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Encyclopedia > Derived set

In music using the twelve tone technique 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.

Rows may 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:

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.

The opposite is partitioning, the use of methods to create segments from entire sets, most often through registral difference.

Results from FactBites:

 Derived set (mathematics) - Wikipedia, the free encyclopedia (331 words) In mathematics a derived set is a construction in point-set topology that consists of taking the set of limit points of a given subset S of a topological space. Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. If we also require that the derived set of a set consisting of a single element be empty, the resulting spaces will be a T1 space.
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