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1 Sets and Nonsets in Schoenberg's Atonal Music Author(s): Allen Forte Reviewed work(s): Source: Perspectives of New Music, Vol. 11, No. 1, Tenth Anniversary Issue (Autumn - Winter, 1972), pp Published by: Perspectives of New Music Stable URL: Accessed: 30/01/ :31 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Perspectives of New Music is collaborating with JSTOR to digitize, preserve and extend access to Perspectives of New Music.

2 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC ALLEN FORTE A consideration of Schoenberg's atonal music ought to refer to a theoretical framework for atonal music in general. The present article, however, will not set forth such a framework, but will focus, instead, on an elementary problem in Schoenberg's atonal oeuvre,1that of distinguishing compositional units or sets from secondary formations or nonsets.2 In order to concentrate on this aspect of Schoenberg's music and to avoid digressions into other interesting aspects, no effort will be made to show the function of sets in the music. No com- plete analyses will be presented, although every musical example quoted is extracted from a complete analysis-and, indeed, could not be convincingly and confidently presented to the reader otherwise. There will also be no comparison of Schoenberg's atonal and twelve-tone procedures. Schoenberg's atonal compositions exhibit a remarkable and innovational kind of structuring, one that cannot be characterized simply, although a matrix of two or more dimensions might serve as an adequate model. With one exception, this aspect of Schoenberg's music has not been documented in the literature.3 The components of these musical structures are pitch-class sets containing from four to eight elements, usually. For any given composition the stock of sets that have structural significance (to be discussed below) is relatively small. As some indication of number, Herzgewichse, Op. 20, is constructed of exactly nineteen sets and their complements.4 An extreme case is the fourth of the Five 1Atonal is used here in the now conventional sense to designate the nontonal, pretwelve-tone music. 2An approach to a theory of atonal music has been made in [1]. A more recent refinement and extension will be found in [2]. 3George Perle, the exception, cites some instances, but his readings are insufficiently detailed and often contain mistakes. 4Some knowledge of now standard terms is assumed. * 43 -

3 PERSPECTIVES OF NEW MUSIC Piano Pieces, Op. 23, which is based upon only three hexachords and their complements. In the course of this exposition, it is hoped that certain notions concerning Schoenberg's use of compositional materials will be laid to rest. Among these are the notion of the "basic cell,"5 the idea that tonalistic formations, such as the triad or augmented triad are important components, the notion that some elements are "independent details,"6 and the idea that familiar configurations, such as the whole-tone scale and "chords in fourths" play significant roles in the music. It is indeed unfortunate that these views have become as widely disseminated as they have, for they tend to obscure Schoenberg's unique genius.7 Before proceeding to the main task, it is necessary to introduce some elementary symbols. All references to pitch-class sets will be by set-names.8 A set-name consists of a digit representing the number of elements in the set, followed by a hyphen, followed by two digits representing the position of the set on a fixed list. (There seems to be no need to reproduce the complete list here, since only a small number of sets are cited and since there is no extended discussion of properties of sets and relations among them.) Thus, for example, 6-14 is the name of a six-element set occupying the fourteenth position on the list. If a Z precedes the position number this means that the set is one of a pair A, B such that A has the same total interval-content as B but is not inversionally or trans- positionally equivalent to it. The bar above a set-name signifies complement. For instance, 5-15 = 7-15 means that 5-15 is the complement of One final convention, which is perhaps more familiar to readers of this journal. Transposition of a set S is defined as the addition (mod 12) of some positive integer t to each element of S. Occasionally this value will be given in connection with the musical examples-for instance, t = 11. It must be emphasized that set-names are only labels; they indicate nothing about the function of a particular instance of a set. This can be dealt with only at a level of analysis higher than that attained in the present endeavor. 5As described in [3]. 6Also in [3]. 7Schoenberg is partly and indirectly responsible for this, of course, by virtue of his discussion of chords in fourths, etc. in the latter part of the Harmonielehre. 8Asin [1] and [2]. *44

