Delta-rational chord

A delta-rational (DR) chord is a chord that has integer ratios between frequency differences of some pair of dyads, called deltas, with the dyads in question assumed to be between successive notes (Δ, capital delta, is often used to denote "difference").

DR chords generalize JI chords, in which all frequency differences of dyads are exactly integer ratios. But unlike JI chords, a DR chord need not have integer ratios between frequencies of notes. For example, the 13edo chord 0-3-8-10\13 (0¢-277¢-738¢-923¢) is close to being delta-rational, because the frequency difference of the dyad 8-10\13 is 0.994 times the frequency difference of the dyad 0-3\13. (In the exactly DR chord 0\13-3\13-8\13-924.159¢, the 3rd and 4th notes have exactly the same frequency difference as the dyad 0-3\13.)

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 (possibly irrational) 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 either fully delta-rational (FDR) or linear.

Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple ratios between dyads (when measured as absolute frequency differences) tend to be perceived as 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 is expected to be weaker in chords with wider voicings, as well as chords played in timbres with loud higher harmonics (because the higher harmonics would make the delta-rational relationships less obvious). The justification for only considering dyads between adjacent notes is that the resulting notes within the dyads could psychoacoustically interfere with the beating of the dyads.

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 most JI chords, a 2/1-equivalent MOS scale can tune a DR chord exactly, provided that the chord's delta signature specifies two integer entries.

Denoting a delta-rational chord

Delta signature

A delta-rational chord is determined by two things:

• the dyad formed by its lowermost two notes;
• its delta signature which has integer ratios, i.e. a list of (scaled) frequency increases between successive notes, their ratios showing the simple rational relationships, with a + before each increase. Note that it is whether the deltas are rationally related to each other that defines DR, not whether the deltas are related to the frequency of the root. If we divide every term by the first term to make the first term 1, the result is called a normalized delta signature.
• Two delta signatures are equivalent if one can be obtained from the other by scaling by a positive real number. For example, +2+e+3 is equivalent to +2φ+eφ+3φ, and both signatures imply a delta-rational chord.

For example, a chord with a +1+2+1 delta signature is a:(a+1):(a+3):(a+4) for some possibly irrational a.

Deltas that are free, i.e. not required to be related to any other deltas are indicated with +?. 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.

If you have some sets of deltas 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 +c +1 +c where c = b/a.

Fully delta-rational chords always have a delta signature with no irrational ratios between terms.

Mathematical definitions

1. 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 the real intervals (α_k1, α_k2) and (α_k3, α_k4) are disjoint and (α_k2 − α_k1)/(α_k4 − α_k3) is rational. Equivalently, a chord is delta-rational if it has a delta signature with some integers showing up.
2. When all dyads are linearly related, equivalently when the chord has a delta signature with all entries integers, we call the chord fully delta-rational (FDR) or linear.
3. A chord that has a delta signature with all entries +1 is called isodifferential.

Due to the aforementioned equivalence of delta signatures under scaling, delta signatures of n terms are really elements of a projective space [math]\displaystyle{ \mathbb{R}\mathbf{P}^{n-1}; }[/math] they are specifically in the subset that is the image of the all-positive orthant of [math]\displaystyle{ \mathbb{R}^n. }[/math]

In practice these terms can loosely refer to approximations of mathematically exact PDR and FDR chords, for example in edo tunings.

Finding a tuning of a MOS scale with an exact DR chord

Let a, b be positive integers and suppose gcd(a, b) = 1. Let E > 1 be the frequency ratio of the equave. Consider a MOS aLbs⟨E⟩ with generator range [math]\displaystyle{ I \subseteq (1, \sqrt{E}) }[/math] (in the linear frequency domain), and consider a pair (u, v) of notes from the root of a given triad in the MOS, 0 (unison) < u < v. Let p, g be a basis formally representing the MOS scale's period and generator. Write

[math]\displaystyle{ \begin{align} \mathbf{u} &= u_p \mathbf{p} + u_g \mathbf{g} \\ \mathbf{v} &= v_p \mathbf{p} + v_g \mathbf{g} \end{align} }[/math]

as elements of [math]\displaystyle{ \mathbb{Z}^2\langle \mathbf{p}, \mathbf{g}\rangle }[/math].

Define the rational function [math]\displaystyle{ r_{\mathbf{u}, \mathbf{v}} : I \to (0,\infty) }[/math] by

[math]\displaystyle{ \displaystyle{r_{\mathbf{u}, \mathbf{v}}(x) = \frac{E^{v_p}x^{v_g}- E^{u_p}x^{u_g}}{E^{u_p}x^{u_g} - 1} }. }[/math]

Then, provided that the positive rational number [math]\displaystyle{ m/n }[/math] lies in the image [math]\displaystyle{ r_{\mathbf{u}, \mathbf{v}}(I), }[/math] we can solve for the frequency ratio [math]\displaystyle{ g \in I }[/math] that satisfies [math]\displaystyle{ r_{\mathbf{u}, \mathbf{v}}(g) = m/n, }[/math] making the specified chord (0, u, v) a +n+m DR chord.

The existence of an exact tuning for a delta signature specification is only guaranteed to hold when we only care about a ratio between two terms in the delta signature being exact. If we want to optimize an arbitrary specified delta signature (with some deltas possibly held free), we can use a least-squares-error solution instead to minimize the error.

Least-squares error measures

Say we want the error of a chord 1:r₁:r₂:...:r_n (in increasing order), with n > 1, in the linear domain as an approximation to a fully delta-rational chord with signature +δ₁ +δ₂ ... +δ_n (where the delta signature is written based on the chord written to have root 1), i.e. a chord

[math]\displaystyle{ 1 : 1 + \delta_1 : \cdots : 1 + \sum_{l=1}^n \delta_l }[/math]

We can replace the 1 with x, vary x and ask, "By at least how much do the deltas have to be off for any x?"

