Editing Set theory (music) (section)

==Basic operations==
{{Main|Transformation (music)}}
[[File:Pitch class 5-6 inversion example.png|upright=0.5|thumb|Pitch class inversion: 234te reflected around 0 to become t9821]]

The basic operations that may be performed on a set are [[transposition (music)|transposition]] and [[Melodic inversion|inversion]]. Sets related by transposition or inversion are said to be ''transpositionally related'' or ''inversionally related,'' and to belong to the same [[set class]]. Since transposition and inversion are [[isometry|isometries]] of pitch-class space, they preserve the intervallic structure of a set, even if they do not preserve the musical character (i.e. the physical reality) of the elements of the set.{{Citation needed|date=July 2019|reason=This claim recently said just the opposite: that the musical character is in fact preserved.}} This can be considered the central postulate of musical set theory. In practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece.

Some authors consider the operations of [[complement (music)|complementation]] and [[multiplication (music)|multiplication]] as well. The complement of set X is the set consisting of all the pitch classes not contained in X.{{sfn|Forte|1973|loc=73–74}} The product of two pitch classes is the product of their pitch-class numbers modulo 12. Since complementation and multiplication are not [[isometries]] of pitch-class space, they do not necessarily preserve the musical character of the objects they transform. Other writers, such as Allen Forte, have emphasized the [[Z-relation]], which obtains between two sets that share the same total interval content, or [[interval vector]]—but are not transpositionally or inversionally equivalent.{{sfn|Forte|1973|loc=21}} Another name for this relationship, used by Hanson,{{sfn|Hanson|1960|loc=22}} is "isomeric".{{sfn|Cohen|2004|loc=33}}

Operations on ordered sequences of pitch classes also include transposition and inversion, as well as [[Permutation (music)|retrograde and rotation]]. Retrograding an ordered sequence reverses the order of its elements. Rotation of an ordered sequence is equivalent to [[cyclic permutation]].

Transposition and inversion can be represented as elementary arithmetic operations. If {{var|x}} is a number representing a pitch class, its transposition by {{var|n}} semitones is written T<sub>{{var|n}}</sub>&nbsp;=&nbsp;{{var|x}}&nbsp;+&nbsp;{{var|n}}&nbsp;mod&nbsp;12. Inversion corresponds to [[Reflection (mathematics)|reflection]] around some fixed point in [[pitch class space]]. If {{var|x}} is a pitch class, the inversion with [[index number]] {{var|n}} is written I<sub>{{var|n}}</sub>&nbsp;=&nbsp;{{var|n}}&nbsp;-&nbsp;{{var|x}}&nbsp;mod&nbsp;12.