View source for Twelve-tone technique

{{Short description|Musical composition method}}
{{Distinguish|12 equal temperament{{!}}twelve-tone equal temperament}}
{{Use dmy dates|date=October 2019}}

[[File:Arnold Schoenberg la 1948.jpg|thumb|upright|[[Arnold Schoenberg]], inventor of the twelve-tone technique]]

{{Listen|type=music
|filename=Hauer - Nomos Op. 19, beginning.mid
|title=Josef Matthias Hauer's "athematic" dodecaphony in ''Nomos'' Op. 19<ref>Whittall 2008, 26.</ref>
|description=[[File:Hauer - Nomos Op. 19, beginning.png|center|350px]]
|filename2=Hauer trope.mid
|title2=Example of Hauer's tropes<ref name="Perle 1991, 145">Perle 1991, 145.</ref>
|description2=[[File:Hauer trope.png|center|300px]]
}}

The '''twelve-tone technique'''—also known as '''dodecaphony''', '''twelve-tone serialism''', and (in British usage) '''twelve-note composition'''—is a method of [[musical composition]]. The technique is a means of ensuring that all 12 notes of the [[chromatic scale]] are sounded equally often in a piece of music while preventing the emphasis of any one note<ref name="Perle 1977, 2">Perle 1977, 2.</ref> through the use of [[tone row]]s, orderings of the 12 [[pitch class]]es. All 12 notes are thus given more or less equal importance, and the music avoids being in a [[Key (music)|key]].

The technique was first devised by Austrian composer [[Josef Matthias Hauer]],{{Citation needed lead|date=February 2020}} 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. Over time, the technique increased greatly in popularity and eventually became widely influential on mid-20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as [[Aaron Copland]] and [[Igor Stravinsky]],{{Clarify|date=December 2016}}<!--How did Copland "actively oppose" a technique he first used in 1927? How did Stravinsky "actively oppose" the technique prior to adopting it in the early 1950s?--> eventually adopted it in their music.

Schoenberg himself described the system as a "Method of composing with twelve tones which are related only with one another".<ref name="Schoenberg 1975, 218">Schoenberg 1975, 218.</ref> It is commonly considered a form of [[serialism]].

Schoenberg's fellow countryman and contemporary Hauer also developed a similar system using unordered [[hexachord]]s or ''[[trope (music)#20th-century music|tropes]]''—independent of Schoenberg's development of the twelve-tone technique. Other composers have created systematic use of the chromatic scale, but Schoenberg's method is considered to be most historically and aesthetically significant.<ref>Whittall 2008, 25.</ref>

==History of use==
{{Listen|type=music
|filename=Schoenberg - Op. 23, mov. 5.mid
|title=Schoenberg's Op. 23, mov. 5, mm. 1–4
|description=The "first 12-note work"<ref>Leeuw 2005, 149.</ref>[[File:Schoenberg - Op. 23, mov. 5.png|center|400px]]
|filename2=Schoenberg - Piano Piece op.33a tone row.mid
|title2=Schoenberg's Piano Piece, Op.&nbsp;33a
|description2=The principal forms, P<sub>1</sub> and I<sub>6</sub>, of Schoenberg's [[Zwei Klavierstücke (Schoenberg)|Piano Piece, Op.&nbsp;33a]], tone row feature hexachordal combinatoriality and contains three perfect fifths each, which is the relation between P<sub>1</sub> and I<sub>6</sub> and a source of contrast between, "accumulations of 5ths", and, "generally more complex simultaneity".<ref>Leeuw 2005, 155–157.</ref> For example, group A consists of B{{music|flat}}–C–F–B{{music|natural}} while the, "more blended", group B consists of A–C{{music|#}}–D{{music|#}}–F{{music|#}}.[[File:Schoenberg - Piano Piece op.33a tone row.png|center|400px]]
}}
The twelve-tone technique is most often attributed to Austrian composer [[Arnold Schoenberg]]. He recalls using it in 1921 and describing it to pupils two years later.<ref>Schoenberg 1975, 213.</ref> Simultaneously, [[Josef Matthias Hauer]] was formulating a similar theory in his writings. In the second edition of his book ''Vom Wesen Des Musikalischen'' (On the Essence of Music, 1923), Hauer wrote that the law of the atonal melody requires all twelve tones to be played repeatedly.<ref>Hauer, Josef Matthias, 1883-1959. ''[https://hdl.handle.net/2027/mdp.39015009463897?urlappend=%3Bseq=57%3Bownerid=13510798884266709-61 Vom Wesen Des Musikalischen: Ein Lehrbuch Der Zwölftöne-musik]''. Berlin-Lichterfelde: Schlesinger (R. Lienau), 1923. 51.</ref>

