marianne kielian-gilbert - relationships of symmetrical pitch-class sets and stravinsky's...

Relationships of Symmetrical Pitch-Class Sets and Stravinsky's Metaphor of PolarityAuthor(s): Marianne Kielian-GilbertSource: Perspectives of New Music, Vol. 21, No. 1/2 (Autumn, 1982 - Summer, 1983), pp. 209-240Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832874 .

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RELATIONSHIPS OF SYMMETRICAL PITCH-CLASS SETS AND STRAVINSKY'S METAPHOR OF POLARITY

MARIANNE KIELIAN-GILBERT

Stravinsky's well-known remarks on polar attraction in Poetics of Music delivered in French in 1939-40 at Harvard University concern both music in general and his own works:

Our chief concern is not so much what is known as tonality as what one might term the polar attraction of sound, of an interval, or even of a complex of tones... All music being nothing but a succession of impulses and repose, it is easy to see that the drawing together and separation of poles of attraction in a way determine the respiration of music.

In view of the fact that our poles of attraction are no longer within the closed system which was the diatonic system, we can bring the poles together without being compelled to conform to the exigencies of tonality.'

Stravinsky's metaphors of polarity are suggestive, yet puzzling, for according to Berger, "polarity" is a word "which cannot accurately be applied (as [Stravinsky] applies it) to one thing without its opposite."2 However, in the context of his works this idea of poles of attraction, or polarity, may be correlated with relationships of paired, transpositionally-related pitch-class structures. Moreover, his metaphor of "the drawing together and separation of poles of attraction" corresponds to changes in the transposition relationships of paired sets or of paired intervals within a set. These changes often feature a developmental "expansion" and "contraction" of the transposition distance of paired elements.

Implicit in the idea of a pole is the dynamic interaction of a pair of equivalent structures (e.g., either extremity of an axis of a sphere, or either of two terminals of an electric cell or the ends of a magnet). The idea of polarity has suggested a "gravitational pull" or a "tonality by assertion" to several writers3 as well as an "equilibrium" often discussed as a harmonic principle (e.g., Pousseur's idea of "maintained tension"4). In characterizing the tritone relationships of the "Petrouchka chord" Berger has proposed that a necessary condition of "polarity" is "the denial of priority to a single pitch-class precisely for the purpose of not deflecting from the priority of a



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whole complexe sonore."5 The aspect of the polarity of a complexe sonore to be developed here is the idea of an "intervallic equilibrium" achieved by a transposition pairing of "single poles" (equivalent interval structures) such that the priority status of one is closely associated with that of the other.

Several Russian and neoclassic works by Stravinsky are examined from the perspective of transposition pairings within or between small pitch-class sets, particularly four-note symmetrical sets.6 Four-note symmetrical pc sets in which two or more interval classes (ICs) occur doubly are featured prominently in all of the selected works (for example, in Three Pieces for String Quartet, No. 2, 0156 symmetrical sets are presented with pairs of IC5s transpositionally related by IC 1 (T1/11) (see Ex. 2A)). By employing symmetrical sets that contain paired interval classes, Stravinsky creates situations in which the pcs of paired structures are in "interval-class equilibrium" within their respective sets yet vary in priority within the surrounding musical context.

Complementation refers to the process of pairing entities on either side of a center of symmetry; complementary subsets of a symmetrical set are corresponding, inversionally-related subsets that balance each other about an axis of symmetry. The axial potential of the twelve pc domain has been described as "inversional balance" or "inversional complementation" and associated with the compositional procedures of Bart6k, Schoenberg, and Berg.7 As shown in Fig. lA, pcs are paired according to their equidistance (or often pitch/registral position) about a center of symmetry (R) which may be present or implied in a musical context. An alternative to the axial treatment of symmetrical sets, Stravinsky's treatment is non- complementary in nature in that transpositional, rather than inversional relationships are emphasized musically. As implied in Fig. IB the pairing of interval classes doubly within a pc set or the pairing of transpositionally-related pc sets is emphasized musically rather than center-of-symmetry relationships.

However, particular transposition pairings of equivalent interval structures favored by Stravinsky often musically do emphasize the "inversional complementation" of invariant pcs within a pair of pc sets as illustrated in Figs. 2A, B, & C. One or more invariant pcs occur in complementary positions in a pair of transpositionally- related pc sets. What is unusual in Stravinsky's works is the emphasis of inversionally-related pc subsets within transpositionally-related structures (see Fig. 2A). Since pcs in complementary positions within



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a series of transpositionally-paired sets are often invariant, connections can be made between pitch events not only through the transposition pairing of equivalent pc sets but also through the inversional pairing of subsets within those sets. The successive inversional complementation of invariant pcs within paired sets also produces a cyclic ordering of pitch classes that may correspond with the ordering of pitch events within a piece." Compositional examples of these aspects of "polarity as intervallic equilibrium" will now be briefly considered.

