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 diaschisma (2048/2025).
All temperaments in the continuum satisfy (15625/15552)n ~ 2048/2025. 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 2.4117…, and temperaments near this tend to be the most accurate ones.
The 17-comma (134217728/129140163) is the characteristic 3-limit comma tempered out in 34edo. It corresponds to a value of n = -1/3.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -3 | 34 & 113 | 152587890625/148769467776 | [-7 -19 16⟩ |
| -2 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| -1 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 0 | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 1 | Wurschmidt | 393216/390625 | [17 1 -8⟩ |
| 2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 3 | Gammic | 95367431640625/95105071448064 | [-28 -11 20⟩ |
| 4 | Quatracot | 1490116119384765625/1479074071160291328 | [-33 -16 26⟩ |
| … | … | … | … |
| ∞ | 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, which is essentially the same thing. The just value of m is 1.2333…
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Mabila | 268435456/263671875 | [28 -3 -10⟩ |
| 0 | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 1 | Hanson/Kleismic | 15625/15552 | [-6 -5 6⟩ |
| 2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 3 | Majvam | 2404631929946112/2384185791015625 | [40 7 -22⟩ |
| … | … | … | … |
| ∞ | Wurschmidt | 393216/390625 | [17 1 -8⟩ |