In music, a permutation of a set is any ordering of the elements of that set.1 Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation (also called rotation), or multiplicative operations (such as the cycle of fourths and cycle of fifths transforms). These may produce reorderings of the members of the set, or may simply map the set onto itself.
The permutations resulting from applying the inversion or retrograde operations are categorized as the prime form's inversions and retrogrades, respectively. Likewise, applying both inversion and retrograde to a prime form produces its retrograde-inversions, which are considered a distinct type of permutation.
Permutation may be applied to smaller sets as well. However, the use of transformation operations to such smaller sets do not necessarily result in permutation of the original set. Here is an example of non-permutation of trichords, using the operations of retrogradation, inversion, and retrograde-inversion, combined in each case with transposition, as found within in the tone row (or twelve tone series) from Anton Webern's Concerto:
B, B♭, D, E♭, G, F♯, G♯, E, F, C, C♯, A
If the first three notes are regarded as the "original" cell, then the next three are its transposed retrograde inversion (backwards and upside down), the next three are the transposed retrograde (backwards), and the last three are its transposed inversion (upside down).3
Restricting the transformations to the usual practice of twelve tone technique before 1950, within all 144 cells, a tone row has a maximum of 48 permutations, including its prime form.citation needed However, not all prime series have so many variations because the transposed and inverse transformations of a tone row may be identical to each other, a phenomenon known as invariance.citation needed
One technique facilitating twelve-tone permutation is the use of number values corresponding with musical letter names. The first note of the first of the primes, actually prime zero (commonly mistaken for prime one), is represented by 0. The rest of the numbers are counted half-step-wise such that: B = 0, C = 1, C♯/D♭ = 2, D = 3, D♯/E♭ = 4, E = 5, F = 6, F♯/G♭ = 7, G = 8, G♯/A♭ = 9, A = 10, and A♯/B♭ = 11.
Prime zero is retrieved entirely by choice of the composer. To receive the retrograde of any given prime, the numbers are simply rewritten backwards. To receive the inversion of any prime, each number value is subtracted from 12 and the resulting number placed in the corresponding matrix cell (see twelve-tone technique). The retrograde inversion is the values of the inversion numbers read backwards.
A given prime zero (derived from the notes of Anton Webern's Concerto):
0, 11, 3, 4, 8, 7, 9, 5, 6, 1, 2, 10
10, 2, 1, 6, 5, 9, 7, 8, 4, 3, 11, 0
0, 1, 9, 8, 4, 5, 3, 7, 6, 11, 10, 2
The retrograde inversion:
2, 10, 11, 6, 7, 3, 5, 4, 8, 9, 1, 0
More generally, a musical permutation is any reordering of the prime form of an ordered set of pitch classes 4 or, with respect to twelve-tone rows, any ordering at all of the set consisting of the integers modulo 12.5 In that regard, a musical permutation is a combinatorial permutation from mathematics as it applies to music. Permutations are in no way limited to the twelve-tone serial and atonal musics, but are just as well utilized in tonal melodies especially during the 20th and 21st centuries, notably in Rachmaninoff's "Variations on the Theme of Paganini" for orchestra and piano.citation needed
Cyclical permutation is the maintenance of the original order of the tone row with the only change being that of the initial pitch-class, with the original order following after.citation needed This is also called rotation.6 A secondary set may be considered a cyclical permutation beginning on the sixth member of a hexachordally combinatorial row. The tone row from Berg's Lyric Suite, for example, is realized thematically and then cyclically permuted (0 is bolded for reference):
5 4 0 9 7 2 8 1 3 6 t e 3 6 t e 5 4 0 9 7 2 8 1
- Allen Forte, The Structure of Atonal Music (New Haven and London: Yale University Press, 1973): 3; John Rahn, Basic Atonal Theory (New York: Longman, 1980), 138
- Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p.97. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).
- George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, fourth edition, revised (Berkeley, Los Angeles, and London: University of California Press, 1977): 79. ISBN 0-520-03395-7.
- Wittlich, Gary (1975). "Sets and Ordering Procedures in Twentieth-Century Music", Aspects of Twentieth-Century Music. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5page needed.
- John Rahn, Basic Atonal Theory (New York: Longman, 1980), 137.
- John Rahn, Basic Atonal Theory (New York: Longman, 1980), 134–34