# Chapter 7 – Pitch Class Set Complements

## 7.1Definitions

• 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.

## 7.2PC Set Complements and Their Interval Vectors

• 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 <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).
• 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 7th chord (4 Pitches) : <004002>

## 7.36-note complements

• 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. ## 7.4PC Set Complements Used in Twelve Tone Composition

• 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, 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 2nd 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.