Identity (music)

Interval-4 family  Play .

Sum-4 family  Play .

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]

"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. C-E belongs to a family [sum-4] 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.

See also

• Klumpenhouwer network

Sources

1. Perle, George (1995). The Right Notes: Twenty-Three Selected Essays by George Perle on Twentieth-Century Music, p.237-238. ISBN 0-945193-37-8.