The method was used during the next twenty years almost exclusively by the composers of the [[Second Viennese School]]—[[Alban Berg]], [[Anton Webern]], and Schoenberg himself. Although, another important composer in this period was [[Elisabeth Lutyens]] who wrote more than 50 pieces using the serial method.<ref>{{Cite journal |date=1983 |title=Elisabeth Lutyens |url=https://www.jstor.org/stable/964096 |journal=The Musical Times |volume=124 |issue=1684 |pages=378 |jstor=964096 |issn=0027-4666}}</ref>

The twelve tone technique was preceded by "freely" [[atonality|atonal]] pieces of 1908–1923 which, though "free", often have as an "integrative element&nbsp;... a minute intervallic [[cell (music)|cell]]" which in addition to expansion may be transformed as with a tone row, and in which individual notes may "function as pivotal elements, to permit overlapping statements of a basic cell or the linking of two or more basic cells".<ref>Perle 1977, 9–10.</ref> The twelve-tone technique was also preceded by "nondodecaphonic serial composition" used independently in the works of [[Alexander Scriabin]], [[Igor Stravinsky]], [[Béla Bartók]], [[Carl Ruggles]], and others.<ref name="Perle 1977, 37">Perle 1977, 37.</ref> Oliver Neighbour argues that Bartók was "the first composer to use a group of twelve notes consciously for a structural purpose", in 1908 with the third of his fourteen bagatelles.<ref>Neighbour 1955, 53.</ref> "Essentially, Schoenberg and Hauer systematized and defined for their own dodecaphonic purposes a pervasive technical feature of 'modern' musical practice, the [[ostinato]]".<ref name="Perle 1977, 37"/> Additionally, John Covach argues that the strict distinction between the two, emphasized by authors including Perle, is overemphasized:

<blockquote>The distinction often made between Hauer and the Schoenberg school—that the former's music is based on unordered hexachords while the latter's is based on an ordered series—is false: while he did write pieces that could be thought of as "trope pieces", much of Hauer's twelve-tone music employs an ordered series.<ref>John Covach quoted in Whittall 2008, 24.</ref></blockquote>

The "strict ordering" of the Second Viennese school, on the other hand, "was inevitably tempered by practical considerations: they worked on the basis of an interaction between ordered and unordered pitch collections."<ref name="Whittall 24">Whittall 2008, 24.</ref>

[[Rudolph Reti]], an early proponent, says: "To replace one structural force (tonality) by another (increased thematic oneness) is indeed the fundamental idea behind the twelve-tone technique", arguing it arose out of Schoenberg's frustrations with free atonality,<ref name="Reti">Reti 1958</ref>{{Page needed|date=June 2009}} providing a "positive premise" for atonality.<ref name="Perle 1977, 2"/> In Hauer's breakthrough piece ''Nomos'', Op. 19 (1919) he used twelve-tone sections to mark out large formal divisions, such as with the opening five statements of the same twelve-tone series, stated in groups of five notes making twelve five-note phrases.<ref name="Whittall 24"/>

[[Felix Khuner]] contrasted Hauer's more mathematical concept with Schoenberg's more musical approach.{{sfn|Crawford and Khuner|1996|loc=28}} Schoenberg's idea in developing the technique was for it to "replace those structural differentiations provided formerly by [[tonality|tonal]] [[harmony|harmonies]]".<ref name="Schoenberg 1975, 218"/> As such, twelve-tone music is usually [[atonality|atonal]], and treats each of the 12 [[semitone]]s of the [[chromatic scale]] with equal importance, as opposed to earlier classical music which had treated some notes as more important than others (particularly the [[tonic (music)|tonic]] and the [[dominant note]]).

The technique became widely used by the fifties, taken up by composers such as [[Milton Babbitt]], [[Luciano Berio]], [[Pierre Boulez]], [[Luigi Dallapiccola]], [[Ernst Krenek]], [[Riccardo Malipiero]], and, after Schoenberg's death, [[Igor Stravinsky]]. Some of these composers extended the technique to control aspects other than the pitches of notes (such as duration, method of attack and so on), thus producing [[serial music]]. Some even subjected all elements of music to the serial process.

