Editing Twelve-tone technique (section)

==Tone row==
{{Main|Tone row}}

{{Listen|type=music|filename=Webern - Sehr langsam.ogg|title="Sehr langsam"|description=Sample of "Sehr langsam" from String Trio Op. 20 by [[Anton Webern]], an example of the twelve-tone technique, a type of [[serialism]]}}

The basis of the twelve-tone technique is the ''[[tone row]]'', an ordered arrangement of the twelve notes of the [[chromatic scale]] (the twelve [[Equal temperament|equal tempered]] [[pitch class]]es). There are four [[postulates]] or preconditions to the technique which apply to the row (also called a ''set'' or ''series''), on which a work or section is based:<ref>Perle 1977, 3.</ref>
# The row is a specific ordering of all twelve notes of the chromatic scale (without regard to [[octave]] placement).
# No note is repeated within the row.
# The row may be subjected to [[Interval (music)|interval]]-preserving [[Transformation (music)|transformations]]—that is, it may appear in ''[[Melodic inversion|inversion]]'' (denoted I), ''[[Retrograde (music)|retrograde]]'' (R), or ''[[Retrograde inversion|retrograde-inversion]]'' (RI), in addition to its "original" or ''prime'' form (P).
# The row in any of its four transformations may begin on any degree of the chromatic scale; in other words it may be freely [[Transposition (music)|transposed]]. (Transposition being an interval-preserving transformation, this is technically covered already by 3.) Transpositions are indicated by an [[integer]] between 0 and 11 denoting the number of semitones: thus, if the original form of the row is denoted P<sub>0</sub>, then P<sub>1</sub> denotes its transposition upward by one semitone (similarly I<sub>1</sub> is an upward transposition of the inverted form, R<sub>1</sub> of the retrograde form, and RI<sub>1</sub> of the retrograde-inverted form).

(In Hauer's system postulate 3 does not apply.)<ref name="Perle 1991, 145"/>

A particular transformation (prime, inversion, retrograde, retrograde-inversion) together with a choice of transpositional level is referred to as a ''set form'' or ''row form''. Every row thus has up to 48 different row forms. (Some rows have fewer due to [[symmetry]]; see the sections on ''derived rows'' and ''invariance'' below.)

===Example===
Suppose the prime form of the row is as follows:

:[[Image:Example tone row.png|400px|B, B{{music|b}}, G, C{{music|#}}, E{{music|b}}, C, D, A, F{{music|#}}, E, A{{music|b}}, F]]

Then the retrograde is the prime form in reverse order:

:[[Image:Retrograde tone row.png|400px|F, A{{music|b}}, E, F{{music|#}}, A, D, C, E{{music|b}}, C{{music|#}}, G, B{{music|b}}, B]]

The inversion is the prime form with the [[Interval (music)|intervals]] [[Inversion (interval)|inverted]] (so that a rising [[minor third]] becomes a falling minor third, or equivalently, a rising [[major sixth]]):

:[[Image:Inversion tone row.png|400px|B, C, E{{music|b}}, A, G, B{{music|b}}, A{{music|b}}, C{{music|#}}, E, F{{music|#}}, D, F]]

And the retrograde inversion is the inverted row in retrograde:

:[[Image:Retrograde inversion tone row.png|400px|F, D, F{{music|#}}, E, C{{music|#}}, A{{music|b}}, B{{music|b}}, G, A, E{{music|b}}, C, B]]

P, R, I and RI can each be started on any of the twelve notes of the [[chromatic scale]], meaning that 47 [[permutation (music)|permutations]] of the initial tone row can be used, giving a maximum of 48 possible tone rows. However, not all prime series will yield so many variations because transposed transformations may be identical to each other. This is known as ''invariance''. A simple case is the ascending chromatic scale, the retrograde inversion of which is identical to the prime form, and the retrograde of which is identical to the inversion (thus, only 24 forms of this tone row are available).