Naive least-squares error

Rewriting a bit, if 1:r₁:r₂:...:r_n has delta signature +ε₁ +ε₂ ... +ε_n (where the chord is 1:1+ε₁:...), let [math]\displaystyle{ D_i = \sum_{k=1}^i \delta_i }[/math] and [math]\displaystyle{ E_i = \sum_{k=1}^i \epsilon_i. }[/math] Then the resulting linear least-squares optimization problem is

[math]\displaystyle{ \displaystyle{ \min_x \sqrt{\sum_{i=1}^n \Bigg( E_ix - D_i \Bigg)^2 } } }[/math]

with solution

[math]\displaystyle{ x = \displaystyle{\frac{\sum_{i=1}^n D_i E_i}{\sum_{i=1}^n E_i^2},} }[/math]

which can be plugged back into the error formula to obtain the error. (We multiply the 1:r₁:r₂:...:r_n chord by x in order to compare it to the target DR chord on the same isodifferential series.)

This error measure is called naive least-squares error (NLS error). This error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.

This error measure was found by Inthar and groundfault.

Symmetric least-squares error

With the same assumptions as above (D₁, ..., D_n and E₁, ..., E_n two lists of cumulative deltas), the symmetric least-squares error (SLS error) is found by solving both

[math]\displaystyle{ \displaystyle{ \min_x \sqrt{ \sum_{i=1}^n \Bigg( E_ix - D_i \Bigg)^2 }} }[/math]

and

[math]\displaystyle{ \displaystyle{ \min_x \sqrt{ \sum_{i=1}^n \Bigg( D_ix - E_i \Bigg)^2 }} }[/math]

and then adding the squares of the solutions and taking the square root. It amounts to solving

[math]\displaystyle{ \displaystyle{ \min_{(x, y) \in \mathbb{R}^2} \sqrt{ \sum_{i=1}^n \Bigg( E_ix - D_i \Bigg)^2 + \sum_{i=1}^n \Bigg( D_iy - E_i \Bigg)^2 }} }[/math] since the Hessian of this function at the minimum is positive.

It satisfies SLSE({D_i}, {E_i}) = SLSE({E_i}, {D_i}).

Open problems

Find a version of the symmetric least-squares error that forms a metric on the set of delta signatures with n terms.

DR triads in small edos

Shown below are approximate DR triads in edos < 24 with symmetric least-squared error < 0.05. All delta signatures are in the 3-integer limit.

