Permutation (music)

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This article is about a concept in musical set theory. For other uses of this word, see permutation (disambiguation).

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:

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, a tone row has a maximum of 48 permutations, including its prime form. However, not all prime series have so many variations because the tranposed and inverse transformations of a tone row may be identical to each other, a phenomenon known as invariance.

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.