[[Image:P-R-I-RI.png|thumb|350px|Prime, retrograde, inverted, and retrograde-inverted forms of the ascending chromatic scale. P and RI are the same (to within transposition), as are R and I.]]

In the above example, as is typical, the retrograde inversion contains three points where the sequence of two pitches are identical to the prime row. Thus the generative power of even the most basic transformations is both unpredictable and inevitable. Motivic development can be driven by such internal consistency.

===Application in composition===
Note that rules 1–4 above apply to the construction of the row itself, and not to the interpretation of the row in the composition. (Thus, for example, postulate 2 does not mean, contrary to common belief, that no note in a twelve-tone work can be repeated until all twelve have been sounded.) While a row may be expressed literally on the surface as thematic material, it need not be, and may instead govern the pitch structure of the work in more abstract ways. Even when the technique is applied in the most literal manner, with a piece consisting of a sequence of statements of row forms, these statements may appear consecutively, simultaneously, or may overlap, giving rise to [[harmony]].

[[File:Schoenberg - Wind Quintet opening.png|thumb|center|upright=2.5|Schoenberg's annotated opening of his [[Wind Quintet (Schoenberg)|Wind Quintet]] Op. 26 shows the distribution of the pitches of the row among the voices and the balance between the hexachords, 1–6 and 7–12, in the principal voice and accompaniment<ref>Whittall 2008, 52.</ref>]]

Durations, dynamics and other aspects of music other than the pitch can be freely chosen by the composer, and there are also no general rules about which tone rows should be used at which time (beyond their all being derived from the prime series, as already explained). However, individual composers have constructed more detailed systems in which matters such as these are also governed by systematic rules (see [[serialism]]).

==== Topography ====
Analyst Kathryn Bailey has used the term 'topography' to describe the particular way in which the notes of a row are disposed in her work on the dodecaphonic music of Webern. She identifies two types of topography in Webern's music: block topography and linear topography.

The former, which she views as the 'simplest', is defined as follows: 'rows are set one after the other, with all notes sounding in the order prescribed by this succession of rows, regardless of texture'. The latter is more complex: the musical texture 'is the product of several rows progressing simultaneously in as many voices' (note that these 'voices' are not necessarily restricted to individual instruments and therefore cut across the musical texture, operating as more of a background structure).<ref>{{Cite book |last=Bailey |first=Kathryn |title=The twelve-note music of Anton Webern: old forms in a new language |date=2006 |publisher=Cambridge University Press |isbn=978-0-521-39088-0 |edition=Digitally printed 1st pbk. version |series=Music in the twentieth century |location=Cambridge [England] New York |pages=31}}</ref>

==== Elisions, Chains, and Cycles ====
Serial rows can be connected through elision, a term that describes 'the overlapping of two rows that occur in succession, so that one or more notes at the juncture are shared (are played only once to serve both rows)'.<ref>{{Cite book |last=Bailey |first=Kathryn |title=The twelve-note music of Anton Webern: old forms in a new language |date=2006 |publisher=Cambridge University Press |isbn=978-0-521-39088-0 |edition=Digitally printed 1st pbk. version |series=Music in the twentieth century |location=Cambridge [England] New York |pages=449}}</ref> When this elision incorporates two or more notes it creates a row chain;<ref>{{Cite journal |last=Moseley |first=Brian |date=2019-09-01 |title=Transformation Chains, Associative Areas, and a Principle of Form for Anton Webern's Twelve-tone Music |url=https://academic.oup.com/mts/article/41/2/218/5514243 |journal=Music Theory Spectrum |language=en |volume=41 |issue=2 |pages=218–243 |doi=10.1093/mts/mtz010 |issn=0195-6167|url-access=subscription }}</ref> when multiple rows are connected by the same elision (typically identified as the same in set-class terms) this creates a row chain cycle, which therefore provides a technique for organising groups of rows.<ref>{{Cite journal |last=Moseley |first=Brian |date=2018 |title=Cycles in Webern's Late Music |url=https://read.dukeupress.edu/journal-of-music-theory/article/62/2/165/136725/Cycles-in-Weberns-Late-Music |journal=Journal of Music Theory |language=en |volume=62 |issue=2 |pages=165–204 |doi=10.1215/00222909-7127658 |s2cid=171497028 |issn=0022-2909|url-access=subscription }}</ref>