Three Pieces for String Quartet

All three pieces feature the pairing of IC5s at particular transposition levels and these T-levels are associated with events of distinct function in each piece. The functional treatment of four-note symmetrical sets containing paired IC5s suggests that an increase and decrease in transposition distance within and between paired structures is an analogue to Stravinsky's metaphor of "the drawing together and separation of poles of attraction." Each piece features a developmental "expansion" and "contraction" of interval-class distance between paired IC5s within four-note pc sets.

Ex. 1 illustrates beginning and ending pairings of IC5s in each piece and their position on a (7,6) interval cycle (an ascending alternation of intervals of 7 and 6 semitones). Each of these collections contains dual IC5-related pcs paired transpositionally by IC 1; these sets have a "higher" hierarchical status in their respective pieces than those of pc sets featuring IC5s paired transpositionally by ICs 2, 3, and 4.

As shown in Fig. 3 there are only six four-note symmetrical sets that contain two IC5s. IC5s occur doubly within each of these sets at variable IC distances such that another IC is doubled as well. Of these sets, IC5 pairings characteristic of a single 0156 and a 0167 set are present throughout Three Pieces, No. 1, IC5 pairings within pc sets 0156, 0257, 0358, and 0127 are functionally differentiated in No. 2, and those of the remaining set, 0158, not emphasized in Nos. 1 or 2, are prominent in the middle section of No. 3. Three Pieces, No. 2: Ex. 2 illustrates these four-note symmetrical

pc sets in the second piece; within each unit of the piece a specific transposition pairing of IC5s is emphasized. In the course of the piece the transposition distance of IC5 pairs increases from IC1 to



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Vertical presentations of two 0156 symmetrical collections are shown in Ex. 2A; each collection presents two IC5-related pcs paired transpositionally by IC 1 (TI/11), A-E/Bb -F and A-E/Ab -E b (or two IC 1-related pcs paired by IC5 (T5/7)), and the two collections themselves are TI-related. The two c units, mm. 13-15 and 17-19, of Section I (mm. 1-24) linearly present a six-note collection of IC5- ICs 2 and 3 and returns again to IC 1. Symmetrical sets containing paired IC5s are shown for the units of the piece in the sketches of Ex. 3A and 3B. Within every discrete unit of the piece one or both of pcs A and Bb are invariant, contextually emphasized, and paired with IC5-related pcs. related pcs, D-A/Eb -Bb /E-B, which is T6-related to that of the vertically presented sets of the a units, Ab-Eb /A-E/Bb -F (see Ex. 2A). The first c unit features IC5 pairs E-B/Eb -Bb (T6-related to the first 0156 vertical collection of the piece, Bb -F/A-E); the c1 unit, mm. 17- 19, presents the previous linear IC5 pairs E-B/Eb -B overlapping with pairs A-D/Bb-E b (a collection T6-related to the second 0156 vertical collection of the piece, E bAb/A-E). The IC5 pairs A-D/Eb-B (0156) that occur within unit c1 also occur as a simultaneity in the e units (see Exs. 2A & 3A). In the two T6-related collections shown in Ex. 2A pcs A and B b are presented in complementary set positions (see blackened noteheads). Hence, in comparison to unit a, pcs A and Bb are inversionally IC5 paired in musical units c1 and e, while pc A is IC5 paired without pc Bb in unit b and pc Bb without pc A in units c and d (see Ex. 3A). The primary function of these units of Section I, mm. 1-24, is thus the presentation of two IC6-related "polar" complexes that exhibit IC5 pairing of pcs A and B b transpositionally at IC 1 and inversionally at IC6.

The 0127 pc collection of the d units is one of only two possible four-note symmetrical sets (the other, 0248) that contain a single pc center of symmetry within the set, in this case about IC6-related pcs F and C b (in all the d units, mm. 15-16, 20-21, and 60-61, pc F is registrally isolated in the cello and C b is positioned between Bb 3 and C4). In Sections I and IV (mm. 48-62) Stravinsky contrasts the axial relationships of the four-note Be-C b-C-F 0127 set of the d units with the other four-note symmetrical sets of the piece -the other sets do not contain a pc center of symmetry.

