Diaschismic–gothmic equivalence continuum

The diaschismic-kleismic equivalence continuum is a continuum of 5-limit temperaments that describes the set of all 5-limit temperaments supported by 34edo.

A reasonable way of defining this continuum equates a number of diaschismas (2048/2025) with the Würschmidt comma (393216/390625), so that (2048/2025)ⁿ ~ 393216/390625. As a result, this may also be called the wurschmidt-diaschismic equivalence continuum, or the diaschismic-gothic equivalence continuum, which is more or less the same thing. The just value of n is 0.5853…, and temperaments near this tend to be the most accurate. The gothic comma (134217728/129140163) is the characteristic 3-limit comma tempered out in 34edo, and it has a value of n = 4. Therefore, one can additionally define k = 4 - n, which has notable advantages - in particular, due to being determined in terms of the 3-limit comma and the comma with the next lowest power of 5, (twice the numerator of) the value of k represents the number of generator steps required to reach the 3rd harmonic.

All temperaments in the continuum also satisfy (15625/15552)^m ~ 393216/390625, for a value of m defined such that 1/n - 1/m = 1; equivalently, we can offset m by 1, and equate a number of kleismas (15625/15552) with the diaschisma, hence the name. Varying m results in different temperaments listed in the second table below. It converges to hanson as m approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.

Temperaments with integer n

n	k	Temperament	Comma
			Ratio	Monzo
4	0	Gothic	134217728/129140163	[27 -17⟩
3	1	Immunity	1638400/1594323	[16 -13 2⟩
2	2	Tetracot	20000/19683	[5 -9 4⟩
1	3	Hanson/Kleismic	15625/15552	[-6 -5 6⟩
0	4	Wurschmidt	393216/390625	[17 1 -8⟩
-1	5	Mabila	268435456/263671875	[28 -3 -10⟩
-2	6	Goldis	549755813888/533935546875	[39 -7 -12⟩
…	…	…	…	…
∞	∞*	Srutal	2048/2025	[11 -4 -2⟩

* in projective tuning space, ∞ = -∞.

We may invert the continuum by setting n such that 1/n - 1/m = 1. The just value of m is 1.4117…, and temperaments near this tend to be the most accurate ones.

Temperaments with integer m

m	k	Temperament	Comma
			Ratio	Monzo
-4	8/3	34 & 113	152587890625/148769467776	[-7 -19 16⟩
-3	5/2	Fifive	9765625/9565938	[-1 -14 10⟩
-2	2	Tetracot	20000/19683	[5 -9 4⟩
-1	∞	Srutal	2048/2025	[11 -4 -2⟩
0	4	Wurschmidt	393216/390625	[17 1 -8⟩
1	7/2	Vishnu	6115295232/6103515625	[23 6 -14⟩
2	10/3	Gammic	95367431640625/95105071448064	[-29 -11 20⟩
3	13/4	Quatracot	1490116119384765625/1479074071160291328	[-35 -16 26⟩
…	…	…	…	…
∞	3	Hanson/Kleismic	15625/15552	[-6 -5 6⟩

Temperaments with fractional m and n

Temperament	m	n	k
Majvam	1/2 = 0.5	1/3 = 0.3	11/3 = 3.6
Chlorine	3/2 = 1.5	3/5 = 0.6	17/5 = 3.4