In post-tonal music theory, identity is similar to identity in

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of stud ...

. 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, unc ...

is a

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

or transformation which transforms a pitch or pitch class set into itself. Generally this requires

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

. For instance, inverting an augmented triad or C4 interval cycle, 048, produces itself. Performing a retrograde operation upon the tone row 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 theory, music theorist. As a composer, his music was largely atonality, atonal, using methods similar to the twelve-tone technique of the Second Viennese School. Th ...

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 . C-E belongs to a family um-4of 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).

References

{{DEFAULTSORT:Identity (Music) Musical set theory

See also

References