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

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.
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.
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 [math]S^{n-1};[/math] this is because they are specifically in the subset that is the image of the all-positive orthant of [math]\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 and 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]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]\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]\mathbb{Z}^2\langle \mathbf{p}, \mathbf{g}\rangle[/math].

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

[math]\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]m/n[/math] lies in the image [math]r_{\mathbf{u}, \mathbf{v}}(I)[/math] and [math]m(E^{u_p}x^{u_g} - 1) = n(E^{v_p}x^{v_g}- E^{u_p}x^{u_g})[/math] yields a nondegenerate polynomial equation after cancelling negative powers, we can solve for the frequency ratio [math]g \in I[/math] that satisfies [math]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

Fully DR

The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: 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] 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?"

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

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

with solution

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

which can be plugged back into the error formula to obtain the error. (We multiply the target DR chord by x in order to compare it to the approximation on the same isodifferential series.) This error measure is invariant under scaling the target delta signature, but not under scaling the approximation, which is why we require the approximating chord to start on 1.

The least-squares error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.

Partially DR

For approximating a target delta signature of the form [math]+\delta_1 +? +\delta_3[/math] with the chord [math]1:(1 + E_1):(1 + E_2):(1 + E_3)[/math] (where the +? is free to vary), the least-squares problem is

[math] \displaystyle {\underset{x,y}{\text{minimize}} \sqrt{(\delta_1 x - E_1)^2 + ((\delta_1 + y)x - E_2)^2 + ((\delta_1 + y + \delta_3) x - E_3)^2 }}. [/math]

We can set the partial derivatives with respect to x and y of the inner expression equal to zero (since the derivative of sqrt() is never 0) and use SymPy to solve the system:

import sympy
x = sympy.Symbol("x", real=True)
y = sympy.Symbol("y", real=True)
d1 = sympy.Symbol("\\delta_{1}", real=True)
d2 = sympy.Symbol("\\delta_{2}", real=True)
d3 = sympy.Symbol("\\delta_{3}", real=True)
E1 = sympy.Symbol("E_1", real=True)
E2 = sympy.Symbol("E_2", real=True)
E3 = sympy.Symbol("E_3", real=True)
err_squared = (d1*x - E1) ** 2 + ((d1 + y)*x - E2) ** 2 + ((d1 + y + d3)*x - E3) ** 2
err_squared.expand()
err_squared_x = sympy.diff(err_squared, x)
err_squared_y = sympy.diff(err_squared, y)
sympy.nonlinsolve([err_squared_x, err_squared_y], [x, y])

We similarly include a free variable to be optimized for every additional +?, after coalescing two consecutive +?'s and omitting the middle note. If two variables are related to each other but not to the integer deltas in the signatures, they have a common variable.

DR chords in small edos

Fully DR triads

Main article: Delta-rational triads in small edos

Partially DR tetrads

Main article: Partially delta-rational tetrads in small edos

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]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]g-g^4/4[/math], and the major third in the same triad has a delta of [math]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]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]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	Chord	Delta signature	Temperament	Generator (cents)	Edos
								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 less accurate, 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; that all inversions of a chord preserve the JI chord's delta signature with low enough error is thus a sensible criterion for a "good" temperament.

For example, take 0.00807 (the least-squares 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 least-squares error 0.0118; the 7:8:12 has an even higher error of 0.0178. 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, the 4:6:7, 6:7:8, and 7:8:12 have errors 0.0108, 0.0106, and 0.00737. If we use the average of the CTE and the perfect +2+1 generator, the errors become 0.00535, 0.0112, and 0.0126.

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), or even a series in the sense of a harmonic series, 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	+1+2+1		{1, 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		+1+1+1
			class ii	3:5:7:9		+2+2+2
			class ii	5:7:9:11		+2+2+2
			class iii	1:4:7:10		+3+3+3
			class iii	2:5:8:11		+3+3+3
			...	...		...
		Not JI, but isodifferential		ɸ:(ɸ+1):(ɸ+2):(ɸ+3)	No, not all or none	+1+1+1
		Not JI, but isodifferential		1:ɸ:(2ɸ-1):(3ɸ-2)		+(ɸ-1)+(ɸ-1)+(ɸ-1)
		Not JI or 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
Not DR				5:τ:7	+(τ-5)+(7-τ)	+1+(7-τ)/(τ-5)	{(7-τ)/(τ-5)}			No, none