Editing Derived row

{{Redirect|Partition (music)||Partition (disambiguation)}}
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'''.

==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.

==Partition and mosaic==
The opposite is partitioning, the use of methods to create segments from entire sets, most often through [[register (music)|registral]] difference.

In music using the [[twelve-tone technique]] a partition is "a collection of disjunct, unordered pitch-class sets that comprise an [[tone row#total chromatic|aggregate]]".{{sfn|Alegant|2001|p=2}} It is a method of creating segments from [[set (music)|sets]], most often through [[register (music)|registral]] difference, the opposite of derivation used in derived rows.

More generally, in musical set theory partitioning is the division of the domain of pitch class sets into types, such as transpositional type, see [[equivalence class]] and [[cardinality]].

Partition is also an old name for types of compositions in several parts; there is no fixed meaning, and in several cases the term was reportedly interchanged with various other terms.

A [[cross-partition]] is, "a two-dimensional configuration of pitch classes whose columns are realized as chords, and whose rows are differentiated from one another by registral, timbral, or other means."<ref name="Alegant 1">{{harvnb|Alegant|2001|p=1}}: "...more accurately described by ''permutation'' rather than ''rotation''. Permutations, of course, include the set of possible rotations."</ref> This allows, "''slot-machine'' transformations that reorder the vertical trichords but keep the pitch classes in their columns."<ref name="Alegant 1" />

A mosaic is "a partition that divides the aggregate into segments of equal size", according to Martino (1961).<ref>{{Cite journal | doi = 10.2307/843226 | last1 = Martino | first1 = Donald | author-link = Donald Martino | year = 1961 | title = The Source Set and its Aggregate Formations | jstor = 843226|journal=[[Journal of Music Theory]]| volume = 5 | issue = 2| pages = 224–273 }}</ref><ref name="Alegant 3">{{harvnb|Alegant|2001|p=3n6}}</ref> "Kurth 1992<ref>{{Cite journal | doi = 10.1525/mts.1992.14.2.02a00040 | last1 = Kurth | first1 = Richard | year = 1992 | title = Mosaic Polyphony: Formal Balance, Imbalance, and Phrase Formation in the Prelude of Schoenberg's Suite, Op. 25 |journal=[[Music Theory Spectrum]]| volume = 14 | issue = 2| pages = 188–208 }}</ref> and Mead 1988<ref>{{Cite journal | doi = 10.2307/833188 | last1 = Mead | first1 = Andrew | year = 1988 | title = Some Implications of the Pitch Class-Order Number Isomorphism Inherent in the Twelve-Tone System – Part One | jstor = 833188|journal=[[Perspectives of New Music]]| volume = 26 | issue = 2| pages = 96–163 }}</ref> use ''mosaic'' and ''mosaic class'' in the way that I use ''partition'' and ''mosaic''", are used here.<ref name="Alegant 3"/> However later, he says that, "the [[#Degree of symmetry|DS]] determines the number of distinct partitions in a ''mosaic'', which is the set of partitions related by transposition and inversion."<ref name="Alegant 2001, p.5">{{harvnb|Alegant|2001|p=5}}</ref>

===Inventory===
The first useful characteristic of a partition, an inventory, is the set classes produced by the [[union (set theory)|union]] of the constituent [[pitch class]] [[set (music)|sets]] of a partition.{{sfn|Alegant|2001|pp=3–4}} For [[trichord]]s and [[hexachord]]s combined see Alegant 1993, [[Milton Babbitt|Babbitt]] 1955, Dubiel 1990, Mead 1994, Morris and Alegant 1988, Morris 1987, and Rouse 1985.{{sfn|Alegant|2001|p=4}}

===Degree of symmetry===
{{See also|Set theory (music)#Symmetry}}

The second useful characteristic of a partition, the degree of symmetry (DS), "specifies the number of operations that preserve the unordered pcsets of a partition; it tells the extent to which that partition's pitch-class sets map into (or onto) each other under transposition or inversion."<ref name="Alegant 2001, p.5"/>

==References==
{{reflist}}
'''Sources'''
* {{cite journal|last=Alegant|first=Brian|date=Spring 2001|title=Cross-Partitions as Harmony and Voice Leading in Twelve-Tone Music|journal=[[Music Theory Spectrum]]|volume=23|number=1|pages=1–40}}

{{Twelve-tone technique}}

[[Category:Twelve-tone technique]]
[[Category:Anton Webern]]