===Properties of transformations===

The tone row chosen as the basis of the piece is called the ''prime series'' (P). Untransposed, it is notated as P<sub>0</sub>. Given the twelve [[pitch class]]es of the chromatic scale, there are 12 [[factorial]]<ref>Loy 2007, 310.</ref> (479,001,600<ref name="Whittall 24"/>) tone rows, although this is far higher than the number of ''unique'' tone rows (after taking transformations into account). There are 9,985,920 classes of twelve-tone rows up to equivalence (where two rows are equivalent if one is a transformation of the other).<ref>Benson 2007, 348.</ref>

Appearances of P can be transformed from the original in three basic ways:
* [[Transposition (music)|transposition]] up or down, giving P<sub>χ</sub>.
* reversing the order of the pitches, giving the ''[[Permutation (music)|retrograde]]'' (R)
* turning each interval direction to its opposite, giving the ''[[Melodic inversion|inversion]]'' (I).

The various transformations can be combined. These give rise to a set-complex of forty-eight forms of the set, 12 transpositions of the ''four'' basic forms: P, R, I, RI. The combination of the retrograde and inversion transformations is known as the ''[[retrograde inversion]]'' (''RI'').

:{| class="wikitable"
|RI is:
|RI of P,
|R of I,
|and I of R.
|-
|R is:
|R of P,
|RI of I,
|and I of RI.
|-
|I is:
|I of P,
|RI of R,
|and R of RI.
|-
|P is:
|R of R,
|I of I,
|and RI of RI.
|}

thus, each cell in the following table lists the result of the transformations, a [[four-group]], in its row and column headers:

:{| class="wikitable"
|P:
|RI:
|R:
|I:
|-
|RI:
|P
|I
|R
|-
|R:
|I
|P
|RI
|-
|I:
|R
|RI
|P
|}

However, there are only a few numbers by which one may ''multiply'' a row and still end up with twelve tones. (Multiplication is in any case not interval-preserving.)

===Derivation===
{{Main|Derived row}}
''Derivation'' is transforming segments of the full chromatic, fewer than 12 pitch classes, to yield a complete set, most commonly using trichords, tetrachords, and hexachords. A [[Derived row|derived set]] can be generated by choosing appropriate transformations of any [[trichord]] except 0,3,6, the [[diminished triad]]{{Citation needed|date=December 2022}}. A derived set can also be generated from any [[tetrachord]] that excludes the interval class 4, a [[major third]], between any two elements. The opposite, ''partitioning'', uses methods to create segments from sets, most often through [[registral difference]].

====Combinatoriality====
{{Main|Combinatoriality}}
[[Combinatoriality]] is a side-effect of derived rows where combining different segments or sets such that the pitch class content of the result fulfills certain criteria, usually the combination of hexachords which complete the full chromatic.

====Invariance====<!--[[Invariance (music)]] redirects directly here.-->
{{Listen|type=music
|filename=Schoenberg - Concerto for Violin - hexachordal invariance.mid
|title=Schoenberg's Concerto for Violin
|description=[[File:Schoenberg - Concerto for Violin - hexachordal invariance.png|center|350px]]Hexachord invariance.<ref>Haimo 1990, 27.</ref> The last hexachord of P<sub>0</sub> (C–C{{music|#}}–G–A{{music|b}}–D–F) contains the same pitches as the first hexachord of I<sub>5</sub> (D–C{{music|#}}–A{{music|b}}–C–G–F).
}}

''Invariant'' formations are also the side effect of derived rows where a segment of a set remains similar or the same under transformation. These may be used as "pivots" between set forms, sometimes used by [[Anton Webern]] and [[Arnold Schoenberg]].<ref>Perle 1977, 91–93.</ref>

''Invariance'' is defined as the "properties of a set that are preserved under [any given] operation, as well as those relationships between a set and the so-operationally transformed set that inhere in the operation",<ref>Babbitt 1960, 249–250.</ref> a definition very close to that of [[Invariance (mathematics)|mathematical invariance]]. [[George Perle]] describes their use as "pivots" or non-tonal ways of emphasizing certain [[pitch (music)|pitches]]. Invariant rows are also [[combinatoriality|combinatorial]] and [[derived row|derived]].

