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Music in the Late Twentieth Century


CHAPTER 3 The Apex
Richard Taruskin
Academicism, American Style

ex. 3-15 Igor Stravinsky, Requiem Canticles, Postlude

Academicism, American Style

ex. 3-16 Igor Stravinsky, Requiem Canticles, the two series as deployed in the Postlude

Academicism, American Style

ex. 3-17 Symmetrical harmonies in Igor Stravinsky, The Firebird

The Requiem Canticles had its first performance (the last Stravinsky premiere) under Robert Craft in October 1966, at a concert that took place on the campus of Princeton University. It was a fitting venue, for over the preceding couple of decades Princeton had become, largely through the efforts of Milton Babbitt, the American stronghold for the theory and practice of serial music. The university administration had actually commissioned the Requiem Canticles in 1965, acting on behalf of the family of an alumnus to whose mother's memory the work was dedicated.

Princetonian theory and practice differed critically from that of Darmstadt, with which it was inevitably compared. The difference had to do, certainly, with the personalities involved; but it also reflected differing institutional structures and a difference in the surrounding intellectual, cultural, and economic climate. As we saw in chapter 1, Darmstadt serialism was the fruit of pessimism, reflecting the “zero hour” mentality of war-ravaged Europe. It thrived on the idea of the cleanest possible break with the past. Princetonian serialism reflected American optimism. It rode the crest of scientific prestige and remained committed to the idea of progress, which implied the very opposite attitude toward the past: namely a high sense of heritage and obligation. Where the two coincided was in the conviction that serious artists lived in only history, not in society, and that fulfilling history's mandate meant resisting the temptation of compromise with social pressures and rewards.

Princetonian serialism—or, more generally, American postwar serialism—reflected the remarkable vision of Milton Babbitt (b. 1916), its leading theorist and (at first) its main practitioner. Trained in mathematics and formal logic as well as music, Babbitt quickly saw the possibility of rationalizing the technique of twelve-tone composition, and generalizing its theoretical foundations, on the basis of what mathematicians call “set theory.” He formulated his new theoretical approach in a paper, “The Function of Set Structure in the Twelve-Tone System” (1946), which he submitted as a Ph.D. thesis to the Princeton music department. Since there was at that time neither a qualified reader on the music faculty nor an officially instituted Ph.D. program there in music theory or composition, Babbitt was not awarded a doctorate even though he had been hired by the department as a professor, eventually holding an endowed chair in recognition of his achievements. This anomalous and frustrating situation spurred Babbitt to lobby actively for recognition of music composition as a legitimate branch of music research.

Meanwhile, his unaccepted Ph.D. dissertation, circulating widely in typescript, became perhaps the most influential unpublished document in the history of twentieth-century music. (Revised sections were eventually published as articles between 1955 and 193, and Babbitt was finally—somewhat jokingly—awarded his long-deferred degree in 1992, by which time he had retired from active teaching, having received several honorary doctorates including one from Princeton.) The precision of its vocabulary and the logical clarity of its presentation had a revolutionizing effect on academic discourse about music, and not only in America.

Several terms that Babbitt coined in his dissertation, particularly “pitch class” (the class of pitches related by octave transposition and designated with a single letter name), quickly became standard parlance even outside the domain of serial theory, for they named musical universals that had previously required cumbersome phrases to define. Another, already used freely in this book on the assumption that readers will understand it, is “aggregate,” meaning the complete set of all twelve pitch-classes. Moreover, Babbitt's appropriation of the mathematical term “combinatorial” made it possible to clarify and rationalize an important concept within serial music that went all the way back to Schoenberg, but had never before been adequately defined or properly understood for lack of a name.

All of these terms originated in set theory, the branch of mathematics, chiefly developed by the German mathematician Georg Cantor (1845–1918), that lies closest to logic. It is basically the study of the relationship between wholes (or aggregates) and parts (or members). A type of music based, like serial music, on the completion of aggregates obviously lends itself to “set-theoretical” description. Every twelve-tone row is an individual ordering of the unchanging aggregate set, so that the most important features of twelve-tone sets are (1) the way in which their particular parts relate to the general whole, and (2) the way in which their parts relate to one another.

Ex. 3-18 is an ingenious analytical table, prepared and first published by George Perle, that summarizes the entire “set complex” on which Babbitt based the first of his Three Compositions for Piano (1947), the first work he composed after formulating his set-theoretic approach to twelve-tone composition. The set complex consists of all the row forms employed in the piece, eight in all: two “primes” (original orderings), two retrogrades, two inversions, and two inverted retrogrades. The interval of transposition between similar row forms is always a tritone (six semitones); and the interval between the inverted and noninverted row forms is a perfect fifth (seven semitones), just as it often is in Schoenberg's music.