[[Charles Wuorinen]] said in a 1962 interview that while "most of the Europeans say that they have 'gone beyond' and 'exhausted' the twelve-tone system", in America, "the twelve-tone system has been carefully studied and generalized into an edifice more impressive than any hitherto known."<ref>Chase 1987, 587.</ref>

American composer [[Scott Bradley (composer)|Scott Bradley]], best known for his musical scores for works like ''[[Tom & Jerry]]'' and ''[[Droopy Dog]]'', utilized the 12-tone technique in his work. Bradley described his use thus:
{{quote|The Twelve-Tone System provides the 'out-of-this-world' progressions so necessary to under-write the fantastic and incredible situations which present-day cartoons contain.<ref>{{cite web|url=http://tralfaz.blogspot.com/2017/01/cartoon-composer-scott-bradley.html|title=Tralfaz: Cartoon Composer Scott Bradley|last=Yowp|date=7 January 2017}}</ref>}}

An example of Bradley's use of the technique to convey building tension occurs in the ''Tom & Jerry'' short "[[Puttin' on the Dog]]", from 1944. In a scene where the mouse, wearing a dog mask, runs across a yard of dogs "in disguise", a chromatic scale represents both the mouse's movements, and the approach of a suspicious dog, mirrored octaves lower.<ref>{{cite book|last=Goldmark|first=Daniel|title=Tunes for 'Toons: Music and the Hollywood Cartoon|url=https://books.google.com/books?id=4mo3DwAAQBAJ&pg=PA71|year=2007|publisher=Univ of California Press|isbn=978-0-520-25311-7|page=71}}</ref> Apart from his work in cartoon scores, Bradley also composed [[tone poem]]s that were performed in concert in California.<ref>{{IMDB name|nm0005973|Scott Bradley}}</ref>

Rock guitarist [[Ron Jarzombek]] used a twelve-tone system for composing [[Blotted Science]]'s [[extended play]] ''[[The Animation of Entomology]]''. He put the notes into a clock and rearranged them to be used that are side by side or consecutive. He called his method "Twelve-Tone in Fragmented Rows."<ref name="ms">{{cite web |last1=Mustein |first1=Dave |title=Blotted Science's Ron Jarzombek: The Twelve-tone Metalsucks Interview |url=https://www.metalsucks.net/2011/11/02/blotted-sciences-ron-jarzombek-the-twelve-tone-metalsucks-interview/ |website=MetalSucks |access-date=19 January 2021|date=2 November 2011}}</ref>

==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>

==Schoenberg's mature practice==
Ten features of Schoenberg's mature twelve-tone practice are characteristic, interdependent, and interactive:<ref name="Haimo">Haimo 1990, 41.</ref>
# [[Hexachord]]al [[inversion (music)|inversional]] [[combinatoriality]]
# [[tone row#total chromatic|Aggregates]]
# Linear [[set (music)|set]] presentation
# [[Derived row|Partitioning]]
# Isomorphic partitioning
# [[#Invariance|Invariants]]
# Hexachordal [[Level (music)|levels]]
# [[Harmony]], "consistent with and derived from the properties of the referential set"
# [[Meter (music)|Metre]], established through "pitch-relational characteristics"
# [[Simultaneity (music)|Multidimensional]] set presentations.

== See also ==
* [[List of dodecaphonic and serial compositions]]
* [[All-interval twelve-tone row]]
* [[All-interval tetrachord]]
* [[All-trichord hexachord]]
* [[Pitch interval]]
* [[List of tone rows and series]]

