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 (Δ, capital delta, is often used to denote "difference"), 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¢-277¢-738¢-923¢) is close to being delta-rational 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-based 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 among the fundamentals and among lower harmonics of the fundamentals; 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 special cases of delta-rational chords, but in these chords all dyads are rationally related in frequency space, which we call fully delta-rational (FDR).

Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicing of JI chords stay JI. However, unlike the case with JI chords, there are mos tunings that can hit a DR chord exactly, if the chord's increment signature has two integer entries.

Notation

Increment signature

Some, but not all, delta-rational chords can be described with an increment signature which has integer ratios, i.e. a list of increments between successive notes, their ratios showing the simple rational relationships and ? for no relationship (or no obvious one). For example saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have almost equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer dyads. The example 13edo chord is approximately +1 +? +1. Note that this notation only considers dyads between successive notes, what we call increments.

If you have some sets of increments related to each other but not to other sets of increments, you could write the related sets with variables a, b, c or use one fewer letter by writing one set with positive integers without variables: an +a +b +a +b chord can also be written +1 +a +1 +a.

Fully delta-rational chords always have an increment signature with no ?'s.

Mathematical definitions

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.
A chord is increment-rational (IR) if it has an increment signature with some integers showing up. Not all PDR chords are increment-rational.

Finding approximate DR chords in edos

Some heuristics for finding delta-rational chords:

If a chord with a given step numbers in an edo is delta-rational it or similar numbers will usually also be delta-rational in nearby edos.

Incomplete list of approximate DR chords in small edos

(Chords whose range is < 1200¢ and only contains ratios of 1's and 2's between related dyads. Decide on error bound and search programmatically)

9edo

0-4-7 (+1 +1)

10edo

0-4-7 (+1 +1)

11edo

0-2-4-7 (+1 +? +2)
0-3-7 (+2 +3)
0-3-6-8 (+1 +? +1)
0-3-6-10 (+1 +? +2)
0-3-7-9 (+1 +? +1)
0-3-8-10 (+1 +? +1)
0-4-7 (+1 +1)
0-4-10 (+1 +2)
0-4-6-9 (+1 +1)
0-5-7 (+2 +1)
0-5-9 (+1 +1)
0-5-8-10 (+3 +? +2 (?))
0-6-9 (+3 +2)
0-6-10 (+1 +1)
0-6-8-10 (+2 +? +1)

13edo

0-2-7 (+1 +3)
0-2-11 (+1 +6)
0-2-4-7 (+1 +? +2)
0-2-4-11 (+1 +? +5)
0-2-5-8 (+1 +? +2)
0-2-6-10 (+1 +? +3)
0-2-7-9 (+2 +? +3)
0-2-7-11 (+1 +? +3)
0-2-8-10 (+2 +? +3)
0-3-8 (+1 +2)
0-3-10 (+1 +3)
0-3-7-9 (+1 +? +1)
0-3-7-11 (+1 +? +2)
0-3-8-10 (+1 +? +1)
0-3-9-11 (+1 +? +1)
0-4-10 (+1 +2)
0-4-6-8 (+3 +? +2)
0-4-6-9 (+1 +? +1)
0-4-7-9 (+3 +? +2)
0-4-7-10 (+1 +? +1)
0-5-7 (+2 +1)
0-5-9 (+1 +1)
0-5-7-9 (+2 +1 +1)
0-6-9 (+3 +2)
0-6-9-11 (+2 +? +1)

14edo

0-3-9-11 (+1 +? +1)
0-5-9 (+1 +1)

18edo

0-3-8 (+1 +2)
0-3-11-13 (+1 +? +1)
0-3-12-14 (+1 +? +1)
0-3-13-15 (+1 +? +1)
0-4-7-10-13 (+1 +? +1 +1)
0-5-7-11 (+1 +? +1)
0-5-9 (+1 +1)
0-5-13-16 (+1 +? +1)
0-5-14-17 (+1 +? +1)
0-6-11 (+1 +1)
0-6-11-15 (+1 +1 +1)
0-7-10 (+2 +1)
0-7-17 (+1 +2)
0-7-9-14 (+1 +? +1)
0-7-10-15 (+2 +1 +2)

21edo

0-3-6-11 (+1 +1 +2)
0-6-11-19 (+1 +1 +2)