Editing Twelve-tone technique (section)

===Derivation===
{{Main|Derived row}}
''Derivation'' is transforming segments of the full chromatic, fewer than 12 pitch classes, to yield a complete set, most commonly using trichords, tetrachords, and hexachords. A [[Derived row|derived set]] can be generated by choosing appropriate transformations of any [[trichord]] except 0,3,6, the [[diminished triad]]{{Citation needed|date=December 2022}}. A derived set can also be generated from any [[tetrachord]] that excludes the interval class 4, a [[major third]], between any two elements. The opposite, ''partitioning'', uses methods to create segments from sets, most often through [[registral difference]].

====Combinatoriality====
{{Main|Combinatoriality}}
[[Combinatoriality]] is a side-effect of derived rows where combining different segments or sets such that the pitch class content of the result fulfills certain criteria, usually the combination of hexachords which complete the full chromatic.

====Invariance====<!--[[Invariance (music)]] redirects directly here.-->
{{Listen|type=music
|filename=Schoenberg - Concerto for Violin - hexachordal invariance.mid
|title=Schoenberg's Concerto for Violin
|description=[[File:Schoenberg - Concerto for Violin - hexachordal invariance.png|center|350px]]Hexachord invariance.<ref>Haimo 1990, 27.</ref> The last hexachord of P<sub>0</sub> (C–C{{music|#}}–G–A{{music|b}}–D–F) contains the same pitches as the first hexachord of I<sub>5</sub> (D–C{{music|#}}–A{{music|b}}–C–G–F).
}}

''Invariant'' formations are also the side effect of derived rows where a segment of a set remains similar or the same under transformation. These may be used as "pivots" between set forms, sometimes used by [[Anton Webern]] and [[Arnold Schoenberg]].<ref>Perle 1977, 91–93.</ref>

''Invariance'' is defined as the "properties of a set that are preserved under [any given] operation, as well as those relationships between a set and the so-operationally transformed set that inhere in the operation",<ref>Babbitt 1960, 249–250.</ref> a definition very close to that of [[Invariance (mathematics)|mathematical invariance]]. [[George Perle]] describes their use as "pivots" or non-tonal ways of emphasizing certain [[pitch (music)|pitches]]. Invariant rows are also [[combinatoriality|combinatorial]] and [[derived row|derived]].