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 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 provides the following example:
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.
