Combinatoriality
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In music using the twelve tone technique combinatoriality is a side-effect of derived rows where combining different segments or sets such that the pitch class content of the result fulfills certain criteria, usually the combination of hexachords which complete the full chromatic.
The term was first described by Milton Babbitt. Hexachordal inversional combinatoriality refers to any two rows, one of which is an inversion and one is not. The first row's first half, or six notes, are the second's last six notes, but not necessarily in the same order. Thus the first half of each row is the others complement, as with the second half, and, when combined, these rows still maintain a fully chromatic feeling and don't tend to reinforce certain pitches as tonal centers as would happen with freely combined rows. Babbitt also described the semi-combinatorial row and the all-combinatorial row, the latter being a row which is combinatorial with any of its derivations and their transpositions. Retrograde Hexachordal combinatoriality is considered trivial, since any set has retrograde hexachordal combinatoriality with itself. Combinatoriality may be use to create an aggregate or all twelve tones, though the term often refers simply to combinatorial rows stated together.
Semi-combinatorial sets are sets whose hexachords are capable of forming an aggregate with one of its basic transformations transposed.
All-combinatorial sets are sets whose hexachords are capable of forming an aggregate with any of its basic transformations transposed. There are six source sets, or basic hexachordally all-combinatorial sets, each hexachord of which may be reordered within itself:
- (A) 0 1 2 3 4 5 // 6 7 8 9 10 11
- (B) 0 2 3 4 5 7 // 6 8 9 10 11 1
- (C) 0 2 4 5 7 9 // 6 8 10 11 1 3
- (D) 0 1 2 6 7 8 // 3 4 5 9 10 11
- (E) 0 1 4 5 8 9 // 2 3 6 7 10 11
- (F) 0 2 4 6 8 10 // 1 3 5 7 9 11