# Equivalence class (music)

*This article is about equivalency in music; for equivalency in mathematics see Equality (mathematics) and equivalence class.*

In music theory, **equivalence class** is an equality (=) or equivalence between sets or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation.^{[1]} "It is not surprising that music theorists have different concepts of equivalence [from each other]..."^{[2]} "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence."^{[1]}

A definition of equivalence between two twelve-tone series that Schuijer describes as informal despite its air of mathematical precision, and that shows its writer considered equivalence and equality as synonymous:

Two sets [twelve-tone series], P and P′ will be considered equivalent [equal] if and only if, for any p_{i,j}of the first set and p′_{i′,j′}of the second set, for all is and js [order numbers and pitch class numbers], if i=i′, then j=j′. (= denotes numeral equality in the ordinary sense).— Milton Babbitt, (1992).The Function of Set Structure in the Twelve-Tone System, 8-9, cited in^{[3]}

Forte (1963, p. 76) similarly uses *equivalent* to mean *identical*, "considering two subsets as equivalent when they consisted of the same elements. In such a case, mathematical set theory speaks of the 'equality,' not the 'equivalence,' of sets."^{[4]}

Other equivalencies in music include:

- Enharmonic equivalency
- Inversional equivalency
- Octave equivalency
- Permutational equivalency
- Transpositional equivalency

## See also

## Sources

- 1 2 Schuijer (2008).
*Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts*, p.85. ISBN 978-1-58046-270-9. - ↑ Schuijer (2008), p.86.
- ↑ Schuijer (2008), p.87.
- ↑ Schuijer (2008), p.89.