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 in #Significance of tetracot. 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;
• Kleismic (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	Kleismic	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	Hemienneatonic	(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 resulting continuum equates a number of immunity commas to the gothic comma, but as the immunity comma is both larger and more complex than the diaschisma, this continuum does not contain as many useful temperaments at simple points which aren't already found by (half-)integer points on the diaschismic-gothmic and kleismic-tetracot continua.

It is worth briefly noting that on this continuum: m = 0 yields gothic, m = 1 yields diaschismic (a.k.a. srutal), m = 2 yields tetracot, m = 3 yields the 34 & 36c temperament occurring at n = -1/2, and the simplest non-integer convergent that approximates the JIP, m = 3/2, yields kleismic. A unique (but not very good) temperament in this continuum is m = 1/2, yielding the 34 & 29c temperament which may also be described as the 34 & 107 temperament, which is essentially complementary (w.r.t. 34et) to the simpler immunity.

We may also examine temperaments that are structurally nontrivial in that they correspond to non-half-integer fractional n and m, presented here for potential insight into meanings of their fractional values of n and m as they relate to the pergen structures of the 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⟩

Note that all of these correspond to half-integer points of either n or k (defined below), hence part of the usefulness of the inversion discussed in the #Kleismic-tetracot continuum subsection.

Significance of tetracot

Tetracot appears as the unique simplest minimal positive integer n which achieves:

1. The simplest comma (compare the monzos, ratios or expressions of gothic (n = 0) and immunity (n = 1)).

2. The simplest temperament mapping (compare the mappings of gothic (which has a whopping 17 periods per octave, but lacks the accuracy of something like chlorine) and immunity which takes slightly more generators to reach the same intervals of tetracot, so initially seems comparable, but whose generator's 5-limit interpretation is questionably damaged and complex compared to tetracot).

3. A characteristic damage of 34et which is not trivial; gothic is trivial in that it is just a subgroup restriction, and immunity, though not trivial and comparable in complexity to tetracot, is not characteristic, because it asks for a flat tuning of 5, so that it is arguably more or equally as characteristic of 63edo's or 97edo's representation of the 5-limit, but the fact that it has so many 3's in it when 34edo does not do so well in consistency of 3's to long distance should be a hint that we can do better. Compare with tetracot, which admits comparatively more and lower damage patent tunings and is clearly a type of imperfect simplification corresponding to a structural awkwardness in 5-limit JI – the minimal diesis – so it has characteristic damage on 10/9 (flatwards) and 9/8 (sharpwards) by exaggerating the difference; this is to say, if you look at 34et's tuning of the 5-limit, its damage is strongly characteristic of tetracot. Exaggerating this difference in this way is also characteristic of immunity, but its general tunings are at odds with those of tetracot's so that they only merge in 34edo, which is arguably a more unusual tuning for immunity than it is for tetracot, where it is clearly characteristic.

4. As aforementioned, a convenient point to invert the scale to define the kleismic-tetracot continuum nicely, discussed below.

Kleismic-tetracot continuum

We may also describe the set of all 5-limit temperaments supported by 34et by expressing the continuum (15625/15552)^k ~ 20000/19683, for a value of k defined such that 1/r + 1/k = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying k (for number of kleismas) results in different temperaments listed in the table below. It converges to kleismic 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. This also suggests that the kleisma is, loosely speaking, a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma.

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	Kleismic	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	Hemienneatonic	[45 -2 -18⟩