4 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC The determination of a significant set, as distinct from a nonset, is not always easy. Some informal guides are: (1) the set occurs consistently throughout-it is not merely "local"; (2) the complement of the set occurs consistently throughout; (3) if the set is a member of a Z-pair, the other member also occurs; (4) the set is an "atonal" set, not a set that would occur in a tonal work. As may be evident from this informal recitation, set-complex structure provides important criteria of "significance," but will not be explicitly invoked here. See [1] and [2]. Most problems concerning the determination of significant sets are resolved only through careful analysis of the entire work. And while no such complete analyses are provided in this article, all the examples are extracted from works that have been completely analyzed, as indicated above. It seems appropriate at this juncture to dispense with further preliminaries, claims and disclaimers, and to proceed to some musical examples. The opening of the first of Schoenberg's Three Piano Pieces, Op. 11 has been cited often, yet nowhere in the literature has its straightforward hexachordal organization been described.9 This is n^~~~~~i..=7.. I* 6-Z1o Z39 (-6-ZIo) nd 6-Z two explicit forms Zio 6-Z39 6-Zio of 6-Zio) Ex. 1 9Perle [3], Ex. 7, discusses the "intervallic cell" and describes the structure in terms of trichords. Brinkmann [4], p. 63, emphasizes what he considers to be "a remarkable accumulation of tonal elements," by which he refers to the fifth (B to E) spanned by the uppermost line and the juxtaposition of major and minor third (G*-G). In this tonalevolutionary posture he emulates von der Null [5], p * 45 -

5 PERSPECTIVES OF NEW MUSIC shown in Ex. 1. A hexachord of type 6-Z10 comprises the upper part, while the lower, "accompanying," part consists of a hexachord of type Subsequent important roles of these hexachords are indicated in the remainder of Ex. 1. Hexachord 6-Z39, the complement of 6-Z10, is featured in the consequent phrase in m. 4; 6-16 is contained in the ascending figure that begins at m. 12, and two forms of that hexachord interlock at m. 13. A more complicated configuration involving both sets occurs at m. 28: 6-16 and 6-Z39 overlap, with 6-16 again being formed between the peak note of the figure and the rising line in the lower register. At m. 29, the culmination of the passage, 6-16 occurs yet again. Within the last portion of the movement, beginning at m. 51, the two hexachords are associated in the clearest possible way. Of the three successive occurrences of the ascending figure in the lower register (Ex. 1), the first two form 6-16, while the final one forms 6-Z39-the latter identical, with respect to pitch and register, to the set in m. 4. The interwoven multiple occurrences of 6-Z10 at m. 34 are most remarkable, but not especially unusual in Schoenberg's atonal music. There in the uppermost part are two ordered transpositions of the melodic theme of the opening. (Note that the invariants exchange positions.) Added to these "explicit" forms of 6-Z10 are two overlapping forms, as shown. Thus, the incipient development is literally saturated with the thematic set, 6-Z10. (Compare Ex. 12 below.) Example 2 provides additional perspective on the aspect of Schoenberg's compositional technique under discussion. The music shown in Ex. 2a is the opening of Herzgewachse, Op. 20 (1911). Langsam i+b--- 1i 1 i Ex. 2a

6 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC Z19 6-Z3 '"6-Z / Z zs Z ^~~ ' ~I/ -- X J., V - - P $ * ~ ^..* 4jF -/l. i= f - M 5-1' !Z29 4-Z15 Ex. 2b Not untypically, large sets are formed, within which there is a good deal of detailed structuring. As shown in the sketch-like representation of Ex. 2b, the two 8-element sets, 8-12 and 8-7, are followed by the hexachords 6-Z19 and 6-Z3. These sets and their complements, as well as all other sets indicated in the example, are significant components over the entire music. Indeed, almost all the set-types utilized in the composition are exemplified here: all seven 4-element sets, all but four 5-element sets (which include two pairs of Z-related sets), and all but one complement-related pair of hexachords. Within the large set 8-12, the hexachord 6-Z44 is formed first. This hexachord type is typical of much of Schoenberg's atonal music, and little wonder, for it is his motto (Es-C-H-B-E-G). Its complement, 6-Z19 (inverted and transposed), subsequently occurs twice in the passage quoted: under 8-7 and immediately following. Observe that 6-Z19 in the latter situation contains the complement of 8-12, 4-12, as well as a new 4-element component, 4-8. The 5-element set, 5-32, under 6-Z44 has no immediate consequences in the passage. However, the set 4-Z15 (one form of the all- * 47

