Integral serialist composers:
The Domain technique of Boulez. Used in Le marteau sans maitre and other subsequent pieces, the domain technique involves a tone-row:
2 4 2 1 3 I 4 2 1 3 2 II 2 1 3 2 4 III 1 3 2 4 3 IV 3 2 4 3 1 VEach permutation generates a "domain" by multiplying these chords and intervals with each other. The multiplication may be understood as transposing one chord on each of the notes of another. If we notate, for example, the chords of domain I with
aa ab ac ad ae ba bb bc bd be ca cb cc cd ce da db dc dd de ea eb ec ed eewhere aa consists of the interval E> - F transposed on E> and F, i.e. the chord: E>, F, D>; ab will consist of the same minor seventh interval (E> - F) built or transposed on each sound of the second chord (D, C#, B>, B), i.e. the chord: D, C, C#, B, B>, A>, (B), A; etc.
Boulez observes that elements ab and ba contain exatly the same sounds and so do ca=ac, de=ed, etc. He calls these total isomorphisms. Partial isomorphisms are created by all the sounds in the same row or in the same column since they all have one element in common. The diagonal: aa, bb, cc, dd, ee is made of unique elements. The composer organizes then the music as a play between these similar, partially similar, and unique groupings.
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