Identity (music) - Post-tonal Theory

Post-tonal Theory

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. For instance, inverting an augmented triad or C4 interval cycle, 048, produces itself, 084. Performing a retrograde operation upon the pitch class set 01210 produces 01210.

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.

George Perle provides the following example:

"C-E, D-F♯, E♭-G, are different instances of the same interval ... other kind of identity...has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:"

C=0, so in mod12:

Thus, in addition to being part of the interval-4 family, C-E is also a part of the sum-4 family.