Permutation (music)

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Prime, retrograde, inverse, and retrograde-inverse permutations. P-R-I-RI.png
Prime, retrograde, inverse, and retrograde-inverse permutations.
Principal forms of Anton Webern's tone row from Variations for piano, op. 27, movement 2. Play (help*info) Webern - Piano Variations op. 27 tone row.png
Principal forms of Anton Webern's tone row from Variations for piano, op. 27, movement 2. Loudspeaker.svg Play  

In music, a permutation (order) of a set is any ordering of the elements of that set. [3] A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, 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.

Order is particularly important in the theories of composition techniques originating in the 20th century such as the twelve-tone technique and serialism. Analytical techniques such as set theory take care to distinguish between ordered and unordered collections. In traditional theory concepts like voicing and form include ordering; for example, many musical forms, such as rondo, are defined by the order of their sections.

The permutations resulting from applying the inversion or retrograde operations are categorized as the prime form's inversions and retrogrades, respectively. Applying both inversion and retrograde to a prime form produces its retrograde-inversions, considered a distinct type of permutation.

Permutation may be applied to smaller sets as well. However, transformation operations of such smaller sets do not necessarily result in permutation the original set. Here is an example of non-permutation of trichords, using retrogradation, inversion, and retrograde-inversion, combined in each case with transposition, as found within in the tone row (or twelve-tone series) from Anton Webern's Concerto: [4]

Permutation (music)

If the first three notes are regarded as the "original" cell, then the next 3 are its transposed retrograde-inversion (backwards and upside down), the next three are the transposed retrograde (backwards), and the last 3 are its transposed inversion (upside down). [5]

Not all prime series have the same number of variations because the transposed and inverse transformations of a tone row may be identical, a quite rare phenomenon: less than 0.06% of all series admit 24 forms instead of 48. [6]

One technique facilitating twelve-tone permutation is the use of number values corresponding with musical letters. 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 technique). The retrograde inversion is the values of the inversion numbers read backwards.


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 set of pitch classes [7] or, with respect to twelve-tone rows, any ordering at all of the set consisting of the integers modulo 12. [8] In that regard, a musical permutation is a combinatorial permutation from mathematics 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.[ citation needed ]

Cyclical permutation (also called rotation) [9] is the maintenance of the original order of the tone row with the only change being the initial pitch class, with the original order following after. 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
Initial statement begins on F(=5), mm. 2-4, cyclical permutation begins on E(=3) in mm. 7-9 (Perle 1996, p.20). Berg's Lyric Suite Mov. I thematic statement.PNG
Initial statement begins on F(=5), mm. 2-4, cyclical permutation begins on E(=3) in mm. 7-9 (Perle 1996, p.20).

See also

Related Research Articles

In music, a tone row or note row, also series or set, is 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.

Atonality Music that lacks a tonal center or key

Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about 1908 to the present day, where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized classical European music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments".

In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were also working to establish serialism as a form of post-tonal thinking. Twelve-tone technique orders the twelve notes of the chromatic scale, forming a row or series and providing a unifying basis for a composition's melody, harmony, structural progressions, and variations. Other types of serialism also work with sets, collections of objects, but not necessarily with fixed-order series, and extend the technique to other musical dimensions, such as duration, dynamics, and timbre.

Twelve-tone technique Musical composition method using all 12 chromatic scale notes equally often & not in a key

The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.

The String Quartet, Op. 28 by Anton Webern is written for the standard string quartet group of two violins, viola and cello. It was the last piece of chamber music that Webern wrote.

Set theory (music) Branch of music theory that categorizes musical objects and describes their relationships by using sets and permutations of pitches and pitch classes, rhythmic onsets, beat classes, etc.

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.

The term "partition" is also French for the sheet music of a transcription.

In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."

Complement (music)

In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.

Retrograde inversion

Retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order." Retrograde reverses the order of the motif's pitches: what was the first pitch becomes the last, and vice versa. This is a technique used in music, specifically in twelve-tone technique, where the inversion and retrograde techniques are performed on the same tone row successively, "[t]he inversion of the prime series in reverse order from last pitch to first."

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. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

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

An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords, when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].

Trope (music) Concepts in music

A trope or tropus may refer to a variety of different concepts in medieval, 20th-, and 21st-century music.

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. It revolves around the idea of 'emancipating dissonance', that is, freeing the structure of music from the familiar harmonic patterns that are derived from natural overtones. As music becomes more complex, dissonance becomes indistinguishable from consonance.

Anton Webern's Concerto for Nine Instruments, Op. 24, Op. 24, written in 1934, is a twelve-tone concerto for nine instruments: flute, oboe, clarinet, horn, trumpet, trombone, violin, viola, and piano. It consists of three movements:

Forte number

In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music. The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.

In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence." Traditionally, octave equivalency is assumed, while inversional, permutational, and transpositional equivalency may or may not be considered.

In music, the "Ode-to-Napoleon" hexachord is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte (1942) by Arnold Schoenberg. Containing the pitch-classes 014589 it is given Forte number 6–20 in Allen Forte's taxonomic system. The primary form of the tone row used in the Ode allows the triads of G minor, E minor, and B minor to easily appear.


  1. Nolan, Catherine. 1995. "Structural Levels and Twelve-Tone Music: A Revisionist Analysis of the Second Movement of Webern's 'Piano Variations Op. 27'", p.49–50. Journal of Music Theory, Vol. 39, No. 1 (Spring), pp. 47–76. For whome 0 = G.
  2. Leeuw, Ton de. 2005. Music of the Twentieth Century: A Study of Its Elements and Structure, p.158. Translated from the Dutch by Stephen Taylor. Amsterdam: Amsterdam University Press. ISBN   90-5356-765-8. Translation of Muziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur. Utrecht: Oosthoek, 1964. Third impression, Utrecht: Bohn, Scheltema & Holkema, 1977. ISBN   90-313-0244-9. For whom 0 = E.
  3. Allen Forte, The Structure of Atonal Music (New Haven and London: Yale University Press, 1973): 3; John Rahn, Basic Atonal Theory (New York: Longman, 1980), 138
  4. Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p.97. New York: Cambridge University Press. ISBN   978-0-521-68200-8 (pbk).
  5. George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, fourth edition, revised (Berkeley, Los Angeles, and London: University of California Press, 1977): 79. ISBN   0-520-03395-7.
  6. Emmanuel Amiot, "La série dodécaphonique et ses symétries", Quadrature 19, EDP sciences[ clarification needed ] (1994).
  7. Wittlich, Gary (1975). "Sets and Ordering Procedures in Twentieth-Century Music", Aspects of Twentieth-Century Music. Wittlich, Gary (ed.). Englewood Cliffs, New Jersey: Prentice-Hall. ISBN   0-13-049346-5 p. 475.
  8. John Rahn, Basic Atonal Theory (New York: Longman, 1980), 137.
  9. John Rahn, Basic Atonal Theory (New York: Longman, 1980), 134