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 not overlap (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 ignoring overlapping dyads 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.
• 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.

Dyads with no rational relationship (or no obvious one) are often 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.

Mathematics of DR

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 [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-squared-error solution instead to minimize the error.

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 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

Finding approximate DR chords in edos

Todo: add more heuristics

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)

12edo

• 0-4-10 (+1 +2)

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)

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.