tone row

P-0 tone row melody from Arnold Schoenberg's Op. 25 Minuet Trio opening^[1]

In music, a tone row or note row (German: Reihe or Tonreihe), also series and set^[2], refers to a non-repetitive ordering of the twelve notes (pitch-classes in musical set theory) of the chromatic scale. 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^[3] and, of course, 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).^[4] 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".^[5] It is clear from Schoenberg's own writings that he must have been aware of this practice. ^[6]

P-6 tone row melody from Schoenberg's Op. 25, P-0 transposed up 6 semitones^[7]

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^[7]

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).

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^[8], 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.

The set-complex is the forty-eight forms of the set generated by stating each "aspect" or transformation on each pitch class.^[2]

An all-interval row is a tone row arranged so that it contains one instance of each interval within the octave, 0 through 11. For example, the first all-interval row, devised by Fritz Heinrich Klein: F, E, C, A, G, D, A♭, D♭, E♭, G♭, B♭, C♭.^[9]

In integers, this row is represented as

0 e 7 4 2 9 3 8 t 1 5 6

with the interval between each note being

 e 8 9 t 7 6 5 2 3 4 1

This row was also used by Alban Berg in his Lyric Suite

The total chromatic (aggregate^[10]) is the set of all twelve pitch classes. An array is a succession of aggregates^[10].

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^[10]

Nonstandard tone rows

Prime form of five note tone row from Igor Stravinsky's In memoriam Dylan Thomas^[11]

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^[12], 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), as used in his Requiem Canticles:

Basic row forms from Stravinsky's Requiem Canticles^[13]: P R I IR

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

Sources

1. ^ Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music, p.2. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).
2. ^ ^{a b} Perle, George. 1977. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern, p.3. Fourth Edition. Berkeley, Los Angeles, and London: University of California Press. ISBN 0-520-03395-7.
3. ^ Discovery Reveals Bach's Postmodern Side. Weekend Edition Sunday, NPR, 6 Sep 2009.
4. ^ Hans Keller, "Strict Serial Technique in Classical Music", Tempo, New Series, no. 37 (Autumn, 1955): 12–24; citations on 14–21.
5. ^ Keller 1955, 22–23.
6. ^ Keller 1955, 23.
7. ^ ^{a b} Whittall 2008, p.5.
8. ^ Whittall 2008, p.97.
9. ^ Whittall 2008, p. 271 and 68-69.
10. ^ ^{a b c} Whittall 2008, p.271.
11. ^ Whittall 2008, p.127.
12. ^ Whittall 2008, p.195.
13. ^ Whittall 2008, p.139.

External links

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