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Music in the Late Twentieth Century


CHAPTER 3 The Apex
Richard Taruskin

These questions, always discomforting, achieved special poignancy after the war—and especially in America, where the sheer research and development of musical technique achieved a prodigious, institutionally supported acme that was never approached in Europe. Stravinsky's location in America colored his serial quest in ways he was probably unaware of, conditioning its slow, cumulative, evolutionary—and yes, somewhat academic—progress (in sharp distinction to the sweeping revolutionary gestures of the Darmstadt school, so reminiscent of Stravinsky's own early maximalist phase).

The Path to the New/Old MusicThe Path to the New/Old Music

ex. 3-8 Igor Stravinsky Cantata, Ricercar II, fourth canon

Stravinsky continued to work with “rows” of varying length and content for a while, before settling on the canonical twelve non-repeating pitch classes. The next work after the Cantata was a Septet for three woodwinds, three strings, and piano. Its middle movement, a Passacaglia, has a sixteen-note ground bass on which Stravinsky erected an intricate canonic structure like the one in “Tomorrow Shall Be My Dancing Day,” replete with inversions, retrogrades, and retrograde inversions. The last movement, called Gigue (but actually a multiple fugue), is obviously modeled on the Gigue in Schoenberg's Suite, op. 29 (1926), for an almost identical instrumental septet. (Craft prepared a performance of Schoenberg's Suite at UCLA while Stravinsky was at work on the piece, and Stravinsky attended all the rehearsals.) Each instrument in this contrapuntal tour de force has its own “row,” as Stravinsky (misleadingly) put it in the score, meaning that every instrument employs an eight-note “unordered collection” or scale abstracted from the Passacaglia's ground bass, each at its own transposition.

“Musicke to heare” (Sonnet VIII), the first of Three Songs from William Shakespeare (1953), gives the serial treatment to a “row” of only four notes; In Memoriam Dylan Thomas (1954) uses a row of five. The two large-scale works that followed—Canticum Sacrum (1955), a cantata in honor of St. Mark composed for performance at the Venice Cathedral bearing his name, and Agon (1957), a “ballet for twelve dancers” commissioned for the choreographer George Balanchine by the New York City Ballet—contain Stravinsky's first compositions using twelve-tone rows of the usual sort. In each, however, the twelve-tone component consists of short, individual, self-contained episodes in what is otherwise a tonally centered, nonserial composition.

At last, with Threni (1958), a thirty-five-minute oratorio on texts from the biblical Lamentations of Jeremiah, Stravinsky turned out a work that was both serial and atonal, and that was composed using twelve-tone rows of standard design throughout. Stravinsky prepared for the task of writing it by working through the exercises in Studies in Counterpoint, a textbook on twelve-tone composition (the first in English), which the newly emigrated Ernst Krenek had published in 1940 while teaching composition at Vassar College. Krenek had completed a setting of verses from the Lamentations himself in 1942, and published it in 1957; Stravinsky studied that, too. From it, he appropriated a clever technique of serial manipulation that fascinated him, and that he would make peculiarly his own.

It was inspired by Webern's symmetrical row structures, which Krenek had been among the first to analyze. Krenek called the technique “rotation.” More precisely, it was a process of cyclic permutation whereby one varied the intervallic structure of a row, or a portion thereof, by starting on each of its constituent notes in turn and transferring the previous starting note to the end. To gain an extra dimension of symmetrical design à la Webern, Krenek worked the technique on the two complementary halves (or “hexachords”) of a row fashioned so that the hexachords were related symmetrically.

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ex. 3-9 “Rotation” (cyclic permutation and transposition) in Ernst Krenek, Lamentatio Jeremiae prophetae

In the unusually scalar row Krenek adopted for his Lamentations setting, the intervallic sequence of the second hexachord reproduced that of the first in retrograde inversion. Ex. 3-9 shows Krenek's “rotation” technique applied to the Lamentations row. The process of cyclic permutation is accompanied in each case by a transposition that keeps the starting pitches for the two respective hexachords the same. In this way the hexachords become “modal scales,” as Krenek put it, as one might derive the whole series of diatonic modes or octave species (Dorian, Phrygian, Lydian, and so on) on a given tonic by applying Krenek's rotation method to a major or minor scale. What evidently attracted him to the idea, when writing a modern version of sacred choral polyphony, was the apparent relationship of his “twelve-tone modal system” to the modal system of medieval music theory.

But in the process of transposition, Krenek's modal system stops being a “twelve-tone technique,” strictly speaking; it introduces pitch duplications between the hexachords, whose sums therefore no longer necessarily exhaust the twelve pitch classes. As Krenek put it in the essay already quoted in chapter 1, “the purpose of the operation was not so much to make the serial design stricter, but rather to relax it, insofar as the wide variety of available six-note patterns made it possible to remain within the frame of reference of the twelve-tone serial technique without constantly having to use complete twelve-tone rows.”17 Indeed, the technique “relaxes” serial design far more than Krenek admits, since it produces a series of varied arrangements of intervals around a pair of constant pitches (here, D♭ and B), which are in a sense promoted to the level of tone centers. Minimizing this contradiction with the bland remark about remaining “within the frame of reference of the twelve-tone serial technique,” Krenek has in effect (unwittingly?) readmitted tonal (or, more neutrally, “centric”) relations into the purportedly serial domain.

