Diaschismic–gothmic equivalence continuum

The diaschismic-gothmic equivalence continuum (or diaschismic-tetracot equivalence continuum) 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 Pythagorean gothma a.k.a. 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 harmonic 3. For example:

• Immunity (n = 1) splits its twelfth in two;
• Tetracot (n = 2) splits its fifth in four;
• Hanson (n = 3) splits its twelfth in six;
• Etc.

The factor of 2 between n and the split of the interval class of 3 has to do with the fact that 34et has two rings of 17et's.

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 half-integer n and r

r	n	Temperament	Comma
			Ratio	Monzo
-2	0	Gothic	134217728/129140163	[27 -17⟩
-1.5	1/2	22c & 34	(30 digits)	[43 -30 2⟩
-1	1	Immunity	1638400/1594323	[16 -13 2⟩
-0.5	3/2	34 & 36c	(22 digits)	[21 -22 6⟩
0	2	Tetracot	20000/19683	[5 -9 4⟩
0.5	5/2	Fifive	9765625/9565938	[-1 -14 10⟩
1	3	Hanson	15625/15552	[-6 -5 6⟩
1.5	7/2	Vishnu	(20 digits)	[23 6 -14⟩
2	4	Würschmidt	393216/390625	[17 1 -8⟩
2.5	9/2	34 & 142	(28 digits)	[45 -2 18⟩
3	5	Mabila	268435456/263671875	[28 -3 -10⟩
3.5	11/2	34 & 166c	(42 digits)	[67 -10 -22⟩
4	6	Goldis	(24 digits)	[39 -7 -12⟩
…	…	…	…	…
∞	∞	Srutal	2048/2025	[11 -4 -2⟩

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. The immunity comma is both larger and more complex than the diaschisma. As such, this continuum does not contain as many useful temperaments.

Temperaments with non-half-integer fractional n and m

n	m	Temperament	Comma
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⟩
11/3 = 3.6	11/8 = 1.375	Majvam	[40 7 -22⟩

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 half-integer k in the kleismic-tetracot continuum

k	n	Temperament	Comma
			Ratio	Monzo
-2	8/3	34 & 113	(24 digits)	[-7 -19 16⟩
-1	5/2	Fifive	9765625/9565938	[-1 -14 10⟩
0	2	Tetracot	20000/19683	[5 -9 4⟩
0.5	1	Immunity	1638400/1594323	[16 -13 2⟩
1	∞	Srutal	2048/2025	[11 -4 -2⟩
1.5	5	Mabila	268435456/263671875	[28 -3 -10⟩
2	4	Würschmidt	393216/390625	[17 1 -8⟩
2.5	11/3	Majvam	(32 digits)	[40 7 -22⟩
3	7/2	Vishnu	(20 digits)	[23 6 -14⟩
3.5	17/5	Chlorine	(48 digits)	[-52 -17 34⟩
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 n, r and k

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