==References==

===Notes===
{{Reflist|20em}}

===Sources===
{{div col|colwidth=45em}}
* Alegant, Brian. 2010. ''The Twelve-Tone Music of Luigi Dallapiccola''. Eastman Studies in Music 76. Rochester, New York: University of Rochester Press. {{ISBN|978-1-58046-325-6}}.
* [[Milton Babbitt|Babbitt, Milton]]. 1960. "Twelve-Tone Invariants as Compositional Determinants". ''[[The Musical Quarterly]]'' 46, no. 2, Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–259. {{doi|10.1093/mq/XLVI.2.246}}. {{JSTOR|740374}}{{Subscription required}}.
* Babbitt, Milton. 1961. "Set Structure as a Compositional Determinant". ''[[Journal of Music Theory]]'' 5, no. 1 (Spring): 72–94. {{JSTOR|842871}}{{Subscription required}}.
* Benson, Dave. 2007 ''[https://books.google.com/books?id=Ko1NsIq4qLIC&pg=PA348 Music: A Mathematical Offering]''. Cambridge and New York: Cambridge University Press. {{ISBN|978-0-521-85387-3}}.
* Brett, Philip. "Britten, Benjamin." ''[[Grove Music Online]]'' ed. L. Macy (Accessed 8 January 2007)
* [[Gilbert Chase|Chase, Gilbert]]. 1987. ''America's Music: From the Pilgrims to the Present'', revised third edition. Music in American Life. Urbana: University of Illinois Press. {{ISBN|0-252-00454-X}} (cloth); {{ISBN|0-252-06275-2}} (pbk).
* {{wikicite|ref={{harvid|Crawford and Khuner|1996}}|reference=Crawford, Caroline and [[Felix Khuner]]. 1996. ''[https://digitalassets.lib.berkeley.edu/rohoia/ucb/text/violinistsjourn00khunrich.pdf Felix Khuner: A Violinist's Journey from Vienna's Kolisch Quartet to the San Francisco Symphony and Opera Orchestras]'', intro. by Tom Heimberg. [[Regional Oral History Office]], [[The Bancroft Library]]. Berkeley: University of California.}}
* Haimo, Ethan. 1990. ''Schoenberg's Serial Odyssey: The Evolution of his Twelve-Tone Method, 1914–1928''. Oxford [England] Clarendon Press; New York: Oxford University Press {{ISBN|0-19-315260-6}}.
* Hill, Richard S. 1936. "Schoenberg's Tone-Rows and the Tonal System of the Future". ''[[The Musical Quarterly]]'' 22, no. 1 (January): 14–37. {{doi|10.1093/mq/XXII.1.14}}. {{JSTOR|739013}}{{Subscription required}}.
* [[Paul Lansky|Lansky, Paul]]; [[George Perle]] and Dave Headlam. 2001. "Twelve-note Composition". ''[[The New Grove Dictionary of Music and Musicians]]'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (musicologist)|John Tyrrell]]. London: Macmillan.
* [[Ton de Leeuw|Leeuw, Ton de]]. 2005. ''Music of the Twentieth Century: A Study of Its Elements and Structure'', 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}}.
* Loy, D. Gareth, 2007. ''Musimathics: The Mathematical Foundations of Music'', Vol. 1. Cambridge, Massachusetts and London: MIT Press. {{ISBN|978-0-262-12282-5}}.
* [[Oliver Neighbour|Neighbour, Oliver]]. 1954. "The Evolution of Twelve-Note Music". ''[[Proceedings of the Royal Musical Association]]'', volume 81, issue 1: 49–61. {{doi|10.1093/jrma/81.1.49}}
* [[George Perle|Perle, George]]. 1977. ''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. {{ISBN|0-520-03395-7}}
* Perle, George. 1991. ''Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern'', sixth edition, revised. Berkeley: University of California Press. {{ISBN|978-0-520-07430-9}}.
* [[Rudolph Reti|Reti, Rudolph]]. 1958. ''Tonality, Atonality, Pantonality: A Study of Some Trends in Twentieth Century Music''. Westport, Connecticut: Greenwood Press. {{ISBN|0-313-20478-0}}.
* [[Josef Rufer|Rufer, Josef]]. 1954. ''Composition with Twelve Notes Related Only to One Another'', translated by [[Humphrey Searle]]. New York: The Macmillan Company. (Original German ed., 1952)
* [[Arnold Schoenberg|Schoenberg, Arnold]]. 1975. ''Style and Idea'', edited by [[Leonard Stein (musicologist)|Leonard Stein]] with translations by Leo Black. Berkeley & Los Angeles: University of California Press. {{ISBN|0-520-05294-3}}.
** 207–208 "Twelve-Tone Composition (1923)"
** 214–245 "Composition with Twelve Tones (1) (1941)"
** 245–249 "Composition with Twelve Tones (2) (c. 1948)"
* Solomon, Larry. 1973. "New Symmetric Transformations". ''[[Perspectives of New Music]]'' 11, no. 2 (Spring–Summer): 257–264. {{JSTOR|832323}}{{Subscription required}}.
* [[Claudio Spies|Spies, Claudio]]. 1965. "Notes on Stravinsky's ''Abraham and Isaac''". ''[[Perspectives of New Music]]'' 3, no. 2 (Spring–Summer): 104–126. {{JSTOR|832508}}{{Subscription required}}.
* [[Arnold Whittall|Whittall, Arnold]]. 2008. ''The Cambridge Introduction to Serialism''. Cambridge Introductions to Music. New York: Cambridge University Press. {{ISBN|978-0-521-86341-4}} (cloth) {{ISBN|978-0-521-68200-8}} (pbk).
{{div col end}}

