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"Reihe" redirects here. For the music journal, see Die Reihe.

In music, a tone row or note row (German: Reihe or Tonreihe), 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, though both larger and smaller sets are sometimes found.

"Mirror forms", P, R, I, and RI, of a tone row (from Arnold Schoenberg's Variations for Orchestra Op. 31, "Called mirror forms because...they are identical".

Contents

Tone row of Karlheinz Stockhausen's Gruppen für drei Orchester, the registrally fixed pitches of which correspond with duration units and metronome marks.

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, like Dmitri Shostakovich's use of twelve-tone rows, "without dodecaphonic transformations."

A tone row has been identified in the A minor prelude, 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'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 also used the technique but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven". 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 and that "Schoenberg repressed his knowledge of classical serialism because it would have injured his narcissism."

"Prime (music)" redirects here. For the prime form, see set (music). For the interval, see unison.
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.

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.

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—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. 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 "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'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 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 (B, A, C, B) and is composed of three tetrachords:

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 of all twelve pitch classes. An "array" is a succession of aggregates. The term is also used to refer to lattices.

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. For example, the row form consisting of two hexachords:

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:

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.

Pierre Boulez's Second Piano Sonata series 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.
Prime form of five-note tone row from Igor Stravinsky's In memoriam Dylan Thomas.

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 which contains a repeated note making it a 'thirteen-tone row':

Thirteen-note tone row from Luciano Berio's Nones, symmetrical about the central tone with one note (D) repeated.

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), as for example in his Requiem Canticles:

Basic row forms from Stravinsky's Requiem Canticles: P R I IR
Unordered sets from the second of Stockhausen's Klavierstücke I–IV which "retained only the rudiments of the 12-note series".
Unordered sets from the third of Stockhausen's Klavierstücke I–IV

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- otonality, outlining a diminished triad.

Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2 and hexachords.
  1. Leeuw 2005, 154. Italics original.
  2. Perle 1977, 3
  3. Leeuw 2005, 174.
  4. Andrew Kirkman and Alexander Ivashkin, Contemplating Shostakovich: Life, Music and Film: Life, Music and Film. (Farnham: Ashgate Publishing, 2013): [unpaginated]. ISBN 9781409472025.
  5. 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.
  6. "Discovery Reveals Bach's Postmodern Side". Weekend Edition Sunday, NPR, 6 September 2009.
  7. Keller 1955, 14–21.
  8. Keller 1955, 22–23.
  9. Keller 1955, 23.
  10. Leeuw 2005, 158.
  11. Perle 1996, 3.
  12. Whittall 2008, 97.
  13. Whittall 2008, 271
  14. Perle 1977, 100.
  15. Perle 1996, 20.
  16. Haimo 1990, 183.
  17. 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.
  18. Leeuw 2005, 166.
  19. Whittall 2008, 127.
  20. Whittall 2008, 195.
  21. Claudio Spies, "Notes on Stravinsky's Abraham and Isaac", Perspectives of New Music 3, no. 2 (Spring–Summer 1965): 104–126, citation on 118.
  22. Joseph N. Strauss, "Stravinsky's Serial 'Mistakes' ", The Journal of Musicology 17, no. 2 (Spring 1999): 231–271, citation on 242.
  23. Whittall 2008, 139
  24. Leeuw 2005, pp. 176–177
  25. 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.

Sources

  • Haimo, Ethan (1990). Schoenberg's Serial Odyssey: The Evolution of his Twelve-Tone Method, 1914–1928. Oxford: Clarendon Press. ISBN 9780193152601.
  • Keller, Hans (Autumn 1955). "Strict Serial Technique in Classical Music". Tempo. New Series (37): 12–24. doi:10.1017/S0040298200055212.
  • 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.
  • Perle, George (1977). Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern (4th ed.). Berkeley, Los Angeles, and London: University of California Press. ISBN 0-520-03395-7.
  • Perle, George (1996). Twelve-Tone Tonality (2nd, revised and expanded ed.). Berkeley: University of California Press. ISBN 0-520-20142-6.
  • Whittall, Arnold (2008). Serialism. Cambridge Introductions to Music. Cambridge University Press. ISBN 9780521863414. OCLC 237192271.

