Equivalence class (music) - Misplaced Pages

(Redirected from Equivalent set) Not to be confused with Equivalence class in mathematics or return period. ${ \override Score.TimeSignature#'stencil = ##f \relative c' { \clef treble \time 4/4 \key c \major <c c'>1 } }$ A perfect octave between two C's; equivalent but not identical Enharmonic equivalence

The notes F♯ and G♭ are enharmonic equivalents

and B

are enharmonic equivalents, both the same as A♮

Enharmonically equivalent key signatures of B♮ and C♭ major, each followed by its respective tonic chord

In music theory, equivalence class is an equality (=) or equivalence between properties of 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 ..." "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 (sequences and modulations are techniques of the common practice period 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:

Two sets , P and P′ will be considered equivalent if and only if, for any p_i,j of the first set and p′_i′,j′ of the second set, for all is and js , 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." 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

^ 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.

v t e Twelve-tone technique and serialism
Fundamentals	Combinatoriality Complementation Derivation Hexachord Interval class Invariance Partition Cross partition Tone row Aggregate List
Permutations	Prime row Retrograde Inversion Retrograde inversion Multiplication
Notable composers	Milton Babbitt Pierre Boulez Josef Matthias Hauer Second Viennese School Alban Berg Arnold Schoenberg Anton Webern Charles Wuorinen ...more...
Related articles	All-interval twelve-tone row All-trichord hexachord Atonality Chromatic scale Duration series Equivalence Formula composition Modernism (music) Punctualism Semitone Set theory Time point Trope
List of dodecaphonic and serial compositions

This music theory article is a stub. You can help Misplaced Pages by expanding it.

Categories:

See also

References