Derived row

In music using the twelve-tone technique, derivation is the construction of a row through segments. A derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern often used derived rows in his pieces. A partition is a segment created from a set through partitioning. Rows may be derived from a sub-set of any number of pitch classes that is a divisor of 12, the most common being the first three pitches or a trichord. This segment may then undergo transposition, inversion, retrograde, or any combination to produce the other parts of the row (in this case, the other three segments). One of the side effects of derived rows is invariance. For example, since a segment may be equivalent to the generating segment inverted and transposed, say, 6 semitones, when the entire row is inverted and transposed six semitones the generating segment will now consist of the pitch classes of the derived segment. Here is a row derived from a trichord taken from Webern's Concerto, Op. 24: { \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 } } P represents the original trichord, RI, retrograde and inversion, R retrograde, and I inversion. The entire row, if B=0, is: 0, 11, 3, 4, 8, 7, 9, 5, 6, 1, 2, 10. For instance, the third trichord: 9, 5, 6 is the first trichord: 0, 11, 3 backwards: 3, 11, 0 and transposed 6 3+6, 11+6, 0+6 = 9, 5, 6 mod 12. Combinatoriality is often a result of derived rows. For example, the Op. 24 row is all-combinatorial, P0 being hexachordally combinatorial with P6, R0, I5, and RI11. The opposite is partitioning, the use of methods to create segments from entire sets, most often through registral difference. In music using the twelve-tone technique a partition is "a collection of disjunct, unordered pitch-class sets that comprise an aggregate".