In

musicMusic 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...

, a

**permutation** of a

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

is any ordering of the elements of that set. Different permutations may be related by

transformationIn 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...

, through the application of zero or more of certain

*operations*, such as

transpositionIn music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...

,

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

,

retrogradationA musical line which is the reverse of a previously or simultaneously stated line is said to be its retrograde or cancrizans. An exact retrograde includes both the pitches and rhythms in reverse. An even more exact retrograde reverses the physical contour of the notes themselves, though this is...

, circular permutation (also called rotation), or multiplicative operations (such as the cycle of fourths and cycle of fifths transforms). These may produce reorderings of the members of the set, or may simply map the set onto itself.

The

*permutations* resulting from applying the

*inversion* or

*retrograde* operations are categorized as the prime form's

*inversions* and

*retrogrades*, respectively. Likewise, applying both

*inversion* and

*retrograde* to a prime form produces its

*retrograde-inversions*, which are considered a distinct type of permutation.

Permutation may be applied to smaller sets as well. However, the use of transformation operations to such smaller sets do not necessarily result in permutation of the original set. Here is an example of non-permutation of trichords, using the operations of retrogradation, inversion, and retrograde-inversion, combined in each case with transposition, as found within in the

tone rowIn music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...

(or twelve tone series) from

Anton WebernAnton Webern was an Austrian composer and conductor. He was a member of the Second Viennese School. As a student and significant follower of Arnold Schoenberg, he became one of the best-known exponents of the twelve-tone technique; in addition, his innovations regarding schematic organization of...

's

*Concerto*Anton Webern's Concerto for Nine Instruments, Op. 24 is a twelve-tone concerto for nine instruments: flute, oboe, clarinet, horn, trumpet, trombone, violin, viola, and piano; containing three movements: I. Etwas lebhaft, II. Sehr langsam, and III...

:

B, B, D, E, G, F, G, E, F, C, C, A

If the first three notes are regarded as the "original" cell, then the next three are its transposed retrograde inversion (backwards and upside down), the next three are the transposed retrograde (backwards), and the last three are its transposed inversion (upside down).

Restricting the transformations to the usual practice of twelve tone technique before 1950, within all 144 cells, a

tone rowIn music, a tone row or note row , also series and set, refers to a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.-History and usage:Tone rows are the basis of...

has a maximum of 48 permutations, including its

*prime form*. However, not all prime series have so many variations because the transposed and inverse transformations of a tone row may be identical to each other, a phenomenon known as invariance.

One technique facilitating twelve-tone permutation is the use of number values corresponding with musical letter names. The first note of the first of the primes, actually prime zero (commonly mistaken for prime one), is represented by 0. The rest of the numbers are counted half-step-wise such that: B = 0, C = 1, C/D = 2, D = 3, D/E = 4, E = 5, F = 6, F/G = 7, G = 8, G/A = 9, A = 10, and A/B = 11.

*Prime zero* is retrieved entirely by choice of the composer. To receive the

*retrograde* of any given prime, the numbers are simply rewritten backwards. To receive the

*inversion* of any prime, each number value is subtracted from 12 and the resulting number placed in the corresponding matrix cell (see

twelve-tone techniqueTwelve-tone technique is a method of musical composition devised by Arnold Schoenberg...

). The

*retrograde inversion* is the values of the inversion numbers read backwards.

Therefore:

**A given prime zero (derived from the notes of Anton Webern's Concerto):**
0, 11, 3, 4, 8, 7, 9, 5, 6, 1, 2, 10

**The retrograde:**
10, 2, 1, 6, 5, 9, 7, 8, 4, 3, 11, 0

**The inversion:**

0, 1, 9, 8, 4, 5, 3, 7, 6, 11, 10, 2

**The retrograde inversion:**
2, 10, 11, 6, 7, 3, 5, 4, 8, 9, 1, 0

More generally, a musical

*permutation* is any reordering of the prime form of an

ordered setIn order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...

of

pitch classIn 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...

es or, with respect to twelve-tone rows, any ordering at all of the set consisting of the integers modulo 12. In that regard, a musical

*permutation* is a

combinatorialCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

permutationIn mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

from

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

as it applies to music. Permutations are in no way limited to the twelve-tone serial and atonal musics, but are just as well utilized in tonal melodies especially during the 20th and 21st centuries, notably in Rachmaninoff's "Variations on the Theme of Paganini" for orchestra and piano.

**Cyclical permutation** is the maintenance of the original order of the tone row with the only change being that of the initial pitch-class, with the original order following after. This is also called

**rotation**. A secondary set may be considered a cyclical permutation beginning on the sixth member of a hexachordally combinatorial row.

The tone row from Berg's

*Lyric Suite*, for example, is realized thematically and then cyclically permuted (0 is bolded for reference):

5 4

**0** 9 7 2 8 1 3 6 t e

3 6 t e 5 4

**0** 9 7 2 8 1

## See also

- Counterpoint
In music, counterpoint is the relationship between two or more voices that are independent in contour and rhythm and are harmonically interdependent . It has been most commonly identified in classical music, developing strongly during the Renaissance and in much of the common practice period,...

- Identity (music)
In music, 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, identity is similar to identity in universal algebra. An identity function is a permutation or transformation which transforms a pitch or pitch...

- Multiplication (music)
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals , it has been used in other ways for twelve-tone technique, and musical set theory...

- Musical set theory
- Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

- Retrograde (music)
A musical line which is the reverse of a previously or simultaneously stated line is said to be its retrograde or cancrizans. An exact retrograde includes both the pitches and rhythms in reverse. An even more exact retrograde reverses the physical contour of the notes themselves, though this is...