ComposerTools.com / Theory / Pitch Class Sets / 7. PC Set Complements

*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:
- How many
**more**pitch classes does the complement have? Call this 'D'. - 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 <I
_{1}, I_{2}, I_{3}, I_{4}, I_{5}, I_{6}> - Then the interval vector for the complement will be:

<I_{1}+D, I_{2}+D, I_{3}+D, I_{4}+D, I_{5}+D, I_{6}+ (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).
- Example:
- 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>

- Some famous complements:
- 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:
- If you take any 12-tone row and divide it up into two pieces at any point, then
- the two pieces will have similar (or exactly the same) interval content.
- 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*, - 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.

Copyright © 2004 by Paul Nelson, all rights reserved.