Father–3 equivalence continuum

The chromatic-diatonic equivalence continuum, despite its name, is a continuum of 5-limit temperaments which equate a number of classical diatonic semitones (16/15) with the Pythagorean minor third (32/27).

All temperaments in the continuum satisfy (16/15)ⁿ ~ 32/27. Varying n results in different temperaments listed in the table below. It converges to father as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 3edo due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of n is approximately 2.63252…, and temperaments having n near this value tend to be the most accurate ones.

32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of 3 in the generator chain. Such an equivalence continuum is more properly called the father-3 equivalence continuum.

Temperaments with integer n

n	Temperament	Comma
		Ratio	Monzo
0	Alteraugment	32/27	[5 -3⟩
1	Yellow	10/9	[1 -2 1⟩
2	Dicot	25/24	[-3 -1 2⟩
3	Augmented	128/125	[7 0 -3⟩
4	Smate	2048/1875	[11 -1 -4⟩
…	…	…	…
∞	Father	16/15	[4 -1 -1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the yellow-3 equivalence continuum, which is essentially the same thing. The just value of m is 1.61255…

Temperaments with integer m

m	Temperament	Comma
		Ratio	Monzo
0	Alteraugment	32/27	[5 -3⟩
1	Father	16/15	[4 -1 -1⟩
2	Dicot	25/24	[-3 -1 2⟩
…	…	…	…
∞	Yellow	10/9	[1 -2 1⟩

Temperaments with fractional n and m

Temperament	n	m
Wesley	7/3 = 2.3	7/4 = 1.75
Magic	5/2 = 2.5	5/3 = 1.6
Würschmidt	8/3 = 2.6	8/5 = 1.6
Isnes	19/7 = 2.714285	19/12 = 1.583
Magus	11/4 = 2.75	11/7 = 1.571428

Some prefer conceptualizing this continuum in terms of k = 1/(n - 2) such that temperaments satisfy (25/24)^k = 16/15. This is the source of the name chromatic-diatonic equivalence continuum, where both chromatic and diatonic refer to the classical versions of semitones. The just value of k is approximately 1.58097…

Temperaments with integer k

k	Temperament	Comma
		Ratio	Monzo
-1	Yellow	10/9	[1 -2 1⟩
0	Father	16/15	[4 -1 -1⟩
1	Augmented	128/125	[7 0 -3⟩
2	Magic	3125/3072	[10 1 -5⟩
3	Wesley	78125/73728	[13 2 -7⟩
4	3 & 33c	1953125/1769472	[16 3 -9⟩
…	…	…	…
∞	Dicot	25/24	[-3 -1 2⟩

3 & 33c

This low-accuracy high-complexity temperament corresponds to n = 9/4 and m = 9/5.

Subgroup: 2.3.5

Comma list: 1953125/1769472

Mapping: [⟨3 2 6], ⟨0 3 1]]

mapping generators: ~125/96, ~5/4

Optimal tunings:

• CTE: ~125/96 = 1\3, ~5/4 = 368.2534 (~25/24 = 31.7466)
• CWE: ~125/96 = 1\3, ~5/4 = 366.8103 (~25/24 = 33.1897)

Optimal ET sequence: 3, …, 33c, 36c, 69cc

Badness: 0.682

Isnes

Isnes is so called because the generator is half of a 8/5 minor sixth, in a similar way that sensi has a generator of half a 5/3. This corresponds to n = 19/7 and m = 19/12.

Subgroup: 2.3.5

Comma list: [41 2 -19⟩

Mapping: [⟨1 8 3], ⟨0 -19 -2]]

mapping generators: ~2, ~1953125/1572864

Optimal tunings:

• CTE: ~2 = 1\1, ~1953125/1572864 = 405.1689
• CWE: ~2 = 1\1, ~1953125/1572864 = 405.1272

Optimal ET sequence: 3, 71b, 74, 77, 157, 548ccc

Badness: 1.30