Diaschismic–gothmic equivalence continuum

The diaschismic-tetracot equivalence continuum (which is the diaschismic-gothmic equivalence continuum with offset 2) is a continuum of 5-limit temperaments describing the set of all 5-limit temperaments supported by 34edo.

All temperaments in the continuum satisfy (2048/2025)ⁿ ~ [27 -17⟩, equating a number of diaschismas (2048/2025) with the gothic comma (134217728/129140163). At n = 2 (which we align with r = 0) we get tetracot, which is an important offset for a number of reasons discussed later. 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.

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 with the tetracot comma (20000/19683), so that (2048/2025)^r ~ 20000/19683. As a result, r = n - 2, and this labeling may also be called the diaschismic-tetracot equivalence continuum. The just value of r is 1.4146…, and temperaments near this tend to be the most accurate.

Temperaments with integer n and r

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

* in projective tuning space, ∞ = -∞.

All temperaments in the continuum also satisfy (15625/15552)^k ~ 20000/19683, for a value of k defined such that 1/r + 1/k = 1. Varying k (for ^{(number of)} kleismas) results in different temperaments listed in the table below. It converges to hanson as k approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by Lériendil. The just value of k is 3.4117…, and temperaments near this tend to be the most accurate.

Temperaments with integer k in the kleismic-tetracot continuum

k	n	Temperament	Comma
			Ratio	Monzo
-2	8/3	34 & 113	152587890625/148769467776	[-7 -19 16⟩
-1	5/2	Fifive	9765625/9565938	[-1 -14 10⟩
0	2	Tetracot	20000/19683	[5 -9 4⟩
1	∞	Srutal	2048/2025	[11 -4 -2⟩
2	4	Würschmidt	393216/390625	[17 1 -8⟩
3	7/2	Vishnu	6115295232/6103515625	[23 6 -14⟩
4	10/3	Gammic	(28 digits)	[-29 -11 20⟩
5	13/4	Quatracot	(38 digits)	[-35 -16 26⟩
…	…	…	…
∞	3	Hanson	15625/15552	[-6 -5 6⟩

Temperaments with fractional r and k

k	r	n	Temperament	Comma
5/2 = 2.5	5/3 = 1.6	11/3 = 3.6	Majvam	[40 7 -22⟩
7/2 = 3.5	7/5 = 1.4	17/5 = 3.4	Chlorine	[-52 -17 34⟩
5/3 = 1.6	5/2 = 2.5	9/2 = 4.5	34 & 142	[45 -2 -18⟩