Equivalence class (music)

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A perfect octave between two C's

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.[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] 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).

— 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]