8. Set-class tables; Forte's set-class names

There is a limited number of set classes: 208 classes of sets with between 3 and 9 pcs. While gathering set data, analysts often keep a table of the classes handy. Such a table is included with this guide. Each entry in this table has up to five columns of information:

1. Forte's set-class names. Though prime forms are often used as class names, these alternative names have also become quite popular among theorists since they were first proposed in Allen Forte's The Structure of Atonal Music (1973), one of the pioneering texts of pc set analysis. In Forte's list, each set class has a hyphenated name (for example, 4-27). The first number is the cardinality of the set class (so trichords all begin with 3-, tetrachords with 4-, and so on). The second number is simply a listing number: the first class in Forte's list of tetrachords (0123) is "4-1," the next (0124) is "4-2", and so on. (The order in which Forte assigned his names is a bit different from the one used in our table. That's why some Forte numbers appear out of order in this table.) A convenient name for class (01256), then, is "class 5-6".

2. Prime forms. Within each of the cardinality groups, the table lists classes in ascending order of pc content [Note]. In some of the larger classes, letter "T" stands for pc 10.

3. Interval-class vector.One of the shared characteristics of all sets in a class is an identical interval-class content. The ic vector is a way of listing this content. The six number positions in the vector stand for ics 1 to 6. The numbers filling those positions show how many times that ic is represented in any set in the class.

4. Invariance. We normally expect twenty-four distinct sets in any set class: twelve that are equivalent under transposition (T_n) and twelve more that are equivalent to the first twelve under inversion-plus-transposition (T_nI). With 81 of the 208 classes, however, some of these twenty-four sets duplicate each other: they have the same (invariant) pc content. This column lists the number of invariant sets under T_n and under T_nI. The former number is always at least 1, since T₀ of a set is naturally invariant. Those classes with entries higher than 1 / 0 display some degree of intervallic symmetry; the greater the symmetry, the greater the amount of invariance among sets.

5. "Z-mates" While most set classes have unique interval-class vectors, some pairs of classes happen to share their ic profiles. By convention, these pairs are called "Z-related" classes, and their Forte names include the "Z". So hexachords 6-Z6 and 6-Z38 are distinct classes, but both share ic vector 421242. It's useful to have the Z-mate of such classes listed in the table, because one sometimes finds concrete embodiments of this seemingly abstract relation in atonal music.

EXERCISE 8-1. The Pc set-class table (not yet available)

Page last modified 25 September 2004 / GRT