A 0156 symmetrical set represents one diatonic collection uniquely; 0156 pcs E-F-A-Bb of unit a occur uniquely in the F diatonic collection. In m. 58 the E-F-A-Bb collection concludes the a unit for the first time in the piece and is followed directly in m. 60 by unit d



213

with F2 in the cello.9 Here the d unit could be construed as realizing the diatonic implications of 0156 pcs E-F-A-B b of the a unit or paradoxically suggesting the final priority of a Bb-F "pole of attraction." A T6-related 0127 set E-F-F#-B, with invariant pcs F and Cb, also occurs earlier as the initial part of the "cadential" gestures of Section II, mm. 33-34, and Section III, mm. 43-45, which precedes the return of the 0156 E-F-A-B b set of the a unit in m. 48 (see Ex. 3B). In the first section of the piece the IC 1-related pcs B b-C b and C b-C within the initial 0127 d units refer to the ICl-related pc sets of units a, c, c1, and e, whereas, the 027-related pcs B b-C-F look ahead to the "ostinato" patterns of Section II, unit Fs (see Ex. 2C and Ex. 3B, mm. 26-28). With the melodic and harmonic materials of Sections II (mm. 25-

35) and III (mm. 35-47) the distance between IC5 pairs expands from IC 1 to ICs 2 and 3 in collections featuring pcs A and Bb. IC 1 (T1/11) is the distance of IC5 pairs of the 0156 collections of Section I (see mm. 1-24, Ex. 3A), IC2 (T2/10) the distance of IC5 pairs within the 0257 collections of Section II (see Exs. 2C and 3A), and IC3 (T3/9) the distance of IC5 pairs within the 0358 collections of Sections II and III (see Exs. 2D and 3B, units Gx, Hx1, and Hx). In Section IV (mm. 48-62) with the return of the 0156 collections of unit a the distance between IC5-related pcs is again IC1 (T1/11).

The expansion of transposition pairing to IC2 occurs in Section II, unit Fs (mm. 26-28), see Ex. 2C) where TI-related 0257 collections are superimposed and linearly presented in preparation for the slower, solo melodic passages of units G (mm. 31-35) and H (mm. 36-46).10 This T1 pairing of two 0257 collections (with T2-related IC5 pairs) is reminiscent of the T1 pairing of the 0156 vertical and linear collections of Section I. Another reference to Section I within Section II involves the complementary positions of ostinato pcs A b and B b (see violin II, mm. 26-28) in the 0257 set E-F-Ab-Bhb, a secondary set within the a units of Section I, and the 0257 set A-B b-Db-E b, a primary set of unit Fs.

Another expansion in transposition pairing occurs with the prominent 0358 collections, A #-C #-D# -F # (x) and C

,-E-F #-A (xl), that are presented linearly in units Gx of Section II, mm. 31-32, and Hx1 and Hx of Section III, mm. 36-47. The IC5 members of these 0358 collections are paired at T3, and the two 0358 collections themselves are T3-related. Moreover, a T3 relationship is characteristic of the IC5 C-G of the B unit, mm. 45-46, when paired with the IC5 A-E of the b unit of Section I, mm. 4-5 (C-G/A-E=0358), and also of



214

the unordered pc collections of units Gz (mm. 33-34) and Hz1 (mm. 43-44)."

The 0358 pc sets of the x units of Sections II and III (see Ex. 2D) contain invariant pcs F # and C #. The remaining IC5 pairs, D#-A # of the 0358 sets of Gx and Hx and A-E of Hx1, have figured previously as pairs in unit Fs of Section II and the IC6-related complexes of Section I. 025 subsets of each 0358 collection also occur prominently in Sections I and IV. An E6-F #5-A5 grace-note pattern with IC5 E-A occurs in the e units concluding Section I (see mm. 22 and 24) and the e unit of Section IV (mm. 53). An E b5-D b5-B b4 pattern with IC5 B -B concludes each of the c units in Section I and Section IV (see c2, m. 52, and c3, mm. 54-55).

The ordering of these 0358 sets as Gx (A#-C#-ID#-F#), Hx1 (C -E-F#-A), and Hx (A#-C#-D#-F #) in mm. 31-44 has the effect of enclosing the A-E of the Hx1 units by the A O-D # of the Gx and Hx units. This same enclosure of A-E by E b-B b is preserved in mm. 52- 55 of Section IV -unit e is now framed by units c2 and c3- and accounts for the particular reordering of the musical units of Section I in Section IV. In addition, the complex of the c2, e, and c3 units of Section IV, mm. 52-55, is framed by the IC6-related complex of the a units, mm. 48-51 and 56-58 (see Ex. 3B). The framing of IC5 A-E by IC5 E -B in Sections III and IV is thus part of a larger process consisting of the succession of the a units by the "polar" IC6-related complex of the c, c', and e units in Section I, the "mixing" of those complexes in Section II, and their separation in Sections III and IV.

