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Music in the Late Twentieth Century


CHAPTER 3 The Apex
Richard Taruskin

But his methods for doing so differed fundamentally from theirs. In works like Boulez's Structures or Krenek's Sestina, described in chapter 1, the “rhythmic series” were derived from the pitch series by arbitrary numerical association. The note C was associated with the number 1, and so was the thirty-second note; C♯D♭ was associated with 2, as was the sixteenth; D = 3 = dotted sixteenth; D♯E♭ = 4 = eighth; and so on. Rests were arbitrary punctuations. Babbitt saw the arbitrariness of the “Darmstadt” method as a weakness. He drew upon his mathematical training to devise demonstrable analogies between the procedures of twelve-tone permutation as applied to a pitch series and the same procedures as applied to a series of durations. In particular, he found a way of systematically applying the process of inversion to duration.

As long as inversion was conceptualized in terms of traditional musical notation (that is, as a reversal of up-and-down contour on an imagined vertical grid), there could be no meaningful analogy with a strictly linear concept of musical time measured (as it has to be in notation) left-to-right along a horizontal grid, its single direction corresponding to the single irreversible “direction” in which real time elapses. But we do not necessarily think of inversion as a literal reversal of contour. We have no trouble thinking of a sixth as the inversion of a third even if they both ascend or descend. It was thinking about intervals in this way that gave Babbitt his clue to generalizing a theory of inversion that treated pitches not as individual frequencies but as “pitch classes,” and that could be applied with equal precision to duration.

Add a perfect fourth to a perfect fifth in the same direction and you get an octave; ditto a major third and a minor sixth, a minor second and a major seventh, two tritones, and so on. Intervals related by inversion always add up to an octave in this way. One of the traditional ways of demonstrating inversion, in fact, is based on this observation: play middle C and the E above and then transpose the C up an octave (or the E down an octave) and the interval between the tones is inverted. To put this in terms of mathematical set functions, one invokes the principle of complementation, or completion to a given sum. Any interval complements its inversion to the constant sum of an octave.

This relationship can be easily generalized into a numerical rule if one represents all the intervals as multiples of the smallest interval, namely the semitone. Thus, to recall the first example in the preceding paragraph, a perfect fourth (five semitones) plus a perfect fifth (seven semitones) equals an octave (twelve semitones). Intervallic inversion is thus reducible to a special case of what mathematicians call “complementation to the sum of 12,” or in professional jargon, “complementation modulo 12” (more colloquially, “complementation mod. 12”). Just as 9 + 3 = 12 is a universally applicable mathematical fact, so a major sixth (nine semitones) is the inversion of a minor third (three semitones); and so it goes.

And if intervallic inversion is regarded as a special case of arithmetic complementation, then complementation of time durations to a given sum may equally be so regarded, so long as both the sum and the units of measurement are constants. Babbitt sets up just such a scheme in the first of his Three Compositions for Piano. The constant unit of value (analogous to the semitone) is the sixteenth note, and the constant sum (analogous to the octave) is six. All that is needed now is a rhythmic “series,” or fixed order of quantities, and the analogy with serial pitch organization will be complete.

Let us look once again at the first pair of measures in Ex. 3-19a. The two “hocketing” voices have identical rhythmic groupings. In both, there is an initial group of five sixteenth-notes (or more precisely five sixteenth-note attacks, since the last note is extended to mark the end of the group), followed by a single sixteenth (set off by a following rest), a group of four, and a concluding group of two. That four-element series—5 1 4 2—is chosen cannily to maximize the analogies with pitch ordering. For one thing, the sum of its constituent units is twelve, so that in its complete form it maps neatly onto one complete statement of a pitch row. And for another, it contains two pairs (5 + 1, 4 + 2) that add up to six, so that complementing the series “mod. 6” will not introduce any new elements into it.