edo	steps	delta signature	naive LS error	symmetric LS error
6	0, 2, 5	+1+2	0.0025	0.0026
6	0, 4, 5	+3+1	0.0043	0.0044
7	0, 1, 2	+1+1	0.0447	0.0449
7	0, 2, 5	+1+2	0.0242	0.0247
7	0, 4, 5	+3+1	0.0273	0.0277
7	0, 4, 6	+3+2	0.0046	0.0047
8	0, 1, 2	+1+1	0.0391	0.0392
8	0, 4, 7	+1+1	0.0060	0.0065
8	0, 2, 5	+1+2	0.0442	0.0450
8	0, 5, 7	+2+1	0.0416	0.0432
8	0, 4, 5	+3+1	0.0443	0.0447
8	0, 4, 6	+3+2	0.0322	0.0325
9	0, 1, 2	+1+1	0.0347	0.0348
9	0, 4, 7	+1+1	0.0089	0.0094
9	0, 5, 7	+2+1	0.0231	0.0238
10	0, 1, 2	+1+1	0.0312	0.0313
10	0, 4, 7	+1+1	0.0207	0.0217
10	0, 5, 9	+1+1	0.0392	0.0427
10	0, 2, 7	+1+3	0.0463	0.0468
10	0, 5, 7	+2+1	0.0085	0.0087
10	0, 6, 9	+3+2	0.0195	0.0197
11	0, 1, 2	+1+1	0.0283	0.0284
11	0, 4, 7	+1+1	0.0303	0.0315
11	0, 5, 9	+1+1	0.0265	0.0283
11	0, 6, 10	+1+1	0.0415	0.0454
11	0, 3, 8	+1+2	0.0454	0.0465
11	0, 4, 10	+1+2	0.0193	0.0202
11	0, 2, 7	+1+3	0.0301	0.0304
11	0, 5, 7	+2+1	0.0034	0.0034
11	0, 8, 10	+3+1	0.0106	0.0108
11	0, 6, 9	+3+2	0.0088	0.0089
12	0, 1, 2	+1+1	0.0260	0.0260
12	0, 4, 7	+1+1	0.0383	0.0395
12	0, 5, 9	+1+1	0.0159	0.0168
12	0, 3, 8	+1+2	0.0321	0.0327
12	0, 4, 10	+1+2	0.0025	0.0026
12	0, 2, 7	+1+3	0.0165	0.0166
12	0, 3, 10	+1+3	0.0310	0.0316
12	0, 2, 3	+2+1	0.0489	0.0490
12	0, 5, 7	+2+1	0.0132	0.0133
12	0, 8, 11	+2+1	0.0125	0.0130
12	0, 8, 10	+3+1	0.0043	0.0044
12	0, 6, 9	+3+2	0.0322	0.0325
13	0, 1, 2	+1+1	0.0240	0.0240
13	0, 2, 4	+1+1	0.0481	0.0485
13	0, 4, 7	+1+1	0.0451	0.0462
13	0, 5, 9	+1+1	0.0070	0.0074
13	0, 7, 12	+1+1	0.0087	0.0095
13	0, 3, 8	+1+2	0.0207	0.0210
13	0, 4, 10	+1+2	0.0119	0.0122
13	0, 5, 12	+1+2	0.0215	0.0225
13	0, 2, 7	+1+3	0.0049	0.0050
13	0, 3, 10	+1+3	0.0145	0.0147
13	0, 2, 3	+2+1	0.0451	0.0452
13	0, 5, 7	+2+1	0.0214	0.0217
13	0, 8, 11	+2+1	0.0005	0.0005
13	0, 8, 10	+3+1	0.0167	0.0170
13	0, 8, 12	+3+2	0.0275	0.0279
14	0, 1, 2	+1+1	0.0222	0.0223
14	0, 2, 4	+1+1	0.0447	0.0449
14	0, 5, 9	+1+1	0.0006	0.0006
14	0, 6, 11	+1+1	0.0401	0.0426
14	0, 7, 12	+1+1	0.0186	0.0201
14	0, 1, 3	+1+2	0.0468	0.0469
14	0, 3, 8	+1+2	0.0110	0.0111
14	0, 4, 10	+1+2	0.0242	0.0247
14	0, 5, 12	+1+2	0.0363	0.0377
14	0, 2, 7	+1+3	0.0050	0.0050
14	0, 3, 10	+1+3	0.0002	0.0002
14	0, 4, 13	+1+3	0.0294	0.0301
14	0, 2, 3	+2+1	0.0418	0.0419
14	0, 5, 7	+2+1	0.0284	0.0287
14	0, 8, 11	+2+1	0.0116	0.0119
14	0, 4, 9	+2+3	0.0461	0.0464
14	0, 7, 9	+3+1	0.0393	0.0397
14	0, 8, 10	+3+1	0.0273	0.0277
14	0, 8, 12	+3+2	0.0046	0.0047
15	0, 1, 2	+1+1	0.0208	0.0208
15	0, 2, 4	+1+1	0.0417	0.0419
15	0, 5, 9	+1+1	0.0071	0.0074
15	0, 6, 11	+1+1	0.0319	0.0336
15	0, 7, 12	+1+1	0.0273	0.0291
15	0, 8, 14	+1+1	0.0149	0.0164
15	0, 1, 3	+1+2	0.0437	0.0437
15	0, 3, 8	+1+2	0.0025	0.0025
15	0, 4, 10	+1+2	0.0349	0.0355
15	0, 2, 7	+1+3	0.0137	0.0137
15	0, 3, 10	+1+3	0.0122	0.0123
15	0, 4, 13	+1+3	0.0134	0.0137
15	0, 2, 3	+2+1	0.0390	0.0391
15	0, 5, 7	+2+1	0.0345	0.0348
15	0, 7, 10	+2+1	0.0415	0.0423
15	0, 8, 11	+2+1	0.0211	0.0216
15	0, 3, 7	+2+3	0.0496	0.0498
15	0, 4, 9	+2+3	0.0294	0.0296
15	0, 5, 11	+2+3	0.0356	0.0359
15	0, 7, 9	+3+1	0.0304	0.0306
15	0, 8, 10	+3+1	0.0364	0.0368
15	0, 8, 12	+3+2	0.0151	0.0152
16	0, 1, 2	+1+1	0.0195	0.0195
16	0, 2, 4	+1+1	0.0391	0.0392
16	0, 5, 9	+1+1	0.0129	0.0132
16	0, 6, 11	+1+1	0.0248	0.0259
16	0, 7, 12	+1+1	0.0348	0.0368
16	0, 8, 14	+1+1	0.0060	0.0065
16	0, 9, 15	+1+1	0.0373	0.0410
16	0, 1, 3	+1+2	0.0410	0.0410
16	0, 3, 8	+1+2	0.0049	0.0050
16	0, 4, 10	+1+2	0.0442	0.0450
16	0, 5, 13	+1+2	0.0386	0.0398
16	0, 6, 15	+1+2	0.0257	0.0268
16	0, 2, 7	+1+3	0.0213	0.0213
16	0, 3, 10	+1+3	0.0231	0.0233
16	0, 4, 13	+1+3	0.0007	0.0008
