Diaschismic–gothmic equivalence continuum: Difference between revisions
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The '''diaschismic-kleismic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]]. | The '''diaschismic-kleismic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] that describes the set of all [[5-limit]] temperaments supported by [[34edo]]. | ||
A reasonable way of defining this continuum equates a number of [[2048/2025|diaschismas (2048/2025)]] with the [[393216/390625|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', 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, | A reasonable way of defining this continuum equates a number of [[2048/2025|diaschismas (2048/2025)]] with the [[393216/390625|Würschmidt comma (393216/390625)]], so that (2048/2025)<sup>''n''</sup> ~ 393216/390625. As a result, this may also be called the ''wurschmidt-diaschismic equivalence continuum'', or the ''diaschismic-gothic equivalence continuum'', which is more or less the same thing. The just value of ''n'' is 0.5853…, and temperaments near this tend to be the most accurate. The [[17-comma|gothic comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo, and it has a value of ''n'' = 4. Therefore, one can additionally define ''k'' = 4 - ''n'', 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, twice the numerator of the value of ''k'' represents the number of generator steps required to reach the 3rd harmonic. | ||
All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second table below. It converges to [[Hanson_and_cata|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. | All temperaments in the continuum also satisfy (15625/15552)<sup>''m''</sup> ~ 393216/390625, for a value of ''m'' defined such that 1/''n'' - 1/''m'' = 1; equivalently, we can offset ''m'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, hence the name. Varying ''m'' results in different temperaments listed in the second table below. It converges to [[Hanson_and_cata|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. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ Temperaments | |+ Temperaments in the diaschismic-gothic continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
| Line 33: | Line 33: | ||
| [[20000/19683]] | | [[20000/19683]] | ||
| {{monzo| 5 -9 4}} | | {{monzo| 5 -9 4}} | ||
|- | |||
| 3/2 | |||
| 5/2 | |||
| [[Fifive]] | |||
| 9765625/9565938 | |||
| {{monzo| -1 -14 10}} | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 39: | Line 45: | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6}} | | {{monzo| -6 -5 6}} | ||
|- | |||
| 2/3 | |||
| 10/3 | |||
| [[Gammic]] | |||
| 95367431640625/95105071448064 | |||
| {{monzo| -29 -11 20 }} | |||
|- | |||
| ? | |||
| 7/5 | |||
| [[Chlorine]] | |||
| [ [[582076609134674072265625/581595589965365114830848|very long;]] equal to ([[25/24]])<sup>17</sup> / [[2/1|2]] ] | |||
| {{monzo| -52 -17 34 }} | |||
|- | |||
| ? | |||
| 7/2 | |||
| [[Vishnu]] | |||
| [[6115295232/6103515625]] | |||
| {{monzo| 23 6 -14}} | |||
|- | |- | ||
| 0 | | 0 | ||
| 4 | | 4 | ||
| [[ | | [[Würschmidt]] | ||
| [[393216/390625]] | | [[393216/390625]] | ||
| {{monzo| 17 1 -8}} | | {{monzo| 17 1 -8}} | ||
| Line 75: | Line 99: | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments with integer ''m'' | |+ Temperaments with integer ''m'' in the kleismic | ||
|- | |- | ||
! rowspan="2" | ''m'' | ! rowspan="2" | ''m'' | ||
Revision as of 22:17, 21 July 2024
The diaschismic-kleismic equivalence continuum is a continuum of 5-limit temperaments that describes the set of all 5-limit temperaments supported by 34edo.
A reasonable way of defining this continuum equates a number of diaschismas (2048/2025) with the Würschmidt comma (393216/390625), so that (2048/2025)n ~ 393216/390625. As a result, this may also be called the wurschmidt-diaschismic equivalence continuum, or the diaschismic-gothic equivalence continuum, which is more or less the same thing. The just value of n is 0.5853…, and temperaments near this tend to be the most accurate. The gothic comma (134217728/129140163) is the characteristic 3-limit comma tempered out in 34edo, and it has a value of n = 4. Therefore, one can additionally define k = 4 - n, 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, twice the numerator of the value of k represents the number of generator steps required to reach the 3rd harmonic.
All temperaments in the continuum also satisfy (15625/15552)m ~ 393216/390625, for a value of m defined such that 1/n - 1/m = 1; equivalently, we can offset m by 1, and equate a number of kleismas (15625/15552) with the diaschisma, hence the name. Varying m results in different temperaments listed in the second 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.
| n | k | 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⟩ |
| 3/2 | 5/2 | Fifive | 9765625/9565938 | [-1 -14 10⟩ |
| 1 | 3 | Hanson/Kleismic | 15625/15552 | [-6 -5 6⟩ |
| 2/3 | 10/3 | Gammic | 95367431640625/95105071448064 | [-29 -11 20⟩ |
| ? | 7/5 | Chlorine | [ very long; equal to (25/24)17 / 2 ] | [-52 -17 34⟩ |
| ? | 7/2 | Vishnu | 6115295232/6103515625 | [23 6 -14⟩ |
| 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, ∞ = -∞.
We may invert the continuum by setting m such that 1/n - 1/m = 1. The just value of m is 1.4117…, and temperaments near this tend to be the most accurate ones.
| m | 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 | [-29 -11 20⟩ |
| 3 | 13/4 | Quatracot | 1490116119384765625/1479074071160291328 | [-35 -16 26⟩ |
| … | … | … | … | … |
| ∞ | 3 | Hanson/Kleismic | 15625/15552 | [-6 -5 6⟩ |
| Temperament | m | n | k |
|---|---|---|---|
| Majvam | 1/2 = 0.5 | 1/3 = 0.3 | 11/3 = 3.6 |
| Chlorine | 3/2 = 1.5 | 3/5 = 0.6 | 17/5 = 3.4 |