Permutation (music)

Applying both inversion and retrograde to a prime form produces its retrograde-inversions, considered a distinct type of permutation.

Therefore: A given prime zero (derived from the notes of Anton Webern's Concerto): The retrograde: The inversion: The retrograde inversion: More generally, a musical permutation is any reordering of the prime form of an ordered set of pitch classes [7] or, with respect to twelve-tone rows, any ordering at all of the set consisting of the integers modulo 12.

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.

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):

Prime, retrograde, inverse, and retrograde-inverse permutations.
Principal forms of Anton Webern 's tone row from Variations for piano , op. 27, movement 2. [ 1 ] [ 2 ] Play ⓘ
Initial statement begins on F(=5), mm. 2-4, cyclical permutation begins on E ♭ (=3) in mm. 7-9 (Perle 1996, p.20).