music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the " rudiments", that are needed to understand music notation (k ...

, equivalence class is an

equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...

( =) or equivalence between properties of sets (unordered) or

twelve-tone row In music, a tone row or note row (german: Reihe or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets ...

s (ordered sets). A relation rather than an operation, it may be contrasted with

derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...

.Schuijer (2008). ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts'', p.85. . "It is not surprising that music theorists have different concepts of equivalence rom 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." Traditionally,

octave equivalency In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

is assumed, while inversional, permutational, and transpositional equivalency may or may not be considered (

sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...

and modulations are techniques of the

common practice period In European art music, the common-practice period is the era of the tonal system. Most of its features persisted from the mid-Baroque period through the Classical and Romantic periods, roughly from 1650 to 1900. There was much stylistic evoluti ...

which are based on transpositional equivalency; similarity within difference; unity within variety/variety within unity). 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 (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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

speaks of the 'equality,' not the 'equivalence,' of sets."Schuijer (2008), p.89. However, equality may be considered identical (equivalent in ''all'' ways) and thus contrasted with equivalence and similarity (equivalent in one or more ways but not all). 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 (C major has F while G major has F). The major third and the minor sixth are not identical but share inversional equivalence (an inverted M3 is a m6, an inverted m6 is a M3). 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.

References

Musical set theory {{music-theory-stub

See also

References