All-interval twelve-tone row
musical tone series
In music, an all-interval twelve-tone row, series, or chord, is a twelve-tone tone row arranged so that it contains one instance of each interval within the octave, 1 through 11 (an ordering of every interval, 0 through 11, that contains each (ordered) pitch-interval class, 0 through 11). A "twelve-note spatial set made up of the eleven intervals [between consecutive pitches]."
There are various ways to count all-interval twelve-tone rows, depending upon one's notion of equivalence. Of the 479,001,600 twelve-tone rows total, 46,272 are all-interval. Counting up to transposition, or equivalently, setting the first pitch class equals 0, brings the number down to 3856. Taking inversion as an equivalence, essentially making the second pitch class 1,2,3,4, or 5, cuts that in half, to 1,928. Retrograde inversion sets the list of intervals running backwards and therefore
cannot leave any all-interval row invariant, but the P0 and R6 of an all interval row can coincide - and do for 88 of the 1,928. So there are 1,008 all-interval row types in the sense of Schoenberg's twelve-tone system.
Going beyond Schoenberg can involve multiplication operators or rotations that split the row at the internal tritone (since the sum of numbers 1 through 11 equals 66, an all-interval row must contain a tritone between its first and last notes, as well as in their middle), and each of these operations approximately halve the count. These sets may be ordered in time or in register.
The 1,928 tone rows have been independently rediscovered several times, their first computation probably was by Andre Riotte in 1961.
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