7 PERSPECTIVES OF NEW MUSIC interval tetrachord), which occurs within 6-Z44, returns to accompany the entrance of the voice (m. 2), at which point it forms a new hexachord, 6-Z3, together with Set 4-Z15 is also contained in 8-7, as shown at the bottom of Ex. 2b. In the same context there occurs the other form of the all-interval tetrachord, 4-Z29. In addition to 6-Z19 and 6-Z44, other complement-related sets are of interest here. In particular, the large set 8-7 is preceded by its complement, 4-7. And set 7-21 (under 8-7) contains its complement, The latter two sets are among those especially favored by Schoenberg in his atonal music. Set 5-21 occurs throughout the orchestral pieces Op. 16, for example, and throughout Pierrot Lunaire. (Set 7-21 is, in fact, the final sonority of the latter work.) Two important observations need to be made in connection with Ex. 2-important because they can be extended to apply to all of Schoenberg's atonal music. First, notice that the slurred melodic formation Gt-A-D-Eb (Ex. 2a) does not form a significant set (Ex. 2b). The Gt is an element of five sets, while A and D belong to 4-13 and Eb belongs to 8-7 and its subcomponents. Similarly, the opening "motive" of the voice, F-G-G-F#t-Ft, is not in itself a structural set, but a secondary formation, a nonset, the elements of which belong to other sets. In general, a given "melodic line" may not necessarily be a discrete structural component.10 This facet of Schoenberg's music has led to much misunderstanding and caused many blunders in the past, particularly where such secondary formations are "chromatic lines" (as in the case of the voice motive here), "whole-tone scales," or other familiar patterns. Schoenberg simply did not compose with these and other well-worn formulae, just as he did not compose with sets that properly belong to the vocabulary of tonal music-although, as will be evident in subsequent examples, such sets may be indirect results of the interaction of significant structural components. The second observation to be made in connection with Ex. 2 is as follows. Every pitch belongs to at least one significant set; there are no "independent details" [3]. This includes pitches written in small notation, as Ab and C in the single occurrence of 4-8 in Ex. 2a or Gt in 6-Z3 (Ex. 2b). Every detail, no matter how minute, is an integral part of the complete musical conception. 10This applies as well to instrumental parts marked Hauptstimme or Nebenstimme. These are not necessarily significant entities in themselves. * 48 -

8 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC Misreadings of structural components abound in the literature on Schoenberg's atonal music. Even a scholar as astute as George Perle has erred in this respect. Example 3a shows his reading of part of the penultimate measure of the fourth of the Five Piano Pieces, Op This, we are told, is an instance of an indirect chromatic progression (a chromatic progression with octave displacement).12 However, as Perle well knows, this composition utilizes a small number of hexachords throughout. Why would a more or less trivial "chromatic" formation be introduced here at the very end of the composition? To be sure, the hexachordal organization of the lower part is somewhat concealed, but certainly the regularity exhibited by the tetrachords should provide a clue. a 6-z f t 5 b 6-z44 \ 6-z Io n,.,/,^m! --- -Z f p, 9:0. =$ 6-Z10 6,-10 \ J- 1b Ex. 3 Example 3b shows a correct reading of the passage in its entirety. The upper part carries 6-Z44, a replica of the first statement of that set, in m. 1. Three disjoint forms of 6-Z10 occur notated on 11In [3],p The reading of this passage is all the more strange in view of the virtually correct reading of the beginning of the piece, in [3], p. 48. * 49 -

9 PERSPECTIVES OF NEW MUSIC the lower staff, the final one a replica of the first statement of that set in m. 1. Finally, 6-Z10 is formed once again between the staves, as indicated. From the foregoing examples it should be apparent that a systematic description of Schoenberg's way of combining sets is feasible-a typology of segmentation for his music. Nothing of the kind will be undertaken here; instead, examples of some basic kinds of structuring will be discussed and illustrated. The embedded single set, which has been shown in Exx. 2 and 3, is, of course, a common type. An especially interesting case is that in which a set contains its complement. (See, for instance, Ex. 6.) Similarly, simple overlapping has been shown (Ex. 1). The combination of embedded sets (inclusion) with overlapping sets (intersection) is an obvious source of complicated configurations. Passages of this kind will be illustrated below. Not at all obvious, however, is a remarkable feature of Schoenberg's music that might be called overlapping in a single dimension.13 Example 4, from Herzgewachse, Op. 20, is typical. The celesta part alone, except for the first three notes, consists of overlapping forms of 4-7, transpositionally related, with t = 11. Notice that the repeated notes in the third form of the set effect an ex- pansion, disrupting what would otherwise be a regular patterning. Here, as in other aspects of his compositional process, Schoenberg eschews the routine and obvious. m.17 Celesta a ~ Voice Celesta 4 L L b -1. I gw We - o I? Ex. 4 13This was first suggested to the author by Steven Gilbert's dissertation [6], in which melodic lines are subjected to a systematic analysis in terms of trichordal subsets. * 50 -

