Equivalence class (music)Wikipedia open wikipedia design.
In music theory, equivalence class is an equality (=) or equivalence between sets (unordered) or twelve-tone rows (ordered sets). 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." Octave equivalency is assumed, while inversional, permutational, and transpositional equivalency are not considered.
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."
|This music theory article is a stub. You can help Wikipedia by expanding it.|