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{{Short description|Sequence of all twelve chromatic tones}}
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[[File:Arnold_Schoenberg_la_1948.jpg|thumb|[[Arnold Schoenberg]], inventor of the twelve-tone technique|230x230px]]
In [[music]], a '''tone row''' or '''note row''' ({{langx|de|Reihe}} or ''{{lang|de|Tonreihe}}''), also '''series''' or '''set''',<ref name="Perle Serial">{{harvnb|Perle|1977|loc=3}}</ref> is a non-repetitive ordering of a set of [[pitch-class]]es, typically of the twelve notes in [[set theory (music)|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 [[Arnold Schoenberg]]'s [[twelve-tone technique]] and most types of [[serialism|serial music]]. Tone rows were widely used in 20th-century contemporary music, like [[Dmitri Shostakovich]]'s use of twelve-tone rows, "without dodecaphonic transformations."<ref>Andrew Kirkman and Alexander Ivashkin, ''Contemplating Shostakovich: Life, Music and Film: Life, Music and Film''. (Farnham: Ashgate Publishing, 2013): {{unpaginated}}. {{ISBN|9781409472025}}.</ref><ref>Stephen C. Brown, "Twelve-Tone Rows and Aggregate Melodies in the Music of Shostakovich," ''Journal of Music Theory'', Vol. 59, No. 2 (Fall 2015): 191–234.</ref>

A tone row has been identified in the A-minor prelude, [[BWV]] 889, from Book II of [[Johann Sebastian Bach|J.S. Bach]]'s ''[[The Well-Tempered Clavier]]'' (1742).<ref>[https://www.npr.org/templates/story/story.php?storyId=112602288&ft=1&f=1039 "Discovery Reveals Bach's Postmodern Side"]. ''[[Weekend Edition]]'' Sunday, [[NPR]], 6 September 2009.</ref> It is also found in works such as [[Wolfgang Amadeus Mozart|Mozart]]'s [[Milanese Quartets (Mozart)#Quartet No. 4 in C major, K. 157|String Quartet in C, K. 157]] (1772), [[String Quartet No. 16 (Mozart)|String Quartet in E{{music|flat}}, K. 428]], [[String Quintet No. 4 (Mozart)|String Quintet in G minor, K. 516]] (1790), and the [[Symphony No. 40 (Mozart)|Symphony No. 40, K. 550]] (1788).{{sfn|Keller|1955|loc=14–21}} A passage from Symphony No. 40 is shown below in which every tone in the [[chromatic scale]] is played except for G (the [[Tonic (music)|tonic]]): {{Clear}}{{Block indent|<score sound="1">
\relative c'' {
 \key g \minor
 \time 2/2
 \set Staff.midiInstrument = #"violin"
 \set Score.tempoHideNote = ##t \tempo 2 = 120
 \partial 4 bes\f( |
 d4) f-. a-. bes-. |
 des2( c4) e,-. |
 as4-. r b,-. r |
 r2 r4 r8 \times 2/3 { b16( c d } |
 es!4-.) r fis,-. r |
 bes!4-. r r2 |
 cis,4-. r r2 |
 f!4-. r gis,-. r |
}
</score>}}

[[Ludwig van Beethoven|Beethoven]] also used the technique but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven".{{sfn|Keller|1955|loc=22–23}} [[Franz Liszt]] used a twelve-tone row in the opening of his ''[[Faust Symphony]]''. [[Hans Keller]] claims that Schoenberg was aware of this serial practice in the [[classical period (music)|classical period]] and that "Schoenberg repressed his knowledge of classical serialism because it would have injured his [[narcissism]]."{{sfn|Keller|1955|loc=23}}

==Theory and compositional techniques==
{{redirect|Prime (music)|the prime form|set (music)|the interval|unison}}Tone rows are designated by letters and subscript numbers (e.g.: RI<sub>11</sub>, which may also appear as RI11 or RI–11). The numbers indicate the initial (P or I) or final (R or RI) pitch-class number of the given row form, most often with ''c'' = 0.

