Derived row

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For other uses, see partition.

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.

[edit] Derivation

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 transposed, 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:

B, Bb, D, Eb, G, F#, G#, E, F, C, C#, A

O 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.

[edit] Partition

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 method of creating segments from sets, most often through 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.