In
music theory
Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
, equivalence class is an
equality
Equality generally refers to the fact of being equal, of having the same value.
In specific contexts, equality may refer to:
Society
* Egalitarianism, a trend of thought that favors equality for all people
** Political egalitarianism, in which ...
(
=) 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.
[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 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 call ...
and
modulations are techniques of the
common practice period
In Western classical music, the common practice period (CPP) was the period of about 250 years during which the tonal system was regarded as the only basis for composition. It began when composers' use of the tonal system had clearly supersede ...
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
Identical may refer to:
* Identical, when two things are the same, see Identity (philosophy)
* Identical (Hopkins novel), ''Identical'' (Hopkins novel), a 2008 young adult novel by Ellen Hopkins
* Identical (Turow novel), ''Identical'' (Turow novel ...
(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
In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that are ...
*
Identity (music)
In post-tonal music theory, identity is similar to identity in universal algebra. An identity function is a permutation or transformation which transforms a pitch or pitch class set into itself. Generally this requires symmetry. For inst ...
*
Invariance (music)
*
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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