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Music in the Early Twentieth Century


CHAPTER 12 In Search of Utopia
Richard Taruskin

Whatever the motivation, Schoenberg’s career was dedicated henceforth to justifying the claims he had made for his method, elaborating and standardizing procedures so that its neoclassical principles could indeed become law. The greatest emphasis was placed on whatever might enhance the status of the row as Grundgestalt, with the result that principles of symmetry and complementation, already noticeable in Schoenberg’s earliest twelve-tone works, would become the basic determinants of structure. Thus twelve-tone music, for all that it descended from expressionism (at least insofar as some leading composers of the one tendency became leading composers of the other), quickly metamorphosed into a haven for technical research and compositional tours de force. Sharing the predilection of all other neoclassicists for abstract or generic forms, twelve-tone composers went further than any others in ordering the content of their work according to rational structural principles, making content in effect tantamount to form.

An important structural principle that Schoenberg hit upon at the time of the Variations (one of the earliest works to exemplify it) could only be described in exceedingly cumbersome terms during his lifetime. Richard S. Hill, for example, a music scholar and librarian who in an article of 1936 gave the first comprehensive description of twelve-tone technique in English, spoke of “pieces in which the ‘row’ is divided into two six-note groups, the first of which in the prime contains the same notes as the second half of the mirror, but in a different order, the other halves being necessarily related similarly.”33

Schoenberg himself, in a letter to Rufer dating from 1950, the year before his death, tried to explain his purposes as well as (still cumbersomely) describing the method: “Personally I endeavour to keep the series such that the inversion of the first six tones a fifth lower gives the remaining six tones. The consequent, the seventh to twelfth tones, is a different sequence of these second six tones. This has the advantage that one can accompany melodic phrases made from the first six tones with harmonies made from the second six tones, without getting doublings.”34 Doublings were to be avoided, in Schoenberg’s oft-stated view, because they strengthened some tones at the expense of others, compromising the implied equality of “twelve tones, related only to each other” under the aegis of emancipated dissonance. Yet we have already seen that Schoenberg’s actual practice admitted tonal hierarchies. They always would. As with most theorist-composers, his stated principles (or at least the principles he stated to outsiders) were purer than his deeds—which is less to accuse him of hypocrisy than to suggest we seek the reasons for the technique in question in his most basic, and therefore unarticulated, assumptions. What may seem at first an anachronistic or extrinsic approach has in fact proved the most revealing.

Nowadays, following a nomenclature first proposed by Milton Babbitt, a mathematically adept American theorist and composer, rows that meet the criteria defined by Hill and Schoenberg are called “combinatorial,” and the technique of contriving and employing them, “combinatoriality.” In mathematics, combinatoriality is a branch of probability theory that analyzes permutations (i.e., the reordering of sets) and combinations (i.e., principles of sampling from sets). It is a useful analogy for studying the properties of twelve-tone rows, since all rows are permutations of a single set (the “aggregate,” as we have been calling it, or more simply but vaguely, the contents of the chromatic scale).

A row is combinatorial if from its various row-forms corresponding samples can be drawn that, when combined, produce invariant relationships that can be exploited as “basic shapes” (in this case harmonic constants) in the composition. In the most consummate instance, noted by Hill and Schoenberg, the corresponding segments of combinatorially related row forms complete the aggregate. Schoenberg sought out such rows because he considered them to offer a basis for a true—that is to say distinctive, consistent, and self-contained—twelve-tone harmonic system.

Look again at Ex. 12-14a, the theme from the Variations for Orchestra, op. 31, together with its harmonization. At the beginning, Po is pitted against I9, the former providing the melody, the latter the harmony. In Ex. 12-15, the two row forms, labeled A and B, are notated abstractly, one atop the other, with their respective six-note segments or hexachords labeled with subscripts 1 and 2. The pitch content of A1 and A2, two halves of a single row, are of course by definition mutually exclusive. But in this particular combination of inversion and transposition, so are A1 and B1. And from this it must follow that A1 and B2 have identical pitch content (as must A2 and B1). Segments A1 and B2, in other words, are permutations of a single (six-note) set. They have the same relationship to one another as the row forms of which they are a part. And—the converse of this relationship—A1 and B1, though mutually exclusive in pitch content, have identical intervallic orders—another combination of constancy and difference that can be compositionally exploited.

When Po is put in counterpoint with I9 all of these reciprocal and complementary relationships can be turned to structural account. Aggregates are completed in two dimensions: horizontal (A1 + A2, B1 + B2) and vertical (A1 + B1, A2 + B2). In addition, the cross pairs A1 and B2 or B1 and A2, being identical in pitch content but different in ordering, can be exploited either as harmonic constants or as variations on a basic shape. With all of these features in play, the motivic consistency of the music—its “relatedness quotient”—is vastly enhanced. Within an esthetic that valued music according to precisely this criterion (even to the point of declaring it an emblem of national supremacy), combinatoriality produced a self-evidently superior music; its discovery constituted incontrovertible musical progress.

