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;
- 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.
| 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.
| 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 (define below), hence part of the usefulness of the inversion discussed in the #kleismic-tetracot continuum subsection. They are 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.
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 gothmic (n=0) and immunity (n=1)).
2. The simplest temperament mapping (compare the mappings of gothmic (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; gothmic is trivial in that it's 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
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.
| 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⟩ |
| 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⟩ |