16	0, 2, 3	+2+1	0.0365	0.0366
16	0, 5, 7	+2+1	0.0398	0.0401
16	0, 7, 10	+2+1	0.0334	0.0340
16	0, 8, 11	+2+1	0.0294	0.0300
16	0, 10, 14	+2+1	0.0416	0.0432
16	0, 11, 15	+2+1	0.0010	0.0010
16	0, 3, 7	+2+3	0.0383	0.0384
16	0, 4, 9	+2+3	0.0148	0.0149
16	0, 5, 11	+2+3	0.0178	0.0179
16	0, 6, 13	+2+3	0.0348	0.0352
16	0, 7, 9	+3+1	0.0227	0.0228
16	0, 8, 10	+3+1	0.0443	0.0447
16	0, 12, 15	+3+1	0.0162	0.0166
16	0, 8, 12	+3+2	0.0322	0.0325
16	0, 10, 15	+3+2	0.0325	0.0330
17	0, 1, 2	+1+1	0.0183	0.0183
17	0, 2, 4	+1+1	0.0367	0.0369
17	0, 5, 9	+1+1	0.0179	0.0184
17	0, 6, 11	+1+1	0.0185	0.0192
17	0, 7, 12	+1+1	0.0414	0.0434
17	0, 8, 14	+1+1	0.0019	0.0020
17	0, 9, 15	+1+1	0.0454	0.0494
17	0, 9, 16	+1+1	0.0327	0.0360
17	0, 1, 3	+1+2	0.0386	0.0386
17	0, 3, 8	+1+2	0.0115	0.0116
17	0, 5, 13	+1+2	0.0281	0.0288
17	0, 6, 15	+1+2	0.0134	0.0139
17	0, 2, 7	+1+3	0.0280	0.0281
17	0, 3, 10	+1+3	0.0328	0.0331
17	0, 4, 13	+1+3	0.0133	0.0135
17	0, 5, 16	+1+3	0.0164	0.0168
17	0, 2, 3	+2+1	0.0344	0.0344
17	0, 5, 7	+2+1	0.0444	0.0447
17	0, 7, 10	+2+1	0.0264	0.0267
17	0, 8, 11	+2+1	0.0366	0.0373
17	0, 10, 14	+2+1	0.0318	0.0328
17	0, 11, 15	+2+1	0.0110	0.0114
17	0, 12, 16	+2+1	0.0477	0.0499
17	0, 3, 7	+2+3	0.0283	0.0284
17	0, 4, 9	+2+3	0.0020	0.0020
17	0, 5, 11	+2+3	0.0021	0.0021
17	0, 6, 13	+2+3	0.0163	0.0164
17	0, 7, 15	+2+3	0.0389	0.0394
17	0, 3, 4	+3+1	0.0499	0.0500
17	0, 7, 9	+3+1	0.0159	0.0160
17	0, 11, 14	+3+1	0.0474	0.0483
17	0, 12, 15	+3+1	0.0053	0.0054
17	0, 13, 16	+3+1	0.0310	0.0318
17	0, 8, 12	+3+2	0.0472	0.0476
17	0, 10, 15	+3+2	0.0133	0.0135
18	0, 1, 2	+1+1	0.0173	0.0173
18	0, 2, 4	+1+1	0.0347	0.0348
18	0, 5, 9	+1+1	0.0224	0.0229
18	0, 6, 11	+1+1	0.0129	0.0134
18	0, 7, 12	+1+1	0.0472	0.0493
18	0, 7, 13	+1+1	0.0430	0.0452
18	0, 8, 14	+1+1	0.0089	0.0094
18	0, 9, 16	+1+1	0.0246	0.0267
18	0, 10, 17	+1+1	0.0145	0.0160
18	0, 1, 3	+1+2	0.0364	0.0365
18	0, 3, 8	+1+2	0.0173	0.0174
18	0, 5, 13	+1+2	0.0186	0.0191
18	0, 6, 15	+1+2	0.0025	0.0026
18	0, 7, 17	+1+2	0.0038	0.0040
18	0, 2, 7	+1+3	0.0340	0.0341
18	0, 3, 10	+1+3	0.0414	0.0417
18	0, 4, 13	+1+3	0.0244	0.0248
18	0, 5, 16	+1+3	0.0028	0.0028
18	0, 2, 3	+2+1	0.0324	0.0325
18	0, 5, 7	+2+1	0.0485	0.0488
18	0, 7, 10	+2+1	0.0201	0.0204
18	0, 8, 11	+2+1	0.0430	0.0437
18	0, 10, 14	+2+1	0.0231	0.0238
18	0, 11, 15	+2+1	0.0198	0.0205
18	0, 3, 7	+2+3	0.0194	0.0194
18	0, 4, 9	+2+3	0.0094	0.0095
18	0, 5, 11	+2+3	0.0119	0.0120
18	0, 6, 13	+2+3	0.0002	0.0002
18	0, 7, 15	+2+3	0.0198	0.0201
18	0, 8, 17	+2+3	0.0453	0.0461
18	0, 3, 4	+3+1	0.0471	0.0472
18	0, 7, 9	+3+1	0.0099	0.0099
18	0, 11, 14	+3+1	0.0380	0.0386
18	0, 12, 15	+3+1	0.0043	0.0044
18	0, 13, 16	+3+1	0.0407	0.0417
18	0, 5, 8	+3+2	0.0495	0.0497
18	0, 10, 15	+3+2	0.0036	0.0036
19	0, 1, 2	+1+1	0.0164	0.0164
19	0, 2, 4	+1+1	0.0329	0.0330
19	0, 3, 6	+1+1	0.0494	0.0498
19	0, 5, 9	+1+1	0.0264	0.0269
19	0, 6, 11	+1+1	0.0079	0.0082
19	0, 7, 13	+1+1	0.0371	0.0387
19	0, 8, 14	+1+1	0.0151	0.0159
19	0, 9, 16	+1+1	0.0174	0.0187
19	0, 10, 17	+1+1	0.0220	0.0239
19	0, 1, 3	+1+2	0.0345	0.0345
19	0, 3, 8	+1+2	0.0225	0.0227
19	0, 4, 11	+1+2	0.0433	0.0439
19	0, 5, 13	+1+2	0.0102	0.0104
19	0, 6, 15	+1+2	0.0073	0.0075
19	0, 7, 17	+1+2	0.0149	0.0155
19	0, 2, 7	+1+3	0.0393	0.0394
19	0, 3, 10	+1+3	0.0492	0.0495
19	0, 4, 13	+1+3	0.0345	0.0349
19	0, 5, 16	+1+3	0.0095	0.0097
19	0, 2, 3	+2+1	0.0307	0.0307
19	0, 7, 10	+2+1	0.0145	0.0147
19	0, 8, 11	+2+1	0.0487	0.0494
19	0, 10, 14	+2+1	0.0154	0.0158
19	0, 11, 15	+2+1	0.0277	0.0285
19	0, 13, 18	+2+1	0.0327	0.0342
19	0, 3, 7	+2+3	0.0115	0.0115