10 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC The lower part of Ex. 4, Ex. 4b, shows the overlapping effected by voice and celesta combined. While the middle part of the configuration is made up of overlapping forms of 4-18 and 4-19, beginning and end yield 4-12 and 4-7, respectively. A clear example of a nonset is provided by the total configuration shown in Ex. 4b: the 10-element "chromatic" set In this case the significance of the whole is entirely dependent upon the significance of its parts. Overlapping of a more complicated kind is shown in Ex. 5, extracted from the ninth piece of Pierrot Lunaire, Op Of the two lines, the upper forms 4-18, the lower 4-8. At the same time, the vertical pairing indicated creates disjoint forms of 4-18 and 4-8, in that order. In the vertical resulting from the notes tied over the barline, the upper hexachord is 6-Z43, a set that occurs as the first hexachord in the piece (and in the same context as its complement, 6-Z17-not shown here). One additional set is marked, 4-13, which is brought in with the three notes F$ -G-A. All three sets, 4-13, 4-8, and 4-18, are stated in the previous music and are significant sets throughout. m Ex. 5 Overlapping is a consistent feature of the vertical dimension as well. Example 6, from Herzgew'achse, illustrates. Not all sets are shown, but only those relevant to the overlapping feature. In particular, note that the overlapping 4-element sets, 4-7 and 4-19, occur here in conjunction with three statements of The last statement of 5-21 is within one of the hexachords fundamental to this music, 6-Z44. Both sets are favored by Schoenberg, as mentioned above. 14Actually the first piece to be composed. See Rufer [7].? 51?

11 PERSPECTIVES OF NEW MUSIC m.24 6-Z _ :-9 I , U Ex. 6 A somewhat similar situation is shown in Ex. 7, from the second of the Six Little Piano Pieces, Op. 19, but one which is more concisely structured. Within the large set, 8-19, occurs the hexachord This hexachord, in turn, is structured by a regular overlapping of two forms of 4-19, the complement of These are inversionally related (t = 9), intersecting in the two common notes, Eb and F#, and embedded within the complement set, 8-19, as already noted. Still another level of overlapping is exhibited by the two forms of 5-21 within 6-20, as shown.15 This latter overlapping, however, is trivial, in the specific sense that 6-20 contains only 5- element subsets of type That is, any selection of five distinct elements from 6-20 is reducible to a set of type L I 8-19 Ex. 7 15It seems doubtful that Travis's characterization [8] of this final sonority as a functioning tonic chord is reasonable, in view of the substructures that have been pointed out. Every component has a specific contextual meaning, in terms of this piece as well as with reference to the entire set of pieces, whereas the "triad," C-E-G, is a nonset. Compare Ex. 9 below. * 52?

12 ,^ fksi^~ SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC Vertical and horizontal overlapping combine to create passages of greater complexity, while retaining a high degree of "strict" organization. Example 8, from the first of the Four Songs for Voice and Orchestra, Op. 22, provides an illustration. In this short passage m _i-^-s^ a,q -is1l -9 t w -: ==$LJFLI Q Ja~~~~~.17 1 \ \ Fb L 6-20 Ex. 8 the total chromatic is deployed in the following way. First, 6-20 is stated as a vertical (by violins) and then is sustained throughout the passage, as indicated by the incomplete ties. The initial statement of 6-20 is followed by another form of the same set (clarinets), related to the first by transposition, with t = 2. At this transposition level (one of two such) complete variance is effectedi. e., the literal complement is produced.16 As is apparent from the notation, the remainder of the passage consists of the horizontal projection of 6-20 in all six voices, with the result that the hexachord permeates both horizontal and vertical dimensions. What is not obvious is that the hexachord is so ordered in both dimensions that all six forms of the tetrachord 4-19 contained in 6-20 are stated as overlapping segments.17 Only two pairs of 4-19 are marked in Ex. 8. Represented in conventional letter-names, all six forms of 4-19 in 6-20 are as follows: Eb-G-Bb-B D-Eb-F -Bb G-B-D-Eb 16Readers will recognize 6-20 as one of Babbitt's all-combinatorial hexachords. 17The ordering of 6-20 is special in another way, as well. It is one of the 24 permutations of that set which produce the basic interval pattern uniquely associated with See [2]. Schoenberg discusses none of these matters in [9] or elsewhere. And, indeed, there is every reason to believe that he would never have explained his compositional selections in any depth. * 53 -