Academicism, American Style

ex. 3-18 George Perle's analytical table summarizing the “set complex” of Milton Babbitt's Three Compositions for Piano, no. 1

The two shorter staves at the bottom of the diagram show what conditioned Babbitt's choice of row forms. In every case, the pitch content of the two constituent half rows (hexachords) reproduces that of the original statement (P-0). The unordered pitch content of the two complementary hexachords making up a row, then, is a constant for this composition, or (in the language of set theory) an “invariant.” The table shows the way in which Babbitt, in keeping with the title of one of the seminal articles spun off from his dissertation, has employed “Twelve-Tone Invariants as Compositional Determinants.”20 The use of such invariants is a way of intensifying the motivic unity of a composition beyond what the mere use of a row guarantees, and that a composing poodle, so to speak, could therefore attain.

Another way of defining the relationship between the row forms is “combinatorial,” since their constituent hexachords can combine interchangeably to produce aggregates. Laying out the unordered pitch content of the hexachords in the summary staves at bottom to form ascending six-note scales reveals an interesting characteristic of combinatorial sets—sets, that is, which can be transposed to produce the sort of “hexachordal complementation” we have been observing. The six-note scales (like all complementary twelve-tone hexachords) are intervallically identical, but also palindromic. Whether read from left to right or vice versa, they produce the same sequence of tones and semitones: T-S-S-S-T. (Another way of observing their symmetry is out from the middle, producing three-note S-S-T groups that mirror one another to the left and to the right.)

Academicism, American Style

ex. 3-19a Milton Babbitt, Three Compositions for Piano, no. 1, mm. 1-8

Academicism, American Style

ex. 3-19b Milton Babbitt, Three Compositions for Piano, no. 1, last two measures

Academicism, American Style

ex. 3-19c Milton Babbitt, Three Compositions for Piano, no. 1, mm. 9-17

In his Three Compositions, Babbitt plays continually with these constants and symmetries, and with “puns” that arise out of the interplay. The first pair of measures in the first of the set, where (in apparent tribute to the opening of Schoenberg's Suite for piano, op. 25) the left hand has P0 and the right has P6, set the tone (see Ex. 3-19a). Each hand completes an aggregate over the length of the pair, but each measure also contains an aggregate formed by the two hands together. The same is also true of mm. 3–4, in which the left hand has RI1 and the right hand has R0; mm. 5–6, in which I7 in the left is pitted against RI7 in the right; and mm. 7–8, in which the combinatorial pair are R6 and I1. In sum, Babbitt has contrived combinatorial pairs of row forms that sum up all the possible relationships between orderings: transposition, inversion, retrograde, and inverted retrograde.

Especially interesting, from the point of view of set theory, is the occasional use of a technique resembling medieval hocket, in which notes played by pianist's two hands alternate in time. The first eight measures, already analyzed for linear and contrapuntal relationships, exemplify this texture as well. When combinatorial sets are in play, the hocketing device allows “secondary sets” (alternative orderings of the aggregate) to emerge like variations on a theme. For example, the notes in the first pair of measures, taken exactly in the order in which they are heard, produce two secondary sets, as follows:

Table 3-1

The use of secondary sets adds another dimension to combinatoriality, since it adds another way in which hexachords with mutually exclusive pitch content may generate aggregates. Throughout the composition, Babbitt uses this principle to guide his choice of successive set forms. The parenthetical indication of the corresponding retrogrades at the end of the table is a reminder that secondary sets may be subjected to the same manipulations as any others; and a glance at the very end of the first piece (Ex. 3-19b) will show how Babbitt used these very retrogrades as if to enclose the entire composition in a palindrome. Like his “post-Webernian” counterparts in Europe, he was fascinated by the symmetries that gave Webern's scores their distinctive profiles; and (again like them) he saw in these patterned interactions between and among multiple set forms the means of creating a truly or “purely” twelve-tone musical syntax. Finally, Babbitt was just as eager as they were to find ways of integrating other “parameters” or measurable variables, such as rhythm and dynamics, into the serial scheme.


(20) The article appears in Lang, Problems of Modern Music, pp. 108–21.

Citation (MLA):
Richard Taruskin. "Chapter 3 The Apex." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 22 Jan. 2020. <https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003006.xml>.
Citation (APA):
Taruskin, R. (n.d.). Chapter 3 The Apex. In Oxford University Press, Music in the Late Twentieth Century. New York, USA. Retrieved 22 Jan. 2020, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003006.xml
Citation (Chicago):
Richard Taruskin. "Chapter 3 The Apex." In Music in the Late Twentieth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 22 Jan. 2020, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003006.xml