![Webern - Piano Variations op](https://upload.wikimedia.org/wikipedia/commons/f/f9/Webern_-_Piano_Variations_op._27_tone_row.png)
In
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
, a permutation (order) of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
, such as
pitch,
dynamics, or
timbre
In music, timbre ( ), also known as tone color or tone quality (from psychoacoustics), is the perceived sound quality of a musical note, sound or musical tone, tone. Timbre distinguishes different types of sound production, such as choir voice ...
. Different permutations may be related by
transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
, through the application of zero or more ''operations'', such as
transposition,
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
,
retrogradation
Retrogradation is the landward change in position of the front of a river delta with time. This occurs when the mass balance of sediment into the delta is such that the volume of incoming sediment is less than the volume of the delta that is lost ...
, circular permutation (also called ''rotation''), or
multiplicative operations (such as the cycle of fourths and
cycle of fifths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...
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
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law o ...
and
serialism
In music, serialism is a method of Musical composition, composition using series of pitches, rhythms, dynamics, timbres or other elements of music, musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, thou ...
. Analytical techniques such as
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
take care to distinguish between ordered and unordered collections. In traditional theory concepts like
voicing and
form
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
...
include ordering; for example, many musical forms, such as
rondo
The rondo is an instrumental musical form introduced in the Classical period.
Etymology
The English word ''rondo'' comes from the Italian form of the French ''rondeau'', which means "a little round".
Despite the common etymological root, 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
In music, a tone row or note row (german: Reihe or '), 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 ar ...
(or twelve-tone series) from
Anton Webern
Anton Friedrich Wilhelm von Webern (3 December 188315 September 1945), better known as Anton Webern (), was an Austrian composer and conductor whose music was among the most radical of its milieu in its sheer concision, even aphorism, and stea ...
's
Concerto
A concerto (; plural ''concertos'', or ''concerti'' from the Italian plural) is, from the late Baroque era, mostly understood as an instrumental composition, written for one or more soloists accompanied by an orchestra or other ensemble. The typi ...
:
:
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).
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.
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 twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law o ...
). 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 set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
of
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
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
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
from
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
Sergei Vasilyevich Rachmaninoff; in Russian pre-revolutionary script. (28 March 1943) was a Russian composer, virtuoso pianist, and conductor. Rachmaninoff is widely considered one of the finest pianists of his day and, as a composer, one o ...
's Variations on the Theme of Paganini for orchestra and piano.
Cyclical permutation (also called rotation)
[John Rahn, ''Basic Atonal Theory'' (New York: Longman, 1980), 134] is the maintenance of the original order of the tone row with the only change being the initial
pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...
, with the original order following after. A
secondary set
In music, a tone row or note row (german: Reihe or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in set theory (music), musical set theory of the chromatic scale, though both larger ...
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
*
Change ringing
Change ringing is the art of ringing a set of tuned bells in a tightly controlled manner to produce precise variations in their successive striking sequences, known as "changes". This can be by method ringing in which the ringers commit to memor ...
*
Counterpoint
In music, counterpoint is the relationship between two or more musical lines (or voices) which are harmonically interdependent yet independent in rhythm and melodic contour. It has been most commonly identified in the European classical tradi ...
*
Identity (music)
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 ...
*
Order (mathematics)
Order in mathematics may refer to:
Set theory
* Total order and partial order, a binary relation generalizing the usual ordering of numbers and of words in a dictionary
* Ordered set
* Order in Ramsey theory, uniform structures in consequence to ...
*
Set theory (music)
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 ...
References
{{DEFAULTSORT:Permutation (Music)
Permutations
Musical set theory
Musical techniques