Diaschismic–gothmic equivalence continuum
The diaschismic-kleismic equivalence continuum is a continuum of 5-limit temperaments which equate a number of kleismas (15625/15552) with the Würschmidt_comma (393216/390625).
All temperaments in the continuum satisfy (15625/15552)n ~ 393216/390625. Equivalently, we can offset n by 1, and equate a number of kleismas with the diaschisma (2048/2025), hence the name. Varying n results in different temperaments listed in the table below. It converges to hanson 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 both commas and thus tempers all combinations of them. The just value of n is 1.4117…, and temperaments near this tend to be the most accurate ones.
An equally reasonable way of defining this continuum equates a number of diaschismas with the Würschmidt comma, so that (2048/2025)m ~ 393216/390625. The value of m is defined such that 1/m - 1/n = 1, and its just value is 0.5853…. The gothic comma (134217728/129140163) is the characteristic 3-limit comma tempered out in 34edo, and it has a value of m = 4. Therefore, one can additionally define k = 4 - m, which 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, the value of k represents the number of generator steps required to reach the 3rd harmonic, even as n comes naturally from examining a chain of commas connected by kleismas.
| n | k | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -4 | 8/3 | 34 & 113 | 152587890625/148769467776 | [-7 -19 16⟩ |
| -3 | 5/2 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| -2 | 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| -1 | ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 0 | 4 | Wurschmidt | 393216/390625 | [17 1 -8⟩ |
| 1 | 7/2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 2 | 10/3 | Gammic | 95367431640625/95105071448064 | [-28 -11 20⟩ |
| 3 | 13/4 | Quatracot | 1490116119384765625/1479074071160291328 | [-33 -16 26⟩ |
| … | … | … | … | … |
| ∞ | 3 | Hanson/Kleismic | 15625/15552 | [-6 -5 6⟩ |
We may invert the continuum by setting m such that 1/m - 1/n = 1. This may be called the wurschmidt-diaschismic equivalence continuum, or the diaschismic-gothic equivalence continuum, which is more or less the same thing.
| m | k | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -2 | 6 | Goldis | 549755813888/533935546875 | [39 -7 -12⟩ |
| -1 | 5 | Mabila | 268435456/263671875 | [28 -3 -10⟩ |
| 0 | 4 | Wurschmidt | 393216/390625 | [17 1 -8⟩ |
| 1 | 3 | Hanson/Kleismic | 15625/15552 | [-6 -5 6⟩ |
| 2 | 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 3 | 1 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 4 | 0 | Gothic | 134217728/129140163 | [27 -17⟩ |
| … | … | … | … | … |
| ∞ | ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
| Temperament | n | m | k |
|---|---|---|---|
| Majvam | 1/2 = 0.5 | 1/3 = 0.3 | 11/3 = 3.6 |
| Chlorine | 5/2 = 2.5 | 5/3 = 1.6 | 7/3 = 2.3 |