===Further reading===
{{div col|colwidth=45em}}
* Covach, John. 1992. "The Zwölftonspiel of Josef Matthias Hauer". ''[[Journal of Music Theory]]'' 36, no. 1 (Spring): 149–84. {{JSTOR|843913}}{{Subscription required}}.
* Covach, John. 2000. "Schoenberg's 'Poetics of Music', the Twelve-tone Method, and the Musical Idea". In ''Schoenberg and Words: The Modernist Years'', edited by Russell A. Berman and Charlotte M. Cross, New York: Garland. {{ISBN|0-8153-2830-3}}
* Covach, John. 2002, "Twelve-tone Theory". In ''The Cambridge History of Western Music Theory'', edited by Thomas Christensen, 603–627. Cambridge: Cambridge University Press. {{ISBN|0-521-62371-5}}.
* [[Ernst Krenek|Krenek, Ernst]]. 1953. "Is the Twelve-Tone Technique on the Decline?" ''[[The Musical Quarterly]]'' 39, no 4 (October): 513–527.
* Šedivý, Dominik. 2011. ''Serial Composition and Tonality. An Introduction to the Music of Hauer and Steinbauer'', edited by Günther Friesinger, Helmut Neumann and Dominik Šedivý. Vienna: edition mono. {{ISBN|3-902796-03-0}}
* Sloan, Susan L. 1989. "[http://www.schoenberg.at/library/index.php/attachments/single/244 Archival Exhibit: Schoenberg's Dodecaphonic Devices]". ''Journal of the Arnold Schoenberg Institute'' 12, no. 2 (November): 202–205.
* Starr, Daniel. 1978. "Sets, Invariance and Partitions". ''[[Journal of Music Theory]]'' 22, no. 1 (Spring): 1–42. {{JSTOR|843626}}{{Subscription required}}.
* [[Charles Wuorinen|Wuorinen, Charles]]. 1979. ''Simple Composition''. New York: Longman. {{ISBN|0-582-28059-1}}. Reprinted 1991, New York: C. F. Peters. {{ISBN|0-938856-06-5}}.
{{div col end}}

==External links==
{{commonscat}}
{{Wikiquote}}
* [http://www.andreaslehmann.com/musik-tools/reihenquadrat/ Twelve tone square] to find all combinations of a 12 tone sequence
* [https://web.archive.org/web/20071115223321/http://solomonsmusic.net/diss8.htm New Transformations: Beyond P, I, R, and RI] by Larry Solomon
* [http://composertools.com/Tools/ Javascript twelve tone matrix calculator and tone row analyzer]
* [http://www.musictheory.net/utilities/html/id98_en.html Matrix generator from musictheory.net] by Ricci Adams
* [http://musike.cmpr.edu/docs/v001/roman-eng.pdf Twelve-Tone Technique, A Quick Reference] by Dan Román
* {{YouTube|4niz8TfY794|Twelve Tones by mathemusician Vi Hart}}
* [http://recherche.ircam.fr/equipes/repmus/mamux/documents/fjPisa.pdf Dodecaphonic Knots and Topology of Words] by {{interlanguage link|Franck Jedrzejewski|fr}}
* [https://web.archive.org/web/20140522213508/http://www.uni-graz.at/~fripert/db/ Database on tone rows and tropes]

{{atonality}}
{{Twelve-tone technique}}
{{Second Viennese School}}
{{Modernism (music)}}
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[[Category:Twelve-tone technique| ]]
[[Category:Arnold Schoenberg]]
[[Category:12 (number)]]
[[Category:Josef Matthias Hauer]]