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Transpositional Combination andAggregate Formation in Debussy MARK MCFARLAND The most conspicuous moments of chromaticism in Debussy's worksarefound in passages unified by sequential motion featuring semitonal transposition. This topic has been termed transpositional combination (TC) by Cohn, and it relates naturally to Debussy's method of aggregate formation. Such passages appear throughout Debussy's output, and serve a variety of functions. Debussy ex- panded his method of aggregate formationin his final works beyond TC to include inversional symmetry as well.The "mystery" behind the similarity in a chord progression common to works by both Debussy and Berg is foundin a traitboth worksshare: aggregate formation via TC. lthough the total chromatic, or aggregate, has been a subject traditionally associated with the Second Viennese School and later serial composers, aggre- gate formation in works of the Classical era has been the subject of several recent studies.' James Baker argues that music of this era provides an ideal environment in which to study the structural functions of chromaticism since there is frequently a strong contrast between diatonic and chromatic material. This distinction was significantly blurred in the 19th century, and thus the rhetorical power obtained through aggregate formation in the Classical style was lost. This, as Baker notes, was "a sacrifice Romantic composers were more than willing to make."2 While Baker's observation is cer- tainly true, it brings up an important question: is it possible for Romantic or even early 20th-century composers to em- ploy aggregate formation in a meaningful way, or was their harmonic language inherently too chromatic for such an event to be perceived? I explore this question through the analysis of works by a composer rarely, if ever, associated with the chromatic aggregate: Claude Debussy. Baker shows that unordered aggregate formation in the works of Mozart often unfolds over the course of a phrase or period. Burnett and O'Donnell limit themselves to the or- dered formation of the aggregate, and therefore the time span of its presentation is greatly expanded. Indeed, the or- dered aggregate they find in a Haydn Divertimento unfolds over nearly the entire sonata-form movement, albeit a short one.3 This study follows Baker's model for unordered aggre- gate formation, while at the same time remaining sensitive to the interval cycles that play a large role in Burnett and I See Baker 1992 and 1993; Burnett and O'Donnell 1996; and Burnett 1998. 2 Baker 1993, 307. 3 Another distinction between the methodology of these scholars lies in their treatment of chromatic pitches: Baker always counts the first ap- pearance of each chromatic pitch class, while Burnett and O'Donnell sometimes only count salient appearances-which may not be the ini- tial appearance-of the chromatic pitch classes. 187

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Transpositional Combination andAggregate Formation in Debussy

MARK MCFARLAND

The most conspicuous moments of chromaticism in Debussy's works are found in passages unified by sequential motion featuring semitonal transposition. This topic has been termed transpositional combination (TC) by Cohn, and it relates naturally to Debussy's method of aggregate formation. Such passages appear throughout Debussy's output, and serve a variety of functions. Debussy ex- panded his method of aggregate formation in his final works beyond TC to include inversional symmetry as well. The "mystery" behind the similarity in a chord progression common to works by both Debussy and Berg is found in a trait both works share: aggregate formation via TC.

lthough the total chromatic, or aggregate, has been a subject traditionally associated with the Second Viennese School and later serial composers, aggre-

gate formation in works of the Classical era has been the subject of several recent studies.' James Baker argues that music of this era provides an ideal environment in which to study the structural functions of chromaticism since there is frequently a strong contrast between diatonic and chromatic material. This distinction was significantly blurred in the 19th century, and thus the rhetorical power obtained through aggregate formation in the Classical style was lost. This, as Baker notes, was "a sacrifice Romantic composers were more than willing to make."2 While Baker's observation is cer- tainly true, it brings up an important question: is it possible for Romantic or even early 20th-century composers to em- ploy aggregate formation in a meaningful way, or was their

harmonic language inherently too chromatic for such an event to be perceived? I explore this question through the analysis of works by a composer rarely, if ever, associated with the chromatic aggregate: Claude Debussy.

Baker shows that unordered aggregate formation in the works of Mozart often unfolds over the course of a phrase or period. Burnett and O'Donnell limit themselves to the or- dered formation of the aggregate, and therefore the time span of its presentation is greatly expanded. Indeed, the or- dered aggregate they find in a Haydn Divertimento unfolds over nearly the entire sonata-form movement, albeit a short one.3 This study follows Baker's model for unordered aggre- gate formation, while at the same time remaining sensitive to the interval cycles that play a large role in Burnett and

I See Baker 1992 and 1993; Burnett and O'Donnell 1996; and Burnett 1998.

2 Baker 1993, 307.

3 Another distinction between the methodology of these scholars lies in their treatment of chromatic pitches: Baker always counts the first ap- pearance of each chromatic pitch class, while Burnett and O'Donnell sometimes only count salient appearances-which may not be the ini- tial appearance-of the chromatic pitch classes.

187

188 MUSIC THEORY SPECTRUM 27 (2005)

O'Donnell's work.4 The following analysis of a passage from Chopin not only demonstrates the methodology to be used in this study, but also reveals a conspicuous method of aggre- gate formation within the context of chromatic harmony.

Example 1 contains the final eight measures of Chopin's Nocturne, op. 9 no. 3. The first five measures of this passage employ a generally stepwise bass line that connects two dominant seventh chords. This passage is unremarkable in terms of its chromatic harmony: two applied dominants are heard; one resolves deceptively. Nevertheless, these applied dominants add three pitch classes (Fx, E#, and D?) to the B-major scale, thus forming in the opening five measures the aggregate minus only two pitch classes. However, these four measures sound far less chromatic than the melodic orna- mentation prolonging the structural dominant that ends this phrase. Once the 4-3 suspension within this chord is re- solved, a strictly sequential passage descends chromatically through almost two octaves and definitively forms the total chromatic. Chopin uses this passage to prolong a single har- mony, while the registral shift between the beginning and end of the sequence is traversed in a manner infinitely more elegant than would be achieved by an abrupt shift between the two extremes.

It is the formation of an aggregate within a single sequen- tial gesture that makes this chromatic flourish stand out so clearly from its harmonic background. This context is crucial in providing an answer to Baker's question cited above: while aggregate formation in a Mozart score is readily perceptible, the increased use of mixture in the 19th century blurs the dis- tinction between diatonic and chromatic harmony. Thus, in the Romantic era, aggregate formation within a gesture uni-

fied by sequential motion is necessary to achieve the same impact as aggregate formation within a phrase or period in Mozart.

While other methods of aggregate formation can be found in works of the 19th and early 20th century, a thorough investigation of this phenomenon would lie outside the scope of this study. The type of aggregate formation used in the Chopin example above was chosen specifically because it provides a clear link to the works of Debussy. It is well known that Debussy admired the works of Chopin, and a passage from the Danse Bohemienne of 1880 suggests that he used Chopin as a model in his earliest works (Example 2). Just prior to the return of the work's main theme, a sequence of chromatically-descending major triads prolongs the domi- nant in the service of a registral shift; the entire gesture forms the aggregate.

While Debussy quickly matured beyond modeling his music on that of Chopin, his works almost exclusively used the method of aggregate formation that he learned from the Polish composer. The technical nature of Debussy's aggre- gate formation developed beyond sequential transposition only in his late works, although the musical reasons for its appearance are varied. In this study, I categorize the numer- ous roles that aggregate formation plays in Debussy's works. I then conclude with an exploration into the central role that aggregate formation plays in several of Debussy's works composed between 1912-15 and the expanded means to achieve these results.

Although there are exceptions, Debussy's aggregate for- mation involves the systematic transposition of a given set class to form a larger pitch collection. Richard Cohn has ex- haustively explored this process, which he calls transposi- tional combination (TC).5 I adopt Cohn's theory of transpo- sitional combination in this study because it relates so naturally to Debussy's method of aggregate formation, and also because it makes aggregate formation explicit in analytic

4 Roy Howat has published an analysis of the prelude "Des pas sur la neige" that is identical in its methodology to that of Burnett and O'Donnell. D Aeolian is established in the opening measures of the work while the chromatic pitch classes are gradually added by ascend- ing perfect fifth, beginning with BT and ending with Eb. See Howat 1988, 83. 5 See Cohn 1986, 1988, 1991a, and 1991b.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY I89

EXAMPLE I. Chopin, Nocturne, op. 9, no. 3 (1830-32),final eight measures.

