Equivalence class (music)


In music theory, equivalence class is an equality or equivalence between properties of 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 ..." "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." Traditionally, octave equivalency is assumed, while inversional, permutational, and transpositional equivalency may or may not be 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:
Forte 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." However, equality may be considered identical and thus contrasted with equivalence and similarity. For example, the C major scale, G major scale, and the major scale in all keys, are not identical but share transpositional equivalence in that the size of the intervals between scale steps is identical while pitches are not. The major third and the minor sixth are not identical but share inversional equivalence. A melody with the notes G A B C is not identical to a melody with the notes C B A G, but they share retrograde equivalence.