12. Inclusion relations: subsets and supersets.

Suppose in our analysis of a piece we have identified the sets given in Example 10-1 below. Can we perceive any relationships among the sets of segments a), b) and c)?

Example 12-1

The relationship of the overall set of segment a) to that of segment b) is a close and obvious one: b) is a subset of a), and a) is a superset of b). That is, set [10,1,2,4], is wholly included within the larger set [1,2,4,7,10]. We can also see that the upper voice of both segments features a shared subset, trichord [10,1,2]. Both [10,1,2,4] and [1,2,4,7,10] are supersets of [10,1,2]. This subset-superset relation is sometimes called the inclusion relation.

Now set c), [5,7,8,11], is clearly not a subset of set a); the two only share a single pitch class. It has, however, a more abstract class inclusion relation. That is, set class 4-12, of which [5,7,8,11] is a member, is a subset of class 5-31, of which [1,2,4,7,10] is a member. (Of course, [5,7,8,11] is also a class equivalent of [10,1,2,4]; both are members of class 4-12. The same can be said of their trichordal subsets, [7,8,11] and [10,1,2], both members of class 3-3.)

Both sets and set classes obviously have many possible subsets and supersets--for example, any pentachord has five tetrachordal and ten trichordal subsets--and most will probably not be musically significant in any given piece of music. At the set level, you should be sure that your segmentations make sense, that any subset-superset relations you identify have real, not contrived, musical importance.

Class inclusion relations might seem too abstract ever to have concrete musical importance. However, class subsets and supersets do share a familial resemblance based on interval-class content. We may be able to group some or many of the sets we find in a piece into families based on inclusion ties. For instance, we might find with more analysis that the piece from which the above segments are taken uses the octatonic scale as its basis for pitch structure. After all, classes 3-3, 4-12, and 5-31 are all subsets of the octatonic class 8-28. (In fact, the combined pc material of sets a), b) and c) is set 8-28 [1,2,4,5,7,8,10,11].)

At other times, shared class subsets may be used to associate two or more families of material. In some of Stravinsky's music, for example, octatonic (class 8-28) and diatonic (class 7-35) materials appear to be linked through shared subsets like the tonal triad (class 3-11), and dominant-seventh-type tetrachord (class 4-27). In fact, the prominence of subsets traditionally associated with diatonicism is one feature that made the octatonic scale attractive to several twentieth-century composers.

Page last modified 10 July 2001 / GRT