Three Pieces, No. 3: Pc sets containing IC5-related pc pairs are differentiated transpositionally at ICs 1, 2, 3, and 6 in Three Pieces, No. 2; the remaining pc set 0158, containing IC5-related pcs at IC4 (T4/8), is prominent in the middle section of No. 3, unit d (mm. 27- 37), as shown in Ex. 4. These 0158 sets containing IC5 pairs at T4 again indicate an interval-class expansion from, and subsequent contraction to, the Ti-related pairs of the opening and closing a and a1 units, respectively. Moreover, the two 0156 collections of the closing unit a1 (mm. 45-46) are related by T4 and contain invariant pcs C and G of the initial unit a. Throughout the third piece intervallic formations that contain paired IC5-related pcs alternate with formations that do not. The formations that do not contain IC5 pairs are characterized by a particular registral distribution of lines as shown in Ex. 5 in which cello and viola lines are consistently IC4- related while those of violins I and II are IC5-related, thus foreshadowing the pairing of IC5s at IC4 in Section II.



215

Two hypotheses have provided the bases of my observations in the Three Pieces: first, that a study of the make-up and relationships of small, rather than large, pitch-class collections can provide insights into orderings of pitch patterns in twentieth-century works, and second, that the pitch-class content of an intervallic collection must be characterized and is as significant as the abstract intervallic properties of that collection. The development process of the pitch organization of the Three Pieces involves primarily an increase and decrease of interval-class distance within small symmetrical pc sets and the invariance of particular pcs, each paired with an IC5-related pc, within those sets. In particular, in the second piece the effect of "polarity as equilibrium" is achieved by the transpositional pairing of symmetrical sets of IC5 pairs and the emphasis of inversionally- related subsets containing pcs A and B b within those sets. Short examples from the "Introduction" of the Rite of Spring and the second movements of the Octet and Symphony of Psalms will be briefly considered to suggest that the metaphor of "intervallic equilibrium" characterizes interval-class pairing relationships of pc sets associated with important thematic material.

Rite of Spring: "Introduction"

The same type of "intervallic equilibrium pairing" also characterizes the beginning and concluding solo bassoon passages of the Rite of Spring Introduction. Structurally similar pc collections are transpositionally paired by T1 and an invariant pc B/C b occupies complementary positions within the pair of pc sets (see Ex. 6, Nos. A, F, & G). Each bassoon passage presents a 024579 diatonic hexachord, a symmetrical set which may be arranged in a cycle-of- fifths ordering12 as shown in Ex. 6G with the invariant pc B/C b in inversionally-complementary set positions. Pc B has a complementary or "polarizing" function in the Introduction in relation to pcs C and B b by virtue of the particular ordering of all three pcs as pedals: pc B is a pedal at Rehearsal Nos. 4-5, B b at Nos. 5-6, C at Nos. 7-8, B b at Nos. 8-9 and B at Nos. 10-12. These pedals emphasize the only semitone- related pcs of each diatonic hexachord: C-B in the hexachord of the first bassoon solo and C b-B in the final hexachord.

The final bassoon passage occurs a semitone lower than the initial passage (i.e., on Ch); thus a large-scale T-1 relationship is aurally prominent (IC5s of motivic paterns B and E are also T-1 related). On



216

the other hand, motivic patterns occurring between the initial and final bassoon passages are all 0257 patterns that prepare for the final T-1 bassoon solo by successively completing or "filling in" the circle- of-fifths cycle of the first bassoon solo (see Ex. 6B-E and G). Thus, a T-1 transposition pairing characterizes motivic patterns of the beginning and end of the Introduction and IC5s paired transpositionally by IC2 are the basis of the melodic materials of the middle portions. In this case the "drawing together" of "poles of attraction" is effected by T-1 pairing of the bassoon solos, the "separation" by the IC2 and IC5-related 0257 melodic fragments of the middle portions of the Introduction.

Octet for Winds: "Tema con Variazioni"

The second variation-form movement of Stravinsky's Octet has a theme and five different variations occurring in the following order: A, B, A, C, D, A and E. The transposition relationships of melodic patterns in this movement are organized so that an "intervallic equilibrium polarity" obtains between the patterns of the "Tema" and first Var. A and those of the final variation, Var. E. In Ex. 7, the melody of the theme is divided into segments x (0134), y (101346) and xy (10236) that recur prominently throughout the variations; and these segments are labeled by their starting pc and shown for Variations A and E of the movement." The "intervallic equilibrium polarity" is achieved by the transposition relationships of the 0134 x segments which are shown above the xy and y segments in Ex. 8A.14 As shown in Ex. 8B, invariant xA segments occur in inversionally- complementary positions within T7-related 015 groups of segments (on pcs A-B ?-D and A-E-F) for "Tema" and Var. A, and Var. E, respectively.

