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:
- 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".
- 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.
- 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.
- Invariance. We normally expect twenty-four distinct sets in
any set class: twelve that are equivalent under transposition (Tn)
and twelve more that are equivalent to the first twelve under inversion-plus-transposition
(TnI). 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 Tn
and under TnI. 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.
- "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.
Key concepts on this page:
- The pc set-class table
- Forte's set-class names
|Page last modified 25
2004 / GRT