In music, a tone row or note row (German: Reihe or Tonreihe), also series and set, 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 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, 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). 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". It is clear from Schoenberg's own writings that he must have been aware of this practice.
Theory and compositional techniques 
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.
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:
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).
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.
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.
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 (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]
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. First used in Babbitt's String Quartet No. 4 (Babbitt). An aggregate may be vertically or horizontally weighted. An all-partition array is created by combining a collection of hexachordally combinatorial arrays.
Nonstandard tone rows 
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':
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.
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.
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.
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