Exhaustive chord progressions and their use in music composition

Abstract

A general approach for the design of mild chord progressions from the n-chords (unordered pitch-class sets of cardinal n) of a p-tonic scale was developed. Four relations between n-chords were considered: equivalence through transformation (transposition or inversion), parsimony (quasi-identity), mildness (equivalence or parsimony) and fuzziness (quasi-equivalence). The results showed that these relations are symmetrical and compatible with any transformation. Therefore, a parsimonious, mild or fuzzy progression of n-chords exists in 48 forms that keep its properties: direct, retrograde, inverted, retrograde inverted and their transpositions. Circular non-redundant exhaustive parsimonious progressions of the n-chords from a p-tonic scale were established for n = 2–5 and p = 2–9. The n-chords and their progressions can be represented in a two-dimensional table of the (n – 1)-chords of the p-tonic scale versus the set classes of its n-chords. Harmony and structural form can thus be deduced from the intrinsic properties of a scale.