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. 43-64 Published by: Perspectives of New Music Stable URL: http://www.jstor.org/stable/832462 . Accessed: 30/01/2012 02:31 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp 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 [email protected]. Perspectives of New Music is collaborating with JSTOR to digitize, preserve and extend access to Perspectives of New Music. http://www.jstor.org 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 fo- cus, instead, on an elementary problem in Schoenberg's atonal oeuvre,1that of distinguishing compositional units or sets from sec- ondary 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 other- wise. There will also be no comparison of Schoenberg's atonal and twelve-tone procedures. Schoenberg's atonal compositions exhibit a remarkable and inno- vational kind of structuring, one that cannot be characterized sim- ply, 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 com- position 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, pre- twelve-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 - 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 signi- ficant 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 num- ber 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 7-15. One final convention, which is perhaps more familiar to readers of this journal. Transposition of a set S is de- fined 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 in- dicate 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 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 comple- ment 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 ex- plicitly 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 mu- sical 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 .=7 n^~~~~~i. .. I* 6-Z1o 6-16 6-Z39 (-6-ZIo) 6-16 6-16 6-166-1 6-Z3 6-16 nd two explicit forms 6-16 616 6-Z39 6-Z 6-16 6-16 6-Zio 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 tonal- evolutionary posture he emulates von der Null [5], p. 102. * 45 - 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 hex- achord of type 6-16. Subsequent important roles of these hexa- chords 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 com- plicated 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 low- er register (Ex. 1), the first two form 6-16, while the final one forms 6-Z39-the latter identical, with respect to pitch and reg- ister, 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 aton- al music. There in the uppermost part are two ordered transpo- sitions of the melodic theme of the opening. (Note that the in- variants 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 mu- sic shown in Ex. 2a is the opening of Herzgewachse, Op. 20 (1911). Langsam 1 |i+b--- 1i i Ex.2a - 46 - SETS AND NONSETS IN SCHOENBERG'S ATONAL MUSIC 6-Z19 6-Z3 8-12 8-7 '"6-Z 4 47 4-13 7-21 / 112 4 8 4-zs5 4-13 5-32 6 Z19 4-Z15 5-21 4-19 4-19 .^~~ ' ~I/ -- XJ.