Set Class and Prime Form

A set class is the family of all pc sets that can be related to one another by transposition (T_n) or inversion followed by transposition (T_nI). For instance, every major triad — C major, F♯ major, B♭ major, and so on — belongs to the same set class, as does every minor triad. Identifying a set's class allows analysts to recognize deep structural similarities across passages that look or sound superficially different.

Prime form is the single representative chosen to name the set class: it is the most compact normal order, packed as far to the left as possible. To derive prime form, compute the normal order of the set and the normal order of its inversion, then choose whichever begins with 0 and has the smallest integers reading left to right. Shift all elements so the first is 0 by subtracting the first element mod 12. The result — a string of integers starting at 0 — is the prime form, written in parentheses without commas: (0, 1, 4) or (0, 3, 7).

Allen Forte catalogued all prime forms and assigned each a Forte name (e.g., 3-11 for the set class containing major and minor triads, where 3 is the cardinality and 11 is an index number). Forte names appear widely in analytical literature. Prime form and Forte names together give analysts a precise, standard shorthand for discussing any collection of pitch classes, regardless of register, spelling, or ordering in the score.