19	0, 4, 9	+2+3	0.0197	0.0197
19	0, 5, 11	+2+3	0.0244	0.0245
19	0, 6, 13	+2+3	0.0150	0.0151
19	0, 7, 15	+2+3	0.0028	0.0028
19	0, 8, 17	+2+3	0.0260	0.0264
19	0, 3, 4	+3+1	0.0446	0.0446
19	0, 7, 9	+3+1	0.0045	0.0045
19	0, 11, 14	+3+1	0.0296	0.0300
19	0, 12, 15	+3+1	0.0128	0.0130
19	0, 5, 8	+3+2	0.0412	0.0413
19	0, 7, 11	+3+2	0.0464	0.0466
19	0, 10, 15	+3+2	0.0187	0.0189
19	0, 12, 18	+3+2	0.0359	0.0365
20	0, 1, 2	+1+1	0.0156	0.0156
20	0, 2, 4	+1+1	0.0312	0.0313
20	0, 3, 6	+1+1	0.0469	0.0472
20	0, 5, 9	+1+1	0.0300	0.0305
20	0, 6, 11	+1+1	0.0035	0.0036
20	0, 7, 13	+1+1	0.0317	0.0330
20	0, 8, 14	+1+1	0.0207	0.0217
20	0, 9, 16	+1+1	0.0109	0.0116
20	0, 10, 17	+1+1	0.0287	0.0310
20	0, 10, 18	+1+1	0.0392	0.0427
20	0, 11, 19	+1+1	0.0036	0.0039
20	0, 1, 3	+1+2	0.0328	0.0328
20	0, 3, 8	+1+2	0.0272	0.0274
20	0, 4, 11	+1+2	0.0369	0.0374
20	0, 5, 13	+1+2	0.0026	0.0026
20	0, 6, 15	+1+2	0.0162	0.0166
20	0, 7, 17	+1+2	0.0250	0.0259
20	0, 7, 18	+1+2	0.0467	0.0485
20	0, 8, 19	+1+2	0.0271	0.0283
20	0, 2, 7	+1+3	0.0442	0.0443
20	0, 4, 13	+1+3	0.0436	0.0440
20	0, 4, 14	+1+3	0.0463	0.0468
20	0, 5, 16	+1+3	0.0206	0.0209
20	0, 6, 19	+1+3	0.0076	0.0078
20	0, 2, 3	+2+1	0.0292	0.0292
20	0, 7, 10	+2+1	0.0095	0.0096
20	0, 10, 14	+2+1	0.0085	0.0087
20	0, 11, 15	+2+1	0.0348	0.0357
20	0, 13, 18	+2+1	0.0238	0.0247
20	0, 14, 19	+2+1	0.0087	0.0091
20	0, 3, 7	+2+3	0.0043	0.0043
20	0, 4, 9	+2+3	0.0289	0.0290
20	0, 5, 11	+2+3	0.0356	0.0358
20	0, 6, 13	+2+3	0.0283	0.0285
20	0, 7, 15	+2+3	0.0125	0.0127
20	0, 8, 17	+2+3	0.0086	0.0087
20	0, 9, 19	+2+3	0.0335	0.0341
20	0, 3, 4	+3+1	0.0423	0.0424
20	0, 7, 9	+3+1	0.0003	0.0003
20	0, 11, 14	+3+1	0.0221	0.0223
20	0, 12, 15	+3+1	0.0205	0.0208
20	0, 5, 8	+3+2	0.0336	0.0337
20	0, 7, 11	+3+2	0.0361	0.0362
20	0, 10, 15	+3+2	0.0322	0.0325
20	0, 12, 18	+3+2	0.0195	0.0197
20	0, 13, 19	+3+2	0.0460	0.0468
21	0, 1, 2	+1+1	0.0148	0.0148
21	0, 2, 4	+1+1	0.0297	0.0298
21	0, 3, 6	+1+1	0.0447	0.0449
21	0, 5, 9	+1+1	0.0333	0.0338
21	0, 6, 11	+1+1	0.0006	0.0006
21	0, 7, 13	+1+1	0.0269	0.0278
21	0, 8, 14	+1+1	0.0257	0.0268
21	0, 9, 16	+1+1	0.0051	0.0054
21	0, 10, 17	+1+1	0.0348	0.0372
21	0, 10, 18	+1+1	0.0325	0.0351
21	0, 11, 19	+1+1	0.0033	0.0036
21	0, 12, 20	+1+1	0.0350	0.0387
21	0, 1, 3	+1+2	0.0312	0.0313
21	0, 3, 8	+1+2	0.0315	0.0317
21	0, 4, 11	+1+2	0.0311	0.0315
21	0, 5, 13	+1+2	0.0043	0.0044
21	0, 6, 15	+1+2	0.0242	0.0247
21	0, 7, 17	+1+2	0.0341	0.0351
21	0, 7, 18	+1+2	0.0372	0.0385
21	0, 8, 19	+1+2	0.0372	0.0388
21	0, 8, 20	+1+2	0.0290	0.0303
21	0, 1, 4	+1+3	0.0476	0.0477
21	0, 2, 7	+1+3	0.0485	0.0486
21	0, 3, 11	+1+3	0.0475	0.0478
21	0, 4, 14	+1+3	0.0378	0.0382
21	0, 5, 16	+1+3	0.0306	0.0311
21	0, 5, 17	+1+3	0.0451	0.0459
21	0, 6, 19	+1+3	0.0042	0.0043
21	0, 2, 3	+2+1	0.0278	0.0278
21	0, 7, 10	+2+1	0.0050	0.0051
21	0, 10, 14	+2+1	0.0023	0.0023
21	0, 11, 15	+2+1	0.0411	0.0421
21	0, 13, 18	+2+1	0.0157	0.0163
21	0, 14, 19	+2+1	0.0169	0.0176
21	0, 15, 20	+2+1	0.0458	0.0480
21	0, 3, 7	+2+3	0.0022	0.0022
21	0, 4, 9	+2+3	0.0372	0.0373
21	0, 5, 11	+2+3	0.0458	0.0460
21	0, 6, 13	+2+3	0.0403	0.0406
21	0, 7, 15	+2+3	0.0264	0.0266
21	0, 8, 17	+2+3	0.0071	0.0072
21	0, 9, 19	+2+3	0.0160	0.0162
21	0, 3, 4	+3+1	0.0403	0.0403
21	0, 7, 9	+3+1	0.0046	0.0046
21	0, 11, 14	+3+1	0.0153	0.0155
21	0, 12, 15	+3+1	0.0273	0.0277
21	0, 16, 20	+3+1	0.0191	0.0196
21	0, 5, 8	+3+2	0.0268	0.0269
21	0, 7, 11	+3+2	0.0268	0.0269
21	0, 9, 14	+3+2	0.0480	0.0483
21	0, 10, 15	+3+2	0.0444	0.0448
21	0, 12, 18	+3+2	0.0046	0.0047
22	0, 1, 2	+1+1	0.0141	0.0141
22	0, 2, 4	+1+1	0.0283	0.0284
