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.

All temperaments in the continuum satisfy (2048/2025)ⁿ ~ [27 -17⟩. Varying n results in different temperaments listed in the table below. It converges to diaschismic 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 out both commas and thus tempers out all combinations of them. The just value of n is approximately 3.41464…, and temperaments having n near this value tend to be the most accurate ones. This is more properly called the diaschismic-gothmic equivalence continuum.

The gothic comma is the characteristic 3-limit comma tempered out in 34edo. Describing the continuum this way 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 n represents the number of generator steps required to reach the interval class of 3.

Another reasonable way of defining this continuum equates a number of diaschismas (2048/2025) with the würschmidt comma (393216/390625), so that (2048/2025)^k ~ 393216/390625. As a result, k = 4 - n, and this may also be called the diaschismic-würschmidt equivalence continuum, which is more or less the same thing. The just value of k is 0.5853…, and temperaments near this tend to be the most accurate.

Temperaments in the diaschismic-gothic continuum

k	n	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	15625/15552	[-6 -5 6⟩
0	4	Würschmidt	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 m such that 1/n + 1/m = 1. The just value of m is 1.41414…, and temperaments near this tend to be the most accurate ones.

Temperaments with integer m

m	Temperament	Comma
		Ratio	Monzo
0	Gothic	134217728/129140163	[27 -17⟩
1	Srutal	2048/2025	[11 -4 -2⟩
2	Tetracot	20000/19683	[5 -9 4⟩
…	…	…	…
∞	Immunity	1638400/1594323	[16 -13 2⟩

Temperaments with fractional m and n

n	m	Temperament	Comma
5/2 = 2.5	5/3 = 1.6	Fifive	[-1 -14 10⟩
13/4 = 3.25	13/9 = 1.4	Quatracot	[-33 -16 26⟩
10/3 = 3.3	10/7 = 1.428571	Gammic	[-29 -11 20⟩
17/5 = 3.4	17/12 = 1.416	Chlorine	[-52 -17 34⟩
7/2 = 3.5	7/5 = 1.4	Vishnu	[23 6 -14⟩
11/3 = 3.6	11/8 = 1.375	Majvam	[40 7 -22⟩
9/2 = 4.5	9/7 = 1.285714	34 & 142	[45 -2 -18⟩

All temperaments in the continuum also satisfy (15625/15552)^l ~ 393216/390625, for a value of l defined such that 1/k - 1/l = 1; equivalently, we can offset l by 1, and equate a number of kleismas (15625/15552) with the diaschisma. Varying l results in different temperaments listed in the table below. It converges to hanson as l 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 l in the kleismic-würschmidt continuum

l	Temperament	Comma
		Ratio	Monzo
-4	34 & 113	152587890625/148769467776	[-7 -19 16⟩
-3	Fifive	9765625/9565938	[-1 -14 10⟩
-2	Tetracot	20000/19683	[5 -9 4⟩
-1	Srutal	2048/2025	[11 -4 -2⟩
0	Würschmidt	393216/390625	[17 1 -8⟩
1	Vishnu	6115295232/6103515625	[23 6 -14⟩
2	Gammic	(28 digits)	[-29 -11 20⟩
3	Quatracot	(38 digits)	[-35 -16 26⟩
…	…	…	…
∞	Hanson	15625/15552	[-6 -5 6⟩

Temperaments with fractional k and l

l	k	Temperament	Comma
1/2 = 0.5	1/3 = 0.3	Majvam	[40 7 -22⟩
3/2 = 1.5	3/5 = 0.6	Chlorine	[-52 -17 34⟩
-1/3 = -0.3	-1/2 = -0.5	34 & 142	[45 -2 -18⟩