
In
music
Music is the arrangement of sound to create some combination of Musical form, form, harmony, melody, rhythm, or otherwise Musical expression, expressive content. Music is generally agreed to be a cultural universal that is present in all hum ...
, a tone row or note row ( 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
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonality, tonal music. Other theorists, such as Allen Forte, further devel ...
of the
chromatic scale, though both larger and smaller sets are sometimes found.
History and usage
Tone rows are the basis of
Arnold Schoenberg
Arnold Schoenberg or Schönberg (13 September 187413 July 1951) was an Austrian and American composer, music theorist, teacher and writer. He was among the first Modernism (music), modernists who transformed the practice of harmony in 20th-centu ...
's
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. The technique is a means of ensuring that all 12 notes of the chromatic scale ...
and most types of
serial music. Tone rows were widely used in 20th-century contemporary music, like
Dmitri Shostakovich
Dmitri Dmitriyevich Shostakovich, group=n (9 August 1975) was a Soviet-era Russian composer and pianist who became internationally known after the premiere of his First Symphony in 1926 and thereafter was regarded as a major composer.
Shostak ...
's use of twelve-tone rows, "without dodecaphonic transformations."
A tone row has been identified in the A minor prelude,
BWV
The (, ; BWV) is a Catalogues of classical compositions, catalogue of compositions by Johann Sebastian Bach. It was first published in 1950, edited by Wolfgang Schmieder. The catalogue's second edition appeared in 1990 and the third edition in ...
889, from book II of
J.S. Bach's ''
The Well-Tempered Clavier'' (1742) and by the late eighteenth century it is found in works such as
Mozart
Wolfgang Amadeus Mozart (27 January 1756 – 5 December 1791) was a prolific and influential composer of the Classical period (music), Classical period. Despite his short life, his rapid pace of composition and proficiency from an early age ...
's
C major String Quartet, K. 157 (1772),
String Quartet in E-flat major, K. 428,
String Quintet in G minor, K. 516 (1790), and the
Symphony in G minor, K. 550 (1788).
Beethoven
Ludwig van Beethoven (baptised 17 December 177026 March 1827) was a German composer and pianist. He is one of the most revered figures in the history of Western music; his works rank among the most performed of the classical music repertoire ...
also used the technique but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven".
Franz Liszt
Franz Liszt (22 October 1811 – 31 July 1886) was a Hungarian composer, virtuoso pianist, conductor and teacher of the Romantic music, Romantic period. With a diverse List of compositions by Franz Liszt, body of work spanning more than six ...
used a twelve-tone row in the opening of his ''
Faust Symphony
''A Faust Symphony in three character pictures'' (), List of compositions by Franz Liszt (S.1 - S.350), S.108, or simply the "''Faust Symphony''", is a choral symphony written by Hungarians, Hungarian composer Franz Liszt inspired by Johann Wolfga ...
''.
Hans Keller claims that Schoenberg was aware of this serial practice in the
classical period and that "Schoenberg repressed his knowledge of classical serialism because it would have injured his
narcissism
Narcissism is a self-centered personality style characterized as having an excessive preoccupation with oneself and one's own needs, often at the expense of others. Narcissism, named after the Greek mythological figure ''Narcissus'', has evolv ...
."
Theory and compositional techniques

Tone rows are designated by letters and subscript numbers (e.g.: RI
11, 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
inversion, a forward-directed upside-down form.
* "R" indicates
retrograde, a backwards right-side up form.
* "RI" indicates
retrograde-inversion, a backwards upside-down form.
*
Transposition is indicated by a ''T number'', for example P8 is a T(4) transposition of P4.
A twelve-tone composition will take one or more tone rows, called the "prime form", as its basis plus their
transformations (inversion, retrograde, retrograde inversion, as well as transposition; 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. The technique is a means of ensuring that all 12 notes of the chromatic scale ...
for details). These forms may be used to construct a melody in a straightforward manner as in Schoenberg's
Piano Suite Op. 25 Minuet Trio, 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, as described at
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. The technique is a means of ensuring that all 12 notes of the chromatic scale ...
.
Initially, Schoenberg required the avoidance of suggestions of
tonality
Tonality is the arrangement of pitch (music), pitches and / or chord (music), chords of a musical work in a hierarchy of perceived ''relations'', ''stabilities'', ''attractions'', and ''directionality''.