Tone row Article Talk Language Watch Edit Reihe redirects here For the music journal see Die Reihe In music a tone row or note row German Reihe or Tonreihe also series or set 2 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 Mirror forms P R I and RI of a tone row from Arnold Schoenberg s Variations for Orchestra Op 31 Called mirror forms because they are identical 1 source source source Contents 1 History and usage 2 Theory and compositional techniques 3 Nonstandard tone rows 4 See also 5 References 6 Further reading 7 External linksHistory and usage Edit Tone row of Karlheinz Stockhausen s Gruppen fur drei Orchester the registrally fixed pitches of which correspond with duration units and metronome marks 3 source source source 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 like Dmitri Shostakovich s use of twelve tone rows without dodecaphonic transformations 4 5 A tone row has been identified in the A minor prelude BWV 889 from book II of J S Bach s The Well Tempered Clavier 1742 6 and by the late eighteenth century it is 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 7 Beethoven also used the technique but on the whole Mozart seems to have employed serial technique far more often than Beethoven 8 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 and that Schoenberg repressed his knowledge of classical serialism because it would have injured his narcissism 9 Theory and compositional techniques Edit Prime music redirects here For the prime form see set music For the interval see unison 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 10 source source source 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 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 11 further explanation needed 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 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 The main tone row of his Violin Concerto hints at this tonality source Audio playback is not supported in your browser You can download the audio file 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 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 s Concerto for Nine Instruments Op 24 shown below 12 source Audio playback is not supported in your browser You can download the audio file 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 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 B A C B and is composed of three tetrachords source Audio playback is not supported in your browser You can download the audio file The set complex is the forty eight forms of the set generated by stating each aspect or transformation on each pitch class 2 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 13 is the set of all twelve pitch classes An array is a succession of aggregates 13 The term is also used to refer to lattices 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 13 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 14 15 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 16 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 17 Nonstandard tone rows Edit Pierre Boulez s Second Piano Sonata series 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 18 source source source Prime form of five note tone row from Igor Stravinsky s In memoriam Dylan Thomas 19 source source source 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 which contains a repeated note making it a thirteen tone row Thirteen note tone row from Luciano Berio s Nones 20 symmetrical about the central tone with one note D repeated source source source 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 21 22 23 as for example in his Requiem Canticles Basic row forms from Stravinsky s Requiem Canticles 23 P R I IR Unordered sets from the second of Stockhausen s Klavierstucke I IV which retained only the rudiments of the 12 note series 24 source source source Unordered sets from the third of Stockhausen s Klavierstucke I IV 24 source source source 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 otonality outlining a diminished triad Primary forms of the just tone row from Ben Johnston s String Quartet No 7 mov 2 25 source source source and hexachords source source source See also EditMusical set theory Unified field Side slipping Pitch interval List of tone rows and seriesReferences Edit Leeuw 2005 154 Italics original a b Perle 1977 3 Leeuw 2005 174 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 Discovery Reveals Bach s Postmodern Side Weekend Edition Sunday NPR 6 September 2009 Keller 1955 14 21 Keller 1955 22 23 Keller 1955 23 Leeuw 2005 158 Perle 1996 3 Whittall 2008 97 a b c Whittall 2008 271 Perle 1977 100 Perle 1996 20 Haimo 1990 183 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 Leeuw 2005 166 Whittall 2008 127 Whittall 2008 195 Claudio Spies Notes on Stravinsky s Abraham and Isaac Perspectives of New Music 3 no 2 Spring Summer 1965 104 126 citation on 118 Joseph N Strauss Stravinsky s Serial Mistakes The Journal of Musicology 17 no 2 Spring 1999 231 271 citation on 242 a b Whittall 2008 139 a b Leeuw 2005 pp 176 177 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 Sources Haimo Ethan 1990 Schoenberg s Serial Odyssey The Evolution of his Twelve Tone Method 1914 1928 Oxford Clarendon Press ISBN 9780193152601 Keller Hans Autumn 1955 Strict Serial Technique in Classical Music Tempo New Series 37 12 24 doi 10 1017 S0040298200055212 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 amp Holkema 1977 ISBN 90 313 0244 9 Perle George 1977 Serial Composition and Atonality An Introduction to the Music of Schoenberg Berg and Webern 4th ed Berkeley Los Angeles and London University of California Press ISBN 0 520 03395 7 Perle George 1996 Twelve Tone Tonality 2nd revised and expanded ed Berkeley University of California Press ISBN 0 520 20142 6 Whittall Arnold 2008 Serialism Cambridge Introductions to Music Cambridge University Press ISBN 9780521863414 OCLC 237192271 Further reading EditHunter David J Hippel Paul T von February 2003 How Rare Is Symmetry in Musical 12 Tone Rows The American Mathematical Monthly 110 2 124 132 doi 10 2307 3647771 JSTOR 3647771 External links EditDatabase on tone rows and tropes Harald Fripertinger Peter Lackner 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