A Directional Interval Class Representation of Chord Transitions

Chords are commonly represented, at a low level, as absolute pitches (or pitch classes) or, at a higher level, as chords types within a given tonal/harmonic context (e.g. roman numeral analysis). The former is too elementary, whereas, the latter, requires sophisticated harmonic analysis. Is it possible to represent chord transitions at an intermediate level that is transposition-invariant and idiom-independent (analogous to pitch intervals that represent transitions between notes)? In this paper, a novel chord transition representation is proposed. A harmonic transition between two chords can be represented by a Directed Interval Class (DIC) vector. The proposed 12-dimensional vector encodes the number of occurrence of all directional interval classes (from 0 to 6 including +/for direction) between all the pairs of notes of two successive chords. Apart from octave equivalence and interval inversion equivalence, this representation preserves directionality of intervals (up or down). Interesting properties of this representation include: easy to compute, independent of root finding, independent of key finding, incorporates voice leading qualities, preserves chord transition asymmetry (e.g. different vector for I→V and V→I), transposition invariant, independent of chord type, applicable to tonal/post-tonal/ atonal music, and, in most instances, constituent chords from a chord transition can be uniquely derived from a DIC vector. DIC vectors can be organised in different categories depending on their content, and distance between vectors can be used to calculate harmonic similarity between different music passages. Preliminary tests are presented using simple tonal chord sequences and jazz sequences. This proposal provides a simple and potentially powerful representation of elementary harmonic relations that may have interesting applications in the domain of harmonic representation and processing.