Tone row

P-0 tone row melody from Arnold Schoenberg's Op. 25 Minuet Trio opening^[1]  Play (help·info)

In music, a tone row or note row (German: Reihe or Tonreihe), also series and set,^[2] refers to 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

Tone row of Stockhausen's Gruppen für drei Orchester Play (help·info) the pitches of which correspond with duration units and metronome marks.^[3]

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, though one has been identified in a 1742 composition of Johann Sebastian Bach,^[4] and by the late eighteenth century was a well-established technique, found in works such as Mozart's C Major String Quartet, K. 156 (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).^[5] Beethoven also used the technique, for example in the finale of his Ninth Symphony but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven".^[6] It is clear from Schoenberg's own writings that he must have been aware of this practice.^[7]

Theory and compositional techniques

Principal forms of Webern's tone row from 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  Play (help·info).^[8]

Tone rows are designated by letters and subscript numbers (ex.: RI₁₁, also may appear 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.^[9]

P-6 tone row melody from Schoenberg's Op. 25, P-0 transposed up 6 semitones^[10]

A twelve-tone or serial 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.

I-0 tone row melody from Schoenberg's Op. 25, P-0 inverted^[10]

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:

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 "It is enough" (Es ist genug) from Bach's cantata no. 60, which opens with consecutive whole tones, is quoted literally in the woodwinds (mostly clarinet).

Mirror forms, P, R, I, and RI, of a tone row (from Schoenberg's Variations for Orchestra op. 31  Play (help·info)): "Called mirror forms because...they are identical.".^[11]

Some tone rows have a high degree of internal organisation. Here is the tone row from Anton Webern's Concerto Opus 24:

Webern's Concerto Op. 24 tone row,^[12] composed of four trichords: P RI R I

B, B♭, D, E♭, G, F♯, G♯, E, F, C, C♯, 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.

Webern's String Quartet Op. 28 tone row, composed of three tetrachords: P I RI, with P = the BACH motif.

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 chord 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]

First array of four aggregates (numbered 1-4 at bottom) from 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]

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 (one in italics and one in bold):

   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]

Pierre Boulez's Second Piano Sonata series  Play (help·info) 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.^[15]

Nonstandard tone rows

Prime form of five note tone row from Igor Stravinsky's In memoriam Dylan Thomas^[16]

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 which 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 Nones,^[17] symmetrical about the central tone with one note (D) repeated

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),^[18][19][20] as for example in his Requiem Canticles:

Basic row forms from Stravinsky's Requiem Canticles^[20]: P R I IR

Series from the second of Stockhausen's Klavierstücke I-IV  Play (help·info) which, "retained only the rudiments of the 12-note series."^[21]

Series from the third of Stockhausen's Klavierstücke I-IV  Play (help·info).^[21]

Ben Johnston uses a just tone row (see just intonation) in works including String Quartet No. 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  Play (help·info) and  Play hexachords (help·info).

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.

Tone row may also be used to describe other musical collections or scales such as in Arabic music.

• Musical set theory
• Unified field
• Side-slipping
• Pitch interval

Sources

1. ^ Arnold Whittall, The Cambridge Introduction to Serialism, Cambridge Introductions to Music (New York: Cambridge University Press, 2008): 2. ISBN 978-0-521-68200-8 (pbk).
2. ^ ^{a b} 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 9053567658. 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 9031302449.
4. ^ Discovery Reveals Bach's Postmodern Side. Weekend Edition Sunday, NPR, 6 Sep 2009.
5. ^ Hans Keller, "Strict Serial Technique in Classical Music", Tempo, New Series, no. 37 (Autumn, 1955): 12–24; citations on 14–21.
6. ^ Keller 1955, 22–23.
7. ^ Keller 1955, 23.
8. ^ Leeuw (2005), p.158.
9. ^ George Perle, Twelve-Tone Tonality, second edition, revised and expanded (Berkeley: University of California Press, 1996): 3. ISBN 0-520-20142-6.
10. ^ ^{a b} Whittall 2008, 5.
11. ^ Leeuw (2005), p.154. Italics original.
12. ^ Whittall 2008, 97.
13. ^ ^{a b c} Whittall 2008, 271.
14. ^ Perle 1977, 100; Perle 1996, 20.
15. ^ Leeuw (2005), p. 166.
16. ^ Whittall 2008, 127.
17. ^ Whittall 2008, 195.
18. ^ Claudio Spies, "Notes on Stravinsky's Abraham and Isaac", Perspectives of New Music 3, no. 2 (Spring-Summer 1965): 104–26. Citation on p. 118.
19. ^ Joseph N. Strauss, "Stravinsky's Serial 'Mistakes'", The Journal of Musicology 17, no. 2 (Spring 1999): 231–71, citation on 242.
20. ^ ^{a b} Whittall 2008, 139.
21. ^ ^{a b} Leeuw (2005), p.176-77.

External links

• "How Rare Is Symmetry in Musical 12-Tone Rows?" by David J. Hunter and Paul T. von Hippel.