190 MUSIC THEORY SPECTRUM 27 (2005)

(0 4 7) T6 T5 T4 T3 T2 T1

Return of Main Theme

1 1 5 8 2 2 6 9 3 3 7 10 4 4 8 11 5 5 9 0 6 6 10 1

0 4 7

Complete Aggregate Formation

EXAMPLE 2. Debussy, Danse Bohimienne (1880), mm. 67-70.

terms. For example, the passage from the Danse Bohimienne transposes major triads from F# to C# by interval 11, or the descending chromatic scale.6 The aggregate formed in this passage is more easily seen using the numerical notation shown below the music in Example 2.7 There is a significant

6 Conversely, interval 1 indicates the ascending chromatic scale. 7 This notation is borrowed from Cohn and requires some explanation.

The set class that is transposed is found under the border on the bot-

tom of this equation, while the transposition levels at which the set class appears are found outside the border running up the left side. The "multiplication" of set class and transposition level appears inside the borders of the equation along the intersecting row. The resulting pitch classes from each multiplication are stacked row upon row, and the complete set class that is formed consists of the numbers that appear within the borders of the equation. Cohn uses the normal form of his set classes, which allows him to develop his theory in significant ways. The normal form of set classes will not be used in this study in order to represent the precise pitch classes that result from this multiplication.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 191

amount of duplication within this aggregate:8 pitch classes 1, 5, 6, 8, 9, and 10 each appear twice, while the remaining six pitch classes only appear once.9 This is common among ag- gregates heard within a tonal context, whether formed by TC in Debussy or by other means in earlier works.

As mentioned above, the passage cited in Example 2 oc- curs immediately before the return of the work's main theme. This is the role that aggregate formation plays most frequently in Debussy's mature works. Two factors help to explain this preference: first, aggregate formation implicitly signals a completion of some type, which may explain why Debussy places such events immediately before important structural divisions.10 Second, there is a satisfying release of tension as the aggregate gives way to the following theme, one that is often strictly diatonic in order to heighten the contrast across the formal boundary. Debussy was hardly the first composer to use aggregate formation in this way; Baker shows that Mozart frequently places aggregates near formal boundaries, and Burnett and O'Donnell find the same prac- tice in works by Haydn, Dittersdorf, J. C. Bach, C. P. E. Bach, Mozart, and Beethoven.

Debussy takes advantage of the instability generated by aggregate formation and also employs such passages as mod- ulating transitions. One such example is found in "Nuages" from the Trois Nocturnes (1897-99). The opening two-part

counterpoint gradually intensifies until a series of dominant major-ninth chords (B9, Ab9, Gk9, Eb9, C9, and Bb9) serves as a transition from the opening theme's B centricity to the Bb centricity of the following music. The previous example transposed a given set through an interval cycle to produce the chromatic aggregate, albeit with duplications. Debussy frequently uses interval cycles 1/11, 2/10, and 3/9, although it is not surprising that a composer with an aversion of for- mulae of any kind also uses transpositional combination without recourse to a specific interval cycle. Such is the case in "Nuages": the chords are transposed by only two intervals -9 and 10-although their asymmetrical distribution (9-10-9-9-10) eliminates the possibility of even a mixed interval cycle.11

This transitional measure appears later in "Nuages," once again serving as a modulatory link, now with the move- ment's central pentatonic theme. However, there are several important differences in this second presentation. First, the transitional measure appears twice, with a new whole-tone motive heard between these statements. Second, the reduc- tion in texture from five to four voices makes it more difficult to form an aggregate here, and indeed the note A is missing (see Example 3). Debussy occasionally omits a single note in his aggregate formation. For example, the missing pitch- class A would have appeared to complete the aggregate had all six chords in this passage been of the same quality. Instead, there are two qualities of chords within this passage: the lowest two notes (G#-D) of the first chord would need to be transposed up a semitone (A-D#) to make all of the chords identical.12 On the downbeat of the second transi- tional measure, the hypothetical A/D# in the lowest voices do not belong to the preceding whole-tone collection while the transposed notes G#/D do. The potential aggregate

8 In his study of Schoenberg's music, Ethan Haimo makes the distinc- tion between chromatic completion, where pitch classes can be re- peated before all twelve appear, and aggregate formation, in which there is no doubling. The term aggregate will be used in this study to describe both types of events since Debussy's use of aggregate forma- tion was entirely different-both technically and aesthetically-from that of Schoenberg. See Haimo 1990, 44 and 183.

9 Cohn's later writings on TC do not allow pitch-class duplications among the transpositionally-combined set classes. This study, like Cohn's disser- tation and first article, does not limit itself by this constraint.

Io It must be noted that Debussy occasionally begins a phrase by forming an aggregate. An instance of this from Debussy's string quartet is dis- cussed below in Example 4.

ii For more information on Debussy's use of interval cycles, see Gary Karpinski 1995, 183-206.

12 This imprecision in TC is indicated in the example by the appearance of pitch-class integers in parentheses.

192 MUSIC THEORY SPECTRUM 27 (2005)

(0 4 7 10) T11* T8 T6 T3 TO T10 (0 4 8) T2 T4 T2 (0 4 7 10) T11* T8 T6 T3 TO T10

Transition #1 Whole-Tone Motive Transition #2

10 10 2 5 8 10 10 2 5 8 0 0 4 7 10 0 0 4 7 10 3 3 7 10 1 3 3 7 10 1 6 6 10 1 4 2 2 6 10 6 6 10 1 4 8 8 0 3 6 4 4 8 0 8 8 0 3 6 11 11 (2) 6 (8) 2 2 6 10 11 11 (2) 6 (8)

0 4 7 10 0 4 8 0 4 7 10

Incomplete aggregate WT Coll. A Incomplete aggregate (missing pitch-class A) (0,2,4,6,8,10) (missing pitch-class A)

*Parentheses indicate imprecise TC

EXAMPLE 3. Debussy, "Nuages, "from the Trois Nocturnes (1897-99), No. 6 + 4-6.

formation here through unblemished TC is sacrificed in order to form a smooth motion between the whole-tone mo- tive and the transitional measure, a fluidity created by three common pitch classes rather than only one.

It is ironic that Debussy took such care to create smooth connections between these measures, since together they ac- tually serve as an unstable and modulating transition. The central whole-tone measure is slightly more stable than the outer measures, and thus the entire transition is not accom- plished in a single motion. Debussy's sensitivity to levels of instability often involves the juxtaposition of passages using symmetric scales-whether whole-tone, octatonic, or chro-

matic (presented as an aggregate formed through TC)-with the resolution of this instability brought about through the appearance of diatonic harmony.

The final movement of Debussy's string quartet of 1893 uses aggregate formation in combination with octatonic and whole-tone harmony to create an extended passage that is remarkable due to the careful control of its levels of instabil- ity over an extended phrase. This passage begins immedi- ately after the slow introduction, which concludes with something quite rare in Debussy's oeuvre: an authentic ca- dence. A motive featuring chromatically-descending minor thirds that encompass seven of the twelve chromatic pitch

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 193

classes first appears in the solo cello.13 The intensity quickly builds as the viola enters, and each of the two measures of this duet forms an aggregate. Once all four instruments enter, the intensity reaches its peak as aggregates are repeat- edly formed through the transposition of a fully-diminished seventh chord. This is shown in Example 4(a).14

The same motive is then heard with subtle modifications in order to accommodate its new setting in an octatonic con- text. These four measures consist of a single repetition of the motive, these first two measures being transposed up by a minor third; between the two statements of this motive, col- lection II is fully stated. This octatonic passage, shown in Example 4(b), is considerably less dense than the preceding aggregate formation, although it certainly does not represent the latter's resolution due to the harsh dissonance formed between the F of the first violin and the F# of the second vi- olin and viola.

