Permutation (music)

In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. 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.

Order is particularly important in the theories of compositional techniques originating in the 20th century such as the twelve-tone technique and serialism. Analytical techniques such as set theory take care to distinguish between ordered and unordered collections. In traditional theory concepts such voicing and form include ordering. For example, many musical forms, such as rondo, are defined by the order of their sections.

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:

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