The ordering 5-1-4-2, coming first, is the equivalent of a “prime” ordering of a pitch row; and sure enough, it corresponds with prime forms in both voices (= hands). Looking again now at Ex. 3-19b, the end of the piece, which incorporates two retrograde rows, we are not surprised to find that in both hands the rhythmic groupings are likewise reversed: 2 sixteenths + 4 sixteenths + 1 sixteenth + 5 sixteenths. The same rhythmic groupings are found in the right hand in mm. 3–4, in which the pitch succession also embodies a reversed row: 2(EG♯)-4(AF♯BG)-1(C♯)-5(CDFE♭B♭). But the left hand in these measures plays a retrograde inversion, and so here Babbitt employs his complementation technique, subtracting each of the numbers in the reversed rhythmic series from six to produce the corresponding sequence of groupings: 4(FD♭CE♭)-2(B♭D)-5(G♯AGEF♯)-1(B). The remaining ordering, 1-5-2-4, coincides with the first sounding of an unreversed inversion (left hand in mm. 5–6: (F)-(CB♭D♭E♭D)-(A♭E)-(AF♯GB)).

So it goes throughout the piece, with variations. Sometimes Babbitt sounds the pitch series in even sixteenth notes in the space of a single measure, using articulation rather than contrasting note-values to mark off the appropriate rhythmic groupings. At other times he sounds the pitch series not as twelve individual notes but as four trichords (three-note chords), assigning to each a duration corresponding to the appropriate member of the rhythmic series. In the second section of the piece (Ex. 3-19c), marked off from the first by a rest and by a new tempo, he uses these two variants in counterpoint. Each ordering of the pitch series as shown in Ex. 3-18 appears in both guises, as running sixteenths and as trichords.

Successive sections of the composition are distinguished by their textures. The one beginning in m. 18 puts the set forms in a sort of canon, those in the left hand starting on the downbeats, and those in the right starting halfway through the measure. Each half-measure, read vertically, exhibits an aggregate produced by a combinatorial “secondary set.” The section beginning at m. 29 presents the linear pitch series in fairly unarticulated form, accompanying them with sharply articulated trichords that express the rhythmic series “pointillistically,” marking not successive durations of sound but successive durations of silence between the chordal articulations.

Just as the various permutations of the rhythmic series are associated on a one-to-one basis with the corresponding permutations of the pitch series, so are dynamics coordinated with the other parameters, albeit in less detail. Throughout the composition, prime forms are marked mezzo piano, retrogrades mezzo forte, inversions forte, and retrograde inversions piano. In the last section, where the original tempo returns and the first section is replayed in a rough palindrome, the whole dynamic scheme is hushed down by two degrees: primes are now pp, retrogrades p, inversions mp, and retrograde inversions ppp.

Anyone who finds beauty in orderliness and control will find it here. Babbitt's achievement was a joyous affirmation of formalism at a time when formalism was beginning its cold war ascendancy in the West, and when artistic merit was defined (according to the “new-critical” classroom shibboleth) as “maximum complexity under maximum control.”21 The whole subsequent course of Babbitt's career as a composer could in this special sense be described as a tireless quest of greater and greater beauty (or “elegance,” as mathematicians use the word), for its commitment to an ever increasing, all-encompassing orderly control of an ever more multifarious and detailed complex of relationships is self-evident.

His early compositions could be viewed as a systematic, quasi-scientific program to expand that control and to generalize the twelve-tone system into a unified theory that incorporated all the achievements of its founding generation. In his Composition for Four Instruments (1948), Babbitt turned his attention to what he called “derived sets”—his term for twelve-tone rows, like the one in Webern's Concerto, op. 24, that could be broken down into four trichords of identical intervallic content, each of which could be made to represent one of the four basic permutations.

An Integrated Musical Time/Space

ex. 3-20a Milton Babbitt, Composition for Four Instruments, end

An Integrated Musical Time/Space

ex. 3-20b Three forms of the basic set: P0, P6, and I5 in Milton Babbitt, Composition for Four Instruments

The composition is scored for flute, violin, clarinet, and cello, four instruments of contrasting range and timbre. The basic row from which the entire composition is derived is stated complete—once only—at the very end. That row, consisting of the last three notes played by each of the instruments in turn (Ex. 3-20a), is shown in Ex. 3-20b. The four trichords are laid out roughly in the order in which they are heard: the cello's trichord is labeled a, the violin's b, the flute's c, and the clarinet's d. It is very easy to see that when they are laid out in this order the resulting row is combinatorial, since the two hexachords (a + b and c + d) divide the chromatic scale into two mutually exclusive registers. The first hexachord contains all the chromatic pitches between E and A, and the second contains all the rest, from B♭ to E♭. Were the set transposed by a tritone or inverted at the fourth above, the hexachords would exchange pitch content and new aggregates (secondary sets) could be formed by combining the row forms in question contrapuntally.

