmetries (Table 5.2c). Indeed, the shift from a
retrograde-transposition to a retrograde-inversion at midpoint forcefully articulates an
axis around which a global symmetry is unfolded.1
'The incomplete last row unfolding rounds off the second section by bringing out the first row unfolding through T,.
193
Table 5.2a. Truncated row unfolding in the second section
A Solid line indicates a structural division. A dotted line indicates an interruption between the truncated row unfoldings.
*RI6P and T6P share a common-tone, pitch class 6, played by the flute. **The second pitch-class, 4, deviates from the pitch-class ordering of the row; it would have been pitch-class 3, which conforms to the pitch-class ordering of RT„P.
194
Table 5.2b. Local symmetry in the truncated row unfolding
Measures Row Unfolding
417-427 (Right Hand)
430-440 (Flute)
440-448 (Left Hand)
463-472 (Flute)
480-489 (Flute)
I4P:-T9P: IUP: T,P: I4P: I8P: T4P:
RT6P:_ RI]0P: -RI6P: -RT3P: -RI,P: -RTUP: RI6P -
T6P:-I.P: -T„P: I9P: -T6P: T2P: I«P:
RI7P: RT3P: RT7P: RI10P: RT0P: RI2P: RT7P:
I5P: T10P: I«P: T2P: I5P: I9P:
RT,
RT,
Table 5.2c. Global symmetry in the truncated row unfolding