The motive next appears close to its original form and harmonic setting, although it is heard at a single transposi- tion level and accompanied by fragments of the descending chromatic scale. Despite the reduction in density, this form of the motive is developed more systematically by TC and is able to form an aggregate within two measures (Example 4(c)). The motive is altered once again in the final three

measures: minor thirds are replaced with major thirds, and these intervals are transposed by both interval 10 and 11. Whole-tone collection A is formed by the initial transposi- tion of these thirds by interval 10, although the following transposition by interval 11 presents half of collection B.15 An uneasy whole-tone context is thus formed (Example 4(d)). This passage is followed by major thirds transposed strictly by interval 11, first by the viola and cello and then the cello alone (Example 4(e)). Within this concluding ges- ture an aggregate is formed minus a single note, G. The re- duction in texture and the rest that immediately precedes the final statement of the thirds strongly signals a reduction in instability. In fact, the five separate passages described above represent a gradual movement from instability towards sta- bility, one that begins after a rapid motion in the opposite direction as the quartet gradually enters. This entire passage serves to delay the introduction of the main theme of the movement, whose tonic pitch-class G completes the aggre- gate left incomplete in the previous measure.

As impressive as the careful handling of the relative insta- bility in this passage is, it also illustrates another important aspect of Debussy's aggregate formation. While the final measures complete an aggregate, these measures are the most stable before the return of diatonic harmony. Because the previous two instances of aggregate formation were both less stable than the final one, Debussy's aggregate formation need not always constitute a striking event in the musical ar- gument. Indeed, the final aggregate formed in this passage was actually more stable than the whole-tone and octatonic passages that preceded it. It was argued above that aggregate formation within a single gesture unified by sequential mo- tion is necessary in order for such an event to stand out from its harmonic background in 19th- and early 20th-century music. An addendum must be added to this rule, one that

13 An additional pitch class appears after the strict transposition of the minor third ends. A different reading of these opening measures is pos- sible, one that consistently treats the second minor third of each chro- matic descent as non-harmonic. This interpretation yields an octatonic reading of these opening measures, one that moves from collection II in measures 1-4 to collection III in measures 5-6. Throughout this article, I will follow van den Toorn's designations of the three octatonic collec- tions. The one that includes the semitone E-F is referred to as collec- tion I, the one that includes the semitone F-F#, collection II, and the last that includes F#-G, collection III. See van den Toorn 1983.

14 Note that this analysis discounts the motive's concluding diminished- seventh chord arpeggiation since this lies outside the TC organization of this passage and therefore stands apart from this unified gesture. The distinction made between TC and its ornamentation will be used throughout this study.

15 The whole-tone scale that includes the whole-step C-D will be re- ferred to as collection A, while the one that includes the whole-step C#-D# will be collection B.

194 MUSIC THEORY SPECTRUM 27 (2005)

Violin I "&

Violin II N

Viola

,!

in

L,,nbI,1 kI, P 4

Violoncello I l!I-.k

I - , , L1IEr"!'TTTTi•,IMwm -"

(03) T11 T10 T9 T8 (06) T2 T5 T1 T4 TO T3 Set 7-1 T8 Complete Aggregate

p p

-J I ., -.I

I

"

P 4 P P

p p (0693) T3 T6 T2 T5 T1 T4

Complete Aggregate

cello cello/viola duet quartet 3 3 9 4 4 10 1 7 0 0 6 1 1 7 10 4

8 8 11 4 4 10 5 5 11 2 8 9 9 0 1 1 7 2 2 8 11 5 10 10 1 5 5 11 6 6 0 3 9 11 11 2 2 2 8 3 3 9 0 6

0 3 0 6 0 6 9 3 Set 7-1 T6 Aggregate heard four times

(a) Debussy, String Quartet (1893), iv, no. 15-15+5

EXAMPLE 4

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 195

Octatonic Collection II (0 3 9 11) T6

Violin I

Violin II i? I

Viola ,op RIC%- --tr

tr-------------------_--- V ioloncello '

. v -

42 _ r- _h._ _---_ _ f______"

(03) TO T11 T9 (0 3 9 11) T9

, r- ?---.?-,.----- .

"- h•

k.., - ?

?-- "- h•- ----- ?

LL L-- -----.

(03) T3 T2 TO

Harmonic TC Melodic TC 0 0 3 2 2 5 3 3 6 9 9 0

9 9 0 6 8 11 11 2 6 6 9 3 5 0 0 3

0 3 9 11 0 3 Octatonic Collection II (5,6,8,9,11,0,2,3)

(b) Debussy, String Quartet (1893), iv, no. 15+6-15+9.

EXAMPLE 4. [continued]

196 MUSIC THEORY SPECTRUM 27 (2005)

Aggregate Completion (0 3) T10 T9 T8 T7 T6 T4 T3 T2 T1 TO

Violin I A%'0I61'"TI

Violin II

f 44 > > > > > > > > 4

Violin II

Viola__ _ _ _ _ _ _ _ _ _ _

4 " > > >

f 4 4 4

0 0 3 1 1 4 2 2 5 3 3 6 4 4 7 6 6 9 7 7 10 8 8 11 9 9 0 10 10 1

0 3

Complete Aggregate Formation

(c) Debussy, String Quartet (1893), iv, no. 15+10-15+11.

EXAMPLE 4. [continued]

accounts for the relative strength of the aggregate formation. In order to make the most striking effect, aggregate forma- tion should occur in the shortest time possible (a chord is felt more strongly than a sequence drawn out over a number of measures); it should involve the greatest number of musi- cal lines in the TC (it should transpose ninth chords, for ex- ample, rather than single intervals); it should feature trans-

position by interval 1/11 (since for Debussy the semitonal motion between chords most immediately invokes the inter- val that characterizes the aggregate); and it should duplicate the greatest number of pitch classes within the aggregate, preferably all to the same degree.

While each of the aggregates found in Classic era works by Baker and Burnett and O'Donnell play an important, and

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 197

Violin I

fUII 7dim.

ViolinI "_ _ _ _ _ di.W k

f f dm. Viola

-, 1&oncello i .

f f dim. (0 4 8) T6 T4 T3

3 3 7 11 4 4 8 0 6 6 10 2

0 4 8 Whole-Tone Collection A (0,2,4,6,8,10) + Half of Whole-Tone Collection B (3,7,11)

(d) Debussy, String Quartet (1893), iv, no. 15+12-15+13.5.

EXAMPLE 4. [continued]

even motivic role, this is not true in Debussy's oeuvre. Transpositional combination within a single gesture is such a common feature in Debussy's scores that the strength of the aggregate formation and the placement of this gesture within its musical context determine if it will be conspicuous or not. However, some of Debussy's last scores consistently use the chromatic aggregate in a conspicuous way, and such passages are often used as cadential gestures. Further, Debussy attaches importance to the distinction between complete and incomplete aggregates by sometimes with- holding pitch classes in order to delay the large-scale musical climax to coincide with true aggregate formation. These works, including Jeux (1912-13), "Feux d'artifice" from the Prl1udes Book II (1910-13), and the Etudes "Pour les cinq

doigts" and "Pour les huit doigts" (1915), represent a signifi- cant development in his use of aggregate formation. Each of these works will be discussed briefly in chronological order to show the development of Debussy's compositional thought.

The music heard before the curtain rises in Jeux is written in a three-part structure: a central scherzando section is framed by mysterious and languid music featuring whole- tone harmony. The scherzando section immediately presents two of the primary motives of the ballet; these are labeled A and B in Example 5.16 The second note of motive B is initially heard as a diatonic, rather than chromatic, upper neighbor,

16 Here and below I follow the motivic designations used in Pasler 1982.

198 MUSIC THEORY SPECTRUM 27 (2005)

Violin I 06-W f

Violin II 06

P

Viola

(08) T2 T6 T1 T5 TO T4

Violoncello _L__"_______-4___________________________

(0 4) T11 T10 T9 T8

Cello/Viola Cello Duet Solo 4 4 0 0 0 8 5 5 1 8 8 0 1 1 9 9 9 1 6 6 2 10 10 2 2 2 10 11 11 3

0 8 0 4

Aggregate with pitch-class G appearing as tonic in the following phrase

(e) Debussy, String Quartet (1893), iv, no. 15+13.5-15+15.

