Diaschismic–gothmic equivalence continuum

The diaschismic-kleismic equivalence continuum is a continuum of 5-limit temperaments which equate a number of kleismas (15625/15552) with the diaschisma (2048/2025).

All temperaments in the continuum satisfy (15625/15552)ⁿ ~ 2048/2025. Varying n results in different temperaments listed in the table below. It converges to hanson as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 34edo 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 2.4117…, and temperaments near this tend to be the most accurate ones.

The 17-comma (134217728/129140163) is the characteristic 3-limit comma tempered out in 34edo. It corresponds to a value of n = -1/3.

Temperaments with integer n

n	Temperament	Comma
		Ratio	Monzo
-3	34 & 113	152587890625/148769467776	[-7 -19 16⟩
-2	Fifive	9765625/9565938	[-1 -14 10⟩
-1	Tetracot	20000/19683	[5 -9 4⟩
0	Srutal	2048/2025	[11 -4 -2⟩
1	Wurschmidt	393216/390625	[17 1 -8⟩
2	Vishnu	6115295232/6103515625	[23 6 -14⟩
3	Gammic	95367431640625/95105071448064	[-28 -11 20⟩
4	Quatracot	1490116119384765625/1479074071160291328	[-33 -16 26⟩
…	…	…	…
∞	Hanson/Kleismic	15625/15552	[-6 -5 6⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the wurschmidt-diaschismic equivalence continuum, which is essentially the same thing. The just value of m is 1.7083…

Temperaments with integer m

m	Temperament	Comma
		Ratio	Monzo
-2	34 & 142	35184372088832/34332275390625	[45 -2 -18⟩
-1	Mabila	268435456/263671875	[28 -3 -10⟩
0	Srutal	2048/2025	[11 -4 -2⟩
1	Hanson/Kleismic	15625/15552	[-6 -5 6⟩
2	Vishnu	6115295232/6103515625	[23 6 -14⟩
3	Majvam	2404631929946112/2384185791015625	[40 7 -22⟩
…	…	…	…
∞	Wurschmidt	393216/390625	[17 1 -8⟩

Temperaments with fractional n and m

Temperament	n	m
Immunity	-1/2 = -0.5	1/3 = 0.3
Chlorine	5/2 = 2.5	5/3 = 1.6
Goldis	1/3 = 0.3	-1/2 = -0.5