The harmonic structure of a chord C composed of rational frequency ratios is studied from its tonal graph. The graph describes harmoncity/periodicity properties of the chord allowing to characterize the chord from several parameters, which are easily visualized by the chord spheroid. Relevant harmonic structures are analyzed, such as self-symmetric graphs associated with harmonically symmetric chords. Recurrence relationships for the lcm(C) in terms of the gcd’s of the powerset of C, and for the gcd(C) in terms of the lcm’s, are derived to unveil how the harmonic quotient lcm(C)/gcd(C) depends on the common undertones and overtones. In particular, for any number of tones, the harmonic quotient can be univocally expressed from the chord’s common undertones. Therefore, although undertones and overtones have not been explicitly taken into account in the current model, the harmonic quotient actually incorporates information about them.
Rafael Cubarsí (Wed,) studied this question.