At the beginning of the movement segment xA is followed by a T3-related xC segment in the theme, and by the T1- and T5-related xB ? and xD segments of the initial Var. A. A T3 or T9 transposition of a 0134 set preserves octatonic collection identity and features two invariant pcs in complementary 0134 set positions; accordingly, T3 and T9 successively-related x segments, as octatonic extensions of each other, occur only in the middle section of the theme and in the middle variations of the movement (within Var. C and between Vars. C and D). Apart from the T3-related segment mentioned above, the transposition levels of x segments in the "Tema" and



217

initial Var. A are the same as those of Var. E. All of the five fugato entries of Var. E begin with 0134 x segments; segment xA appears with segments xE and xF. Segments xE and xF occur without segment xA only in Var. B. Thus, segment xA is invariant in the segment complex of the theme and initial Var. A and that of Var. E as shown in the diagram of Fig. 4 and Ex. 8B.

The contextual rivalry of T1-related pcs A and B b within the first x segment (pcs A-Bb-C-C#) and within the theme itself15 is also evident in the succession of segment xA of the theme by the T4-related segments xB b & D of the initial Var. A. Segment xA is transformed from its subordinate position with segment xB b in the D-A-B b segment group of the theme and initial Var. A to a priority position among segments xE and xF in Var. E. The large-scale conflict of segment xA as T3-related to segment xC or T5-related to segment xD in the middle variations is a working out of the conflict of the xA segment within the theme as "subordinate" in a D diatonic system on one hand, or third-related to segment xC and octatonically "independent" on the other. In Var. E, the only variation in which an xA segment is successively T7-related to xE, segment xA achieves its independent diationic status implied in the theme and asserted in Var. D. This transformation in the priority status of segment xA facilitates a transition to the C diatonic Finale which follows without pause.

Var. E achieves a summary yet transformation of previous thematic relationships by the occurrence of an invariant subset (pcs A-Bg,-C-C#) in a complementary position within a transpositionally- equivalent pitch-class structure. In addition, as the A variations were the only variations to contain T4-related xB & D segments with invariant pc D in inversionally-complementary 0134 set positions, so, at the conclusion of the movement, Var. E was the only variation to contain T8-related xA & F segments with invariant pc A in complementary set positions (see Ex. 8B). Thus the Octet second movement illustrates Stravinsky's use of invariant subsets in complementary positions within transpositionally-related structures, that is, an "intervallic equilibrium polarity" between invariant pc sets at the beginning and end of the movement.



218

Symphony of Psalms, II

The final example concerning the two fugue subjects of the second movement of the Symphony of Psalms, demonstrates that the different "intervallic equilibrium" of the first and second subjects is directly related to the ordering and transposition levels of the subjects. As shown in Ex. 9 the first subject is characterized by a 0134 pc set that occurs throughout the movement as successive pitches of an instrumentally-presented four-note motive which is derived from the basic cell of the work, "two minor thirds joined by a major third."6 On the other hand, the structural diatonic pcs of the vocally-presented second fugue subject (see Rehearsal No. 5) are 027- and 0257-related pcs (IC5 pairing by IC2) as shown in Ex. 10.

The IC 1 and IC2 relationships embedded in the two fugue subjects respectively, serve as the basis for the transposition levels of the remaining fugal entries indicated by open noteheads in Ex. 11. Within the first and second subject presentations the entries are paired by IC5 (T7) and the entry pc pairs of the first subject are T3- related to those of the second subject. The two subject areas are followed by vocal and instrumental stretto statements at Nos. 10 and 12 that are based on the second and first subjects, respectively. As shown in Ex. 11, the entry pitches of statements in the first stretto section (No. 10) are all IC5-related and each entry pitch is one of the 0257-related pcs of Subject II (see Ex. 9). Moreover, the first stretto entry begins on F5, a major second (IC2) higher than the first entry pitch of the second fugue subject. The second stretto (No. 12) consists of four statements of the 0134 motive of the first fugue subject but their transposition levels are organized in IC2 pairs that are IC 1-related, B b2-A ~ 3/C b3-A4. These last three entries, when grouped with the following trombone entry on C3, are IC3 pairs that are IC1-related forming a 0134 set, Ab3-C 3/A4-C3. At No. 13 the first fugue motive stated on C3 "ascends" in imitations on F3 and F#3 (solo trombone) while a motive, associated with the first counter- subject and stated on G6, "descends" in imitations on G4 and C4 (solo flute, oboe, and horn). As illustrated in Ex. 11, the transposed 0134 statement on F# forms ICs 1 and 6 (a 016 pattern) with the combined statements on C, F, and G (a 027 pattern) and marks the end of a major section and the convergence of the two fugue subjects.