===Cross partition===<!--[[Cross-partition]] redirects directly here.-->
[[File:Aggregate Von Heute auf Morgen.png|thumb|Aggregates spanning several local set forms in Schoenberg's ''[[Von heute auf morgen]]''.<ref>Haimo 1990, 13.</ref>]]
{{See also|Derived row#Partition and mosaic}}

A ''cross partition'' is an often monophonic or homophonic technique which, "arranges the pitch classes of an aggregate (or a row) into a rectangular design", in which the vertical columns (harmonies) of the rectangle are derived from the adjacent segments of the row and the horizontal columns (melodies) are not (and thus may contain non-adjacencies).<ref>Alegant 2010, 20.</ref>

For example, the layout of all possible 'even' cross partitions is as follows:<ref name="Alegant 21">Alegant 2010, 21.</ref>
:{|
|style="width:30px;"|6<sup>2</sup>
|style="width:30px;"|4<sup>3</sup>
|style="width:30px;"|3<sup>4</sup>
|style="width:30px;"|2<sup>6</sup>
|-
|**||***||****||******
|-
|**||***||****||******
|-
|**||***||****
|-
|**||***
|-
|**
|-
|**
|}

One possible realization out of many for the ''order numbers'' of the 3<sup>4</sup> cross partition, and one variation of that, are:<ref name="Alegant 21"/>
 0 3 6 9 0 5 6 e
 1 4 7 t 2 3 7 t
 2 5 8 e 1 4 8 9

Thus if one's tone row was 0 e 7 4 2 9 3 8 t 1 5 6, one's cross partitions from above would be:
 0 4 3 1 0 9 3 6
 e 2 8 5 7 4 8 5
 7 9 t 6 e 2 t 1

Cross partitions are used in Schoenberg's [[Zwei Klavierstücke (Schoenberg)|Op. 33a ''Klavierstück'']] and also by [[Alban Berg|Berg]] but [[Luigi Dallapiccola|Dallapicolla]] used them more than any other composer.<ref>Alegant 2010, 22, 24.</ref>

===Other===
In practice, the "rules" of twelve-tone technique have been bent and broken many times, not least by Schoenberg himself. For instance, in some pieces two or more tone rows may be heard progressing at once, or there may be parts of a composition which are written freely, without recourse to the twelve-tone technique at all. Offshoots or variations may produce music in which:
* the full chromatic is used and constantly circulates, but permutational devices are ignored
* permutational devices are used but not on the full chromatic

Also, some composers, including Stravinsky, have used [[cyclic permutation]], or rotation, where the row is taken in order but using a different starting note. Stravinsky also preferred the [[Inverse retrograde|inverse-retrograde]], rather than the retrograde-inverse, treating the former as the compositionally predominant, "untransposed" form.<ref>Spies 1965, 118.</ref>

Although usually atonal, twelve tone music need not be—several pieces by Berg, for instance, have tonal elements.

One of the best known twelve-note compositions is ''[[Variations for Orchestra (Schoenberg)|Variations for Orchestra]]'' by [[Arnold Schoenberg]]. "Quiet", in [[Leonard Bernstein]]'s ''[[Candide (musical)|Candide]]'', satirizes the method by using it for a song about boredom, and [[Benjamin Britten]] used a twelve-tone row—a "tema seriale con fuga"—in his ''Cantata Academica: Carmen Basiliense'' (1959) as an emblem of academicism.<ref>Brett 2007.</ref>