Editing Twelve-tone technique (section)

====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]].