Combinatoriality

In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates (all twelve tones).^[1] Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."^[1]

Combinatoriality is a side effect of derived rows, where the initial segment or set may be combined with its transformations (T,R,I,RI) to create an entire row. "Derivation refers to a process whereby, for instance, the initial trichord of a row can be used to arrive at a new, 'derived' row by employing the standard twelve-tone operations of transposition, inversion, retrograde, and retrograde-inversion."^[2]

Combinatorial properties are not dependent on the order of the notes within a set, but only on the content of the set, and combinatoriality may exist between three tetrachordal and between four trichordal sets, as well as between pairs of hexachords,^[3] and six dyads.^[4] A complement in this context is half of a combinatorial pitch class set and most generally it is the "other half" of any pair including pitch class sets, textures, or pitch range.

^ ^a ^b Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p. 272. New York: Cambridge University Press. ISBN 978-0-521-86341-4 (hardback) ISBN 978-0-521-68200-8 (pbk).
^ Christensen, Thomas (2002). The Cambridge History of Western Music Theory, ^{[unpaginated]}. Cambridge. ISBN 9781316025482.
^ George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, fourth edition, revised (Berkeley, Los Angeles, London: University of California Press, 1977), 129–131. ISBN 0-520-03395-7
^ Peter Westergaard, "Some Problems Raised by the Rhythmic Procedures in Milton Babbitt's Composition for Twelve Instruments", Perspectives of New Music 4, no. 1 (Autumn–Winter 1965): 109–118. Citation on 114.

[Whittall-1] Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p. 272. New York: Cambridge University Press. ISBN 978-0-521-86341-4 (hardback) ISBN 978-0-521-68200-8 (pbk).

[Christensen-2] Christensen, Thomas (2002). The Cambridge History of Western Music Theory, ^{[unpaginated]}. Cambridge. ISBN 9781316025482.

[3] George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, fourth edition, revised (Berkeley, Los Angeles, London: University of California Press, 1977), 129–131. ISBN 0-520-03395-7

[4] Peter Westergaard, "Some Problems Raised by the Rhythmic Procedures in Milton Babbitt's Composition for Twelve Instruments", Perspectives of New Music 4, no. 1 (Autumn–Winter 1965): 109–118. Citation on 114.

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