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. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. P[itch]C[lass] set theory, however, has adhered to formal definitions of equivalence."
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 pi,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).
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."
Other equivalencies in music include:
- Enharmonic equivalency
- Inversional equivalency
- Octave equivalency
- Permutational equivalency
- Transpositional equivalency
- 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.