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C - Hexachordal Combinatoriality

Published online by Cambridge University Press:  17 February 2023

Robert Morris
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Summary

Hexachordal combinatoriality depends the properties of the set A, the first six notes of a row:

  • 1. If A can be transposed and inverted so the result has no tones in common with A, I-type combinatoriality is possible.

  • 2. If A can be transposed so the result has no tones in common with A, T-type combinatoriality is possible.

  • 3. If A can be transposed and inverted so the result is the same set as A, RI-type combinatoriality is possible.

  • 4. If A can be transposed so the result is the same set as A, R-type combinatoriality is possible. Here the transposition can always be by 0 semitones, so all rows can produce this type. However, there are some A sets that can produce identity under a nonzero transposition.

We can write the four cases algebraically, where B is the complement of A; that is, A and B have no tones in common, and A and B comprise the aggregate:

1. I-type: B = TnIA for at least one value of n.

2. T-type: B = TqA for at least one value of q.

3. RI-type: A = TsIA for at least one value of s.

4. R-type: B = TuA for at least one value of u.

An all-(hexachordal)-combinatorial row has a set A that fulfills each case.

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The Whistling Blackbird
Essays and Talks on New Music
 , pp. 367
Publisher: Boydell & Brewer
Print publication year: 2010

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