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.
| 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.
| 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⟩ |
| 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 | 2404631929946112/2384185791015625 | [40 7 -22⟩ |
| 3 | 7/2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 3.5 | 17/5 | Chlorine | (24 digits; equal to (25/24)17 / 2) | [-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⟩ |