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music Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...
, 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 The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
, 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-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...
's twelve-tone technique and most types of serial music. Tone rows were widely used in 20th-century contemporary music, like Dmitri Shostakovich'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 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. An abbreviated version of that second edition, known as BWV ...
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 17565 December 1791), baptised as Joannes Chrysostomus Wolfgangus Theophilus Mozart, was a prolific and influential composer of the Classical period. Despite his short life, his rapid pace of composition r ...
'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. Beethoven remains one of the most admired composers in the history of Western music; his works rank amongst the most performed of the classic ...
also used the technique but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven".
Franz Liszt Franz Liszt, in modern usage ''Liszt Ferenc'' . Liszt's Hungarian passport spelled his given name as "Ferencz". An orthographic reform of the Hungarian language in 1922 (which was 36 years after Liszt's death) changed the letter "cz" to simpl ...
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 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 interest in one's physical appearance or image and an excessive preoccupation with one's own needs, often at the expense of others. Narcissism exists on a co ...
."


Theory and compositional techniques

Tone rows are designated by letters and subscript numbers (e.g.: RI11, 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 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 * ...
, 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 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. Initially, Schoenberg required the avoidance of suggestions of
tonality Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is cal ...
—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 sm ...
, however, sometimes incorporated tonal elements into his twelve-tone works. The main tone row of his
Violin Concerto A violin concerto is a concerto for solo violin (occasionally, two or more violins) and instrumental ensemble (customarily orchestra). Such works have been written since the Baroque music, Baroque period, when the solo concerto form was first dev ...
hints at this tonality: : This tone row consists of alternating minor and major triads starting on the
open strings ''Open Strings'' is an album by French jazz fusion artist Jean-Luc Ponty, released in 1971 on vinyl by the MPS label. Track listing All songs written by Jean-Luc Ponty, except where noted. Side one #"Flipping, Pt.1" – 4:40 #"Flipping, Pt.2 ...
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 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 of the ...
"
Es ist genug "" ("It is enough") is a German Lutheran hymn, with text by Franz Joachim Burmeister, written in 1662. The melody, Zahn No. 7173, was written by Johann Rudolph Ahle who collaborated with the poet. It begins with a sequence of three consecutiv ...
" (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 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 ste ...
'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 ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency pr ...
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, a succession of notes important or characteristic to a piece, ''B flat, A, C, B natural''. In German musical nomenclature, in which the note ''B natural'' is named ''H'' and the ''B flat'' named ...
(B, A, C, B) and is composed of three
tetrachord In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency pr ...
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 In music, an all-interval twelve-tone row, series, or chord, is a twelve-tone tone row arranged so that it contains one instance of each interval within the octave, 1 through 11 (an ordering of every interval, 0 through 11, that contains each ...
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 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 hexachords. 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 Milton Byron Babbitt (May 10, 1916 – January 29, 2011) was an American composer, music theorist, mathematician, and teacher. He is particularly noted for his serial and electronic music. Biography Babbitt was born in Philadelphia to Albert E ...
'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 work (in particular his 1968 composition ''Sinfonia'' and his series of virtuosic solo pieces titled '' Sequenza''), and for his pioneering wo ...
'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, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...
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 ''The Journal of Musicology'' is a quarterly peer-reviewed academic journal of musicology published by University of California Press. The journal was established in 1982 by Marian C. Green Marian may refer to: People * Mari people, a Finno-Ug ...
'' 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 the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...
) 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 (also known as the minor flatted fifth) is a triad consisting of two minor thirds above the root. It is a minor triad with a lowered ( flattened) fifth. When using chord symbols, it may be indicated by the ...
.


See also

* Musical set theory * Unified field * Side-slipping * Pitch interval * List of tone rows and series


References

Sources * * * * * *


Further reading

*


External links


Database on tone rows and tropes
Harald Fripertinger, Peter Lackner {{DEFAULTSORT:Tone Row Twelve-tone technique Musical symmetry