ComposerTools.com
/ Theory / Pitch Class Sets / 6. Subsets
and Supersets

# 6
Subsets and Supersets

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 by Paul Nelson, all rights reserved.