Chapter 6 – Subsets and Supersets

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Any of the larger PC Sets can be divided into pieces. These pieces are, of course, also PC Sets in their own right. The smaller PC Sets are said to be "subsets" of the larger PC Set, which is the "superset".

  • For Example, the superset  (0,1,2,6,7,8) is quite dissonant and has the interval vector <420423>. It contains the following subsets:

Subsets 1:  =>  [0,2,7] + [1,6,8]           = two quintal/quartal triads

Subsets 2:  =>  [1,8] + [0,2,6,7]           = a simple fifth + a complex chord (a dominant+tonic sound)

Subsets 3:  =>  [1,7] + [0, 2, 6, 8]        = a tritone + whole-tone-scale fragment

Subsets 4:  => [6,7,8] + [0,1,2]            = two chromatic clusters

Subsets 5:  =>  [0,6] + [1,7] + [2,8]     = three tritone intervals

PC subsets and supersets are a very useful compositional technique. Be sure to explore all of the subsets for PC Sets that you use (see ComposerTools.com). This will help you to use, space, and manipulate your harmonies.

  • Other things to experiment with:
    • Use subsets for growth;  i.e. restrict sections of your music to use only portions of a larger PC set and then grow the PC set over time, making your harmonies denser and more complex.
    • Put the sub-sets in different registers to emphasize their unique sounds (see examples below).
    • Construct melodies from sub-sets which can be combined together to create

6.1          Definition: Transpositional Combination of Two Common Subsets

  • Transpositional combination: When a superset is created from two equal subsets, where one is transposed.
  • Example 1:
    [0,1,2] + [0,1,2]{transposed by 6 halfsteps}
    =>  [0,1,2] + [6,7,8]
    =>  [0,1,2,6,7,8]
  • Example 2:
    [0,2,7] + [0,2,7]{transposed by 6 halfsteps}
    =>  [0,2,7] + [6,8,1]
    =>  [0,1,2,6,7,8]

6.2          Definition: Inversional Combination of Two Common Subsets

  • Inversional combination:  When a superset is created from two equal subsets, where one is inverted (and possibly transposed)
  • Example:
    [0,1,6] + [0,1,6]{invert and transpose by 8 half steps}
    =>  [0,1,6] + [12-0+8, 12-1+8, 12-6+8]
    =>  [0,1,6] + [8,19,14]  =>  [0,1,6] + [8,7,2]  =>  [0,1,2,6,7,8]
  • Note:  The result of an inversional combination will always be "inversionally symmetric"  (see below for a discussion of inversional symmetry)

Copyright © 2004-2017 by Paul Nelson, all rights reserved.

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