EXAMPLE 4. [continued]

making the TC in its first appearances imprecise. By no. 3, however, the motive is normalized into a descending chro- matic trichord, each member of which is embellished with a chromatic upper neighbor. Once this occurs, the harmonic support of this motive is transposed systematically. Within the four-measure gesture beginning at no. 3, TC in both the melody and its accompaniment form set-class 10-1 (the ag- gregate minus a [01] dyad). At no. 3+4 the harmonic accom- paniment inverts the motivic trichord from descending to

ascending,17 and the melody is transformed into a two- measure idea through the addition of a concluding chro- matic trichord. The statement of this two-measure idea forms the exact collection of pitch classes as the previous four-measure gesture. However, the missing pitch classes A6 and A? are supplied when the second measure of this idea is

17 This is indicated in the example by pitch-class integers appearing in italics.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 199

S (0 11 10) T22* T1

(0 1110) T! T1 TO Motive A Motive B Motive A Motive B Motive A Motive B Motive A

(0 1110) T2* (0 1110) T5I (0 11) T4I (0 11 10) T7I I IA L W

11 10) T 1

(0 1110)T1 (0 11) T2I (0 1110) T5I ' (0 11 10) TO

Motive B ,(Motive

B 'Motive A Motive B Motive A Motive B Motive A Motive B

(0 11 10) T2* T1 T1 (0 11 10) T

TO TO 7 011 10) T1

Set 10-1 T10/T7I

(0 1110) T4 T5 T1 T4 T5 T6

(0 11 10) TO S Motive B

(0 11 10) T7I T1I T4I T7I TI

(-.1 '(o 11 10) T8I

Set 10-1 T10/T7I Complete Aggregate Formation I Complete Aggregate Formation

no. 1 no. 2 no. 2+2 no. 2+4 5 5 4 3

2 (3) 1 0 1 1 0 11 2 (3) 1 0 2 (3) 1 0 1 1 0 11 0 0 11 10 1 1 0 11 1 1 0 11

0 11 10 0 11 10 0 11 10 0 11 10

EXAMPLE 5. Debussy,Jeux (1912-13), nos. 1-4 (reduction).

200 MUSIC THEORY SPECTRUM 27 (2005)

no. 3

7 7 6 5 5 5 4 3 4 4 3 2 2 1 1 1 0 11 0 0 11 10

0 11 10

Set-class 10-1

no. 3+4

5 5 4 3 4 4 3 2 1 1 0 11 0 0 11 10 10 10 9 8 7 7 6 5 4 4 3 2

0 11 10

Complete aggregate formation

no. 3+7 6 6 5 4 5 5 4 3 4 4 3 2 1 1 0 11 0 0 11 10 10 10 9 8 8 8 7 6 7 7 6 5 4 4 3 2

0 11 10

Complete aggregate formation

*Parentheses indicate imprecise TC Italics indicate inverted statement of melodic set

EXAMPLE 5. [continued]

repeated a semitone higher, thus completing the aggregate. The two-measure idea is then repeated at its original trans- position level, and the subsequent transposition of the sec- ond measure not once, but three times, further emphasizes the aggregate formation within this gesture. This is the cli- max of the ballet's introduction, one that is abruptly juxta- posed with the framing whole-tone music. Aggregate forma- tion therefore relates to the formal discontinuity here as it does elsewhere in the ballet, including the following section.

As the curtain rises "on an empty park," motives C and D are quickly introduced (see Example 6). Motive B disrupts the diatonic calm of these two motives at no. 6+5 by present- ing set-class 6-1[012345], which is possible now that its two intercalated chromatic trichords are related by a minor third. The motion between chromatic and diatonic music quickens as motives B and D alternate beginning at no. 7. Each of the first two statements of motive B and its TC accompaniment here forms an aggregate minus a single pitch class, E. This note does appear as the final melodic pitch in this gesture,

although the lack of the minor third immediately after this note and the increase in rhythmic activity encourages it to be heard as a passing note to the opening D# of motive D.18 The same alternation of motives appears in the following six measures, although a diatonic harmonization is main- tained throughout this passage. The final measures of this section beginning at no. 8 serve as its climax; motive B is ac- companied here by chromatically descending triads, and an

18 As indicated in Example 6, I analyze this measure as four descending statements of a chromatic trichord from To, T3, T7 and T10. The melodic pitch-class E is not part of any of these statements, and so set- class 11-1 is formed rather than the chromatic aggregate. However, it is possible to analyze motive B as set-class [0, 3] presented at T7, T6, T5, and (partially) at T4. This interpretation ignores the importance of the chromatic trichord in these measures, but yields a complete aggregate. Whichever interpretation is preferred by the reader, the smooth con- nection between motives B and D, formed by the continuation of the TC into the diatonic measure, significantly weakens the impact of these chromatic set classes.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 201

Motive B Set 6-1

(0 11 10)T10

Motive C Motive D (0 11 10) T7 Motive D

Motive B Motive D Motive B Motive D Motive B Motive D F (0 1110) T10 T10 T10

(0 11 10)T7 T7 T7

(0 1110) T3 T3

(0 11 10) TO TO II I

Aggregate Minus Pitch-class E Aggregate Minus Pitch-class E

Motive B (0 11 10) T10 T10 T10 ET10 T6 T2

(o 11 10) T7 T7 T7 T7 T3 T11

.

T

,(011109)TO

T8 T4

(0 11 10 9) T4 TO T8

Aggregate Aggregate Aggregate

EXAMPLE 6. Debussy, Jeux (1912-13), nos. 6+2-8+2.

202 MUSIC THEORY SPECTRUM 27 (2005)

no. 6+5 no. 7 10 10 9 8 7 7 6 5

10 10 9 8 3 3 2 1 7 7 6 5 0 0 11 10

0 11 10 0 11 10

Set 6-1[012345] Aggregate minus pitch-class E

no. 8 no. 8+1 no. 8+2 10 10 9 8 6 6 5 4 2 2 1 0 7 7 6 5 3 3 2 1 11 11 10 9 0 0 11 10 9 8 8 7 6 5 4 4 3 2 1 4 4 3 2 1 0 0 11 10 9 8 8 7 6 5

0 11 10 9 0 11 10 9 0 11 10 9

Complete Aggregate Formation

EXAMPLE 6. [continued]

aggregate is formed in each of these three measures.19 The conclusion of this gesture serves as the abrupt end of this section, which is then juxtaposed with new music accompa- nying the appearance of the tennis ball on stage.

Aggregate formation is used in this form-defining role again between nos. 51 and 78, from when the young man separates the heads of the two young girls to their triple kiss. At this point in the ballet, the jealousy of the three charac- ters has finally been overcome, and the stage directions tell us "everything encourages them to give free rein to their fancy." It is for this reason that the triple dance is the most continuous passage of the entire ballet. Thus, the aggregate

formations before nos. 53, 58-4, 65-4, and 70 do not create discontinuities within the score, but simply create formal punctuations within this extended passage. However, the final aggregate formation of the ballet is similar to those at its opening. In fact, three passages appear in quick succession just prior to no. 78 to mark decisively the arrival of the bal- let's climax with the triple kiss. The first two of these pas- sages each create an aggregate through the rapid juxtaposi- tion of complete diatonic collections-formed first in scalar runs (Example 7(a)), and then within a chain of chords (Ex- ample 7(b))-one an exact transposition of the other by semitone. The third passage uses smaller sets in its TC orga- nization and is missing the single note G# (Example 7(c)).20

19 The final melodic note of motive B is again heard as a passing tone since it is not fully incorporated into the TC organization of this mo- tive. Yet while the rhythm of the accompanying triads could cause the last of them to be labeled a passing chord, its notes are included in the aggregate tally because this final chord is fully incorporated into the TC organization of this gesture.

20 Debussy's initial version of this passage did not include this final passage of TC, and Diaghilev suggested that he rewrite it. Debussy obliged, and wrote to his publisher "the few measures requested by Diaghilev have obliged me to modify-quite happily-the end of Jeux

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 203

(0 2 3 5 7 8 10) To (02 3 5 7 8 10) T1

A-

_Ai _A l _

i.l

1 1 3 4 6 8 9 11 0 0 2 3 5 7 8 10

0 2 3 5 7 8 10

Juxtaposition of C and C# Aeolian

Complete Aggregate Formation

(a) Debussy, Jeux (1912-13), no. 75+4.

