6.11: Set Theory - Complements

The literal complement of a pitch-class set is every pitch not included in that set. For example, the complement of the seven-note C major scale is the five-note pentatonic scale: F-sharp, G-sharp, A-sharp, C-sharp, and D-sharp.

The complement of any n-chord is always a 12-n chord. Thus, a trichord’s compliment will be a nonachord, a tetrachords an octachord, and so on.

Below you’ll see the trichord [0,1,2]. Its literal complement is all of the notes not a part of it: [3,4,5,6,7,8,9,T,E]. When we put both of those pitch-class sets in prime form, the two are said to be abstract complement:

On the set-class list, _abstract complements _are listed next to one another, and they have a very interesting intervallic relationship, as you can see by comparing their IC vectors. Complementary set classes have a similar “distribution” of intervals. Below, you’ll see that the set (012345678) has exactly 6 more of each type of interval class than does its complement (012). That is, except for the tritone. It has only 3 more.

---