Editing Derived row (section)

==Derivation==
Rows may be derived from a sub-[[Set theory (music)|set]] of any number of [[pitch class]]es that is a [[divisor]] of 12, the most common being the first three pitches or a [[trichord]]. This segment may then undergo [[Transposition (music)|transposition]], [[Melodic inversion|inversion]], [[Permutation (music)|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 (music)|invariance]]. For example, since a segment may be [[equivalence class (music)|equivalent]] to the generating segment inverted and transposed, say, 6 [[semitone]]s, 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 [[Anton Webern|Webern]]'s [[Concerto (Webern)|Concerto]], Op. 24:<ref>{{cite book|last=Whittall|first=Arnold|author-link=Arnold Whittall|year=2008|title=Serialism|series=Cambridge Introductions to Music|page=97|location=New York|publisher=Cambridge University Press|isbn=978-0-521-68200-8|type=pbk.}}</ref>

:<score sound="1" lang="lilypond">
{
\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>

[[File:Webern - Concerto Op. 24 tone row Boulez symmetry diagram.png|thumb|Symmetry diagram of Webern's Op. 24 row, after [[Pierre Boulez]] (2002).<ref>[[Daniel Albright|Albright, Daniel]] (2004). ''Modernism and Music'', p. 203. {{ISBN|0-226-01267-0}}.</ref>]]
[[File:Webern - Concerto Op. 24 tone row squares.svg|thumb|The mirror symmetry may clearly be seen in this representation of the Op. 24 tone row where each trichord (P RI R I) is in a rectangle and the axes of symmetry (between P & RI and R & I) are marked in red.]]

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 [[modular arithmetic|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.