* "P" indicates prime, a forward-directed, right-side-up form.
* "I" indicates [[Melodic inversion|inversion]], a forward-directed, upside-down form.
* "R" indicates [[Retrograde (music)|retrograde]], a backwards, right-side-up form.
* "RI" indicates [[Retrograde inversion|retrograde-inversion]], a backwards, upside-down form.
* [[Transposition (music)|Transposition]] is indicated by a ''T number'', for example P8 is a T(4) transposition of P4.{{sfn|Perle|1996|loc=3}}{{Explain|reason=|date=September 2019}}

A twelve-tone composition will take one or more tone rows, called the "prime form", as its basis plus their [[Transformation (music)|transformations]] (inversion, retrograde, retrograde inversion, as well as transposition). These forms may be used to construct a melody in a straightforward manner as in the fifth movement from Schoenberg's [[Suite for Piano (Schoenberg)|Piano Suite, Op. 25]], where P-0 is used to construct the opening melody and later varied through transposition, as P-6, and also in articulation and dynamics. It is then varied again through inversion, untransposed, taking form I-0. However, rows may be combined to produce melodies or harmonies in more complicated ways, such as taking successive or multiple pitches of a melody from two different row forms.[[File:Schoenberg - Variations for Orchestra op. 31 tone row mirror forms.png|thumb|"Mirror forms", P, R, I, and RI, of a tone row (from [[Arnold Schoenberg]]'s [[Variations for Orchestra (Schoenberg)|Variations for Orchestra, Op. 31]], "Called ''mirror forms'' because... they are identical".<ref>{{harvnb|Leeuw|2005|loc=154}}. Italics original.</ref><br>[[File:Schoenberg - Variations for Orchestra op. 31 tone row mirror forms.mid]]|center|700x700px]]

Initially, Schoenberg required the avoidance of suggestions of [[tonality]]—such as the use of consecutive imperfect consonances (thirds or sixths)—when constructing tone rows, reserving such use for the time when the dissonance is completely [[emancipation of the dissonance|emancipated]]. [[Alban Berg]], however, sometimes incorporated tonal elements into his twelve-tone works. The main tone row of his [[Violin Concerto (Berg)|Violin Concerto]] hints at this tonality:

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c' {
        \time 12/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        g1 bes d fis a c e gis b cis ees f
    }
}
</score>

This tone row consists of alternating minor and major [[Triad (music)|triads]] starting on the [[Open string (music)|open strings]] of the violin, followed by a portion of an ascending [[whole tone scale|whole-tone scale]]. This whole-tone scale reappears in the second movement when the [[chorale]] "[[Es ist genug]]" ("It is enough") from J.S. Bach's cantata [[O Ewigkeit, du Donnerwort, BWV 60|''O Ewigkeit, du Donnerwort'', BWV 60]] is quoted literally in the woodwinds (mostly clarinet).

{{anchor|Mirror forms|Mirror form}}

Some tone rows have a high degree of internal organization. An example is the tone row from [[Anton Webern]]'s [[Concerto for Nine Instruments (Webern)|Concerto for Nine Instruments]] Op. 24, shown below.{{sfn|Whittall|2008|loc=97}}

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c'' {
        \time 3/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        b1 bes d
        es, g fis
        aes e f
        c' cis a
    }
}
</score>

In this tone row, if the first three notes are regarded as the "original" [[Cell (music)|cell]], then the next three are its retrograde inversion, the next three are retrograde, and the last three are its inversion. A row created in this manner, through variants of a [[trichord]] or [[tetrachord]] called the [[Generating set of a group|generator]], is called a ''[[derived row]]''.

The tone rows of many of Webern's other late works are similarly intricate. The tone row for Webern's [[String Quartet (Webern)|String Quartet, Op. 28]] is based on the [[BACH motif]] (B{{music|flat}}, A, C, B{{music|natural}}) and is composed of three [[tetrachord]]s:

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c'' {
        \time 4/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        bes1 a c b!
        dis e cis d
        ges, f aes g
    }
}
</score>

The "set-complex" is the forty-eight forms of the set generated by stating each "aspect" or transformation on each pitch class.<ref name="Perle Serial" />

The [[all-interval twelve-tone row]] is a tone row arranged so that it contains one instance of each interval within the octave, 0 through 11.<!--[[Secondary set]], [[Set complex]], and [[Weighted aggregate]] redirect directly here.-->{{anchor|Total chromatic}}

The <span id="total_chromatic">"total chromatic"</span><!--Preceding span is a link target; disturb only with care--> (or "aggregate") is the [[set (music)|set]] of all twelve [[pitch class]]es. An "array" is a succession of aggregates.<ref name="Whittall 271">{{harvnb|Whittall|2008|loc=271}}</ref> The term is also used to refer to [[lattice (music)|lattices]].

