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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.[further explanation needed] 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.
- "C-E, D-F♯, E♭-G, are different instances of the same interval [interval-4]...[an] other kind of identity...has to do with axes of symmetry. C-E belongs to a family [sum-4] 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.
- Perle, George (1995). The Right Notes: Twenty-Three Selected Essays by George Perle on Twentieth-Century Music, p.237-238. ISBN 0-945193-37-8.