13 PERSPECTIVES OF NEW MUSIC Ft-G-Bb-D B-Eb-F# -G Bb-B-D-Ft As a final comment on Ex. 8 it might be pointed out that the "augmented triads" visible from the notation are not significant structural components and, in fact, are a fortuity of notation or, perhaps, a notational concealment-which would not be untypical of Schoenberg.18 Example 11 below shows another instance. The point that a total set may be significant only in terms of its subsets was made above in connection with Ex. 4. Also common in Schoenberg's music is the situation in which a subset is significant only in terms of a superset. An amusing case is shown in Ex. 9, m Z15 6-Z44 Ex. 9 from the last movement of Pierrot Lunaire, Op. 21: the nonset B- E-Gt that occurs at the end of the opening passage. This "triad" is meaningful only as a subset of 6-Z44 (Schoenberg's motto again) and 4-19, as shown in Ex. 9. Further, its two lower notes combine with Db and F to form 4-Z15. Both tetrachords are prominent in the opening phrase. There can be little doubt that the use of a 6 chord here is a bit of Schoenbergian whimsy. Indeed, the entire opening, with its surface "simplicity," may have been intended to dupe some unsuspecting critic.19 Thus far, some basic types of set usage have been examined and illustrated and some general comments have been made. More complex passages will now be considered, passages in which inclusions and overlappings are compounded to create elaborate textures. 18Nor is it untypical of Stravinsky. Similar passages abound, for example, in The Rite of Spring. 19There have been a few willing victims. H. H. Stuckenschmidt, for example, writes: "The impression of simplicity is strengthened by presenting the theme in thirds. It is in E major, with a 6/4 chord in the third and sixteenth bars." [10], p. 71. * 54?

14 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC Example 10a provides the notation for the opening phrase of the third of the Three Piano Pieces, Op. 11, a complicated work that has perplexed several writers, and not without reason.20 The underlying set structure of the music is difficult to sort out and almost as difficult to represent. Example 10b is a representation of the sets that occur in the opening "thematic" statement. Only sets of five or more elements are shown. The 4-element sets, which are of considerable importance in the music, will be discussed below. The passage divides naturally into two parts, corresponding to the rhythmic articulation on the fifth sixteenth-note. The large set formed in the first part, 8-Z15, is followed by the 7-element set, 7-Z18, which contains its complement, 5-Z18, as indicated. These two large "top to bottom" sets are almost matched in importance by the two large "left to right" sets, 6-Z44 and 8-18; but, whereas 8-Z15 and 7-Z18 are disjoint, 6-Z44 and 8-18 overlap. The complement of 6-Z44, 6-Z19, is formed at the very outset within 8-Z15, and the first hexachord within 7-Z18 is again 6-Z44. The latter overlaps, as shown, with 6-5, and 6-5 here is the same pitch-class set as it is when it occurs as the melodic theme of the second movement. Within 6-5 are statements of 5-7 and 5-Z18, the latter embedded within its complement, as mentioned above. These two 5-element sets pair off against 5-7 and 5-21 in 8-Z15, and all three 5-element sets interact throughout the movement. Although additional, related sets are introduced subsequently, the six sets shown here are fundamental components of the entire composition. Still a further level of complexity is introduced by the 4-element sets, as shown in Ex. 10c. Three of these are obvious: 4-Z15, the opening motive in the lowest voice; 4-19, in the uppermost voice; and 4-19 again, as the first four-note vertical. The large "left to right" set, 8-18, decomposes into overlapping tetrachords of four types, and the last of these, 4-16, occurs twice, as indicated. Only the first (4-19) and last (4-16) verticals are significant sets. The line beginning on Ct brings in additional sets by overlapping, as shown separately at the bottom of the main part of Ex. 1 c. All subsequently have various roles in the music. 20See, for example, Perle [3], Ex. 19, p. 21. Brinkmann [4], p. 121, has little of significance to say about the opening of the movement, but is concerned with the octave doublings, the motivic manipulations (motivische Arbeit), and the rhythmic "acceleration" that begins in m. 2. * 55?

15 PERSPECTIVES OF NEW MUSIC Bewegt 6-Z19 6-Z Z44 6-Z44 E E E A A C G# b 8-18 \ C C C F F Ab D C1 DI FI GI D C# A A GI D C C C D D IC 8-Z 15 7-Z ZlI Z \ E E A G# c C C C F F Ab DI Cl D# Fl G# D C# A D# F# G# D C# A 4-Z / 4Ex. 10 Ex. 10 * 56?

16 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC It would be gratifying to be able to assert that in Schoenberg's atonal music there is a discernible increase or decrease in complexity depending upon the date of composition of a particular work. No such fluctuation is evident, however; the degree of complexity seems to be more or less uniform over the entire corpus of music, and, moreover, is independent of medium. The next example, Ex. 11, the opening of the fourth movement of the Five Orchestral Pieces, Op. 16, is a case in point. 6-Z37 Molto Allegro 6 - Z-_- 3 6-Z37 ---!... W a _ Y _.j. ^ a Lif= Z4 (-6-Z37) formed by every such triple of "thirds" C I t Is = 6-20 formed by every such pair of "augmented triads" d 1 A k b b 6-Z6 formed tri7pl-eof by every such "f ourth 6-Z6 formed by every such triple of "fourths" e? C-ft- kt 4 Z ev fo d _ - u Z19 formed by every such pair of "minor triads" Ex. 11 As indicated in Ex. 11, the dramatic opening melodic line forms 6-Z37, its peak note intersecting with a 5-element set (not significant in itself) to form the first statement of the hexachord * 57?

