7. Sizes of sets


How many notes we group into a segment is a decision we make in analyzing music. The number of different pcs in a set is another matter. The maximum number is, of course, twelve -- and there is obviously only one set containing all twelve pcs. Since some composers liked to avoid hinting at tonality by using all the pcs democratically, the aggregate, as the 12-pc set is called, can play a central role in atonal pitch structure. (It's also the set used in 12-tone music, where the order in which the pcs appear becomes important.)

Sets containing just 2 different pcs and those containing 10 or 11 also can be significant in pc set analysis, but most attention is usually given to sets with from 3 to 9 pcs. This is partly because there are enough different set classes with these numbers of pcs that classifying the sets we find becomes analytically worthwhile.

The number of pcs a set contains is called its cardinality. Below are listed the usual names for the sets of various cardinalities, along with the number of different set classes there are for each cardinality.

cardinality name number of distinct set classes
2 dyad 6   (the interval classes from 1 to 6)
3 trichord 12
4 tetrachord 29 (including one Z-related pair)
5 pentachord 38 (including three Z-related pairs)
6 hexachord 50 (including fifteen Z-related pairs)
7 septachord 38 (including three Z-related pairs)
8 octachord 29 (including one Z-related pair)
9 nonachord 12
10 decachord 6
11 undecachord    1
12 the aggregate 1
In all there are 208 distinct set classes of cardinalities 3 to 9.

You've probably noticed that the list above has a pattern. The number of trichord and nonachord classes, the number of tetrachord and octachord classes, and the number of pentachord and septachord classes is the same. This is not an accident but is a natural property of the 12-pc universe.


Key concepts on this page:

  • the aggregate set
  • cardinality
  • names for set cardinalities
Page last modified 8 August 2001 / GRT