In music, identity may refer to two different concepts, one in post-tonal theory and one in tuning theory.
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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. 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[1]:
| D | D♯ | E | F | F♯ | G | G♯ | ||||||
| D | C♯ | C | B | A♯ | A | G♯ |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||||||||
| + | 2 | 1 | 0 | 11 | 10 | 9 | 8 | |||||||
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
C=0, so in mod12:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||||||
| - | 9 | 10 | 11 | 0 | 1 | 2 | 3 | |||||||
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
Thus, in addition to being part of the interval-4 family, C-E is also a part of the sum-4 family.
In musical tuning, an identity is each of the odd numbers below and including the limit in a tuning. For example, the identities included in 3-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the harmonic series and may thus be considered an identity:
C C G C E G B C D E F G ... 1 2 3 4 5 6 7 8 9 10 11 12 ...
"The number 9, though not a prime, is nevertheless an identity in music, simply because it is an odd number".[2] Partch defines "identity" as "one of the correlatives, 'major' or 'minor', in a tonality; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".[3]