17 PERSPECTIVES OF NEW MUSIC Upon first inspection, the subsequent motion is straightforward. Each successive pair of verticals in sixteenth-notes states 7-21 and, as shown, 7-21 is partitioned into sets 5-21 (its complement) and 6-31, creating the regular sequence indicated. (The three successive occurrences of 7-21 are related by transposition, with t = 10, a value that yields minimal invariance.) An ordered transposition of the initial 6-Z37 follows, culminating again in The latter, however, does not have the same vertical arrangement as its counterpart on the downbeat of m. 2. About the diverging "chordal streams" in m. 2, it can be said, without qualification, that (once again) the "augmented triads," "minor triads," and "chromatic lines" so congenial to the amateur analyst are of no significance in themselves.21 Instead, four additional hexachords, which subsequently become more explicit struc- tural components of the music, are introduced. Examples 1 lb through lie set out the patterns. Overlapping triples of thirds (Ex. 1lb) create a succession of six forms of 6-Z4, a set of immediate local significance, since it is the complement of 6-Z37, the set formed by the opening melodic theme. Each pair of "augmented triads" (both disjoint and overlapping) brings in a new form of 6-20 (Ex. llc), and the remaining hexachords, 6-Z6 and 6-Z19, are formed in a similar way (Ex. 1id and Ex. lie). This is perhaps as clear an illustration of multiple dimensions in Schoenberg's music as one could find, although it is by no means exceptional. As a final example of complexity, Ex. 12 displays the sets in a passage from the fourth of the Five Piano Pieces, Op. 23. Here the reader's indulgence is requested; it is extremely difficult to devise clear ways of showing the sets that make up a complicated texture of this kind. In the present instance six staves below the ordinary notation have been aligned to show the component sets in expanded form. The piece is extraordinary for a number of reasons, not the least of which, as mentioned above, is the fact that it is based entirely upon three hexachords and their complements: 6-Z10 and its complement 6-Z39, 6-Z19 and its complement 6-Z44, and Only 6-Z39 is not stated in the passage under consideration. 21Robert Craft [11] has characterized the sixteenth-note passage in m. 2 as "A trumpet 'smear,' consisting of seven augmented triads..." (p. 21). The minor triads (trombones) are not discussed. * 58 -

18 - - _ --- I.. SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC m.19 A, A,.,~~ K ^frsrr~\ -~>' '-r p ---= ' ' ~v~~~~ A, A, 9=L:B4ft w _:-~1~51 b-, i2 a ( _ ~~~~~~~~~~~I ----~~~~~~~~~~~ rb~= Ex. 12 A somewhat deeper analysis of relations among the sets in this passage will, it is hoped, provide clarification. First, however, it should be pointed out that here, as is often the case elsewhere in Schoenberg's atonal music, notation is not a reliable guide to the underlying structure. For instance, in the formation labeled A1 the beamed group B-A-Ab-G does not form a significant set (4-2), nor, in B1 does the beamed group C-B-Bb-A. And, again, the various "triads" are nonsets. These and other formations on the surface of the music result from a more intricate and systematic deployment of the basic hexachordal components. This deployment is perhaps most easily described with reference to the subscripted groups shown in Ex. 12. The first two forms of 6-Z10 in A1 are inversionally related, with t = 8, a value of t that produces four invariants-more than can be * 59?

19 PERSPECTIVES OF NEW MUSIC obtained for any value of t under transposition alone. In A2 the first two forms of 6-Z10 are again inversionally related, with t = 4. Now, if the natural ordering effected by inverting and transposing the first pair had been preserved, the middle dyads of both forms would have retained the positions they now occupy, i. e., G Ft F Eb D A Db Ft Eb F D B but the outer dyads would have occupied reversed positions. Schoenberg's compositional ordering, however, is such that the invariants are assigned to the same position throughout, i. e., G Ft F Eb D A B Ft F Eb D Db This accounts for the intersection of the two forms of 6-Z10, an intersection which obscures to a large extent the discrete identity of each. An analogous situation exists with respect to the first two forms of 6-Z10 in A2. Since the value of t for these two forms is the inverse of the value of t for the first two in A1, the two pairs have the same number of invariants.22 The entire group A2 is a transposition of A1, with t = 4. This value of t produces the maximum number of invariants, seven, under transposition. And the invariant set is 7-33, the complement of the 5-element "whole-tone" set. Thus, the invariants between the two formations account for a secondary "whole-tone" feature at the surface level. Within and between groups A3 and A4 the relations are analogous to those of groups A1 and A2. One, perhaps minor, feature should be pointed out however: although the values of t in A2 for the two forms of 6-Z10 are inverse-related, they are not 8 and 4, as in A1, but 2 and 10. Groups B1 and B2 exhibit a marked change in surface pattern with respect to that of the A groups. This change is also evident in 22See Babbitt's fundamental discussion of invariance [12], which includes a discussion of the results of inverse-related values of t under inversion. * 60 -

