In music

Music

Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, identity may refer to two different concepts, one in post-tonal theory and one in tuning theory.

Post-tonal theory

In post-tonal music theory

Post-tonal music theory

Post-tonal music theory is the set of theories put forward to describe music written outside of, or 'after', the tonal system of the common practice period.-Overview:...

, identity is similar to identity in universal algebra

Universal algebra

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....

. An identity function

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

 is a permutation

Permutation (music)

In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...

 or transformation

Transformation (music)

In music, a transformation consists of any operation or process that may apply to a musical variable in composition, performance, or analysis. Transformations include multiplication, rotation, permutation In music, a transformation consists of any operation or process that may apply to a musical...

 which transforms a pitch

Pitch (music)

Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

 or pitch class

Pitch class

In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...

 set into itself. For instance, inverting

Inversion (music)

In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...

 an augmented triad

Augmented triad

In music, an augmented triad is a triad, or chord, consisting of two major thirds . The term augmented triad arises from an augmented triad being a three note chord, or triad, whose top note is raised, or augmented...

 or C4 interval cycle

Interval cycle

In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached In music, an interval cycle is a collection...

, 048, produces itself, 084. Performing a retrograde operation upon the pitch class set

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects...

 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

George Perle

George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...

 provides the following example:

"C-E, D-F, E-G, are different instances of the same interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

 [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.

Tuning

In musical tuning

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

, an identity is each of the odd numbers below and including the limit

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

 in a tuning. For example, the identities included in 3-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the harmonic series

Harmonic series (music)

Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...

 and may thus be considered an identity:
C C G C E G B C D E F G ...
1 2 3 4 5 6 7 8 9 10 11 12 ...

"The number 9, though not a prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, is nevertheless an identity in music, simply because it is an odd number". Partch defines "identity" as "one of the correlatives, 'major

Otonality and Utonality

Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones or "undertones" of a given fixed tone. For example: 1/1, 2/1, 3/1,.....

' or 'minor

Major and minor

In Western music, the adjectives major and minor can describe a musical composition, movement, section, scale, key, chord, or interval.Major and minor are frequently referred to in the titles of classical compositions, especially in reference to the key of a piece.-Intervals and chords:With regard...

', in a tonality

Tonality

Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...

; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".

See also

• Numerary nexus
  Numerary nexus
  In musical tuning, a numerary nexus is an identity shared by two or more interval ratios in their numerator or denominator, with different identities in the other...
  
  
• Otonality and Utonality
  Otonality and Utonality
  Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones or "undertones" of a given fixed tone. For example: 1/1, 2/1, 3/1,.....
  
  
• Tonality diamond
  Tonality diamond
  In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality...
  
  
• Tonality flux
  Tonality flux
  Tonality flux is Harry Partch's term for the kinds of subtle harmonic changes that can occur in a microtonal context from notes moving from one chord to another by tiny increments of voice leading. For instance, within a major third G-B, there can be a minor third G 1/4-tone sharp to B 1/4-tone...
  
  
• Klumpenhouwer network
  Klumpenhouwer network
  A "Klumpenhouwer Network", named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is, "any network that uses T and/or I operations [ transposition or inversion] to interpret interrelations among pcs" [ pitch class sets]...