Set Theory (music) - Basic Operations

Basic Operations

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, and hence its musical character. 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).) However, 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 sharing the same total interval content, or interval vector, but which are not transpositionally or inversionally equivalent (Forte 1973, 21). Another name for this relationship, used by Howard Hanson (1960), 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 T_n = x + n (mod12). 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 I_n = n - x (mod12).