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:"
| 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.
Read more about this topic: Identity (music)
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