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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).〔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).〕 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."〔 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."〔Christensen, Thomas (2002). ''The Cambridge History of Western Music Theory'', . Cambridge. ISBN 9781316025482.〕 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,〔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), 129–31. ISBN 0-520-03395-7〕 and six dyads.〔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): 109–18. Citation on 114.〕 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.
==Definition==
Most generally complementation is the separation of pitch-class collections into two complementary sets, one containing the pitch classes not in the other.〔 More restrictively complementation is "the process of pairing entities on either side of a center of symmetry".〔Kielian-Gilbert, Marianne (1982–83). "Relationships of Symmetrical Pitch-Class Sets and Stravinsky’s Metaphor of Polarity", ''Perspectives of New Music'' 21: 210. .〕
The term, "'combinatorial' appears to have been first applied to twelve-tone music by Milton Babbitt" in 1950,〔Whittall, 245n8〕 when he published a review of René Leibowitz's books ''Schoenberg et son école'' and ''Qu'est ce qu la musique de douze sons?''〔Milton Babbitt, Untitled review, ''Journal of the American Musicological Society'' 3, no. 1 (Spring 1950): 57–60. The discussion of combinatoriality is on p. 60.〕 Babbitt also introduced the term ''derived row''.〔Christensen, Thomas (2002). ''The Cambridge History of Western Music Theory'', . Cambridge. ISBN 9781316025482.〕
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