![Set 3-1 has three possible versions: [01112T], [011TE1], and [0TT1E1], where subscripts indicate adjacency intervals. The normal form is the smallest "slice of pie" (shaded) or most compact form; in this case, [01112T]. Set theory 3-1 in the chromatic circle fix.svg](http://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Set_theory_3-1_in_the_chromatic_circle_fix.svg/500px-Set_theory_3-1_in_the_chromatic_circle_fix.svg.png)
This is a list of set classes , by Forte number. [1] In music theory, a set class (an abbreviation of pitch-class-set class) is an ascending collection of pitch classes, transposed to begin at zero. For a list of ordered collections, see this list of tone rows and series.
Contents
Sets are listed with links to their complements. For unsymmetrical sets, the prime form is marked with "A" and the inversion with "B"; sets without either are symmetrical. Sets marked with a "Z" refer to a pair of different set classes with identical interval class content unrelated by inversion, with one of each pair listed at the end of the respective list when they occur. ("Z" is derived from the prefix zygo-, from Ancient Greek ζυγόν, "yoke". Hence: zygosets.) "T" and "E" are conventionally used in sets to notate ten and eleven, respectively, as single characters. Since, for any given set, its interval-class vector is independent of the version (cyclic permutation) considered, for any cardinality the ordering of sets in the list (except for Z-related sets, as explained below) is based on the string of numerals in the interval vector treated as an integer, decreasing in value, following the strategy used by Forte in constructing his numbering system.
There are two slightly different methods of obtaining the prime form—an earlier one by Allen Forte and a later but now generally more popular one by John Rahn—both often confusingly described as "most packed to the left". However, a more precise description of the Rahn spelling is to select the version most dispersed from the right, whereas the precise description of the Forte spelling is to select the version most packed to the left within the smallest span. [a] In the lists here, the Rahn spelling is used for the 17 out of 352 set classes where the two methods yield different results; the alternative Forte spellings are listed in the footnotes. [3] [4]
Before either (1960–67), Elliott Carter had produced a numbered listing of pitch class sets, or "chords", as Carter referred to them, for his own use. [5] [6] Donald Martino had produced tables of hexachords, tetrachords, trichords, and pentachords for combinatoriality in his 1961 article, "The Source Set and its Aggregate Formations". [7]
The magnitude of the difference between the interval-class vector of a set and that of its complement is ⟨X, X, X, X, X, X/2⟩, where (in base-ten) X = 12 – 2C, where C is the smaller set's cardinality. In nearly all cases, complements of unsymmetrical sets are related by inversion—i.e., the complement of an "A" version of a set of cardinality C is (usually) the "B" version of the respective complementary set of cardinality 12 – C. The most significant exceptions are the sets 4-14/8-14, 5-11/7-11, and 6-14, which are all closely related in terms of subset/superset structure.
According to Allen Forte's own rule for numbering (for sets sharing the same interval content, the prime form is the version that is most packed to the left within the smallest span), this rule is also usually applied to determine which zygote appears in the main list and which appears added at the end (with a much larger index number). For example, of the two all-interval tetrads, 4-Z15 and 4-Z29, the former has a minimum span of 6 semitones and the latter a minimum span of 7 semitones. Similarly, 5-Z12 has a minimum span of 6 semitones and 5-Z36 a minimum span of 7 semitones. Sets 5-Z17 and 5-Z37 both have a minimum span of 8 semitones, but 5-Z17 is more packed to the left. Sets 8-Z15 and 8-Z29 both have a minimum span of 9 semitones, but 8-Z15 is more packed to the left. With one clear mistake on Forte's part (noted below regarding 6-Z28 and 6-Z49), this rule is otherwise applied to the hexads, as well.
However, 7-Z12 (the complement of 5-Z12) has a minimum span of 9 semitones and 7-Z36 (the complement of 5-Z36) a minimum span of 8 semitones—which is the reverse of the above rule. Also, 7-Z17 (the complement of 5-Z17) has a minimum span of 9 semitones and 7-Z37 (the complement of 5-Z37) a minimum span of 8 semitones—which also violates the rule. Similarly, set 7-Z18 (the complement of 5-Z18) has a minimum span of 9 semitones and 7-Z38 (the complement of 5-Z38) a minimum span of 8 semitones—again the reverse of the rule. These heptads have clearly been assigned index numbers corresponding to their pentad complements instead of following the general rule applied in (most of the) other cases.
The hexad anomaly in Allen Forte's book concerns the numbering of the pair 6-Z28, [0,1,3,5,6,9], and 6-Z49, [0,1,3,4,7,9]. They both have the same span, but, within that span, the hexad [0,1,3,4,7,9] is clearly more packed to the left than [0,1,3,5,6,9] and therefore, according to Forte's own rule, the set [0,1,3,4,7,9] should have been assigned the lower number 6-Z28, with [0,1,3,5,6,9] given the higher number 6-Z49.
To avoid confusion, the original Forte numbering system is retained here.