22	0, 3, 6	+1+1	0.0426	0.0429
22	0, 5, 9	+1+1	0.0362	0.0367
22	0, 6, 11	+1+1	0.0042	0.0043
22	0, 7, 13	+1+1	0.0225	0.0232
22	0, 8, 14	+1+1	0.0303	0.0315
22	0, 8, 15	+1+1	0.0461	0.0481
22	0, 9, 16	+1+1	0.0003	0.0003
22	0, 10, 17	+1+1	0.0403	0.0428
22	0, 10, 18	+1+1	0.0265	0.0283
22	0, 11, 19	+1+1	0.0096	0.0104
22	0, 12, 20	+1+1	0.0415	0.0454
22	0, 12, 21	+1+1	0.0182	0.0201
22	0, 1, 3	+1+2	0.0298	0.0298
22	0, 3, 8	+1+2	0.0354	0.0355
22	0, 4, 11	+1+2	0.0259	0.0261
22	0, 5, 13	+1+2	0.0106	0.0108
22	0, 6, 15	+1+2	0.0315	0.0321
22	0, 6, 16	+1+2	0.0454	0.0465
22	0, 7, 17	+1+2	0.0424	0.0435
22	0, 7, 18	+1+2	0.0286	0.0295
22	0, 8, 19	+1+2	0.0465	0.0482
22	0, 8, 20	+1+2	0.0193	0.0202
22	0, 9, 21	+1+2	0.0459	0.0481
22	0, 1, 4	+1+3	0.0455	0.0455
22	0, 3, 11	+1+3	0.0415	0.0418
22	0, 4, 14	+1+3	0.0301	0.0304
22	0, 5, 16	+1+3	0.0398	0.0404
22	0, 5, 17	+1+3	0.0357	0.0363
22	0, 6, 19	+1+3	0.0151	0.0154
22	0, 2, 3	+2+1	0.0265	0.0265
22	0, 7, 10	+2+1	0.0010	0.0010
22	0, 9, 13	+2+1	0.0478	0.0485
22	0, 10, 14	+2+1	0.0034	0.0034
22	0, 11, 15	+2+1	0.0469	0.0478
22	0, 12, 17	+2+1	0.0457	0.0469
22	0, 13, 18	+2+1	0.0085	0.0088
22	0, 14, 19	+2+1	0.0242	0.0251
22	0, 3, 7	+2+3	0.0080	0.0081
22	0, 4, 9	+2+3	0.0448	0.0449
22	0, 7, 15	+2+3	0.0390	0.0393
22	0, 8, 17	+2+3	0.0214	0.0216
22	0, 9, 19	+2+3	0.0000	0.0000
22	0, 10, 21	+2+3	0.0240	0.0244
22	0, 3, 4	+3+1	0.0384	0.0384
22	0, 7, 9	+3+1	0.0085	0.0085
22	0, 11, 14	+3+1	0.0092	0.0093
22	0, 12, 15	+3+1	0.0335	0.0339
22	0, 15, 19	+3+1	0.0415	0.0423
22	0, 16, 20	+3+1	0.0106	0.0108
22	0, 17, 21	+3+1	0.0171	0.0175
22	0, 5, 8	+3+2	0.0207	0.0207
22	0, 7, 11	+3+2	0.0184	0.0184
22	0, 9, 14	+3+2	0.0372	0.0374
22	0, 12, 18	+3+2	0.0088	0.0089
22	0, 14, 21	+3+2	0.0384	0.0391
23	0, 1, 2	+1+1	0.0135	0.0135
23	0, 2, 4	+1+1	0.0271	0.0272
23	0, 3, 6	+1+1	0.0408	0.0410
23	0, 5, 9	+1+1	0.0389	0.0394
23	0, 6, 11	+1+1	0.0076	0.0077
23	0, 7, 13	+1+1	0.0185	0.0190
23	0, 8, 14	+1+1	0.0345	0.0357
23	0, 8, 15	+1+1	0.0415	0.0431
23	0, 9, 16	+1+1	0.0051	0.0053
23	0, 10, 17	+1+1	0.0453	0.0478
23	0, 10, 18	+1+1	0.0210	0.0223
23	0, 11, 19	+1+1	0.0153	0.0164
23	0, 11, 20	+1+1	0.0448	0.0484
23	0, 12, 21	+1+1	0.0118	0.0129
23	0, 13, 22	+1+1	0.0176	0.0195
23	0, 1, 3	+1+2	0.0285	0.0285
23	0, 3, 8	+1+2	0.0389	0.0391
23	0, 4, 11	+1+2	0.0211	0.0213
23	0, 5, 13	+1+2	0.0163	0.0166
23	0, 6, 15	+1+2	0.0381	0.0388
23	0, 6, 16	+1+2	0.0385	0.0393
23	0, 7, 18	+1+2	0.0207	0.0213
23	0, 8, 20	+1+2	0.0105	0.0109
23	0, 9, 22	+1+2	0.0059	0.0062
23	0, 1, 4	+1+3	0.0435	0.0435
23	0, 3, 11	+1+3	0.0360	0.0362
23	0, 4, 14	+1+3	0.0230	0.0232
23	0, 5, 16	+1+3	0.0482	0.0488
23	0, 5, 17	+1+3	0.0271	0.0275
23	0, 6, 19	+1+3	0.0250	0.0254
23	0, 6, 20	+1+3	0.0403	0.0412
23	0, 7, 22	+1+3	0.0012	0.0013
23	0, 2, 3	+2+1	0.0253	0.0253
23	0, 7, 10	+2+1	0.0028	0.0028
23	0, 9, 13	+2+1	0.0428	0.0433
23	0, 10, 14	+2+1	0.0085	0.0086
23	0, 12, 17	+2+1	0.0392	0.0401
23	0, 13, 18	+2+1	0.0019	0.0020
23	0, 14, 19	+2+1	0.0309	0.0320
23	0, 16, 22	+2+1	0.0199	0.0208
23	0, 3, 7	+2+3	0.0134	0.0134
23	0, 8, 17	+2+3	0.0344	0.0347
23	0, 9, 19	+2+3	0.0146	0.0148
23	0, 10, 21	+2+3	0.0079	0.0080
23	0, 3, 4	+3+1	0.0367	0.0367
23	0, 7, 9	+3+1	0.0121	0.0121
23	0, 11, 14	+3+1	0.0036	0.0037
23	0, 12, 15	+3+1	0.0392	0.0396
23	0, 15, 19	+3+1	0.0338	0.0344
23	0, 16, 20	+3+1	0.0028	0.0028
23	0, 17, 21	+3+1	0.0249	0.0255
23	0, 5, 8	+3+2	0.0150	0.0151
23	0, 7, 11	+3+2	0.0107	0.0107
23	0, 9, 14	+3+2	0.0274	0.0276
23	0, 12, 18	+3+2	0.0210	0.0213
23	0, 14, 21	+3+2	0.0240	0.0244
23	0, 15, 22	+3+2	0.0325	0.0331

