Delta-rational chord

A delta-rational (DR) chord is a chord with dyads which are close to having simple integer ratios between frequency differences of dyads (hence the "delta" part of the name), but not necessarily integer ratios between frequencies of notes, as in JI chords. For example, the 13edo chord 0\13-3\13-8\13-10\13 (0¢-185¢-738¢-923¢) approximates the property because the dyad 8\13-10\13 in the chord has a frequency difference 0.994 times the frequency difference of the dyad 0\13-3\13. (In 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes would have exactly the same frequency difference as the dyad 0\13-3\13.) Delta-rational chords provide a non-JI approach to concordance, since chords that are delta-rational with simple ratios between dyads are more concordant than other chords. This acoustic effect is thought to be caused by synchronized interference beating between the fundamentals and lower harmonics of the fundamental; the effect may be more or less pronounced depending on register, timbre, the complexity of the linear relationship, etc. For example, the delta-rational acoustic effect would be weaker in chords with very spaced-out voicing, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious).

JI chords and chords that are subsets of isodifferential chords (these correspond to all chords of the form α : α + k₁ : ... : α + k_n for any positive number α and integers k₁, ..., k_n) are a special case of delta-rational chords, but in these chords all dyads are rationally related in frequency space.

Mathematical definition

Mathematically, a chord C = α₁:...:α_n is delta-rational (DR) or partially delta-rational (PDR) when the chord has two distinct dyads α_k1:α_k2 and α_k3:α_k4 such that (α_k2 − α_k1)/(α_k4 − α_k3) is rational. When all dyads are linearly related, i.e. when the chord is of the form (α + k₁):...:(α + k_n), we call the chord fully delta-rational (FDR). In practice these terms can loosely refer to approximations of mathematically PDR and FDR chords.