An aggregate may be achieved through [[complement (music)|complementation]] or [[combinatoriality]], such as with [[hexachord]]s.

A "secondary set" is a tone row which is derived from or, "results from the reversed coupling of hexachords", when a given row form is immediately repeated.{{sfn|Perle|1977|loc=100}}{{sfn|Perle|1996|loc=20}} For example, the row form consisting of two hexachords:
    0 1 2 3 4 5 / 6 7 8 9 t e
when repeated immediately results in the following succession of two aggregates, in the middle of which is a new and complete aggregate beginning with the second hexachord:
    0 1 2 3 4 5 / 6 7 8 9 t e / 0 1 2 3 4 5 / 6 7 8 9 t e
 secondary set: [6 7 8 9 t e / 0 1 2 3 4 5]

A "weighted aggregate" is an aggregate in which the twelfth pitch does not appear until at least one pitch has appeared at least twice, supplied by segments of different set forms.{{sfn|Haimo|1990|loc=183}} It seems to have been first used in [[Milton Babbitt]]'s [[String Quartet No. 4 (Babbitt)|String Quartet No. 4]]. An aggregate may be vertically or horizontally weighted. An "all-partition array" is created by combining a collection of hexachordally combinatorial arrays.<ref>Evan Allan Jones, ''Intimate Voices: The Twentieth-Century String Quartet''. Volume 2: ''Shostakovich to the Avant-garde. Dmitri Shostakovich: The String Quartets'' (Rochester: University of Rochester Press, 2009): 228. {{ISBN|9781580463225}}.</ref>

== Examples ==

The principal forms of the tone row of [[Anton Webern]]'s [[Variations for piano (Webern)|Variations for piano, Op. 27]] are shown below. In this tone row, each [[hexachord]] fills in a chromatic fourth, with B as the pivot (end of P1 and beginning of IR8), and thus linked by the prominent [[tritone]] in the center of the row.{{sfn|Leeuw|2005|loc=158}}[[File:Webern - Piano Variations op. 27 tone row.png|thumb|center|[[File:Webern - Piano Variations op. 27 tone row.mid]]|380x380px]]The first array of four aggregates (numbered 1–4 at bottom) from [[Milton Babbitt]]'s ''[[Composition for Four Instruments]]'' are shown. In this row, each vertical line (four trichords labeled a–d) is an aggregate while each horizontal line (four trichords labeled a–d) is also an aggregate<ref name="Whittall 271" />[[File:Array - Babbitt's Composition for Four Instruments.png|thumb|upright=2|center|252x252px]]Shown below is the tone row for [[Karlheinz Stockhausen]]'s ''[[Gruppen|Gruppen für drei Orchester]]''. Here, the registrally-fixed pitches correspond to duration units and metronome marks.{{sfn|Leeuw|2005|loc=174}}[[File:Stockhausen Gruppen für drei Orchester series.png|thumb|[[File:Stockhausen Gruppen für drei Orchester series.mid]]|center|550x550px]]

=== Nonstandard tone rows ===
Schoenberg specified many strict rules and desirable guidelines for the construction of tone rows such as number of notes and intervals to avoid. Tone rows that depart from these guidelines include the above tone row from Berg's Violin Concerto which contains triads and tonal emphasis, and the tone row below from [[Luciano Berio]]'s ''[[Nones (Berio)|Nones]]'' which contains a repeated note making it a 'thirteen-tone row':

[[File:Berio - Nones thirteen-tone row.png|thumb|center|Thirteen-note tone row from [[Luciano Berio]]'s ''[[Nones (Berio)|Nones]]'',{{sfn|Whittall|2008|loc=195}} symmetrical about the central tone with one note (D) repeated.[[File:Berio - Nones thirteen-tone row.mid]]|380x380px]]

[[Igor Stravinsky]] used a five-tone row, chromatically filling out the space of a major third centered tonally on C (C–E), in one of his early serial compositions, ''In memoriam Dylan Thomas''.

