Inmusic using thetwelve-tone technique,derivation is the construction of a row through segments. Aderived row is atone 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. Apartition is a segment created from a set throughpartitioning.
Rows may be derived from a sub-set of any number ofpitch classes that is adivisor of 12, the most common being the first three pitches or atrichord. This segment may then undergotransposition,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 isinvariance. For example, since a segment may beequivalent to the generating segment inverted and transposed, say, 6semitones, 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 atrichord taken fromWebern'sConcerto, Op. 24:[1]
P represents the original trichord, RI, retrograde and inversion, R retrograde, and I inversion.
The entire row, if B=0, is:
For instance, the third trichord:
is the first trichord:
backwards:
and transposed 6
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 throughregistral difference.
In music using thetwelve-tone technique a partition is "a collection of disjunct, unordered pitch-class sets that comprise anaggregate".[3] It is a method of creating segments fromsets, most often throughregistral 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, seeequivalence class andcardinality.
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.
Across-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."[4] This allows, "slot-machine transformations that reorder the vertical trichords but keep the pitch classes in their columns."[4]
A mosaic is "a partition that divides the aggregate into segments of equal size", according to Martino (1961).[5][6] "Kurth 1992[7] and Mead 1988[8] usemosaic andmosaic class in the way that I usepartition andmosaic", are used here.[6] However later, he says that, "theDS determines the number of distinct partitions in amosaic, which is the set of partitions related by transposition and inversion."[9]
The first useful characteristic of a partition, an inventory, is the set classes produced by theunion of the constituentpitch classsets of a partition.[10] Fortrichords andhexachords combined see Alegant 1993,Babbitt 1955, Dubiel 1990, Mead 1994, Morris and Alegant 1988, Morris 1987, and Rouse 1985.[11]
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."[9]
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