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The Forte number is used to identify a set based on its prime form, and consists of the set's cardinal number, the number of elements in the set, and a number denoting its order in the listing created by A. Forte. The interval vector represents the interval content of a set: it represents the number of times the six interval classes (denoted by "ic") occur in a set, where an interval class is given by the difference between any two elements in the pitch-class set. The first number in an ic corresponds to the number of times ic1 occurs, the second number ic2, and so forth.

The transposition at level of a set is produced when an integer is added to each element of the set. Normal form and prime form give the most "compact" form of a set.

A set is in normal form if it is in order and the differences between the first element and each of the proceeding elements is minimized.

Prime form is the transposition of the normal form such that the first element of the set is 0.

The inversion of set is given by for each element of . Thus, , , and so on.

The matrix of a set is generated as follows: the set is represented in the first row and column, and the values of each row and column cell are added and adjusted modulo 12. It illustrates combinatorality, which is satisfied when has the same elements as the original set. is the transpose at level of the inverse of the original set. The integer is the number that appears in the matrix with a frequency equal to the cardinality of the original set.

Snapshot 1: any set of cardinality 12 has an "uninteresting" interval vector with 12 of each interval class except ic6, and a prime ordering which forms the chromatic scale

Snapshot 2: set 8-1 is combinatorial, as shown by the matrix

Snapshot 3: set 4-19 is not combinatorial; there is no element in the matrix that appears the same number of times as the cardinality of the set

References

[1] A. Forte, The Structure of Atonal Music, New Haven: Yale University Press, 1973.

[2] J. Tomlin. "All About Set Theory." (Jan 10, 2014) www.jaytomlin.com/music/settheory/help.html.