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tone row
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{{Use dmy dates|date=July 2012}}In music, a tone row or note row ( or '), also series or set',George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern'', fourth Edition (Berkeley, Los Angeles, and London: University of California Press, 1977): 3. {{ISBN|0-520-03395-7}}. is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.

History and usage

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- Stockhausen Gruppen für drei Orchester series.png -
Tone row of Karlheinz Stockhausen's Gruppen für drei Orchester{{audio|Stockhausen Gruppen für drei Orchester series.mid|Play
the registrally fixed pitches of which correspond with duration units and metronome marks.(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), 174. {{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}}.)Tone rows are the basis of Arnold Schoenberg's twelve-tone technique and most types of serial music. Tone rows were widely used in 20th-century contemporary music, e.g. Dmitri Shostakovich's use of twelve-tone rows, "without dodecaphonic transformations", Andrew Kirkman and Alexander Ivashkin, Contemplating Shostakovich: Life, Music and Film: Life, Music and Film. (Farnham: Ashgate Publishing, 2013): {{unpaginated}}. {{ISBN|9781409472025}}.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. though one has been identified in the A minor prelude from book II of The Well-Tempered Clavier (1742) by J. S. Bach,Discovery Reveals Bach's Postmodern Side. Weekend Edition Sunday, NPR, 6 September 2009. and by the late eighteenth century was a well-established technique, found in works such as Mozart'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).Hans Keller, "Strict Serial Technique in Classical Music", Tempo, New Series, no. 37 (Autumn, 1955): 12–24; citations on 14–21. Beethoven also used the technique but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven".Keller 1955, 22–23. 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."Keller 1955, 23.

Theory and compositional techniques

{{redirects|Prime (music)||Prime form (music)}}
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- Webern - Piano Variations op. 27 tone row.png -
center|Principal forms of the tone row of Anton Webern's Variations for piano, op. 27. 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 {{audio|Webern - Piano Variations op. 27 tone row.mid|Play
Anton WebernTone 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, 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.George Perle, Twelve-Tone Tonality, second edition, revised and expanded (Berkeley: University of California Press, 1996): 3. {{ISBN|0-520-20142-6}}.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 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—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, however, sometimes incorporated tonal elements into his twelve-tone works, and the main tone row of his Violin Concerto hints at this tonality:
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- Berg vn conc tone row.png -
center|G, B{{music|b}}, D, F{{music|#}}, A, C, E, G{{music|#}}, B, C{{music|#}}, E{{music|b}}, F. {{audio|Berg vn conc tone row.mid|Play
]]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 "Es ist genug" (It is enough) from Johann Sebastian Bach's cantata O Ewigkeit, du Donnerwort, BWV 60, which opens with consecutive whole tones based on Johann Rudolf Ahle's melody from 1662, is quoted literally in the woodwinds (mostly clarinet).{{anchor|Mirror forms|Mirror form}}
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- Schoenberg - Variations for Orchestra op. 31 tone row mirror forms.png -
"Mirror forms", P, R, I, and RI, of a tone row (from Arnold Schoenberg's Variations for Orchestra (Schoenberg)|Variations for Orchestra]] op. 31 {{audio|Schoenberg - Variations for Orchestra op. 31 tone row mirror forms.mid|Play
(): "Called mirror forms because...they are identical.".Leeuw 2005, 154. Italics original.)Some tone rows have a high degree of internal organisation. Here is the tone row from Anton Webern's Concerto for Nine Instruments Opus 24:
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- Webern - Concerto Op. 24 tone row.png -
center|Tone row of Webern's Concerto Op. 24,Whittall 2008, 97. composed of four trichords: P RI R I. {{audio|Webern - Concerto Op. 24 tone row.mid|Play
trichord
B, B{{music|b}}, D, E{{music|b}}, G, F{{music|#}}, G{{music|#}}, E, F, C, C{{music|#}}, A
If the first three notes are regarded as the "original" cell, then the next three are its retrograde inversion (backwards and upside down), the next three are retrograde (backwards), and the last three are its inversion (upside down). A row created in this manner, through variants of a trichord or tetrachord called the generator, is called a derived row. The tone rows of many of Webern's other late works are similarly intricate.
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- Webern String Quartet tone row.png -
center|Webern's String Quartet Op. 28 tone row, composed of three tetrachords: P I RI, with P = the BACH motif. {{audio|Webern String Quartet tone row.mid|Play
BACH motifThe "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.{{anchor|Total chromatic}}The "total chromatic" (or "aggregate")Whittall 2008, 271. is the set of all twelve pitch classes. An "array" is a succession of aggregates.
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- Array - Babbitt's Composition for Four Instruments.png -
center|First array of four aggregates (numbered 1–4 at bottom) from Milton Babbitt's Composition for Four Instruments, each vertical line (four trichords labeled a–d) is an aggregate while each horizontal line (four trichords labeled a-d) is also an aggregate
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.Perle 1977, 100; Perle 1996, 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.Haimo 1990, 183. 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.Evan Allan Jones, Intimate Voices: The Twentieth-Century String Quartet. Volume2: Shostakovich to the Avant-garde. Dmitri Shostakovich: The String Quartets (Rochester: University of Rochester Press, 2009): 228. {{ISBN|9781580463225}}.
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- Pierre Boulez - Second Piano Sonata series.png -
Pierre Boulez's Second Piano Sonata series {{audio|Pierre Boulez - Second Piano Sonata series.mid|Play
consists of three cells: A) an ascending perfect fifth followed by a tritone and a perfect fourth, B) a descending perfect fifth followed by an ascending major second and a descending augmented fifth, and B1) B inverted.Leeuw 2005, 166.]]

