7 PC Set Complements

7.1 Definitions

Literal Complement: When one PC Set contains all of the Pitch Classes not in some other PC Set.

Example: [0,1,4,7] and [2,3,5,6,8,9,10,11] are literal complements of each other

Abstract Complement: When two PC Sets would be complements of each other, except that one is transposed or inverted from the other. When someone says that a PC Set is the complement of some other PC Set, it usually means that they are Abstract Complements of each other.

Example: (0,1,4,7) and (0,1,2,3,5,6,8,9) are abstract complements of each other

The prime forms of abstract complements are listed side-by-side in the PC Set table found in the Appendix (except for the sets of 6 pitch classes).
Note that the Forte designation for a PC Set and it's complement will always have the same PC Set ID number (after the dash). For example, 4-18 and 8-18 are abstract complements of each other.

A Pitch Class Set and its complement will have very similar interval vectors.
In fact, there is a simple formula for computing the interval vector of a complement:

Note: If the original PC set has X pitch classes, it's complement will have (12-X) pitch classes, and the difference between the two will be: D = (12-X)-X = (12-X*2)
For example, if the original PC set has 5 pitches, the complement will have (12-5) pitches (i.e. 7 pitch classes) and the difference (D) between 5 and 7 is (12-5*2) = 2.

If the interval vector for the original PC Set is <I₁, I₂, I₃, I₄, I₅, I₆>
Then the interval vector for the complement will be:
<I₁+D, I₂+D, I₃+D, I₄+D, I₅+D, I₆+ (D/2) >
Note that the tritone is special because it divides the 12-tone chromatic scale exactly in half. For this reason, its interval vector grows by D/2.
Also note that D is always an even number {0, 2, 4, 6, 8, 10}, and so D/2 will always be an integer number (never a fraction).

4-18:(0,1,4,7) has 4 pitch classes and an interval vector of <102111>
It's complement is 8-18:(0,1,2,3,5,6,8,9)
8 - 4 = D = 4
The complement's interval vector is: <1+4, 0+4, 2+4, 1+4, 1+4, 1+(4/2)> = <546553>

Pentatonic Scale (5 Pitches) : <032140> ó Diatonic Scale (7 Pitches) : <254361>
Octatonic Scale (8 Pitches) : <448444> ó doubly-diminished 7^th chord (4 Pitches) : <004002>

The complement of a set with 6 Pitch Classes will itself have 6 Pitch Classes
Therefore, the difference in number of Pitch Classes is always 0 (zero).
Therefore, a 6-note complement will always have the same interval vector as it's complement!
True!

Since all PC Sets with 6 pitches have a complement with the same interval vector, there are only two ways that one of these PC Sets can be related to its complement:

The set is "self complementary", that is, the set and it's complement have the same prime form.
The set and its complement are Z-related: Two sets with the same interval vector but which can not be reduced to the same Prime Form by transposition or inversion.

PC Set complements are critically important when composing music with 12-tone rows, because:

This is one of the reasons why a 12-tone composition has a "built-in" harmonic cohesiveness.
For example, consider the following 12-tone row:

By definition, the last 6 notes of a 12-tone row are the PC Set complement of the first 6 notes
For more harmonic cohesiveness, make the first and last 6 notes of the row the same PC Set, i.e. self complementary and possibly inversionally related.
For more harmonic variety, make the first and last 6 notes of the row Z-related PC-sets.
This is the first step towards hexachordal combinatoriality: where a 12-tone row is made up of two similar halves, for example, where the 2^nd half is a transposed inversion of the first half (further discussion is beyond the scope of this tutorial). This is a favored technique of late Schoenberg compositions.