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In post-tonal music theory, identity is similar to /nowiki>reflection_symmetry_rather_than_interval_families'_rotational_symmetry.html" ;"title="reflection_symmetry.html" ;"title="/nowiki>reflection symmetry">/nowiki>reflection symmetry rather than interval families' rotational symmetry">reflection_symmetry.html" ;"title="/nowiki>reflection symmetry">/nowiki>reflection symmetry rather than interval families' rotational symmetry]. C-E belongs to a family [sum-4] of symmetrically related dyads as follows:"
C=0, so in mod12, the interval-4 family:
Thus, in addition to being part of the sum-4 family, C-E is also a part of the interval-4 family (in contrast to sum families, interval families are based on difference).
In post-tonal music theory, identity is similar to identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
in universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
. An identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
is a permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
or transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
which transforms a pitch or pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
set into itself. Generally this requires symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. For instance, inverting an augmented triad
Augment or augmentation may refer to:
Language
* Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
*Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...
or C4 interval cycle In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class.Whittall, Arnold. 2008. ''The Cambridge Introduction to Serialism'', p. 273-74. New York: Cambridge University Press. (pbk). In other ...
, 048, produces itself. Performing a retrograde operation upon the 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 ...
01210 produces 01210. Doubling the length of a rhythm while doubling the tempo produces a rhythm of the same durations as the original.
In addition to being a property of a specific set, identity is, by extension, the "family" of sets or set forms which satisfy a possible identity. These families are defined by symmetry, which means that an object is invariant to any of various transformations; including reflection and rotation.
George Perle
George Perle (6 May 1915 – 23 January 2009) was an American composer and music theorist. As a composer, his music was largely atonal, using methods similar to the twelve-tone technique of the Second Viennese School. This serialist style, and ...
provides the following example:Perle, George (1995). ''The Right Notes: Twenty-Three Selected Essays by George Perle on Twentieth-Century Music'', p.237-238. .
:"C-E, D-F, E-G, are different instances of the same interval nterval-4.. nother kind of identity...has to do with axes of symmetry See also
* Klumpenhouwer network *Point reflection
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
*Derived row
In music using the twelve-tone technique, derivation is the construction of a row through segments. A derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern often ...
* Twelve-tone technique#Invariance
References
{{DEFAULTSORT:Identity (Music) Musical set theory