Nonstandard tone rows

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- Stravinsky - In memoriam Dylan Thomas five-tone row.png -
Prime form of five-note tone row from Igor Stravinsky's In memoriam Dylan Thomas.Whittall 2008, 127. {{audio|Stravinsky - In memoriam Dylan Thomas five-tone row.mid|Play
Igor StravinskySchoenberg 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 which contains a repeated note making it a 'thirteen-tone row':
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- Berio - Nones thirteen-tone row.png -
center|Thirteen-note tone row from Luciano Berio's Nones,Whittall 2008, 195. symmetrical about the central tone with one note (D) repeated. {{audio|Berio - Nones thirteen-tone row.mid|Play
]]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.In his twelve-tone practice Stravinsky preferred the inverse-retrograde (IR) to the retrograde-inverse (RI),Claudio Spies, "Notes on Stravinsky's Abraham and Isaac", Perspectives of New Music 3, no. 2 (Spring-Summer 1965): 104–26, citation on 118.Joseph N. Strauss, "Stravinsky's Serial 'Mistakes'", The Journal of Musicology 17, no. 2 (Spring 1999): 231–71, citation on 242.Whittall 2008, 139. as for example in his Requiem Canticles:
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- Stravinsky - Requiem Canticles basic row forms.png -
center|Basic row forms from Stravinsky's Requiem Canticles: P R I IR
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- Stockhausen - Klavierstücke I-IV 2 series.png -
Unordered sets from the second of Stockhausen's (Klavierstücke (Stockhausen)#Klavierstücke I–IV: from point to group composition|Klavierstücke I–IV) {{audio|Stockhausen - Klavierstücke I-IV 2 series.mid|Play
which, "retained only the rudiments of the 12-note series."Leeuw 2005, 176–77.]]
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- Stockhausen - Klavierstücke I-IV 3 series.png -
Unordered sets from the third of Stockhausen's Klavierstücke I–IV {{audio|Stockhausen - Klavierstücke I-IV 3 series.mid|Play
.]]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 or utonality on A+ utonality, C otonality and utonality, and E{{music|b}}- otonality, outlining a diminished triad.
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- Ben Johnston String Quartet No. 7, mov. 2 just tone row.png -
center|Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2John 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. {{audio|Ben Johnston String Quartet No. 7, mov. 2 just tone row.mid|Play
Ben Johnston

See also

A literary parallel of the tone row is found in Georges Perec's poems which use each of a particular set of letters only once.{{Citation needed|date=November 2017}}"Tone row" may also be used to describe other musical collections or scales, such as in Arabic music.{{Citation needed|date=November 2017}}

Sources

External links

{{Twelve-tone technique}}{{Authority control}}

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