Английская Википедия:Identity (music)
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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. Generally this requires symmetry. For instance, inverting an augmented triad or C4 interval cycle, 048, produces itself. Performing a retrograde operation upon the tone row 01210 produces 01210. Doubling the length of a rhythm while doubling the tempo produces a rhythm of the same durations as the original.
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. These families are defined by symmetry, which means that an object is invariant to any of various transformations; including reflection and rotation.
George Perle provides the following example:[1]
- "C-E, D-FШаблон:Music, EШаблон:Music-G, are different instances of the same interval [interval-4]...[an] other kind of identity...has to do with axes of symmetry Шаблон:Bracket. C-E belongs to a family [sum-4] of symmetrically related dyads as follows:"
| D | CШаблон:Music | C | B | AШаблон:Music | A | GШаблон:Music | ||||||||
| D | DШаблон:Music | E | F | FШаблон:Music | G | GШаблон:Music | ||||||||
| 2 | 1 | 0 | e | 9 | 8 | 7 | ||||||||
| + | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||||||
| 4 | 4 | 4 | 4 | 4 | 4 | 4 |
C=0, so in mod12, the interval-4 family:
| C | CШаблон:Music | D | DШаблон:Music | E | F | FШаблон:Music | G | GШаблон:Music | A | AШаблон:Music | B | |||||||||||||
| GШаблон:Music | A | AШаблон:Music | B | C | CШаблон:Music | D | DШаблон:Music | E | F | FШаблон:Music | G | |||||||||||||
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | t | e | |||||||||||||
| − | 8 | 9 | 10 | 11 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||||||||||
| 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
Thus, in addition to being part of the sum-4 family, C-E is also a part of the interval-4 family (in contrast to sum families, interval families are based on difference).
See also
References
- ↑ Perle, George (1995). The Right Notes: Twenty-Three Selected Essays by George Perle on Twentieth-Century Music, p.237-238. Шаблон:ISBN.
