Equivalence class (music)
Encyclopedia
In music theory
, equivalence class is an equality (=
) or equivalence
between 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 [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. P[itch]C[lass] set theory, however, has adhered to formal definitions of equivalence."
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."
Other equivalencies in music include:
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, equivalence class is an equality (=
Equals sign
The equality sign, equals sign, or "=" is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. The equals sign is placed between the things stated to have the same value, as in an equation...
) or equivalence
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
between sets
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
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 [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. P[itch]C[lass] set theory, however, has adhered to formal definitions of equivalence."
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."
Other equivalencies in music include:
- Enharmonic equivalencyEnharmonicIn modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
- Inversional equivalencyInversion (music)In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
- Octave equivalencyOctaveIn music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
- Permutational equivalencyPermutation (music)In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...
- Transpositional equivalency