DR and RTT

One may be able to tune a rank-2 regular temperament in such a way that a triad of interest exactly "inherits" its delta signature from a simple JI preimage thereof. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. The value to be solved for is the generator's frequency ratio (not its cent value).

If we want to optimize a 4:5:6 triad in Meantone, for instance, we want a +1+1 delta signature, or equivalently a 1:1 ratio of frequency deltas between the major third and minor third. Fixing any frequency as the triad's root and letting [math]\displaystyle{ g }[/math] be the frequency ratio for the perfect fifth generator for meantone, the minor third in the tempered 4:5:6 triad has a delta of [math]\displaystyle{ g-g^4/4 }[/math], and the major third in the same triad has a delta of [math]\displaystyle{ g^4/4-1 }[/math]. Therefore to ensure that the two deltas form a 1:1 ratio, we must find the appropriate root of the polynomial [math]\displaystyle{ g^4-2g-2 }[/math] (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.4945, or about 695.6 cents.

Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: [math]\displaystyle{ g^4+2g-8=0. }[/math] The latter equation has solution g = 1.4960 = 697.3c.

Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the relevant DR polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients, etc. In the case of negative coefficients, only the absolute value is considered.

Coefficients of terms											Chord	Delta signature	Temperament	Generator (cents)	Edos
g^10	g^9	g^8	g^7	g^6	g^5	g^4	g^3	g^2	g^1	g^0
								1	-1	-1	4:5:6	+1+1	[1 -2 1⟩	833.09 (phi)	36
								2	-1	-2	4:5:6	+1+1	Father	428.42	14
							3	-2		-2	6:7:9	+1+2	Beep	258.65	33, 42, 51
						1			-1	-2	4:5:6	+1+1	Mavila	523.66	23, 39
						1			-2	-2	4:5:6	+1+1	Meantone	695.63	19, 31, 50
					1			2		-4	4:5:6	+1+1	Porcupine	160.89	15
					1	-4				12	5:6:9	+1+3	Mavila	674.90	16, 25
				1					-4	-4	4:5:6	+1+1	Avila	660.23	20
				1	-2					2	4:5:6	+1+1	Hanson	317.96	19
			1						-2	-4	4:5:6	+1+1	Uncle	467.46	18
		3			-4					-16	4:5:7	+1+2	Mabila	527.66	25
	1					-1				-1	4:5:6	+1+1	Tetracot	176.54	34
	1		-1							-4	4:5:6	+1+1	Sensi	442.74	19, 65
1							2			-8	4:5:6	+1+1	Orson	271.51	22, 31, 53

Limitations

Interpreting simple JI chords as signatures/templates for delta-rational chords is reasonable, as the psychoacoustic effect of DR is more robust to detuning than that of JI for many people, but it is no panacea. If the temperament is too inaccurate, other inversions and voicings of a given JI chord will not have the DR signatures preserved acceptably, and compromising will make all of them inaccurate, though this of course depends on what error tolerance you prefer. This is because DR simplicity is only preserved by inversion and revoicing if the chord in question is low-complexity JI.

For example, take 0.0570 (the LSD error of 0-2\11-4\11 as +1+1, approximately equalized 7:8:9) as a somewhat arbitrary but reasonable upper limit of acceptable error. Consider Semaphore temperament (i.e. 2.3.7[14 & 19]). Using a gen of 260.346c, 0-679c-940c is a Semaphore tuning of 4:6:7 that is perfectly +2+1, but inverting the chord yields 0-260c-521c as our 6:7:8, with LSD error 0.0681; the 7:8:12 has an even higher error of 0.123. This is also evident by the fact that we had to use an extreme tuning of Semaphore, which has CWE generator 249.311c and CTE generator 248.126c. If we use the CTE generator, all three inversions are barely acceptable to unacceptable as simple-DR approximations: 4:6:7 has LSD error 0.0574 from +2+1, 6:7:8 has LSD error 0.0652 from +1+1, and 7:8:12 has LSD error 0.0570 from +1+4.

Higher-order differences of frequency

Generalizing, one could consider chords where differences between its frequency deltas (as Tom Price has called them, precessions) are rationally related, while the deltas themselves may not be. This corresponds to chords where differences between various interference beatings go in and out of sync in a periodic manner. One precession-rational chord is 5:5.4142...:6.8284...:9.2426..., a +(sqrt(2) − 1) +sqrt(2) +(sqrt(2) + 1) chord.

Precession being the second-order difference (Δ²) of frequency, we similarly have the theoretical notions of Δ³-rationality, Δ⁴-rationality, and so on. The practical consequences of higher-order differences are as of yet speculative, though a few people have reported finding precession psychoacoustically meaningful.

Isodifferential chord

In an isodifferential chord (known variously by linear chord, equal-hertz chord, equal-beating chord, and proportional-beating chord), the frequencies of the pitches are in an arithmetic sequence, or in other words, there is an equal difference in cycles per second between successive pitches.

