In post-tonal music theory, the concept of identity bears a resemblance to its definition in universal algebra. An identity function is defined as a permutation or transformation that maps a pitch or a pitch class set onto itself. This phenomenon typically necessitates the presence of symmetry. For instance, an augmented triad or a C4 interval cycle (048) remains invariant under inversion. Similarly, applying a retrograde operation to the tone row 01210 yields the original sequence. Furthermore, a rhythm retains its original durations if its length is doubled concurrently with a doubling of the tempo.
Beyond its role as an attribute of a particular set, identity also extends to encompass a "family" of sets or set forms that fulfill a potential identity condition. These families are delineated by symmetry, which implies that an object remains invariant under various transformations, including reflection and rotation.
George Perle illustrates this concept with the following example:
- "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 [reflection symmetry rather than interval families' rotational symmetry]. C-E belongs to a family [sum-4] of symmetrically related dyads as follows:"
Considering C as 0, within modulo 12 arithmetic, the interval-4 family is defined as:
Consequently, C-E is a constituent of both the sum-4 family and the interval-4 family (where interval families are differentiated from sum families by their reliance on intervallic differences).
- Klumpenhouwer network
- Derived row
- References