Editing Tone row (section)

==Theory and compositional techniques==
{{redirect|Prime (music)|the prime form|set (music)|the interval|unison}}
[[File:Webern - Piano Variations op. 27 tone row.png|thumb|center|upright=2|Principal forms of the tone row of [[Anton Webern]]'s [[Variations for piano (Webern)|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.{{sfn|Leeuw|2005|loc=158}}[[File:Webern - Piano Variations op. 27 tone row.mid]]]]

Tone rows are designated by letters and subscript numbers (e.g.: RI<sub>11</sub>, 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 [[Melodic inversion|inversion]], a forward-directed upside-down form.
* "R" indicates [[Retrograde (music)|retrograde]], a backwards right-side up form.
* "RI" indicates [[Retrograde inversion|retrograde-inversion]], a backwards upside-down form.
* [[Transposition (music)|Transposition]] is indicated by a ''T number'', for example P8 is a T(4) transposition of P4.{{sfn|Perle|1996|loc=3}}{{Explain|reason=|date=September 2019}}

A twelve-tone composition will take one or more tone rows, called the "prime form", as its basis plus their [[Transformation (music)|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 [[Suite for Piano (Schoenberg)|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 [[emancipation of the dissonance|emancipated]]. [[Alban Berg]], however, sometimes incorporated tonal elements into his twelve-tone works. The main tone row of his [[Violin Concerto (Berg)|Violin Concerto]] hints at this tonality:

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c' {
        \time 12/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        g1 bes d fis a c e gis b cis ees f
    }
}
</score>

This tone row consists of alternating minor and major [[Triad (music)|triads]] starting on the [[Open string (music)|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|''O Ewigkeit, du Donnerwort'', BWV 60]] is quoted literally in the woodwinds (mostly clarinet).

{{anchor|Mirror forms|Mirror form}}

Some tone rows have a high degree of internal organization. An example is the tone row from [[Anton Webern]]'s [[Concerto for Nine Instruments (Webern)|Concerto for Nine Instruments]] Op. 24, shown below.{{sfn|Whittall|2008|loc=97}}

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c'' {
        \time 3/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        b1 bes d
        es, g fis
        aes e f
        c' cis a
    }
}
</score>

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 [[Generating set of a group|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 (Webern)|String Quartet Op. 28]] is based on the [[BACH motif]] (B{{music|flat}}, A, C, B{{music|natural}}) and is composed of three [[tetrachord]]s:

:<score sound="1">
{
\override Score.TimeSignature
#'stencil = ##f
\override Score.SpacingSpanner.strict-note-spacing = ##t
  \set Score.proportionalNotationDuration = #(ly:make-moment 3/2)
    \relative c'' {
        \time 4/1
        \set Score.tempoHideNote = ##t \tempo 1 = 60
        bes1 a c b!
        dis e cis d
        ges, f aes g
    }
}
</score>

The "set-complex" is the forty-eight forms of the set generated by stating each "aspect" or transformation on each pitch class.<ref name="Perle Serial" />

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.<!--[[Secondary set]], [[Set complex]], and [[Weighted aggregate]] redirect directly here.-->{{anchor|Total chromatic}}

The <span id="total_chromatic">"total chromatic"</span><!--Preceding span is a link target; disturb only with care--> (or "aggregate") is the [[set (music)|set]] of all twelve [[pitch class]]es. An "array" is a succession of aggregates.<ref name="Whittall 271">{{harvnb|Whittall|2008|loc=271}}</ref> The term is also used to refer to [[lattice (music)|lattices]].

[[File:Array - Babbitt's Composition for Four Instruments.png|thumb|upright=2|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<ref name="Whittall 271"/>]]

An aggregate may be achieved through [[complement (music)|complementation]] or [[combinatoriality]], such as with [[hexachord]]s.

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.{{sfn|Perle|1977|loc=100}}{{sfn|Perle|1996|loc=20}} 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.{{sfn|Haimo|1990|loc=183}} It seems to have been first used in [[Milton Babbitt]]'s [[String Quartet No. 4 (Babbitt)|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.<ref>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}}.</ref>