In the main body of the composition, however, this governing set is replaced by four derived sets, each assigned to one of the four instruments. As shown in Ex. 3-21, the set assigned to the clarinet is derived from trichord a, the set assigned to the flute is derived from trichord b, the one assigned to the cello is derived from trichord c, and the one assigned to the violin is derived from trichord d. In each case the “prime” form of the trichord (the one taken directly from Ex. 3-20b) is followed by I, RI, and R. These derived sets maintain the same distribution of pitch content into hexachords as the original set, which means that they have inherited all of its combinatorial properties. Thus, in his “precompositional” work, Babbitt has managed to combine or synthesize the main structural innovations of both Schoenberg (combinatoriality) and Webern (derivation) into a “set of derived sets,” which in his later theoretical writings Babbitt would call a trichordal array.

An Integrated Musical Time/Space

ex. 3-21 The four derived sets in Milton Babbitt, Composition for Four Instruments

The composition is laid out in fifteen sections, corresponding to the possible groupings of the constituent instruments (four solos, four trios, six duos, one tutti). Easiest to analyze, of course are the unaccompanied solos, in which each instrument uses only “its” derived set (clarinet at the beginning, cello at m. 139, violin at m. 229, flute at m. 328). These solos are written in a manner reminiscent of Bach's suites and sonatas for unaccompanied violin or cello, in which the division of a single line into distinct registers suggests counterpoint. In Ex. 3-22, the beginning of the fugue from Bach's Fifth Suite for unaccompanied cello (in which differing registers are used to mark off the various subjects and answers) is juxtaposed with the first fifteen measures from the opening clarinet solo in Babbitt's Composition for Four Instruments, with notes occurring in three different registers grouped by enclosing them in boxes.

An Integrated Musical Time/Space

ex. 3-22a J. S. Bach, Fugue from Cello Suite no. 5

An Integrated Musical Time/Space

ex. 3-22b Milton Babbitt, Composition for Four Instruments, mm. 1-15, with registers delineated

The analysis in Ex. 3-22 shows Babbitt's clarinet line to consist of a series of “secondary sets” (mm. 1–6, 7–9, 9–12, 12–16), each containing a different shuffling, so to speak, of the four trichords that make up the clarinet's derived row. The trichords are distinguished by register, which permits their notes to be intermixed without loss of identity. The first RI (BE♭C), presented intact at the outset to establish the pattern, sounds next in the upper “voice” in m. 7, in the middle “voice” in mm. 9–11, and again in the upper voice in mm. 14–15. The prime (D♭B♭D) is stretched across mm. 2–6 in the middle voice, reappears in the bottom voice in mm. 7–8, in the upper voice in mm. 11–12, and again in the lower voice in mm. 13–14. The retrograde (A♭EG) is at bottom in mm. 2–5, at the top in mm. 8–9, at bottom again in mm. 10–12, and once again at the top in mm. 13–14. The remaining trichord, the inversion (G♭AF), is stretched across the top in mm. 2–6, touches bottom in mm. 8, comes all together in the upper voice at the middle of m. 12, and ends up scraping the bottom again in mm. 15–16.

The clarinet is succeeded in m. 36 by the other three instruments (Ex. 3-23), in a texture contrived so that each individual line is confined to its own derived row, presented trichord by trichord, completing the aggregate with every four. But at the same time the composite of the three lines, taken note by note, completes another series of aggregates (secondary sets). The fifteen sections that make up the piece continually replay this process, the derived sets continuing to complete aggregates in one “dimension” while the secondary sets formed by the composite texture complete them in another. As already hinted, all four instruments come together only in the last section.

