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. Pitch class 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).— Milton Babbitt, (1992). The Function of Set Structure in the Twelve-Tone System, 8-9, cited in
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.