In the final section of the movement (Nos. 14-17) patterns associated with each of the two fugue subjects are developed simultaneously; material derived from the second subject is presented vocally while



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that from the first subject is presented instrumentally. The remaining T-levels of the 0134 motive of the first subject continue to reflect the contrast of IC 1- and IC2-related structures of the movement (see Ex. 11). 0134 motives in the upper voices (trumpets, No. 14ff) are IC2- related, whereas, the 0134 motives in the lower voices are presented for the first time as successive IC4 pairs on pitches G2-B2/F#3-D2. The movement concludes with a final IC3 pairing of 014 motives on pcs C and Eh and an IC5 pairing of two 0257 sonorities presented simultaneously (F-Bh/G-C, flute, and F-C/Eb-B&, oboe, see No. 17 and the final two measures of the movement). In addition, a synthesis of the C and E minor tonal implications of the first and second fugue subjects, respectively, also occurs in the final section of the movement: the 0134 motives on pitches B2, F#3, and D2 are each followed by an IC 1-related C minor triad pitch (C3, m. 74; G3, m. 78; and E b 3, m. 81) while the soprano vocal line outlines E b minor triad pitches (Bb4, m. 71; E

,5, m. 71; and G 5, mm. 72, 74, and 77, see

Ex. 11). Thus, the interval-class components of the two fugue subjects, distinct at the beginning of the movement, are combined in the transposition patterns of the remaining subject fragments. The culmination of this synthesis process is the simultaneous presentation of subject I fragments (with IC3s paired by IC 1) and subject II fragments (with IC5s paired by IC2) in the final measures of the movement.

The pairing of transpositionally-related structures, often symmetrical, and the invariance of pcs in complementary positions within those structures result in an "intervallic equilibrium" of the paired structures that is resolved in various ways within the course of a work. In these works, changes in interval distance between paired structures are a measure of the progression and closure of musical ideas. Theoretical models that describe relationships within and among pc sets, especially invariant aspects of small pc sets, deserve further investigation. The models proposed above suggest that one of Stravinsky's unique compositional practices, his treatment of transpositional relationships within and between paired interval structures, can be correlated with his idea of "the drawing together and separation of poles of attraction."



220

Fig. 1 Dispositions of symmetrical pc relationships

A. Inversional complementation, inversional balance

R

PT ' P Q S T - R

QS. R

I

B. Transpositional pairing

(R)

I PQ PS P Q I S T or

ST QT

(R)

I



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Fig. 2 Invariant pcs in complementary set positions equidistant from a center of symmetry

T = invariant complementary pcs

A. 015 pc sets

B. 0134 pc sets

C. 0358 pc sets

V 9 t



222

Ex. 1 Paired IC5-related pcs, Three Pieces, Nos. 1, 2, and 3 and a (7,6) interval cycle

A. (7,6) Interval cycle, Three Pieces, No. 1 0167

B. (7,6) Cycle: No. 3 k - er - 0156 mmn. .1-2

10 '

No. 2 0156, mm. 45-46 0156

No. 1 mm. 1-12, 48-51, & 56-58 00156 sets in alternation

0156

0167 ("linear")

I., mm. 43-44 II., m. 1 III., mm. 1-2 45-46

scivet

rv oq CemC. Pa P



223

Fig. 3 Four-note symmetrical sets containing two IC5 pc pairs

IC Forte Prime ICs Represented Remaining Category Name Form Doubly in Set ICs

V. 4-8 0156 5 & 1 4 & 6

4-23 0257 5 & 2 3 & 5

4-26 0358 5 & 3 2 & 4

4-20 0158 5 & 4 1 & 3

4-9 0167 5 & 6 & 1

4-6 0127 5 & 1 (3 pcs) 2 & 6



224

Ex. 2 Three Pieces, No. 2, IC5 pairing and formal units

Ex. 2A unit a, mm. 1-3, 6-12, unit c, 48-51, 56-58 mm. 13-14

M.M. J78 V r atouche .

rt.

(012567) 0156

unit a 0156

unit c1, unit e, mm. 17-19 mm. 22, 24, & 53

sJ-.. .

sur Is touche

. ..

....7

.

| .

)

sutrLoucheo

*piss.

. (is-.)

0156)-0(012567) unit c

05 -

unit c 0156

= invariant pcs in complementary set positions



225

Ex. 2 Three Pieces, No. 2, IC5 pairing and formal units (continued)

Ex. 2B

unit d, mm. 15-16, 20-21, 23, 60-61

Allegro (J..e)

Ntllr I J D

unit d 0127

Ex. 2C unit Fs, mm. 26-28

Stm.

(secondary set)

S= invariant pcs in complementary set positions



226

Ex. 2 Three Pieces, No. 2, IC5 pairing and formal units (continued)

Ex. 2D unit Gx, m. 31

Tempo I J:76)

71 s urI o nhe. .