EXAMPLE 7

The release of passion in the triple kiss is reflected in an abrupt change of rhythm, texture, and motive, and the languorous presentation of motive C that begins on G# com- pletes this final aggregate.

The sections of the ballet analyzed above represent its opening and conclusion. This is not to imply that aggregate formation does not occur in the interior of the ballet; rather, stage action controls the formal organization of this music rather than the punctuating appearance of the total chro- matic. Stage action was not an issue in the next work that

Debussy composed, the piano prelude "Feux d'artifice," and aggregate formation and its deliberate suppression are con- sistently used to form the climaxes of this work and to create formal divisions.21

... it is better in place now and the voluptuousness oozes out freely (the Russians are like Syrian cats)." See Orledge 1982, 165.

21 Orledge explores the genesis of this ballet in detail. He writes that Debussy composed Jeux in July and August of 1912, made some revi- sions to its ending at the request of Diaghilev in September, and fin- ished its orchestration between March and April of the following year. See Orledge 1982, 164-65. The chronology of the piano preludes is much more difficult to determine. However, based on extent letters and manuscript evidence, Roy Howat writes that both "Feux d'artifice" and "Les tierce alternees" were the last of the preludes to be composed, at the latest by the beginning of 1913. See Howat 1985, xv.

204 MUSIC THEORY SPECTRUM 27 (2005)

(0 2 5 9) T6 T4 T7 T5

5 5 7 10 2 7 7 9 0 4 4 4 6 9 1 6 6 8 11 3

0 2 5 9

Juxtaposition of B and C Mixolydian Complete Aggregate Formation

(b) Debussy, Jeux (1912-13), no. 76.

EXAMPLE 7. [continued]

The introduction of this prelude features the ostinato F-G-A-Bb-Ab-Gk that creates set-class 6-1[012345], or exactly half of the aggregate. Because this ostinato is heard four times in each of the opening sixteen measures, "Feux d'artifice" begins with a chromatic blur that foreshadows complete aggregate formations later in the prelude.22 The

work's main theme appears soon after the double bar lines of measure 24 signal the end of the introduction. This theme opens with the germinal cell C-G (motive A) that is then embellished with an upper neighbor to form C-A-G (mo- tive B). The arpeggiation of a five-note chord G-BK-C- D-E forms the harmonic setting against which the theme is heard (see Example 8). Motive B and its harmonic accompa- niment are both developed via TC, each with slight changes. Motive B rotates its first note to its final position at the end of this passage, while only four of the five notes of the origi- nal arpeggio are transposed, the second statement in inverted form.23 In spite of these changes, the transpositional connec- tions among the three statements of this music are immedi- ately audible. Slight changes such as these are what keep in- terest in this musical machinery, which David Lewin has described as "almost too parodistic."24 These measures con- tain the first statement of aggregate completion in the pre- lude; this placement seems deliberate, for its only other ap- pearance occurs at the climax of the work in measures 85-87. The intervening phrases all end with cadences simi- larly formed through TC, yet with at least a single pitch class missing from the aggregate. The method by which Debussy withholds pitch classes from these cadences varies, as shown in the following analysis.

The next statement of the main theme withholds the most number of pitches from the chromatic aggregate; this is because the theme has been modified to fit into a whole- tone context (F-CW/F-Db-C6), and it is only transposed once by interval 2.25 Whole-tone collection B is fully stated, and thus half of the total chromatic is withheld in this 22 This opening ostinato, F-G-A-Bb-Ab-Gb, favors the application of IS

over TC since the temporal ordering of its notes suggests an ascending trichord followed by its descending inversion transposed up by a semi- tone. However, throughout "Feux d'artifice" there is a rise-and-fall mo- tion in the accompaniment with programmatic intent (namely, fire- works). Because the set classes that are presented in this manner all have the TC but not the IS property, the compositional surface that favors an IS reading is simply a byproduct of the motion used to depict the flight of the fireworks. Therefore I will use TC, rather than IS, in the analysis of this prelude. More on the distinction between the appli-

cation of either TC or IS is found in the discussion of the etude "Pour les huit doigts" below. For another interpretation of this ostinato, see Lewin 1993, 106.

23 This is indicated by integers in bold type in the example. 24 Lewin 1993, 102. 25 The harmonic accompaniment remains the same throughout this the-

matic statement.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 205

a --va--

- - - - - -

"Triple Kiss"

----------G as Centric Pitch

T34a (0 2 3 4) T

T7

T3 A

(0 2 3 4) T10 T7 \S

Aggregate Minus Pitch-class G#

3 3 5 6 7 1 1 3 4 5 10 10 0 1 2 7 7 9 10 11

0 2 3 4

Aggregate with pitch-class G# appearing as tonic in following phrase (c) Debussy, Jeux (1912-13), no. 77+4.

EXAMPLE 7. [continued]

phrase. The following phrase more clearly reveals Debussy's intent to subvert complete aggregate formation (see Ex- ample 9(a)). The main theme is again modified (B%-EB/ BW-F-EB), and, aside from the rotation within motive B, its subsequent development by TC (B%-G?-FR/G?-F?-B?) is not strict: in order to make this transposition exact, the first note of its initial presentation would need to be A rather than Bb. A similar problem is found in the harmonic support of this passage. The TC in the accompaniment differs from that of the melody: rather than rising strictly by semitone, a five-note subset of the original chord first descends by step; then, only two notes of this chord join the melody in its semitonal ascent. Voice leading is therefore less than obvious

in this passage. Nevertheless, it would have been easy for Debussy to transpose the Ab of the ascending chord in the final measure up by semitone. Not doing so makes the pitch- class A? the only note missing from the aggregate formation here.

The central scherzando section features a motive intro- duced in the preceding measures, one that is now developed through transpositional combination (see Example 9(b)). As in the previous phrase, a single pitch class is withheld from the formation of the total chromatic; the opening pitch-class G of measures 49-50 should be G# to make the TC strict, and it is the note G# that is missing from the aggregate here. The following diatonic phrase ends with clouded octatonic

206 MUSIC THEORY SPECTRUM 27 (2005)

(0 10 8 6 3) T4

va - - (0 10 8 6) T5I (0 10 8 6) T6 8va..

-- - ---- 29

(0 9 7) TO (0 9 7) T1 T2/ - _Motive B

Motive BTC R. H. TC 2 2 11 9 6 6 4 2 0 1 1 10 8 5 5 3 1 11 0 0 9 7 4 4 2 0 10 7

0 9 7 0 10 8 6 3

Complete Aggregate Formation Italics indicate inversion of melodic set ?Bold type indicates rotation of note within Motive B

EXAMPLE 8. Debussy, "Feux d'artifice"from Prdludes Book 1 (1910-13), mm. 27-30.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 207

8va -1 42

.

(07 10 4592) T9

(0 8 6) T9* "a1 44 T10 T11?

(0 7 10 2 4) T6 (0 3) T5I

Motive B TC L. H. TC 11 11 7 5 7 2 5 10 10 6 4 6 6 1 4 10 8 9 (10) 5 3 8 8 3 6 0 10 1 5

0 8 6 0 7 10 4 2 5 9

Aggregate minus pitch-class A *Parentheses indicate imprecise TC Italics indicate inversion of melodic set ?Bold type indicates rotation within motive B

(a) Debussy, "Feux d'artifice, "mm. 42-44.

EXAMPLE 9

208 MUSIC THEORY SPECTRUM 27 (2005)

49 (0 10) T8* T9 T8* T2 T8* T9 T8* T2

S(03 8)T TOT TO* T6 T* T T* T6

(0 2) T9 T10 T9 T3 T9 T10 T9 T3

Melodic TC R. H. Harmonic TC L. H. Harmonic TC 2 2 0 6 6 9 2 3 3 5 9 9 7 1 1 4 9 10 10 0 8 (7) 6 0 0 3 (7) 9 9 11

0 10 0 3 8 0 2 *Parentheses indicate imprecise TC

Aggregate minus pitch-class G#

(b) Debussy, "Feux d'artifice," mm. 49-50.