An Integrated Musical Time/SpaceAn Integrated Musical Time/Space

ex. 3-23 Milton Babbitt, Composition for Four Instruments, mm. 36-59

The Composition is a model, in this regard, of a formalistically conceived work of art. It takes its shape, and has its reason for being, in the exhaustive working out of its own material's potential for elaboration. To quote Babbitt's pupil and exegete, the music theorist Andrew Mead, the work is a “watershed of twelve-tone compositional practice” because of its “multidimensional use of purely self-referential structures.”22 Or to paraphrase Adorno, the work is impelled throughout by the “inherent tendency of musical material.” (One can apply Adorno's concept to the genesis of an individual work without necessarily endorsing the global application to the history of music that Adorno intended.) But of course the tendency is “inherent” only because the author has made it so (as Mead's term “self-referential” already implies). Even if we prefer to see the rules as coming from the material (or from nature, or from God), it is we who have made them and derive meaning and satisfaction from them—or not.

Babbitt's rules, in the Composition for Four Instruments, apply to durations as well as pitch, as indeed they had to. Once he had hit on his method for integrating duration into the serial scheme, it too became part of his music's “inherent” tendency. Put another way, Babbitt regarded composing technique the way most scientists of his generation regarded knowledge: as something that accumulates, a sum total to which each experiment (or composition) adds its mite. Having integrated rhythm into the serial domain, to un-integrate it would be a form of backsliding. Indeed, not to attempt its extension would be irresponsible. Progress imposes obligations.

And so the durational series in Babbitt's Composition for Four Instruments—1-4-3-2, to be complemented mod. 5—is applied not only in terms approximating retrogression and inversion, but according to a new technique simulating transposition as well. Going back to the opening clarinet solo (Ex. 3-22b), we may observe the rhythmic series in its “prime” or primitive form at the very outset. The opening B is 1 sixteenth in duration, the following E♭ lasts 4 sixteenths, the C lasts 3 (counting the sixteenth rest that follows the note), and the fourth note, D♭, lasts 2. The fact that rests can be reckoned along with sound within a duration means that, according to Babbitt's rules at this stage of their formulation, a duration is the time that elapses between attacks. What is measured is not the length of sounds, but the distance between what Babbitt would later call “time points.” The next four notes (G♭, A♭, E, B♭) are one quarter note, a whole note, a dotted half note, and a half note in duration, respectively. These long values are those of the first four notes multiplied by four. So the first four durations are “really” (that is, conceptually) 1 × 1, 4 × 1, 3 × 1, and 2 × 1 sixteenths, and the next four are 1 × 4, 4 × 4, 3 × 4, and 2 × 4. It is easy enough to predict, and then confirm by looking, that the next four durations will be 1 × 3, 4 × 3, 3 × 3, and 2 × 3; and that the next four after that will be 1 × 2, 4 × 2, 3 × 2, and 2 × 2. In short, the whole series has been multiplied (or “transposed”) by each of its constituent elements in turn.

This process gets us as far as the downbeat of m. 8. At this point the rhythm becomes a composite of all the remaining forms of the rhythmic series—retrograde (2-3-4-1), inversion (4-1-2-3), and retrograde inversion (3-2-1-4)—each multiplied by itself the way the prime form had been. The distribution of set forms (1 vs. 3) mirrors the distribution of instruments (solo clarinet vs. remaining three) in the first two sections of the Composition. Ex. 3-24, borrowed from Andrew Mead's analysis, shows how the three superimposed rhythmic series conspire together to produce the rhythms on the musical surface in mm. 8–15, while the pitch sequence continues to traverse aggregates both in direct succession (through secondary sets) and as partitioned into registers (derived sets). Thus every note played by the clarinet in these measures participates simultaneously in one or two durational rows and two pitch rows. By the end of the Composition, when the full texture is employed, all four instruments are doing all of this at the same time.


(21) David Littlejohn, The Ultimate Art: Essays Around and About Opera (Berkeley and Los Angeles: University of California Press, 1992), p. 40.

(22) Andrew Mead, An Introduction to the Music of Milton Babbitt (Princeton: Princeton University Press, 1994), p. 55.

Citation (MLA):
Richard Taruskin. "Chapter 3 The Apex." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 7 Oct. 2022. <https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003007.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 7 Oct. 2022, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003007.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 7 Oct. 2022, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003007.xml