6. Set classes and interval-class content

What relates all the pc sets in a set class is that they are transpositional and/or inversional equivalents of each other. These, we must remember, are the two axiomatic properties that define a set class. There is also, however, another attribute that all sets in the same class share: they all have the same interval-class content. A set's interval class content is the complete inventory of the interval classes that the set contains.

Let's consider sets [1,2,3,6,7] and [10,11,2,3,4] from class (01256) again and take inventory of their interval classes. Each pc in a set forms an interval with every other pc, so a set with five pcs will have ten such intervals. In the tables below, we subtract the first integer from the second to measure the interval. We then list the classes to which these intervals belong.

 [1, 2, 3, 6, 7] ic 1 2 1 1 3 2 1 6 5 1 7 6 2 3 1 2 6 4 2 7 5 3 6 3 3 7 4 6 7 1
 [10, 11, 2, 3, 4] ic 10 11 1 10 2 4 10 3 5 10 4 6 11 2 3 11 3 4 11 4 5 2 3 1 2 4 2 3 4 1
We see that both of these sets have the same interval-class profile: each set features three instances of ic 1, two instances of ics 4 and 5, and one instance of ics 2, 3, and 6. These same interval classes are also found among the pcs of the other twenty-two sets in class (01256).

This shared ic profile is not quite as abstract as it might seem. A set's ic content tends to give the set a particular sound quality no matter how the set is disposed in the music. It gives all the sets in a class a similar quality. (Again, we know this from our experience with tonal music. What is it that makes all major triads sound much alike? Their interval-class content, which features single intervals of a m3/M6, M3/m6, and P4/P5 (ics 3, 4, and 5) and lacks any m2/M7, M2/m7, and tritone intervals (ics 1, 2, and 6). Diminished triads have a different ic profile and sound markedly different.)

Interval-class vectors

All sets in the same class have the same ic content (and hence the same sound quality). Sets in different classes usually have different ic contents (and different sound qualities). To compare ic profiles easily, we need a standard way of writing them, and for this purpose we use the interval-class vector. The ic vector is a simple array of the interval classes from 1 to 6, with a listing of how often each class is represented. For example, we've just seen that, in all sets of class (01256), there are three instances of ic 1, two instances of ics 4 and 5, and one instance of ics 2, 3, and 6. When we array these in an ic vector we get

 ics: 1 2 3 4 5 6 number of occurrences: 3 1 1 2 2 1

We commonly say, then, that set class (01256) has an interval-class vector of 311221. The digits represent occurrances of the ics from 1 to 6. The vector for the major triad mentioned above, which features only single intervals of ics 3, 4, and 5, and none of ics 1, 2, and 6, is 001110.

 Exercise 6-1. Interval-class content and ic vectors

The Z-relation

Now here's one of the curious facts about the world of pc set classes. What generally distinguishes sets of a particular class is a unique interval-class profile. In several cases, however, sets which cannot be mapped onto each other by transposition or inversion -- that is, sets of different classes -- nonetheless display the same profile! The sets of class (0123479) and those of class (0123568), for instance, all share a particular ic profile, 444342. Despite the fact that sets of these classes cannot be mapped onto each other, they do share the same sound quality. (By the way, you should already have found a pair of sets in Exercise 6-1 that share the same ic vector--and a curious vector at that!)

In the conventions of pc set analysis, these set classes are considered distinct (not equivalent), but closely related. For lack of a better label, the relation is called the Z-relation; sets classes (0123479) and (0123568) are said to be "Z-related". Remarkably, what seems to be a pretty abstract level of relatedness sometimes yields surprising concrete embodiments in atonal music.