20 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC the underlying set-structure. Here, instead of one set-type, as in the A groups, there are three. The members of each pair of sets of the same type are transpositionally related, with t = 4. In the case of 6-Z44 and 6-14, this is the maximum amount of invariance obtainable under transposition, with four invariants-and the same four for both types-between those sets in B1 and B2. There is, however, no temporal coincidence of invariants as in the A groups. Moreover, the amount of temporal coincidence of invariants between different set-types is less here than it is between sets of the same type in the A groups. For example, in B1 6-Z44 and 6-14 share no fewer than five pitch-classes: B, C, Eb, and G, but the invariants coincide temporally only in the case of E, G, and Eb. Nonetheless, the intersections of the three set-types are sufficient to conceal effectively the underlying structure of the final two B groups. Any ambiguity generated by this development passage, however, is immediately resolved by the subsequent music, which is a modified reprise of the opening.23 A number of nonsets have been discussed in connection with the passage under examination. To these should be added the 7-element chromatic set that closes each of the B groups and the whole-tone and "diatonic" formations that occur on all the verticals except those which state 6-Z44 and 6-Z19 (first beat of A3 and first beat of B2, respectively). The previous paragraph concludes the discussion of specific examples. Some attention will now be given to reviewing general aspects, with additional comments as required. Of several general observations that have been made during the discussion of specific examples, two are worth repeating here. First, although Schoenberg uses and develops motives consisting of two or three notes, these components are usually not fundamental constituents, but may, in fact, be made up of elements of one or more significant sets. (See, for instance, the opening melodic motive of Herzgewachse, Ex. 2b.) Within the underlying structure of the music, Schoenberg's complex matrix-like configuration, collections of 23As mentioned earlier, Perle [3], Ex. 65, gives a virtually correct reading of the opening, assigning letters A through E to constituent sets. His set D, however, is a transposition of C (6-Z10), with t = 1, and the latter set is not the first occurrence of 6-Z10. This intersects with the first four notes of his A (6-Z44) and is, in normal order: Ab-Bb-B- C-D-DO. Further, his E (7-16) is a nonset, and several other forms of the basic hexachords occur in the opening passage that he quotes. * 61?

21 PERSPECTIVES OF NEW MUSIC fewer than four or more than eight elements are nonsets, for the most part. Second, there is a specific limitation on the vocabulary of sets. Sets with diatonic implications such as 6-32 (e. g., C-D-E-F- G-A) are not components of the music, nor are whole-tone or chromatic formations.24 Exceptions to the preceding statement are to be found in the first of Schoenberg's atonal compositions, Fifteen Poems from "The Book of the Hanging Gardens" by Stefan George, Op. 15. In particular, the second, third, fifth, eighth, and fourteenth songs contain short whole-tone passages. These result from procedures that are residual with respect to the "transitional" late tonal works, such as the First Chamber Symphony, Op. 9. The third of the Five Orchestral Pieces, Op. 16 also contains a recurrent component that combines diatonic and whole-tone elements. This however, is a very special case that can be interpreted only with reference to the music in its entirety.25 One general aspect of the mode of occurrence of a set can easily be inferred from the examples, but should perhaps be made explicit at this point. A set almost invariably consists of contiguous elements; each element of a set intersects with or is adjacent to at least one other element of the set.26 This does not rule out the possibility that sets may be formed over rests, as in Ex. 11. An element may be repeated within a set. Such repetitions are not arbitrary, but effect overlapping (either in the vertical or horizontal dimension) without change of pitch-class content, hence immediate temporal expansion of the set. Schoenberg's remarkable creative and conceptual powers are evident at every stage in the compositional process, but perhaps most extraordinarily so at the fundamental level at which sets are structured to form the basic matrix of the music. Each newly composed configuration may affect more than one dimension, creating new 24In this regard, Schoenberg's music differs markedly from that of Stravinsky. The latter, for example, in The Rite of Spring, does not hesitate to intermingle sets of the familiar type with sets that belong to the new (at that time) vocabulary of atonal music. 25See [2] for an analysis of this movement. 26In the later atonal music, however, there is a tendency to use noncontiguous formations. One such has already been cited, without comment (Ex. 3b). Noncontiguous sets at the surface level, such as the thematic restatement in m. 34 of the first piece of the Three Piano Pieces, Op. 11, provide additional and obvious exceptions. * 62?

