Tone row

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

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.

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History and usage

Tone row of Karlheinz Stockhausen's Gruppen fur drei Orchester
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the registrally fixed pitches of which correspond with duration units and metronome marks. Stockhausen Gruppen fur drei Orchester series.png
Tone row of Karlheinz Stockhausen's Gruppen für drei Orchester Loudspeaker.svg Play   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." [4] [5]

A tone row has been identified in the A minor prelude from book II of J.S. Bach's The Well-Tempered Clavier (1742) [6] and by the late eighteenth century, their use was a well-established technique,[ citation needed ] 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

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
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. Webern - Piano Variations op. 27 tone row.png
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 Loudspeaker.svg Play  .

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:

Tone row

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]

Tone row

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:

Tone row

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 Array - Babbitt's Composition for Four Instruments.png
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. [14] 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. [15] 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. [16]

Nonstandard tone rows

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

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.
Play (help*info) Berio - Nones thirteen-tone row.png
Thirteen-note tone row from Luciano Berio's Nones , symmetrical about the central tone with one note (D) repeated. Loudspeaker.svg 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), [20] [21] [22] as for example in his Requiem Canticles :

Basic row forms from Stravinsky's Requiem Canticles: P R I IR Stravinsky - Requiem Canticles basic row forms.png
Basic row forms from Stravinsky's Requiem Canticles: P R I IR
Unordered sets from the second of Stockhausen's Klavierstucke I-IV
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which, "retained only the rudiments of the 12-note series." Stockhausen - Klavierstucke I-IV 2 series.png
Unordered sets from the second of Stockhausen's Klavierstücke I–IV Loudspeaker.svg Play   which, "retained only the rudiments of the 12-note series."
Unordered sets from the third of Stockhausen's Klavierstucke I-IV
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. Stockhausen - Klavierstucke I-IV 3 series.png
Unordered sets from the third of Stockhausen's Klavierstücke I–IV Loudspeaker.svg 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- otonality, outlining a diminished triad.

Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2
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and
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. Ben Johnston String Quartet No. 7, mov. 2 just tone row.png
Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2 Loudspeaker.svg Play   and Loudspeaker.svg Play hexachords  .

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 ]

"Tone row" may also be used to describe other musical collections or scales, such as in Arabic music.[ citation needed ]

Sources

  1. Leeuw 2005, 154. Italics original.
  2. 1 2 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.
  3. 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.
  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. Hans Keller, "Strict Serial Technique in Classical Music", Tempo, New Series, no. 37 (Autumn, 1955): 12–24; citations on 14–21.
  8. Keller 1955, 22–23.
  9. Keller 1955, 23.
  10. Leeuw 2005, 158.
  11. George Perle, Twelve-Tone Tonality, second edition, revised and expanded (Berkeley: University of California Press, 1996): 3. ISBN   0-520-20142-6.
  12. Whittall 2008, 97.
  13. 1 2 3 Whittall 2008, 271.
  14. Perle 1977, 100; Perle 1996, 20.
  15. Haimo 1990, 183.
  16. 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.
  17. Leeuw 2005, 166.
  18. Whittall 2008, 127.
  19. Whittall 2008, 195.
  20. Claudio Spies, "Notes on Stravinsky's Abraham and Isaac", Perspectives of New Music 3, no. 2 (Spring-Summer 1965): 104–26, citation on 118.
  21. Joseph N. Strauss, "Stravinsky's Serial 'Mistakes'", The Journal of Musicology 17, no. 2 (Spring 1999): 231–71, citation on 242.
  22. 1 2 Whittall 2008, 139.
  23. 1 2 Leeuw 2005, 176–77.
  24. 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.

Related Research Articles

Atonality Music that lacks a tonal center or key

Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about 1908 to the present day, where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized classical European music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments".

In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were also working to establish serialism as a form of post-tonal thinking. Twelve-tone technique orders the twelve notes of the chromatic scale, forming a row or series and providing a unifying basis for a composition's melody, harmony, structural progressions, and variations. Other types of serialism also work with sets, collections of objects, but not necessarily with fixed-order series, and extend the technique to other musical dimensions, such as duration, dynamics, and timbre.

Twelve-tone technique musical composition method using all 12 chromatic scale notes equally often & not in a key

The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or even actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.

In music, the mystic chord or Prometheus chord is a six-note synthetic chord and its associated scale, or pitch collection; which loosely serves as the harmonic and melodic basis for some of the later pieces by Russian composer Alexander Scriabin. Scriabin, however, did not use the chord directly but rather derived material from its transpositions.

The term "partition" is also French for the sheet music of a transcription.

In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."

Complement (music)

In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.

Permutation (music) in music

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<i>Lyric Suite</i> (Berg)

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Retrograde inversion

Retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order." Retrograde reverses the order of the motif's pitches: what was the first pitch becomes the last, and vice versa. This is a technique used in music, specifically in twelve-tone technique, where the inversion and retrograde techniques are performed on the same tone row successively, "[t]he inversion of the prime series in reverse order from last pitch to first."

Otonality and Utonality

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An Otonality is that set of pitches generated by the numerical factors (...identities)...over a numerical constant in the denominator. Conversely, a Utonality is the inversion of an Otonality, a set of pitches with a numerical constant in the numerator over the numerical factors...in the denominator.

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A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords, when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].

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<i>Composition for Four Instruments</i>

Composition for Four Instruments (1948) is an early serial music composition written by American composer Milton Babbitt. It is Babbitt’s first published ensemble work, following shortly after his Three Compositions for Piano (1947). In both these pieces, Babbitt expands upon the methods of twelve-tone composition developed by Arnold Schoenberg. He is notably innovative for his application of serial techniques to rhythm. Composition for Four Instruments is considered one of the early examples of “totally serialized” music. It is remarkable for a strong sense of integration and concentration on its particular premises—qualities that caused Elliott Carter, upon first hearing it in 1951, to persuade New Music Edition to publish it.

Suite for Piano (Schoenberg) twelve tone piece for piano composed between 1921 and 1923

Arnold Schoenberg's Suite for Piano, Op. 25, is a 12-tone piece for piano composed between 1921 and 1923. The work is the earliest in which Schoenberg employs a row of "12 tones related only to one another" in every movement: the earlier 5 Stücke, Op. 23 (1920–23) employs a 12-tone row only in the final waltz movement, and the Serenade, Op. 24 uses a single row in its central Sonnet. The basic tone row of the suite consists of the following pitches: E–F–G–D–G–E–A–D–B–C–A–B.

Anton Webern's Concerto for Nine Instruments, Op. 24, Op. 24, written in 1934, is a twelve-tone concerto for nine instruments: flute, oboe, clarinet, horn, trumpet, trombone, violin, viola, and piano. It consists of three movements:

In music, the "Ode-to-Napoleon" hexachord is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte (1942) by Arnold Schoenberg. Containing the pitch-classes 014589 it is given Forte number 6-20 in Allen Forte's taxonomic system. The primary form of the tone row used in the Ode allows the triads of G minor, E minor, and B minor to easily appear.

The String Quartet No.1 is a piece for two violins, viola and cello, composed by Robert Gerhard between 1951 and 1955, premiered at Dartington in 1956. This work marks a turning point in Gerhard's style and composition processes, because in one hand, he recovers some old techniques such as the sonata form in the first movement, along with others not as old like the 12-tone technique. Gerhard brilliantly develops, combines and transforms these resources along with new systematic processes created by himself, so that it leads to a new and broad theoretical framework that will be essential to his music thereafter.