An Algorithm for the Direct Generation of Set Class Representatives in Any Pitch Class Space

Abstract

When spaces with greater than 20 pitch classes are considered, the problem of generating equivalence classes (set classes) with respect to operations like transposition and inversion becomes increasingly difficult. The brute force approach of enumerating all possible pitch class sets and ignoring those that fall into already selected set classes becomes computationally intractable. Some improvements can be made using Read’s Orderly Algorithm, and the essential features are seen to be the binary coding of pitch class sets and an augmentation operation. A further refined algorithm is described that makes use of a stack technique to directly generate, straight to a text file, all equivalence class representatives of given cardinality within any pitch class universe. This supports the mathematical and compositional exploration of much larger pitch class spaces than hitherto.