page_title: Set theory (music)
section_title: Basic operations
hierarchical_section_title: Set theory (music) / Basic operations
caption_attribution_description: English: Example of pitch class inversion, set 5-6.
context_page_description: Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.
  One branch of musical set theory deals with collections of pitches and pitch classes, which may be ordered or unordered, and can be related by musical operations such as transposition, melodic inversion, and complementation. Some theorists apply the methods of musical set theory to the analysis of rhythm as well.
context_section_description: The basic operations that may be performed on a set are transposition and 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 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. 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 complementation and multiplication as well. The complement of set X is the set consisting of all the pitch classes not contained in X (Forte 1973, 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 (Forte 1973, 21). Another name for this relationship, used by Howard Hanson (1960, 22), is "isomeric" (Cohen 2004, 33).
  Operations on ordered sequences of pitch classes also include transposition and inversion, as well as 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 x is a number representing a pitch class, its transposition by n semitones is written Tₙ = x + n mod 12. Inversion corresponds to reflection around some fixed point in pitch class space.  If x is a pitch class, the inversion with index number n is written Iₙ = n - x mod 12.
caption: Pitch class inversion: 234te reflected around 0 to become t9821
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