EXAMPLE 9. [continued]

harmony:26 in four separate measures, tritone-related major triads from different octatonic collections appear, failing to es- tablish accountability to a single collection and also falling just short of complete aggregate formation (see Example 9(c)).27

The following reappearance of the main theme in mea- sure 65 would seem to signal a return to aggregate forma- tion, as in its initial presentation. In this statement as well as the following three appearances of this theme, however, the concluding development is unaccompanied, and so even the extended TC heard at the end of the second and fourth of these phrases is insufficient to form aggregates. After deny- ing aggregate formation for so long, the climactic passage of the prelude is reached in measure 85: the motive from the scherzando section is reharmonized and transposed along interval 1 to form an aggregate, as do the following simulta- neous black- and white-note glissandi.

26 These measures are only influenced, rather than controlled, by octa- tonic harmony since no single collection is established. The cadenza is strictly octatonic, accountable to collection II. Despite the fact that these two passages are separated, there is an audible connection between them, one that moves from octatonic influence to control. The same type of motion is found in the preceding phrase, where whole-tone in- fluence eventually leads to whole-tone control. This harmonic motion strengthens the connection between these non-adjacent passages, a connection that invokes Edward T. Cone's concept of interlock. I ex- plore the application of Cone's theory of stratification-interlock- synthesis to Debussy's works in "Debussy: The Origins of a Method," Journal of Music Theory, forthcoming.

27 In order to highlight the black-versus-white idea that plays out over the course of this prelude, Debussy changes the quality of the opening triad

in these measures in order to contrast the white of C major with the black of E6 minor. This change is minor compared to the glissandi, not shown in the above example, which cover all the white keys in the first measure and all the black keys in the second. For more on this opposi- tion, see Howat 1988, 86-87.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 209

Octatonic Collection: III I III I III I III I

(0 4 7) TO T6 T4 T10 TO T6 T4 T10 T7 T3* T9 T7 T1 T3* T9 T7 Ti

z,'

di "A"

Aggregate Minus Pitch-classes E6 &A Aggregate Minus Pitch-class C White- and black-note glissandi at the beginning of each measure eliminated for clarity

mm. 61-62 7 7 11 2 10 10 2 5 4 4 8 11 6 6 10 1 0 0 4 7

0 4 7 Aggregate minus EB and A

mm. 63-64

1 1 5 8 7 7 11 2 9 9 1 4 3 3 (6) 10

0 4 7 Aggregate minus pitch-class C

*Parentheses indicate imprecise TC

(c) Debussy, "Feux d'artifice, " mm. 61-64.

EXAMPLE 9. [continued]

The final measures bring the prelude to rest in Dk, al- though the manner by which this is achieved is remarkable. Throughout the prelude, TC has clouded the establishment of a centric pitch class and has formed unstable sets at cli- mactic moments or cadences. In these final measures, how- ever, Debussy employs TC in precisely the opposite manner.

The bass pedals Db-A? are sustained until the penultimate measure, while above this pedal both the main theme and La Marseillaise are heard in C major. Just as La Marseillaise frag- ment ends, motive B is transposed up a semitone. Thus, TC eliminates the tension between these two competing pitch- class centers by bringing the tonality of both the bass pedal

2IO MUSIC THEORY SPECTRUM 27 (2005)

and the melody into agreement and establishing Db as the key of the prelude. This is an important point, as it suggests that Debussy used TC in "Feux d'artifice" to both form the chromatic aggregate at climaxes as well as to logically arrive at the final triadic sonority. It is this duality of purpose given to TC that imparts such power to the final measures.

Like "Feux d'artifice," the dtudes "Pour les cinq doigts" and "Pour les huit doigts" also consistently treat aggregate formation as cadential or climatic events. Yet both works, written only two years after the prelude, reveal Debussy's ex- ploration of new methods besides TC to create aggregates. In "Pour les cinq doigts," this is accomplished simply through the juxtaposition of two diatonic collections related by semitone. Because these juxtaposed diatonic collections are not related by transposition on the compositional surface, Debussy finally exceeded the bounds of TC that he had tested in previous works.28 Once this boundary was crossed, he began to explore new methods of aggregate formation in "Pour les huit doigts."

This study opens with pure diatonic writing in the first measure; by the second measure, Debussy begins to form large subsets of the aggregate, beginning with set-class 8-1 (an eight-note chromatic collection) followed by set-class 10-1 in the next measure. The alternation between diatonic and chromatic harmony continues, although the passage in measures 7-8 differs from the preceding chromatic material in an important way: both indicate the application of a dif- ferent analytic method. In measures 2-4, the aggregate sub- sets were formed by the semitonal transposition of the first beat to the second, albeit with slight modifications (see Example 10(a)).29 Thus, the compositional surface clearly favors the application of TC here. In measures 7-8, two ges-

tures create set-classes 8-1 and 7-1, but these sets are formed by adjacent pentachords (see Example 10(b)). Because of the contrary motion between the pentachord pairs, and the lack of a transpositional relation between the pentachords that form set class 8-1, the compositional surface favors the ap- plication of IS over TC.30 The following measure features two pentachords that are neither transpositionally nor inver- sionally related, a situation which is found again in measures 40-41 (Example 10(c)).31 The choice behind Debussy's se- lection of pentachords is nevertheless clear; together they form a tightly spaced subset of the aggregate, just as in the previous two chromatic passages. That these three set-class relations share a common compositional goal in this etude- while this goal of aggregate formation had previously been exclusively associated with TC-suggests that these new methods grow out of Debussy's use of TC.

Once inversional symmetry substitutes for transpositional combination to form aggregate subsets in measures 7-8, Debussy begins to explore IS in different harmonic settings. For example, Debussy links tetrachords to form scales; dia- tonic scales were formed in measures 5-6 and in the final measure of this opening section, while chromatic scales, rather than chromatic clusters, first appear in measures 10-11. Debussy fills out an octave here with both transpositionally- and inversionally-related tetrachords; the former measure

28 The passages in question include the juxtaposition of G and Ab major in mm. 42-44 to form the complete aggregate, and the final measures that feature incomplete aggregate formation with the juxtaposition of the Db-major scale and the closing C-major triads.

29 These imprecisions in TC are noted in the example: in mm. 2-3, pitch- class B is held as the opening note of each gesture, while in m. 4 the

change from minor to major pentachords is due to a play between black and white keys similar to that noted above in "Feux d'artifice."

30 IS refers to inversional symmetry, an analytic technique most often ap- plied to the music of Bart6k. IS implies a texture in which two voices mirror one another around a central pitch axis. See Perle 1955, Treitler 1959, and Antokoletz 1984.

31 The relation between the tetrachords in measure 9 is similar to the modification of TC found in measures 2-3. However, the tetrachords in measure 9 are not part of a transpositional process, and thus they appear to be neither transpositionally- nor inversionally-related. The passage in measures 40-41, which in no way can be related to TC, suggests that one criterion for tetrachord selection here is the ability to form aggre- gate subsets.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 2II

A

,(0

1197) T6 T5 (0 10 8 7) T3 T2* T1*

(0 2 4 6) T11 T10* (0 2 3 5) T8 T7* T6*

mm. 2-3: set-class 8-1 T m. 4: set-class 10-1 T L.H. R.H. L.H. R.H.

6 6 8 (10) 11 1 1 11 (10) 8 10 (11) 0 2 4 5 5 4 2 0 7 7 9 (11) 0 2 2 0 (11) 9 11 11 1 3 5 6 6 5 3 1 8 8 10 11 1 3 3 1 11 10

0 2 4 6 0 11 9 7 0 2 3 5 0 10 8 7

*Parentheses indicate imprecise TC

(a) Debussy, "Pour les huit doigts, "from Etudes Book II, mm. 2-4.

