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. • "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). These forms may be used to construct a melody in a straightforward manner as in the fifth movement from Schoenberg's
Piano Suite, Op. 25, 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.'s
Variations for Orchestra, Op. 31, "Called
mirror forms because... they are identical".|center|700x700px 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: : { \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 } } 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. : { \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 } } 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: : { \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 } } The "set-complex" is the forty-eight forms of the set generated by stating each "aspect" or transformation on each pitch class. The term is also used to refer to
lattices. 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. 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. 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. == Examples ==