Diaschismic–gothmic equivalence continuum: Difference between revisions
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All temperaments in the continuum satisfy (15625/15552)<sup>''n''</sup> ~ 393216/390625. Equivalently, we can offset ''n'' by 1, and equate a number of kleismas with the [[2048/2025|diaschisma (2048/2025)]], hence the name. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Hanson_and_cata|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. | All temperaments in the continuum satisfy (15625/15552)<sup>''n''</sup> ~ 393216/390625. Equivalently, we can offset ''n'' by 1, and equate a number of kleismas with the [[2048/2025|diaschisma (2048/2025)]], hence the name. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Hanson_and_cata|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)<sup>''m''</sup> ~ 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. | An equally reasonable way of defining this continuum equates a number of diaschismas with the Würschmidt comma, so that (2048/2025)<sup>''m''</sup> ~ 393216/390625. The value of ''m'' is defined such that 1/''m'' - 1/''n'' = 1, and its just value is 0.5853…. The [[17-comma|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. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
Revision as of 13:47, 21 July 2024
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 |