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c'' {
        \time 5/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        e1 es c cis d
    }
}
</score>

In his twelve-tone practice, Stravinsky preferred the inverse-retrograde (IR) to the retrograde-inverse (RI),<ref>Claudio Spies, "Notes on Stravinsky's ''Abraham and Isaac''", ''[[Perspectives of New Music]]'' 3, no. 2 (Spring–Summer 1965): 104–126, citation on 118.</ref><ref>Joseph N. Strauss, "Stravinsky's Serial 'Mistakes{{'"}}, ''[[The Journal of Musicology]]'' 17, no. 2 (Spring 1999): 231–271, citation on 242.</ref><ref name="Whittall 139">{{harvnb|Whittall|2008|loc=139}}</ref> as for example in his ''[[Requiem Canticles]]'':

[[File:Stravinsky - Requiem Canticles basic row forms.png|thumb|center|upright=2|Basic row forms from Stravinsky's ''Requiem Canticles'':<ref name="Whittall 139"/> P, R, I, and IR]]

[[Ben Johnston (composer)|Ben Johnston]] uses a "just tone row" (see [[just intonation]]) in works including String Quartets Nos. 6 and 7. Each permutation contains a just chromatic scale, however, transformations (transposition and inversion) produce pitches outside of the primary row form, as already occurs in the inversion of P0. The pitches of each hexachord are drawn from different [[Otonality and Utonality|otonality]] or utonality on A+ utonality, C otonality and utonality, and E{{music|b}}- otonality, outlining a [[diminished triad]].

[[File:Ben Johnston String Quartet No. 7, mov. 2 just tone row.png|thumb|center|Primary forms of the just tone row from [[Ben Johnston (composer)|Ben Johnston]]'s ''String Quartet No. 7'', mov. 2<ref>[[John Fonville]], "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", ''[[Perspectives of New Music]]'' 29, no. 2 (Summer 1991): 106–137, citation on 127.</ref>[[File:Ben Johnston String Quartet No. 7, mov. 2 just tone row.mid]] and hexachords.[[File:Ben Johnston String Quartet No. 7, mov. 2 just tone row chords.mid]]|650x650px]]

==See also==
* [[Set theory (music)|Musical set theory]]
* [[Unified field]]
* [[Side-slipping]]
* [[Pitch interval]]
* [[List of tone rows and series]]

==References==
{{reflist}}

'''Sources'''
* {{cite book|last=Haimo|first=Ethan|year=1990|title=Schoenberg's Serial Odyssey: The Evolution of his Twelve-Tone Method, 1914–1928|location=Oxford|publisher=Clarendon Press|isbn=9780193152601}}
* {{cite journal|last=Keller|first=Hans|author-link=Hans Keller|title=Strict Serial Technique in Classical Music|journal=[[Tempo (journal)|Tempo]]|series=New Series|number=37|date=Autumn 1955|pages=12–24|doi=10.1017/S0040298200055212}}
* {{wikicite|ref={{harvid|Leeuw|2005}}|reference=[[Ton de Leeuw]], ''Music of the Twentieth Century: A Study of Its Elements and Structure'', translated from the Dutch by Stephen Taylor (Amsterdam: Amsterdam University Press, 2005). {{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}}.}}
* {{cite book|last=Perle|first=George|author-link=George Perle|title=Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern|edition=4th|location=Berkeley, Los Angeles, and London|publisher=University of California Press|year=1977|isbn=0-520-03395-7}}
* {{cite book|last=Perle|first=George|title=Twelve-Tone Tonality|edition=2nd, revised and expanded|location=Berkeley|publisher=University of California Press|year=1996|isbn=0-520-20142-6}}
* {{cite book|last=Whittall|first=Arnold|author-link=Arnold Whittall|year=2008|title=Serialism|series=Cambridge Introductions to Music|publisher=Cambridge University Press|isbn=9780521863414|oclc=237192271}}

==Further reading==
* {{cite journal|last1=Hunter|first1=David J.|last2=Hippel|first2=Paul T. von|title=How Rare Is Symmetry in Musical 12-Tone Rows?|journal=[[The American Mathematical Monthly]]|volume=110|number=2|date=February 2003|pages=124–132|jstor=3647771|doi=10.2307/3647771|ref=none}}

==External links==
* [https://web.archive.org/web/20140522213508/http://www.uni-graz.at/~fripert/db/ Database on tone rows and tropes], Harald Fripertinger, Peter Lackner

{{Twelve-tone technique}}
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{{DEFAULTSORT:Tone Row}}
[[Category:Twelve-tone technique]]
[[Category:Musical symmetry]]