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)
Famous quotes containing the word theory:
“There never comes a point where a theory can be said to be true. The most that one can claim for any theory is that it has shared the successes of all its rivals and that it has passed at least one test which they have failed.”
—A.J. (Alfred Jules)