Diaschismic–gothmic equivalence continuum
The diaschismic-gothic equivalence continuum is a continuum of 5-limit temperaments that describes 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. 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 würschmidt comma (393216/390625), so that (2048/2025)k ~ 393216/390625. As a result, k = 4 - n, and this may also be called the diaschismic-würschmidt equivalence continuum, which is more or less the same thing. The just value of k is 0.5853…, and temperaments near this tend to be the most accurate.
| k | n | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| 4 | 0 | Gothic | 134217728/129140163 | [27 -17⟩ |
| 3 | 1 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 2 | 2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| 1 | 3 | Hanson | 15625/15552 | [-6 -5 6⟩ |
| 0 | 4 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| -1 | 5 | Mabila | 268435456/263671875 | [28 -3 -10⟩ |
| -2 | 6 | Goldis | 549755813888/533935546875 | [39 -7 -12⟩ |
| … | … | … | … | … |
| ∞* | ∞ | Srutal | 2048/2025 | [11 -4 -2⟩ |
- * in projective tuning space, ∞ = -∞.
| n | k | Temperament | Comma |
|---|---|---|---|
| 5/2 = 2.5 | 3/2 = 1.5 | Fifive | [-1 -14 10⟩ |
| 13/4 = 3.25 | 3/4 = 0.75 | Quatracot | [-33 -16 26⟩ |
| 10/3 = 3.3 | 2/3 = 0.6 | Gammic | [-29 -11 20⟩ |
| 17/5 = 3.4 | 3/5 = 0.6 | Chlorine | [-52 -17 34⟩ |
| 7/2 = 3.5 | 1/2 = 0.5 | Vishnu | [23 6 -14⟩ |
| 11/3 = 3.6 | 1/3 = 0.3 | Majvam | [40 7 -22⟩ |
| 9/2 = 4.5 | -1/2 = -0.5 | 34 & 142 | [45 -2 -18⟩ |
All temperaments in the continuum also satisfy (15625/15552)m ~ 393216/390625, for a value of m defined such that 1/k - 1/m = 1; equivalently, we can offset m by 1, and equate a number of kleismas (15625/15552) with the diaschisma, giving rise to the name diaschismic-kleismic equivalence continuum. Varying m results in different temperaments listed in the table below. It converges to hanson as m approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas.
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -4 | 34 & 113 | 152587890625/148769467776 | [-7 -19 16⟩ |
| -3 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| -2 | Tetracot | 20000/19683 | [5 -9 4⟩ |
| -1 | Srutal | 2048/2025 | [11 -4 -2⟩ |
| 0 | Würschmidt | 393216/390625 | [17 1 -8⟩ |
| 1 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 2 | Gammic | (28 digits) | [-29 -11 20⟩ |
| 3 | Quatracot | (38 digits) | [-35 -16 26⟩ |
| … | … | … | … |
| ∞ | Hanson | 15625/15552 | [-6 -5 6⟩ |
| m | k | Temperament | Comma |
|---|---|---|---|
| 1/2 = 0.5 | 1/3 = 0.3 | Majvam | [40 7 -22⟩ |
| 3/2 = 1.5 | 3/5 = 0.6 | Chlorine | [-52 -17 34⟩ |
| -1/3 = -0.3 | -1/2 = -0.5 | 34 & 142 | [45 -2 -18⟩ |