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Music in the Late Twentieth Century


CHAPTER 3 The Apex
Richard Taruskin
Full Realization

ex. 3-24 Milton Babbitt, Composition for Four Instruments, superimposed rhythmic series

As its very title promises, Babbitt's next composition, Composition for Twelve Instruments (1948, revised 1954), is an attempt to extend the controlling techniques even further, integrating durations even more systematically into the serial texture by creating a complete durational analogue to a full chromatic tone row. This is done by assigning to every member of a row two numbers, the first denoting its order within the set, the second its pitch measured in semitones from an arbitrary “zero” If the first pitch in the row is taken as the “zero” pitch, then that pitch will be defined by the numerical pair (0, 0) and the rest will be computed from it. Once assigned to the pitches of P0, the pitch numbers are treated as constants or absolutes throughout the composition. Applied to the row in the Composition for Twelve Instruments (given together with its combinatorially related inversion at the perfect fifth), the assignment of dual designations is shown in Ex. 3-25.

Full Realization

ex. 3-25 P6 and I7 from Milton Babbitt's Composition for Twelve Instruments (pitched on F), with order and pitch numbers

The rhythmic series conforms to the same dual numerical catalogue. Its twelve order positions carry, as always, the numbers from 0 to 11, and the pitch numbers are converted into duration numbers by substituting sixteenth notes for semitones. Thus the series of pitch numbers in Ex. 3-25—0, 1, 4, 9, 5, 8, 3, 10, 2, 11, 6, 7—would translate into a series of durations as shown in Ex. 3-26. It begins with a dotted half note (= 12 sixteenths) because when doing arithmetic with twelve as the modulus, 12 = 0 (as 13 = 1, 14 = 2, and so on.) The pitches of P0 are retained in Ex. 3-26 to show how directly a pitch interval translates into a time interval when using this system.

Full Realization

ex. 3-26 Durational series derived from P0 (Milton Babbitt, Composition for Twelve Instruments)

A glance at the exceedingly sparse or pointillistic score, of which the first eighteen measures are given in Ex. 3-27, shows that the first presentation of the row is made, both in terms of pitch and in terms of rhythm, at a transposition of two semitones/sixteenths, evidently because Babbitt did not want to begin with a note lasting a whole measure. So the first note heard, in the harp, is G (F plus two semitones), which means the first time interval between note-attacks will be an eighth note (0 + 2 sixteenths). The whole series of pitch/duration numbers at this transposition (i.e., constant addition to the original set of numbers) will thus be 2, 3, 6, 11, 7, 10, 5, 0 (= 12), 4, 1, 8, 9. These numbers are easily traced in terms of elapsed time between note attacks. For example, the twelfth note sounded is the clarinet D♭ in m. 6; the time interval before the next attack [bassoon G in m. 7] is exactly nine sixteenths, the last number in the durational series; all the intervening durations can be readily verified.

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ex. 3-27 Milton Babbitt, Composition for Twelve Instruments, mm. 1-18

But D♭ is not the last note of P2, the prime pitch series starting on G. Something more complicated is going on in the pitch domain. The first three notes in the score conform both to the pitches and the implied rhythms of P2, but the fourth note, horn D, is not the expected note. (The expected note, E, does not appear until m. 6, in the flute part.) And that is because Babbitt is starting up another row form from the D, namely RI9. To complicate matters even further, another prime (P3) starts with the violin A♭ in m. 1, which is simultaneously the second note of P2 as traced from the beginning.

In fact the complications have only begun to be accounted for. It turns out on analysis that every one of the instrumental parts, while participating in the generally unfolding complex of series that we have been describing, simultaneously plays its own unique version of the row. The first twelve notes in the flute part at the top of the score, for example, enunciate yet another simultaneous prime form (P11, starting on E), the first twelve in the oboe part running beneath it make up R6, the clarinet unfolds I8, and the bassoon RI7.

So it goes, all the way down the page, so that a total of twelve individual set forms (three Ps, three Is, three Rs, and three RIs) are simultaneously in play, each one starting on a different note of the chromatic scale so that the first notes in all parts create yet another aggregate (as do the second notes, the third notes and so forth). The completion of the first set of twelve simultaneous series occurs in m. 36. Anyone who wishes to expend the time and effort may verify the facts just enumerated and observe in full detail how every single note in the score fulfills multiple functions, simultaneously participating in the completion of at least three (and sometimes four) pitch aggregates, as well as taking its place in a rhythmic series (and also a series of dynamics that has not been described).

Citation (MLA):
Richard Taruskin. "Chapter 3 The Apex." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 30 Nov. 2023. <https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003008.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 30 Nov. 2023, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003008.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 30 Nov. 2023, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003008.xml