DKW theory

DKW theory is a theory developed by Lériendil of the efficacy of the representation of three-prime subgroups of JI by edos and rank-2 temperaments, though generalizations should be possible for multi-prime subgroups. The fundamental element of DKW theory is the diharmonic tonality diamond (DTD), the simplest case of generalized tonality diamond with two prime harmonics.

The structure of DTDs

Given an equave E and two harmonics P and Q such that 1 < P < Q < E, their tonality diamond consists of the consonances {P, Q, E/P, E/Q, Q/P, EP/Q} in some order, placed between 1 and E. 12 orderings are possible:

• F-I: 1 < E/Q < P < EP/Q < Q/P < E/P < Q < E
• F-II: 1 < P < E/Q < EP/Q < Q/P < Q < E/P < E
• F-III: 1 < P < Q/P < Q < E/Q < EP/Q < E/P < E
• F-IV: 1 < E/Q < Q/P < E/P < P < EP/Q < Q < E
• F-V: 1 < Q/P < E/Q < E/P < P < Q < EP/Q < E
• F-VI: 1 < Q/P < P < Q < E/Q < E/P < EP/Q < E
• N-I: 1 < E/Q < P < Q/P < EP/Q < E/P < Q < E
• N-II: 1 < P < E/Q < Q/P < EP/Q < Q < E/P < E
• N-III: 1 < P < Q/P < E/Q < Q < EP/Q < E/P < E
• N-IV: 1 < E/Q < Q/P < P < E/P < EP/Q < Q < E
• N-V: 1 < Q/P < E/Q < P < E/P < Q < EP/Q < E
• N-VI: 1 < Q/P < P < E/Q < Q < E/P < EP/Q < E

Note that by construction, the diamond is in all cases symmetric under equave complementation. Furthermore, under the isomorphisms (P, Q) → (E/Q, E/P), (P, Q) → (Q/P, Q), (P, Q) → (P, EP/Q), one can redefine any tonality diamond with a certain F-type ordering as one that has any other F-type ordering with the same set of consonances, and the same can be done for N-type orderings. Therefore, there are only really 2 types of diharmonic tonality diamond.

Signature

If a diamond can be ordered as 1 < X₁ < X₂ < ... < X_n < X_n+1 < ... < X_2n-1 < < X_2n < E, such that X_i × X_2n+1-i = E, we can define the signature of the diamond to be the set of intervals X₂/X₁, X₃/X₂, ... X_n+1/X_n.

For a diharmonic diamond, the signature consists of three intervals, which we can call A, B, and C in this order. It turns out that either X₁ = B, or X₁ = BC; and that the former is the case for all F-type diamonds, while the latter is the case for all N-type diamonds. Therefore, the intervals of the signature partition the equave into 7 parts (B:A:B:C:B:A:B) for an F-type diamond, or into 9 parts (C:B:A:B:C:B:A:B:C) for a N-type diamond. We can therefore equate how well a given three-prime subgroup of JI is represented in a given tuning system to how well the signature is represented in that tuning system.

The signature basis for commas

Any interval in a three-prime subgroup can be written as a vector with basis elements being the elements of the group's signature. These often serve as a much more useful basis for commas than using the primes directly as the basis, as many useful commas are formed by ratios of powers of two signature intervals (as discussed below), and measuring complexity this way prioritizes a set of intervals more usable as commas.

Ratios of signature intervals and DKW coordinates

Given a signature for a particular tonality diamond (not precisely the same as a particular subgroup), there are three logarithmic ratios of its intervals. The closer the tuning system represents these ratios, the closer it represents the signature, the partition of the octave, and therefore the consonances of the diamond.

Setting any of the ratios C:B, C:A, and B:A to a particular value implies that a comma is tempered out - e.g. C:A = 3:1 means the comma C/A³ is tempered. We call the ratios C:B, C:A, and B:A diaschismian, kleismian, and interdiesian ratios, or simply D-ratios, K-ratios, and W-ratios, with corresponding D-commas, K-commas, and W-commas (these names derive from the specific triad of commas of this type tempered out in 34edo's 5-limit approximation, the diaschisma, the kleisma, and the wurschmidt comma). In this way, projective tuning space in any three-prime subgroup is given a grid consisting of these three families of fundamental commas, with all other commas being expressible in terms of these families, and points corresponding to edos lying at intersections of particular commas of these families.

If A, B, and C are expressed in cents or other logarithmic units, we can define the DKW coordinates of the subgroup's representation to be D = (C-B)/(C+B), K = (C-A)/(C+A), and W = (B-A)/(B+A). The use of these fractions is so that a ratio of x:y will be as negative as y:x is positive. Note that only any two of these ratios are linearly independent from one another, meaning that in theory the third coordinate is redundant; it is included here for the sake of symmetry.

With that in mind, we can define the DKW error of a given tuning with particular valuations for the primes within the subgroup (e.g. a val for an equal temperament, or the tempered-primes definition of a tuning of a regular temperament) as the squared distance between the DKW coordinates of the representation of the signature in this valuation, and the just DKW coordinates (though it is just as well possible to target a non-just valuation for the primes such as might be obtained from a rank-3 temperament that one is working within) - and in the case of rank-2 temperaments or equave stretches of an EDO, an optimal value for the one free parameter as the one that minimizes DKW error.

DKW theory in the 5-limit

By far the most important three-prime subgroup is the 5-limit, i.e. the 2.3.5 subgroup. It turns out that the 2.3.5 tonality diamond's ordering is type N-VI, and that its signature is C = 9/8 : B = 16/15 : A = 25/24.

Below are commas with simple expressions in terms of these intervals, with "complexity" measured as the sum of the absolute values of powers of A, B, and C. Note all commas considered here (aside from complexity-1) are those that contain at least one minus sign in their signature expression.

Comma	Defined temperament	Complexity	Smonzo (9/8.16/15.25/24)
9/8	Antitonic	1	[1 0 0⟩
16/15	Father	1	[0 1 0⟩
25/24	Dicot	1	[0 0 1⟩
27/25	Bug	2	[1 0 -1⟩
135/128	Mavila	2	[1 -1 0⟩
128/125	Augmented	2	[0 1 -1⟩
648/625	Dimipent	3	[1 0 -2⟩
2048/2025	Diaschismic	3	[-1 2 0⟩
3125/3072	Magic	3	[0 -1 2⟩
81/80	Meantone	3	[1 -1 -1⟩