Krenek may eventually have come to see this feature of his rotation technique as an experimental flaw, or perhaps a feature suitable only for writing updated Palestrina. In any case he did not pursue the method in later works. But it gave Stravinsky just what he was looking for, namely a strategy that enabled him to wangle from the twelve-tone technique a familiar sort of material that suited his longstanding creative predilections. We have seen Copland doing similarly: for both composers it was worth the trouble to find a new and tortuous way back to their customary terrain because for their varying reasons they each felt the need to operate, or to be seen as operating, “within the frame of reference of the twelve-tone serial technique.” Beginning with Movements for piano and orchestra (1959–60), the next work after Threni, Stravinsky introduced a new wrinkle into the technique Krenek had pioneered. He began extracting the pitches from each successive vertical column in his hexachord arrays, and using the groups of pitches thus extracted—he called them “verticals”—as chords. Stravinsky once wrote out a demonstration of the method (shown in Fig. 3-4), using for the purpose the first hexachord in the inverted row of his Variations for orchestra (1964), and accompanied it with an idiosyncratic verbal explanation in his version of English:

Some stressed octaves and fifths and doubled intervals which could be found in this score shouldn't contradict the serial (and not harmonical) basis of the composition; the origin of it lies not in a horizontal contrapunctical accord of different voices but in a vertical similtaneous [sic] sounding of several notes belonging to a certain number of forms played together.18

Despite the octave doublings on A and C♯, in other words, the chord shown below is not a traditional triad with double-inflected third, of a kind Stravinsky often used during his “neoclassical” period, but an authentic artifact of the serial system—at least his serial system—since both A and C♯ occur twice in the outlined column of which the chord is a summary.

Stravinsky's squeamishness is revealing, but also concealing. In fact his whole “verticals” technique was designed to give him access to chords like the one outlined in this demonstration. Fig. 3-4 is amplified in Ex. 3-10, which shows all the verticals that can be derived from the array. (Since the first column yields nothing but a single pitch, that column is customarily designated “zero” when doing analyses of this kind.) Ex. 3-11, from the Variations, shows the whole panoply of verticals deployed as chords in the actual composition, punctuating a counterpoint of three trombones that is derived from the rotations of the row's complementary hexachord. (Those looking closely will notice that the chord representing vertical [1] contains an A♯ instead of an F♯; such discrepancies are often noted by analysts of Stravinsky's serial works; opinions differ as to whether they are adjustments for the sake of euphony or slips to be corrected by an editor.)

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fig. 3-4 Stravinsky's analytical demonstration of his “verticals” technique.

And what is the special property of all verticals (not just the ones that contain triads) that made them so attractive to Stravinsky? Another look at Ex. 3-9 or Ex. 3-10 will confirm that the successive transpositions of the hexachords exactly mirror the intervals between the consecutive pitches in the original form, or more simply, that the hexachord is being transposed by its own interval content, inverted. Because of this built-in inversional symmetry, the pitch-content of the verticals produced by any hexachordal “rotation” (permutation-plus-transposition) will invariably be disposed symmetrically around the generating (“zero”) pitch, the latter thus assuming, in the most literal sense of the word, the role of a tone center.

Thus, in Ex. 3-10, verticals 1 + 5, 2 + 4, and 3 (which is necessarily self-inverting) can all be symmetrically displayed around D, the zero pitch. For Stravinsky, who had long ago cut his compositional teeth on symmetrically disposed harmonies mined from the whole-tone and octatonic scales, Krenek's rotation technique, when expanded to include the extraction of verticals, offered systematic access to a greatly expanded vocabulary of symmetrical harmonic constructions, including (but not limited to) all the Stravinskian perennials like the “major-minor triad,” which shows up in the array in Ex. 3-10 not only in position 2 (as demonstrated by Stravinsky himself in Fig. 3-4), but also in position 4, the complementary location.

The same “center” pitches that function as axes of symmetry for the verticals, moreover, continually recur as the “tonics” of the “modal scales” produced by the rotations. By emphasizing these relations, Stravinsky was well aware that he was not abandoning the “tonal system” but rather (as he put it in an interview with Craft) composing in an alternative tonal system (“my tonal system”) inspired by and related to, but not entirely congruent with, the more strictly contrapuntal twelve-tone idiom employed by Schoenberg or Webern.

The Path to the New/Old Music

ex. 3-10 Igor Stravinsky, Variations, full tabulation of “verticals” from the hexachord rotated in Fig. 3-4

For in an important sense the verticals technique is not a twelve-tone technique at all, or even a serial one. As harmonic constructs Stravinsky's verticals are only very tenuously related to the original row, or “series.” In particular they have little or nothing to do with the series qua series—that is, a temporally unfolding sequence or succession of intervals. Instead, that temporal unfolding has been frozen (or “hypostatized”) into a static vertical equilibrium, like so many of the harmonies in The Firebird or The Rite of Spring. In his serial music Stravinsky sought what he had always sought. The new technique made it more difficult to find, but that only added to the virtuousness of adopting the method. Working against greater resistance, according to the existentialist work ethic, gave one's creative products greater “authenticity.” They had to be truly chosen from among consciously weighed alternatives, not merely inherited from unthinking adherence to habit, or dictated by conditioned reflex.

The Path to the New/Old MusicThe Path to the New/Old Music

ex. 3-11 Igor Stravinsky, Variations, mm. 73-85


(17) Ernst Krenek, “Extents and Limits of Serial Technique,” in Problems of Modern Music, ed. Lang (New York: Norton), p. 75.

(18) Reproduced in Robert Craft, A Stravinsky Scrapbook, 1940–1971 (London: Thames and Hudson, 1983), p. 120.

Citation (MLA):
Richard Taruskin. "Chapter 3 The Apex." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 21 Sep. 2020. <https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003004.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 21 Sep. 2020, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003004.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 21 Sep. 2020, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003004.xml