A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.[2]
A set by itself does not necessarily possess any additional structure, such as an ordering. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis. [3]
In the theory of serial music, however, some authors (notably Milton Babbitt[4]) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a twelve-tone row) used to structure a work. These authors speak of "twelve tone sets", "time-point sets", "derived sets", etc. (See below.) This is a different usage of the term "set" from that described above (and referred to in the term "set theory").
For these authors, a set form (or row form) is a particular arrangement of such an ordered set: the prime form (original order), inverse (upside down), retrograde (backwards), and retrograde inverse (backwards and upside down) [2].
A derived set is one which is generated or derived from consistent operations on a subset, for example Webern's Concerto, Op.24, in which the last three sets are derived from the first [6]:
B B♭ D E♭ G F♯ G♯ E F C C♯ A
Represented numerically as the integers 0 to 11:
0 11 3 4 8 7 9 5 6 1 2 10
The first set (B B♭ D) being:
0 11 3 prime-form, interval-string = <-1 +4>
The second set (E♭ G F♯) being the retrograde-inverse of the first, transposed up one semitone:
3 11 0 retrograde, interval-string = <-4 +1> mod 12 3 7 6 inverse, interval-string = <+4 -1> mod 12 + 1 1 1 ------ = 4 8 7
The third set (G♯ E F) being the retrograde of the first, transposed up (or down) six semitones:
3 11 0 retrograde + 6 6 6 ------ 9 5 6
And the fourth set (C C♯ A) being the inverse of the first, transposed up one semitone:
0 11 3 prime form, interval-vector = <-1 +4> mod 12 0 1 9 inverse, interval-string = <+1 -4> mod 12 + 1 1 1 ------- 1 2 10
Each of the four trichords (3-note sets) thus displays a relationship which can be made obvious by any of the four serial row operations, and thus creates certain invariances. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music.
A time-point set is a duration set where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes [8].
See also
- Permutation (music)
- Set (folk music)
References
- ^ Whittall, p.165.
- ^ a b DeLone et al. (Eds.) (1975). Aspects of Twentieth-Century Music, p.475. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.
- ^ R. Morris, Composition With Pitch-Classes. Yale University Press, 1987. ISBN 0-300-03684-1. p. 27.
- ^ See any of his writings on the twelve-tone system, virtually all of which are reprinted in The Collected Essays of Milton Babbitt, S. Peles et. al, eds. Princeton University Press, 2003. ISBN 0-691-08966-3.
- ^ Whittall, Arnold. 2008. The Cambridge Introduction to Serialism, p.127. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).
- ^ Delone, p.474
- ^ Whittall 2008, p.97
- ^ Delone, p.476
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