AN INTEGRATED MUSICAL TIME/SPACE

Chapter:

CHAPTER 3 The Apex

Source:

MUSIC IN THE LATE TWENTIETH CENTURY

Author(s):

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.

Citation (MLA):

Richard Taruskin. "Chapter 3 The Apex." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 8 Dec. 2024. <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 8 Dec. 2024, 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 8 Dec. 2024, from https://www.oxfordwesternmusic.com/view/Volume5/actrade-9780195384857-div1-003007.xml