In this hierarchy, the single pitch or ...
—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
emancipated.
Alban Berg
Alban Maria Johannes Berg ( ; ; 9 February 1885 – 24 December 1935) was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with the twelve-tone technique. Although he left a relatively sma ...
, however, sometimes incorporated tonal elements into his twelve-tone works. The main tone row of his
Violin Concerto hints at this tonality:
:
This tone row consists of alternating minor and major
triads starting on the
open strings of the violin, followed by a portion of an ascending
whole tone scale. This whole tone scale reappears in the second movement when the
chorale
A chorale is the name of several related musical forms originating in the music genre of the Lutheran chorale:
* Hymn tune of a Lutheran hymn (e.g. the melody of " Wachet auf, ruft uns die Stimme"), or a tune in a similar format (e.g. one o ...
"
Es ist genug" (It is enough) from J.S. Bach's cantata
''O Ewigkeit, du Donnerwort'', BWV 60 is quoted literally in the woodwinds (mostly clarinet).
Some tone rows have a high degree of internal organization. An example is the tone row from
Anton Webern
Anton Webern (; 3 December 1883 – 15 September 1945) was an Austrian composer, conductor, and musicologist. His music was among the most radical of its milieu in its lyric poetry, lyrical, poetic concision and use of then novel atonality, aton ...
's
Concerto for Nine Instruments Op. 24, shown below.
:
In this tone row, if the first three notes are regarded as the "original" 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
In music theory, a tetrachord (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...
called the
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 Op. 28 is based on the
BACH motif
In music, the BACH motif is the motif (music), motif, a succession of note (music), notes important or characteristic to a musical composition, piece, ''B flat, A, C, B natural''. In Letter notation, German musical nomenclature, in whi ...
(B, A, C, B) and is composed of three
tetrachord
In music theory, a tetrachord (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...
s:
:
The "set-complex" is the forty-eight forms of the set generated by stating each "aspect" or transformation on each pitch class.
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.
The
"total chromatic" (or "aggregate") is the
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 ...
of all twelve
pitch classes. An "array" is a succession of aggregates.
The term is also used to refer to
lattices.
An aggregate may be achieved through
complementation or
combinatoriality, such as with
hexachord
In music, a hexachord (also hexachordon) is a six- note series, as exhibited in a scale ( hexatonic or hexad) or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial t ...
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. 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. It seems to have been first used in
Milton Babbitt's
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.
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
Luciano Berio (24 October 1925 – 27 May 2003) was an Italian composer noted for his experimental music, experimental work (in particular his 1968 composition ''Sinfonia (Berio), Sinfonia'' and his series of virtuosic solo pieces titled ''Seque ...
's ''
Nones'' which contains a repeated note making it a 'thirteen-tone row':
Igor Stravinsky
Igor Fyodorovich Stravinsky ( – 6 April 1971) was a Russian composer and conductor with French citizenship (from 1934) and American citizenship (from 1945). He is widely considered one of the most important and influential 20th-century c ...
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''.
In his twelve-tone practice, Stravinsky preferred the inverse-retrograde (IR) to the retrograde-inverse (RI),
[Joseph N. Strauss, "Stravinsky's Serial 'Mistakes, '' The Journal of Musicology'' 17, no. 2 (Spring 1999): 231–271, citation on 242.] as for example in his ''
Requiem Canticles'':
Ben Johnston uses a "just tone row" (see
just intonation
In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
) 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 or utonality on A+ utonality, C otonality and utonality, and E- otonality, outlining a
diminished triad
In music theory, a diminished triad is a triad (music), triad consisting of two minor thirds above the root (chord), root. It is a Minor chord, minor triad with a lowered (flat (music), flattened) Fifth (chord), fifth. When using Chord names and ...
.
See also
*
Musical set theory
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonality, tonal music. Other theorists, such as Allen Forte, further devel ...
*
Unified field
*
Side-slipping
In jazz musical improvisation, improvisation, outside playing describes approaches where one plays over a scale, mode or chord that is harmonically distant from the given chord (music), chord. There are several common techniques to playing outside, ...
*
Pitch interval
*
List of tone rows and series
References
Sources
*
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*
*
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*
Further reading
*
External links
Database on tone rows and tropes Harald Fripertinger, Peter Lackner
{{DEFAULTSORT:Tone Row
Twelve-tone technique
Musical symmetry