22 SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC sets or completing sets already partially formed. The result is a strictly controlled, albeit often concealed texture of multiple dimensions in contrast to which the surface of the music gives the appearance of utmost flexibility and freedom. Flawed descriptions of that surface and failure to recognize its underlying structure are responsible for the frequent misunderstanding of Schoenberg's atonal works and for the downgrading of the artistic achievement they represent.27 It is tempting to compare Schoenberg's detailed way of segmenting the musical continuum with the traditional diminutions of tonal music. Both are hierarchic, but they are essentially different with respect to the concept of musical space. Accordingly, the traditional notions of harmony and counterpoint take on radical new meanings in his atonal music, if, indeed, they are applicable at all. It appears that Schoenberg developed his new procedures during a very short period of time, probably during the latter part of 1908, at age thirty-four. Certainly they emerge almost completely formulated in the first of the Three Pieces for Piano, Op. 11 (early 1909). As has been emphasized in the course of this paper, this development is characterized not only by the use of "new harmonies" without reference to a tonal basis,28 but also by the complex de- ployment of those components, creating a new dimensionality appropriate to the new music. Indeed, the very complexity of those procedures suggests an obvious reason for the fact that Schoenberg's way of composing atonal music was not successfully adopted by anyone other than his students, Webern and Berg.29 Others simply did not understand what was going on in the music. In this connection, and in conclusion, it can be said that Webern's procedures are closer to those of Schoenberg, although not as complex, while Berg is by no means 27Consider, for example, Boulez [13], p. 270, on the early atonal music: "Here I observe a preponderance of intervals if the unfolding is horizontal, or of chords if it is coagulated vertically, which are the least native to the classic harmony based upon superimposed thirds." 28Many of the new harmonies of course occurred in the late tonal music. For example, the set 6-Z44 occurs in m. 2 of the first of the Two Songs for Voice and Piano, Op. 14 ( ) and as the first hexachord in the bass line of the second song from the same work. 29There is room for speculation here. As far as one can ascertain from existing evidence, it was not Schoenberg, but Webern who composed the first atonal work, his Five Songs from "The Seventh Ring" by Stefan George, Op. 3 ( ). * 63 -

23 PERSPECTIVES OF NEW MUSIC as thorough or consistent as either Schoenberg or Webern, both with respect to his selection of sets and with respect to the way in which they are composed. Of course, both Webern and Berg developed their own idiosyncratic methods, the discussion of which is not within the scope of this paper. REFERENCES [1] Forte, Allen: "A Theory of Set Complexes. "Journal of Music Theory, Winter, [2] Forte, Allen: The Structure of Atonal Music. Yale University Press, forthcoming. [3] Perle, George: Serial Composition and Atonality. University of California Press, Berkeley, [4] Brinkmann, Reinhold: Arnold Sch8nberg: DreiKlavierstiicke Op. 11. Franz Steiner Verlag, Wiesbaden, [5] von der Nill, Edwin: Moderne Harmonik. Leipzig, [6] Gilbert, Steven E.: "The Trichord: An Analytic Outlook for Twentieth- Century Music." Yale University Ph. D. Dissertation, [7] Rufer, Josef, comp.: Das Werk Arnold Schoenbergs. Barenreiter, Kassel, [8] Travis, Roy: "Directed Motion in Schoenberg and Webern."Perspectives of New Music, Spring-Summer, [9] Schoenberg, Arnold: "Analysis of the Four Orchestral Songs Opus 22." Tr. by Claudio Spies. Perspectives of New Music, Spring-Summer, [10] Stuckenschmidt, H. H.: Arnold Schoenberg. Tr. by E. T. Roberts and H. Searle, Grove Press, New York, [11] Craft, Robert: "Schoenberg's Five Pieces for Orchestra." Perspectives on Schoenberg and Stravinsky, ed. by Benjamin Boretz and Edward T. Cone, W. W. Norton & Co., Inc., New York, [12] Babbitt, Milton: "Twelve-Tone Invariants as Compositional Determinants." Problems of Modern Music, ed. by Paul Henry Lang, W. W. Norton & Co., Inc., New York, [13] Boulez, Pierre: Notes of an Apprenticeship. Tr. by Herbert Weinstock, Alfred A. Knopf, New York, * 64?