Isoharmonic chord

An isoharmonic chord is a specific type of isodifferential chord, where the ratios between the notes are rational numbers, and therefore the chord is in just intonation. Such a chord can be built by successive jumps up the harmonic series by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape—with diminishing step size as one ascends.

An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but the word "chord" is used here for consistency.

Classification

Class i

The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:

harmonic	1		2		3		4		5		6		7		8		9		10		11		12		13		14		15		16
cents diff		1200		702		498		386		316		267		231		204		182		165		151		139		128		119		112

Some "scales" built this way: otones12-24, otones20-40...

Class ii

The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the Bohlen-Pierce tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):

harmonic	1		3		5		7		9		11		13		15		17		19		21		23		25		27		29		31
cents diff		1902		884		583		435		347		289		248		217		193		173		157		144		133		124		115

Class iii

Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:

harmonic	1		4		7		10		13		16		19		22		25		28		31		34		37		40		43		46
cents diff		2400		969		617		454		359		298		254		221		196		176		160		146		135		125		117

harmonic	2		5		8		11		14		17		20		23		26		29		32		35		38		41		44		47
cents diff		1586		814		551		418		336		281		242		212		189		170		155		142		132		122		114

Class iv

harmonic	1		5		9		13		17		21		25		29		33		37		41		45		49		53		57		61
cents diff		2786		1018		637		464		366		302		257		224		198		178		161		147		136		126		117

harmonic	3		7		11		15		19		23		27		31		35		39		43		47		51		55		59		63
cents diff		1467		782		537		409		331		278		239		210		187		169		154		141		131		122		114

Class v

harmonic	1		6		11		16		21		26		31		36		41		46		51		56		61		66		71		76
cents diff		3102		1049		649		471		370		306		259		225		199		179		162		148		136		126		118

harmonic	2		7		12		17		22		27		32		37		42		47		52		57		62		67		72		77
cents diff		2169		933		603		446		355		294		251		219		195		175		159		146		134		125		116

harmonic	3		8		13		18		23		28		33		38		43		48		53		58		63		68		73		78
cents diff		1698		841		563		424		341		284		244		214		190		172		156		143		132		123		115

harmonic	4		9		14		19		24		29		34		39		44		49		54		59		64		69		74		79
cents diff		1404		765		529		404		328		275		238		209		186		168		153		141		130		121		113

Notation

Some complex isoharmonic chords can be expressed with an offset from a simpler isoharmonic chord, so it is useful to notate them in a compact and readable way. For example, 41:51:61 is very similar to 4:5:6, so it can be notated as (4:5:6)[+0.1]. Similarly, (20:22:24:27:30:33:36)[+0.339] can be expanded to 20339:22339:24339:27339:30339:33339:36339.

Irrational isodifferential chords can be expressed with the same notation by using irrational numbers within the square brackets. For example, the chord (1:2:3)[+φ] can be expanded to (1+φ):(2+φ):(3+φ), which is approximately equal to 1.618:2.618:3.618.

Categorization of DR chords

Here is a table which uses the "delta ratio set" — the set of unique undirected ratios between the deltas of a chord's delta signature — to categorize chords.

• How to tell a DR chord from a non-DR chord: a DR chord has at least one rational number in its delta ratio set.
• Within DR chords, how to tell an FDR chord from a non-fully DR chord: a FDR chord has only rational numbers in its delta ratio set.
• Within FDR chords, how to tell an isodifferential chord from a non-isodifferential chord: an isodiffential chord has only 1 in its delta ratio set.

All JI chords are FDR chords, because JI chords are rational, and therefore their delta ratio sets will include only rational numbers.

If an FDR chord is both JI and isodifferential, then it is an isoharmonic chord.

chord type				illustrative examples
				actual chord		deltas			delta ratio set
				frequency ratio	are items all integers?	delta signature	reduced delta signature (class)	are items all the same?	unique undirected ratios between the deltas	are items all rational?
DR	FDR	JI, not isodifferential		4:5:7:8	yes, all	+1+2+1	+1+2+1	no, not all	{ 1, 2 }	yes
				3:5:9:11		+2+4+2
				3:4:7:9		+1+3+2	+1+3+2		{ 3/2, 2, 3 }
		isoharmonic (JI and isodifferential)	class i	4:5:6		+1+1	+1+1	yes, all	{ 1 }
				4:5:6:7		+1+1+1	+1+1+1
				3:4:5:6
			class ii	3:5:7:9		+2+2+2
				5:7:9:11
			class iii	1:4:7:10		+3+3+3
				2:5:8:11
			...	...		...
		not JI, but isodifferential		ɸ:(ɸ+1):(ɸ+2):(ɸ+3)	no, not all or none	+1+1+1
				1:ɸ:(2ɸ-1):(3ɸ-2)		+(ɸ-1)+(ɸ-1)+(ɸ-1)
		not JI and not isodifferential		ɸ:(ɸ+1):(ɸ+3)		+1+2	+1+2	no, not all	{ 2 }
	(incompletely) DR			4:5:τ:7:9		+1+(τ-5)+(7-τ)+2	+1+(τ-5)+(7-τ)+2	(irrelevant for categorization)	{ (7-τ)/(τ-5), 7-τ, τ-5, 2/(τ-5), 2, 2/(7-τ) }	no, but at least one
				5:τ:8:(3+τ)		+(τ-5)+(8-τ)+(τ-5)	+1+(8-τ)/(τ-5)+1		{ 1, (8-τ)/(τ-5) }
				1:(1+a):(1+a+b):(1+a+2b):(1+3a+2b), with a/b irrational		+a+b+b+2a	+a+b+b+2a		{ a/b, 1, 2, 2a/b }
not DR				4:5:τ:7		+1+(τ-5)+(7-τ)	+1+(τ-5)+(7-τ)		{ (7-τ)/(τ-5), 7-τ, τ-5 }	no, none
				5:τ:7	+(τ-5)+(7-τ)	+1+(7-τ)/(τ-5)	{ (7-τ)/(τ-5) }