EXAMPLE IO

features a descending [0, 2, 3, 5] tetrachord and completes the octatonic scale, while the latter measure uses an ascend- ing [0, 2, 4, 5] tetrachord to form Messiaen's sixth mode of limited transposition.32

The next section of the work (mm. 13-41) begins with a two-note trill-like motive. This change in motive also coin- cides with a change in harmony, since neither scales nor tightly spaced aggregate subsets are initially formed. Instead, the trill motive is expanded from a major second through the minor and major thirds to a perfect fourth, and all four of these intervals are freely transposed and combined from one measure to the next. Because of this, neither TC nor IS seem appropriate for the analysis of these measures. However,

a large-scale voice-leading pattern, one highlighted by De- bussy's use of tenutos, shows that this music initially builds towards measures 24-25, both of which feature sets that are transpositionally and inversionally related (Example 11).33 A case could be made for the application of both theories: TC is uniquely capable of revealing the method by which the concluding whole-tone harmony is formed, while IS high- lights the axis of symmetry D/A1 in measure 25 (both notes of which are implied in the score), this tritone serving as ten- dency tones of Eb major, the ostensible key of this section.

32 In the "Technique of My Musical Language," Messiaen spells his sixth mode of limited transposition as C-D-E-F-F#-G#-A#-B-C. See Messiaen 1956, 62 and Example 350.

33 Measures 21-22 reach a climax with the opening chord of the final measure, one that shares both TC and IS properties. However, the next chord has neither property. The repetition of this music in mm. 23-24 adds a single accidental so that both final chords share the TC and IS properties. The harmonic goal of these final measures is found in the added accidental, one that highlights the dual property of its conclud- ing set-classes.

212 MUSIC THEORY SPECTRUM 27 (2005)

mm. 7-8: set-class 8-1 T5 axis 2/3-8/9 set-class 7-1 T8 axis 5/11

(b) Debussy, "Pour les huit doigts, "from Etudes Book II, mm. 7-8.

EXAMPLE IO. [continued]

Example 12 shows that the music then gradually builds to the climax of the first half of the study in measures 31-34: the first measure forms set 10-1, the next 11-1, and the fol- lowing measure begins the alternating black- and white-note glissandi that complete the aggregate.34

After a repetition of the work's opening material, the final section opens with three two-measure phrases (mm. 54-59) that reveal Debussy's remarkable control of aggregate forma- tion and represent his most thoroughgoing exploration of in-

versionally related sets. These six measures are the climax of the second half of the dtude, and each of its three two- measure phrases gradually decreases the level of chromatic saturation to balance the chromatic climax of mm. 31-34: the first two phrases complete the chromatic aggregate while the third phrase omits two pitch classes (E and B).35 Debussy's

34 Once the aggregate appears, a transition back to the return of the open- ing section begins. This transition uses the trill figure from the second section, although harmonically this music is closer to the opening sec- tion; it forms tightly-spaced subsets of the aggregate that increase in size from set-class 4-1 to set-class 8-1.

35 The decrease in chromatic saturation is even more gradual if the final B of measure 57 is read as a BM (as it appears in most scores). In this read- ing, the first phrase completes the aggregate, the second phrase omits a single pitch class (B), and the third phrase omits two pitch classes (E and B). The analysis above follows the Debussy Complete Edition, which confirms that there is no natural in front of the B of m. 56 in either the complete autograph manuscript or the first edition, and con- cludes that m. 56 is "probably a repeat of bar 54." See Helffer 1991, 109.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 213

m. 9: set-class 7-1 T1

mm. 40-41: set-class 8-1 T1

pp leggierissimo ancorapi? pp

(c) Debussy, "Pour les huit doigts, "from Etudes Book II, mm. 9, 40-41.

EXAMPLE 10. [continued]

control of the pitch-class content in these measures is matched by the consistency of the axes of symmetry that are projected. The four-note figures in each hand of the first phrase are inversionally-related for the first two beats of mea- sure 54, as are many of the tetrachord pairs in this passage. Inversional symmetry is clearly an important aspect of this music, although it is not a feature common to each of the tetrachord pairs.

The full extent to which IS controls these measures is re- vealed when trichords, rather than tetrachords, are analyzed. The opening note of each tetrachord pair is the same, separated by an octave; the lower note is emphasized by a tenuto line, marking a slight distinction between these initial octaves and the rest of the tetrachord. The following trichords,

with few exceptions, form set classes that are either transposi- tionally- and inversionally-related, or simply inversionally- related.36 While the harmonic goal of this passage is the axis of symmetry with the index number of 3,37 the axes leading to this climax lie along the diminished seventh chord F#-A- C-EB (see Example 13).38 These four pitch classes are used

36 The exceptions that do occur are placed exclusively on the second beat of each measure, as if to metrically de-emphasize their appearance.

37 More precisely, the goal here is an axis of symmetry with inversion around C#/D and G/G#.

38 This excludes the trichord pairs that are neither transpositionally- nor inversionally-related (m. 55, beat 2, and m. 58, beat 2), that have the TC property alone (m. 57, beat 2), or that have an odd-numbered axis (m. 55, beat 3).

214 MUSIC THEORY SPECTRUM 27 (2005)

m. 24/1: axis 0/1 - 6/7 m. 24/2: axis 5/11 m. 25: axis 2/8 m. 26: axis 8/2

(0 4) T8 T1 T6 TO (0 2) T10 T4

Whole-tone collection A

m. 24/1 24/2 25 1 1 5 0 0 4 4 4 6 8 8 0 6 6 10 10 10 0

0 4 0 4 0 2 Whole-tone collection A: 0 (2) 4 6 (8) 10

EXAMPLE II. Debussy, "Pour les huit doigts," mm. 21-25.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 215

Set-class 10-1 T4 Set-class 11-1 (missing pitch class 5)

(0 9) T1 TO (0 8) T8 T9I T10 T11I TO T1I T4 T3 T10 T11I TO TI T2 T3I

Total Chromatic

m. 31: set-class 10-1 T m. 32: set-class 11-1 T6 m. 34: chromatic aggregate

3 3 11 2 2 10

1 1 9 1 1 9 0 0 8 0 0 8 11 11 7 11 11 7 10 10 6 10 10 6

0 0 9 9 9 5 3 3 11 1 1 10 8 8 4 4 4 0 9 0 8 0 8

Italics indicate inverted statement of melodic set

EXAMPLE 12. Debussy, "Pour les huit doigts, " mm. 31-34.

216 MUSIC THEORY SPECTRUM 27 (2005)

mm. 54/1 & 56/1 mm. 54/1 & 56/2 mm. 54/1 & 56/3 m. 55/1 m. 55/2

axis: 3/9 0/6 3/9 3/9

I I

m. 55/3 m. 57/1 m. 57/2 m. 57/3 m. 58/1

axis: 3/9 - 4/10 3/9 3/ 0/6

m.58/2 m.58/3 m.59/1 m.59/2 59/

2 ais: /8 2/8

i 7 6

EXAMPLE 13. Debussy, "Pour les huit doigts, " mm. 54-59.

prominently as the opening notes of each tetrachord pair; in- deed, the third phrase begins with the motion C-E , the opening phrase arpeggiates from A to Eb, using D as a pass- ing note, while the second phrase arpeggiates through the entire chord, again using D as a passing note. These opening notes also display inversional symmetry in the third and final phrase; the last note D simultaneously completes the inver- sional symmetry of measures 58-59 (C-Ek-DW/ C-E&-D?)

while at the same time confirming the modulation from the first axis to the last.39

39 The pitch-class axes formed by the opening note of these tetrachords (1/2 and 7/8) overlap with, but do not exactly match, the axis formed by the trichord on the third beat of measure 59 (2/8). However, the overlap between these axes is represented by the D? mentioned in the analysis above.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 217

