Identity (music)

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048 equals itself when transposed by 4 or 8 or when inverted

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:<ref>Perle, George (1995). The Right Notes: Twenty-Three Selected Essays by George Perle on Twentieth-Century Music, p.237-238. Template:ISBN.</ref>

"C-E, D-FTemplate:Music, ETemplate:Music-G, are different instances of the same interval [interval-4]...[an] other kind of identity...has to do with axes of symmetry Template:Bracket. C-E belongs to a family [sum-4] of symmetrically related dyads as follows:"

		D		CTemplate:Music		C		B		ATemplate:Music		A		GTemplate:Music
		D		DTemplate:Music		E		F		FTemplate:Music		G		GTemplate: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		CTemplate:Music		D		DTemplate:Music		E		F		FTemplate:Music		G		GTemplate:Music		A		ATemplate:Music		B
		GTemplate:Music		A		ATemplate:Music		B		C		CTemplate:Music		D		DTemplate:Music		E		F		FTemplate: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

• Klumpenhouwer network
• Point reflection
• Derived row
• Twelve-tone technique#Invariance