In music, a permutation of a set is a transformation of its prime form by applying zero or more of certain operations, specifically transposition, inversion, and retrograde.
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.
Here is an example of permutation usage in the tone row (or twelve tone series) from Anton Webern's Concerto:
B, Bb, D, Eb, 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 retrograde inversion (backwards and upside down), the next three are retrograde (backwards), and the last three are its inversion (upside down).
In the twelve tone technique, within all 144 cells, a tone row has a maximum of 48 permutations, including its prime form. 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.
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#/Db = 2, D = 3, D#/Eb = 4, E = 5, F = 6, F#/Gb = 7, G = 8, G#/Ab = 9, A = 10, and A#/Bb = 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, one number value is subtracted from 12 and that number placed in the corresponding matrix cell (see Twelve-tone Technique). The retrograde inversion is the values of the inversion numbers read backwards.
Therefore:
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
The retrograde:
10, 2, 1, 6, 5, 9, 7, 8, 4, 3, 11, 0
The inversion:
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 (DeLone et al. (Eds.), 1975, chap. 6). 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.
See also
References
- DeLone et al. (Eds.) (1975). Aspects of Twentieth-Century Music. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5, Ch. 6.
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