Editing Combinatoriality (section)

{{distinguish|Combinatorics}}
In [[music]] using the [[twelve tone technique]], '''combinatoriality''' is a quality shared by twelve-tone [[tone row]]s whereby each section of a row and a proportionate number of its transformations combine to form [[tone row#total chromatic|aggregates]] (all twelve tones).<ref name="Whittall">[[Arnold Whittall|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).</ref> 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 [[hexachord]]s of each, and the second hexachords of each, respectively."<ref name="Whittall"/> 

Combinatoriality is a side effect of [[derived row]]s, where the initial segment or [[set (music)|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 (music)|transposition]], [[Melodic inversion|inversion]], [[Retrograde (music)|retrograde]], and [[Retrograde inversion|retrograde-inversion]]."<ref name="Christensen">Christensen, Thomas (2002). ''The Cambridge History of Western Music Theory'', {{unpaginated}}. Cambridge. {{ISBN|9781316025482}}.</ref> 

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 [[Tetrachord#Uses|tetrachordal]]{{Broken anchor|date=2024-11-09|bot=User:Cewbot/log/20201008/configuration|target_link=Tetrachord#Uses|reason= The anchor (Uses) [[Special:Diff/566438102|has been deleted]].}} and between four [[trichord]]al sets, as well as between pairs of hexachords,<ref>[[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}}</ref> and six [[Dyad (music)|dyads]].<ref>[[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.</ref> A [[complement (music)|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.