Gx 0358

unit Hx1, m. 36 unit Hx1, Hx m. 40 m. 42

, r'ie sone i lri'i

Vn i " J -

S invariant pcs in complementary

set positions



227

Ex. 3A Three Pieces, No. 2, symmetrical PC sets with two IC5s

mm. 4 6 13 15 17 20 Section I o156 units: a a + b c d d

0257 Q 27 0127 0127 0127

0156 0156 Section II00127 mm, 22 23 24 25 26 29 units: e d e

Fs Ft 027 0167 ra 0127

..j L - 0156 27 0156 0257 I

0156

0127 6-V



228

Ex. 3B Three Pieces, No. 2, symmetrical pc sets with two IC5s

Section II (cont.) Section III

units: Gx Gy Gz Hx1 mm. 31 33 34 36 027

0257 0167 0358 0167 27

40 0156

Hx 0 Hx Hz Hy Section IV 54 56 60 0358 027 0358 0127 44 45 48 252 53 3

0156 b a c 0156 e c a d

A. o0156 0127

. .

0167 I 0167

I / 1 0156 0156 5/

0156 0358

(012567) (012378) (complement)



229

Ex. 4 Three Pieces, No. 3, IC5 pairing and formal units

Sections: I II units: a d mm. 1-2 27-28

Tutti fs.sez efre I.reet e domfe sl e I sngaN r

v Uo

p crase. -o 1s reels

mm. 27-29 m. 37

(01458) (01478)

0156 0158

(IC5 pairing by IC1) (IC5 pairing by IC4)

III a

45-46

Tutti sul tasto

0156

(IC5 pairing by ICI) (set pairing by IC4) pose



230

Ex. 5 Three Pieces, No. 3, registral districution of lines: IC5-related pcs superimposed on IC4-related pcs (see brackets)

n S:40 Tutti sul tasto

_-___I___m(V) (M) IC5s

Crese. PP PP _ __

pp crs. Fpp 1IC4s , ,-=V,, Tutti sul tasto

A V. V.\ VC5s

m

-]r IC4s

I I l i I w



231

Ex. 6 Rite of Spring, "Introduction," motivic patterns and IC5 pairs

bas on solo e

A. 024579 1 m. after cl. in

B. W7 . , I F "

0257

2 mm. after lute

D.

0257

gG.c inO

11bassoon Isolo

D. .

Interval 7 cycle D . ._.



232

Ex. 7 Octet, second movement, Var. E: x, y, and xy melodic segments (x=0134, y=I01346, xy=I0236)

Tema

41r= 92 A A E

I xA x

1"-I" - x 2 Y1 --1-

xC xyA !

- yE ,

xBb_ x Bb

SxBb2 ,IIF If

SVar.D 1) Var. B

i ... .. ....1[491

Pr126

iwxy

#2) ;, Var.C['~I 13) ] Var. Ex



233

Ex. 7 Octet, second movement, Tema & Var. A: x, y, and xy melodic segments (x=0134, y=I01346, xy=10236)

Var. E

A5A1 8nxoA(no Bbj- r- X 21 r

E m F#)

noE __

ba 1 -Am t, -1

I INI P IIIAMPISOI 1L fAll ;

I I I i4

L f'-- dxA. InoVBb r---xf 2I r-"f

I

-Ill,----EI. . x

xA -. x xy

? = B--y a (no Bb) A E 1531

, -- , , I I , ,

-u iA et1 1 % L IF - xy- Ax XE Y E

G# F xy x

5-51M5 #s 44' w ec.



234

Ex. 8A Octet, second movement, Tema & Var. A, Var. E: x (0134), y (101346), and xy (10236) melodic segments, transposition relationships

Tema Var. A T5

x se nts = appears in T1e

m ts ..s T3 9T o

01ft

. -.,m av hisi i, MM 1- ISMW S

& xy) segments

Var. E x segments T8 T Finale IF1

,

W T7 I oofT7

.

xy xy xy xy xy

Ex. 8B Summary: Tema & Var. A (and Var. D & Var. A); Var. E

Tema, Var. D // Var. A Var. E

= invariant pcs or pc sets in inversionally complementary set positions



235

Fig. 4 x segment successions

"Tema" // Var. A Var. E (fugato entries) A C Bb D A E A E F x -- () // x --x x--x --x --x --x

0 I I I

015 015



236

Ex. 9 Symphony of Psalms, II, mm. 1-4, IC3 pairing by IC1

First fugue subject

TIrempo - eO

OberIV _m

I

0134



237

Ex. 10 Symphony of Psalms, II, IC5 pairing by IC2 (embedded)

Second fugue subject

027

IC5 IC3 027 0257IC3

027 0257



238

Ex. 11 Symphony of Psalms, II: transposition levels of the subject entries

Subject I 0

0134

Subj.

