In
music theory, equivalence class is an
equality (
=) or
equivalence
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
between properties of
sets (unordered) or
twelve-tone rows (ordered sets). A relation rather than an operation, it may be contrasted with
derivation.
[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 ...
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 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.
See also
*
Enharmonic equivalency
*
Identity (music)
*
Invariance (music)
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law ...
*
Set theory (music)
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed th ...
*
Similarity relation (music)
References
Musical set theory
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