Set theory (music)

Some theorists apply the methods of musical set theory to the analysis of rhythm as well.

[4] More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates).

[5] The elements of a set may be manifested in music as simultaneous chords, successive tones (as in a melody), or both.

For example, a piece (whether tonal or atonal) with a clear pitch center of F might be most usefully analyzed with F set to zero (in which case {0,1,2} would represent F, F♯ and G. (For the use of numbers to represent notes, see pitch class.)

Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g.

Rahn),[11] octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.

[citation needed] 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.

Since complementation and multiplication are not isometries of pitch-class space, they do not necessarily preserve the musical character of the objects they transform.

[15] Operations on ordered sequences of pitch classes also include transposition and inversion, as well as retrograde and rotation.

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.

[18] Thus the chromatic trichord {0, 1, 2} belongs to set-class 3–1, indicating that it is the first three-note set class in Forte's list.

Western tonal music for centuries has regarded major and minor, as well as chord inversions, as significantly different.

Ignoring the physical reality of sound is an obvious limitation of atonal theory.

Inversionally symmetrical chords are invariant under reflections in pitch class space.

This means that the chords can be ordered cyclically so that the series of intervals between successive notes is the same read forward or backward.