But there is an even more particular consistency in play here. The row forms in Ex. 12-15 have been contrived and selected to produce harmonic symmetry at another level of combinatoriality. A1 begins with a tritone, B♭E, that is automatically mirrored by GC♯ in B1, the inversion. But since a tritone contains six semitones, and B1 is at a transposition of 9 semitones from A1, the /0 6/ tritone in A1 is answered by /3 9/, giving a sum of /03 6 9/, a symmetrical division of the octave known to us since the days of Liszt as the circle of minor thirds. But Schoenberg has also included the reciprocal tritone (C♯G) in the second hexachord of the Po, so that the /0 3 6 9/ coincidence happens twice when the two row forms are put in counterpoint, forming another harmonic constant.


ex. 12-15 Combinatorial row forms in Arnold Schoenberg, Variations for Orchestra, Op. 31

What is true of Po and I9 must necessarily also be true of the next pairing in the harmonized theme, Ro and RI9 (the same two row forms reversed). But now the reciprocal tritones will be found in the reciprocal order positions: 4–5 and 11–12 rather than 1–2 and 8–9. This kind of relationship can be exploited for its punning resemblances to more familiar, functional harmonies: by combining Po and I9 at the very outset, Schoenberg contrives to begin the Variations on a “diminished seventh chord,” a traditional evoker of portent or suspense. But the relationship also produces an abstract and internal consistency: every phrase of the theme as harmonized displays the same basic shape (the constant C♯EGB♭, or sum of two tritones) and surrounds it with a different intervallic configuration. It is this extreme (and extremely controllable) consistency, which Schoenberg did not attempt to explain to his radio audience in 1931, that led him to prefer (or to argue in favor of) the twelve-tone harmonization of his theme over the “tonal” one. It is secured by combinatoriality.

Having discovered these possibilities, Schoenberg went back to the laboratory, so to speak, and gave them a concentrated investigation in a pair of piano pieces, op. 33a and 33b, that resembled the ones in op. 23 and opus 25 except that now he felt secure enough in his new structural principles not to tie the pieces to familiar or archaic genres, preferring, in the use of the neutral designation Klavierstück, to imply that the working out of the row relationships sufficed to generate the form. Ex. 12-16 shows the first page of each piece (the first published in Vienna in 1929, the second published in San Francisco by Henry Cowell’s New Music Edition in 1932), together with the combinatorial row forms that in their interaction have engendered the musical shapes.

It was in these pieces that Schoenberg made it a rule of his own combinatorial practice, as stated in his letter to Rufer, to have the prime and inverted row forms stand at the distance of a perfect fifth. As we have seen in the case of the Variations, where the interval of transposition was a minor third, transposition by a fifth is not the only one that can produce the desired complementation. As Schoenberg confided to Rufer, he wished to ally his twelve-tone practice with what he considered to be “an acoustical law of nature—that between a note and its strongest and most frequent overtone.”35 Nor can we fail to recall that the fifth relation, as embodied in harmonically defined binary structures like “sonata form,” is the one that traditionally governed the form of “tonal” music.

This is symptomatic of Schoenberg’s postwar ambivalence. It has been apparent since our first comparison of his twelve-tone music with his rival Hauer’s that Schoenberg’s immediate inclination was to synthesize the novel technique with as many aspects of traditional practice as possible (and that he regarded multiplying its points of contact with tradition as technical progress), while Hauer, in his very primitiveness, displayed a tendency that was (even if only superficially or trivially) far more radical than Schoenberg’s.


ex. 12-16 Opening of Arnold Schoenberg, Klavierstücke, Op. 33a and 33b with a summary of its combinatorial row forms

Hauer’s Nomos pieces are nothing if not formally idiosyncratic and antitraditional. Seeking to place them in relation to a relevant cultural context, Gregory Dubinsky associates them not with any other contemporary music but with the public poetry readings of the time, highly declamatory affairs at which revolutionary sentiments were often given veiled expression. Dubinsky compares Hauer’s monophonic or primitively homophonic compositions with declaimed poems, each musical phrase corresponding with a line of poetry, larger groupings with stanzas. The paucity of performance directions in Hauer’s Nomos pieces, such as dynamics, articulation, and tempo markings, were an invitation, Dubinsky suggests, “to deliver a natural, affecting declamation of Hauer’s lines of music”36 according to the rhythms of heightened public speech.

But of course nothing could have been further from Schoenberg’s expressive purposes than that sort of staged improvisation. Schoenberg saw in twelve-tone music an instrument for ever-greater control over those “organic” shaping functions that had always defined the greatness of the German musical tradition, and that Schoenberg saw epitomized in his own technique of developing variation. These principles are given a bravura display in the progressively denser, more elaborate combinatorial relations on which the piano pieces of opus 33 are based.

The beginning of op. 33a, for example, consists of a series of six four-note chords, the first three (m. 1) representing a harmonic segmentation of Po and the second a segmentation of its combinatorial mate, I5. The next time the chordal idea is sounded, Po and I5 are juxtaposed vertically (m. 10) and answered by their retrogrades, Ro and RI5, in m. 11. Measures 1 and 2 each contain a single aggregate. Measures 10 and 11 contain four aggregates apiece: one in the right hand, one in the left hand, and two formed by the combinations A1 + B1 and A2 + B2. Each note in these measures has a double function, participating in two aggregate-completions, one horizontal and the other vertical.


(33) Richard S. Hill, “Schoenberg’s Tone-Rows and the Tonal System of the Future,” Musical Quarterly XXII (1936): 14–37.

(34) Arnold Schoenberg to Josef Rufer, 8 April 1950; Josef Rufer, Composition with Twelve Notes Related Only to One Another, trans. Humphrey Searle (2nd ed., London: Barrie & Jenkins, 1961), p. 95.

(35) Rufer, Composition with Twelve Tones, p. 95.

(36) Gregory Dubinsky, Six Essays, Chap. 1.

Citation (MLA):
Richard Taruskin. "Chapter 12 In Search of Utopia." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 21 Oct. 2019. <https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-012007.xml>.
Citation (APA):
Taruskin, R. (n.d.). Chapter 12 In Search of Utopia. In Oxford University Press, Music in the Early Twentieth Century. New York, USA. Retrieved 21 Oct. 2019, from https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-012007.xml
Citation (Chicago):
Richard Taruskin. "Chapter 12 In Search of Utopia." In Music in the Early Twentieth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 21 Oct. 2019, from https://www.oxfordwesternmusic.com/view/Volume4/actrade-9780195384840-div1-012007.xml