Subj. I1 fragment

023

027

Stretto I 1 (from Subj. II)

0257

Stretto II 5T? 0126 Countersubj. I (from Sub. I) 126

0134 *L 27A

0123 IC2 IC2 Subj. I fragments 016

Vocal line from 0257 S ub

, II.-

0257

-!o f-14 -15116

Suij. I fragments A

025 Z014 0125 I 014

IC4 IC4 _ _0 015 014 j 0124



NOTES

1 Igor Stravinsky, Poetics of Music, trans. Arthur Knodel and Ingolf Dahl (Cambridge, Mass.: Harvard University Press, 1970), pp. 48-49.

2 Arthur Berger, "Problems of Pitch Organization in Stravinsky," Perspectives of New Music 2 (Fall-Winter 1963): 25. 1 One of the earliest writers to use the metaphor of polarity in this way was Paul Collaer in his book Stravinsky (Brussels: Editions Equilibres, 1930), pp. 75-76.

This same interpretation also was adopted by Pierre Boulez in Notes of an Apprenticeship, trans. Herbert Weinstock (New York: Alfred A. Knopf, 1968), p. 248: "It is certain that in the amplified tonal-modal language this 'polarization' of a note restored considerable strength to the idea of tonic-a sort of hyperdegree- a strength that it had lost rapidly after Wagner, a strength that it had possessed to this degree only at the inception of the tonal language."

4 Henri Pousseur, "Stravinsky by Way of Webern: The Consistency of a Syntax (II)," trans. Marcelle Clements, Perspectives of New Music 11 (Fall- Winter 1972): 136: ".. . [the] tensions, far from being constantly reabsorbed... are obstinately and rigorously maintained (to which even the polarizations concur, as a result of their balanced antagonisms)."

5 Berger, p. 25. 6 See Berger, pp. 11-42; William E. Benjamin, "Tonality without Fifths:

Remarks on the First Movement of Stravinsky's Concerto for Piano and Wind Instruments," In Theory Only 2 (1977): 53-70 and 3 (1978): 9-31; and Pieter Van den Toorn, "Some Characteristics of Stravinsky's Diatonic Music," Perspectives of New Music 14 (Fall-Winter 1975): 104-38 and 15 (Spring-Summer 1977): 58-95.

7 See George Perle, "Berg's Master Array of the Interval Cycles," Musical Quarterly 63 (January 1977): 1-30 and Twelve-Tone Tonality (Berkeley: University of California Press, 1977); Paul Lansky, "Pitch Class Consciousness," Perspectives of New Music 13 (Spring-Summer 1975): 30- 56; and David Lewin, "Inversional Balance as an Organizing Force in Schoenberg's Music," Perspectives of New Music 6 (Spring-Summer 1968): 1-21.

8 The relevance of cyclic pitch patterns in Stravinsky's music has been noted by Pousseur with respect to Agon and portions of The Rite of Spring. See Pousseur, p. 139.

9 Forte accounts for the succession of unit a by unit d at the close of the piece by the complement relationship of sets 6-Z6 (012567), E ,-E-F-A ?-A-Bn, and 6-Z38 (012378), A-B?-B-C-E-F. However, the complement relationship in this case is an abstract relationship that does not fully account for the specific pc content of the six-note sets, namely, the invariance of the initial four-note set of the work E-F-A-B6 within each set. See Allen Forte, The Structure of Atonal Music (New Haven and London: Yale University Press, 1973), pp. 130-137.

10 An account of the pulsation speeds of the piece is found in Robert Cogan and Pozzi Escot, Sonic Design (Englewood Cliffs, NJ: Prentice-Hall, 1976), pp. 276-283. Cogan and Escot note only general characteristics of the two pulse strata and do not correlate their account of pulse successions with aspects of pitch organization of the piece. Throughout the piece IC5 pairs



240

containing pcs A and Bb are vertical in stratum MM= 228 and linear in stratum MM= 152.

1 The unordered sets comprising the last seven notes of unit Gz (mm. 34-36 pcs A-G#-G-F#-Eb-D-C#) and the first seven notes of Hzl(mm. 43- 44, pcs C-B-Bb -A-F #-F-E) form T3-related 0123678 pc sets.

12 George Perle discusses the diatonic elements of portions of the Introduction as segments of the cycle of fifths in his article "Berg's Master Array of the Interval Cycles," pp. 10-12.

13 Segments marked with the same letter have the same interval content. Superscripts signify the pc content of each segment and refer to the "starting" of the segment (e.g., xA=A-B -C-C x, xC=C-C --D#-E). Subscripts characterize the pc order presentation of each segment (e.g., xo refers to the pc order of mm. 1-2, C#-A-C#-B -C, x1 of mm. 3-4).

14 The 0134 set serves multiple functions in the work as a subset of octatonic and diatonic collections. See Berger, pp. 30-32.

15 See Berger, p. 31. 16 See Berger, p. 32.



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