Despite the fact that Lockspeiser observed that "the abyss between Debussy and Schoenberg is indeed terrifying," the identification of aggregate formation in Debussy and the chronology of his works raise interesting questions regarding the artistic paths of the French composer and the Second Viennese School.40 Similarly, the appearance of both TC and IS in Debussy begs the question of whether the appear- ance of these properties in the works of Bart6k was the result of influence. As tantalizing as these questions are, they lie outside the scope of the current study. Yet aggregate forma- tion formed through TC can help to explain one stylistic issue involving the remarkable similarity between passages in "Pour la danseuse aux crotales" from the Six epigraphes an- tiques and the last of Berg's op. 2 lieder first identified by Stuckenschmidt (see Example 14).41 Berg's song was pub- lished in a volume of Der Blaue Reiter in 1912, while Debussy composed the passage in question only in 1914. In light of the numerous derogatory comments he made regarding German music, it is highly unlikely that Debussy was influ- enced by one of Schoenberg's students, as Stuckenschmidt suggests.42 Instead, the chord progression common to both works systematically transposes set [0, 6, 11) by interval 11 to form the chromatic aggregate.43

The mystery behind the similarity of these works is found in the trait they share: aggregate formation. A comparison

between both complete works reveals the radically different role the common progression plays in each: Debussy's aggre- gate appears only once in "Pour la danseuse aux crotales" and is used in the French composer's typical role as formal punc- tuation, and is thus unexceptional (having been composed just after both Jeux and "Feux d'artifice"). By contrast, Berg's aggregate is the product of the composer's first steps towards atonality by way of his systematic exploration of interval cy- cles in his final student works. Debussy's use of the aggregate does not reveal a hidden Germanophilic side to his art. Rather, it suggests that he instinctively viewed the aggregate as the harmonic totality implied by any diatonic or symmet- rical scale. Debussy's common placement of the aggregate at formal divisions supports this speculation; the fact that the Classical-era composers cited above used the aggregate in much the same way indicates that perhaps this musical in- stinct is universal.

The initial survey of Debussy's use of the aggregate formed through TC led to an investigation of the develop- ment of this idea in his late works. This topic was then offered as a possible explanation for the similarity between works by Debussy and Berg. While Debussy is generally regarded as the father of 20th-century music, this status rests primarily on his use of scales other than major and minor. In its identification of aggregate formation through both TC and IS in Debussy's works, it is hoped that this study has shown that the abyss that separates Debussy from his younger contemporaries is slightly smaller than once thought.

REFERENCES

Antokoletz, Elliott. 1984. The Music ofBela Bartdk. Berkeley, University of California Press.

Baker, James M. 1993. "Chromaticism in Classic Music," in Music Theory and the Exploration of the Past, ed. Christopher Hatch and David W. Bernstein. Chicago: University of Chicago Press, 233-307.

40 Lockspeiser 1965, 149-50. 41 Stuckenschmidt 1965, 453-59. 42 Howat proposes a common source for both passages in mm. 23-26 of

"Le gibet" from Ravel's Gaspard de la nuit (1908), which includes a nearly identical progression. This reading seems to make musical sense: it is well-known that Berg loved the music of the Impressionists, and the falling out between Debussy and Ravel was over which of the two composers was the first to use new harmonies in their work. Only the chronology of these works argues against this interpretation; it is un- clear whether Berg wrote "Warm die Ltifte" after hearing "Le gibet" since the exact dating of this Lied is unknown. See Howat 1988, 88.

43 Debussy's progression is one chord shorter and is missing pitch-class A, which was the root of the chord immediately preceding the progres- sion in question.

218 MUSIC THEORY SPECTRUM 27 (2005)

Poco rubato

&3 c3 3 3 S3 V r

A as centric pitch class . . (0 6 11) T8 T7 T6 T5 T4 T8 T7 T6 T5 T4

Aggregate minus pitch-class A

4 4 10 3 5 5 11 4 6 6 0 5 7 7 1 6 8 8 2 7

0 6 11

Aggregate with pitch-class A appearing as tonic in previous phrase

(a) Debussy, "Pour la danseuse aux crotales, "from Six Epigraphes Antiques (1914), mm. 29-32.

EXAMPLE 14

•-. 1992. "Chromaticism in Mozart's 'Jupiter' Sym-

phony," Mozart-Jahrbuch 1991, ed. Rudolph Angermiiller, Dietrich Berke, Ulrike Hofmann, and Wolfgang Rehm. Kassel: BRirenreiter, 1050-55.

Burnett, Henry. 1998. "Levels of Chromatic Ordering in the First Movements of Haydn's London Symphonies: A New Hypothesis." The International Journal of Musicology 7: 113-65.

Burnett, Henry and Shaugn O'Donnell. 1996. "Linear Ordering of the Chromatic Aggregate in Classical Sym- phonic Music." Music Theory Spectrum 18.1: 22-49.

Cohn, Richard. 1991a. "Properties and Generability of Transpositionally Invariant Sets." Journal ofMusic Theory 35.1/2: 1-32.

- . 1991b. "Bart6k's Octatonic Strategies: A Motivic Approach." Journal of the American Musicological Society 44.2: 262-300.

.. 1988. "Inversional Symmetry and Transpositional Combination in Bart6k." Music Theory Spectrum 10: 19-42.

.. 1986. Transpositional Combination in Twentieth- Century Music. Ph.D. dissertation, Eastman School of Music, University of Rochester.

Haimo, Ethan. 1990. Schoenberg's Serial Odyssey: The Evo- lution of his Twelve-Tone Method, 1914-1928. Oxford: Clarendon Press.

Helffer, Claude. 1991. "Critical Notes" in Etudes: Livre I & Livre II, Ser. 1, vol. 6 of the Oeuvres complates de Claude Debussy. Paris: Editions Costallat & Durand.

TRANSPOSITIONAL COMBINATION AND AGGREGATE FORMATION IN DEBUSSY 219

Der Ei- ne stirbt, da- ne-ben der An-dre lebt:

(0 6 11) T8 T7 T6 T5 T4 T3

8vb ------

3 3 9 2 4 4 10 3 5 5 11 4 6 6 0 5 7 7 1 6 8 8 2 7

0 6 11

Complete Aggregate Formation

(b) Berg, "Warm die Lfifte, " op. 2, no. 4, mm. 20-22.

EXAMPLE 14. [continued]

Howat, Roy. 1988. "Modes and Semitones in Debussy's Preludes and Elsewhere." Studies in Music 22: 81-104.

----. 1985. "Forward," in Preludespourpiano, Ser. 1, vol. 5 of the Oeuvres completes de Claude Debussy. Paris: Editions Costallat & Durand.

Karpinsky, Gary. 1995. "Structural Functions of the Interval Cycles in Early Twentieth-Century Music." International Journal of Musicology 4: 183-206.

Lewin, David. 1993. "A Transformational Basis for Form and Prolongation in Debussy's 'Feux d'artifice'," in Mu-

sical Form and Transformation. New Haven and London: Yale University Press.

Lockspeiser, Edward. 1965. "Quelques Aspects de la Psychologie de Debussy," in Debussy et I'Evolution de la Musique au XXe Sidcle, ed. Edith Weber. Paris: Editions du Centre National de la Recherche Scientifique, 149-50.

Messiaen, Olivier. 1956. The Technique of My Musical Language. Paris: Alphonse Leduc.

Orledge, Robert. 1982. Debussy and the Theatre. Cambridge: Cambridge University Press.

220 MUSIC THEORY SPECTRUM 27 (2005)

Pasler, Jann. 1982. "Debussy, Jeux: Playing with Time and Form." Nineteenth-Century Music 6.1: 60-75.

Perle, George. 1955. "Symmetrical Formations in the String Quartets of Bdla Bart6k." Music Review 16: 300-12.

Stuckenschmidt, Hans-Heinz. 1965. "Debussy or Berg?: The Mystery of a Chord Progression." Musical Quarterly 51.3: 453-59.

Treitler, Leo. 1959. "Harmonic Procedures in the Fourth Quartet of Bdla Bart6k." Journal of Music Theory 3/2: 292-98.

van den Toorn, Pieter. 1983. The Music of Igor Stravinsky. New Haven: Yale University Press.

Music Theory Spectrum, Vol. 27, Issue 2, pp. 187-220, ISSN 0195-6167, electronic ISSN 1533-8339. ? 2005 by The Society for Music Theory. All rights reserved. Please direct all requests for permission to photo- copy or reproduce article content through the University of California Press's Rights and Permissions website, at http://www.ucpress.edu/ journals/rights.htm.