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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]] which equate a number of [[15625/15552|kleismas (15625/15552)]] with the [[2048/2025|diaschisma (2048/2025)]].
The '''diaschismic-kleismic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[15625/15552|kleismas (15625/15552)]] with the [[393216/390625|Würschmidt_comma (393216/390625)]].


All temperaments in the continuum satisfy (15625/15552)<sup>''n''</sup> ~ 2048/2025. 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 2.4117…, 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.


The [[17-comma]] (134217728/129140163) is the characteristic [[3-limit]] comma tempered out in 34edo. It corresponds to a value of ''n'' = -1/3.
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments with integer ''n''
|+ Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 14: Line 16:
! Monzo
! Monzo
|-
|-
| -3
| -4
| 8/3
| 34 & 113
| 34 & 113
| 152587890625/148769467776
| 152587890625/148769467776
| {{monzo| -7 -19 16}}
| {{monzo| -7 -19 16}}
|-
|-
| -2
| -3
| 5/2
| [[Fifive]]
| [[Fifive]]
| 9765625/9565938
| 9765625/9565938
| {{monzo| -1 -14 10}}
| {{monzo| -1 -14 10}}
|-
|-
| -1
| -2
| 2
| [[Tetracot]]
| [[Tetracot]]
| [[20000/19683]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
| {{monzo| 5 -9 4}}
|-
|-
| 0
| -1
| ∞
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
| {{monzo| 11 -4 -2}}
|-
|-
| 1
| 0
| 4
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[393216/390625]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
| {{monzo| 17 1 -8}}
|-
|-
| 2
| 1
| 7/2
| [[Vishnuzmic_family#Vishnu|Vishnu]]
| [[Vishnuzmic_family#Vishnu|Vishnu]]
| [[6115295232/6103515625]]
| [[6115295232/6103515625]]
| {{monzo| 23 6 -14}}
| {{monzo| 23 6 -14}}
|-
|-
| 3
| 2
| 10/3
| [[Gammic_family|Gammic]]
| [[Gammic_family|Gammic]]
| 95367431640625/95105071448064
| 95367431640625/95105071448064
| {{monzo| -28 -11 20}}
| {{monzo| -28 -11 20}}
|-
|-
| 4
| 3
| 13/4
| [[Ragismic_microtemperaments#Quatracot|Quatracot]]
| [[Ragismic_microtemperaments#Quatracot|Quatracot]]
| 1490116119384765625/1479074071160291328
| 1490116119384765625/1479074071160291328
| {{monzo| -33 -16 26}}
| {{monzo| -33 -16 26}}
|-
|-
| …
| …
| …
| …
| …
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|-
|-
| ∞
| ∞
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| [[15625/15552]]
Line 65: Line 77:
|}
|}


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 more or less the same thing. The just value of ''m'' is 1.7083…
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.


{| class="wikitable center-1"
{| class="wikitable center-1"
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|-
|-
! rowspan="2" | ''m''
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 78: Line 91:
|-
|-
| -2
| -2
| 34 & 142
| 6
| 35184372088832/34332275390625
| [[User:Userminusone/Goldis_comma|Goldis]]
| {{monzo| 45 -2 -18}}
| 549755813888/533935546875
| {{monzo| 39 -7 -12}}
|-
|-
| -1
| -1
| 5
| [[Mabila_family#Mabila|Mabila]]
| [[Mabila_family#Mabila|Mabila]]
| 268435456/263671875
| 268435456/263671875
Line 88: Line 103:
|-
|-
| 0
| 0
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| 4
| [[2048/2025]]
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| {{monzo| 11 -4 -2}}
| [[393216/390625]]
| {{monzo| 17 1 -8}}
|-
|-
| 1
| 1
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| [[15625/15552]]
Line 98: Line 115:
|-
|-
| 2
| 2
| [[Vishnu]]
| 2
| [[6115295232/6103515625]]
| [[Tetracot]]
| {{monzo| 23 6 -14}}
| [[20000/19683]]
| {{monzo| 5 -9 4}}
|-
|-
| 3
| 3
| [[High_badness_temperaments#Majvam|Majvam]]
| 1
| 2404631929946112/2384185791015625
| [[Immunity]]
| {{monzo| 40 7 -22}}
| [[1638400/1594323]]
| {{monzo| 16 -13 2}}
|-
| 4
| 0
| [[Gothic]]
| [[134217728/129140163]]
| {{monzo| 27 -17}}
|-
|-
| …
| …
| …
| …
| …
Line 113: Line 139:
|-
|-
| ∞
| ∞
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| ∞
| [[393216/390625]]
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| {{monzo| 17 1 -8}}
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
|}
|}


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|+ Temperaments with fractional ''n'' and ''m''
|+ Temperaments with fractional ''n'' and ''m''
|-
|-
! Temperament !! ''n'' !! ''m''
! Temperament !! ''n'' !! ''m'' !! ''k''
|-
| [[Immunity]] || -1/2 = -0.5 || 1/3 = 0.{{overline|3}}
|-
| [[Gothic]] || -1/3 = -0.{{overline|3}} || 1/4 = 0.25
|-
| [[User:Userminusone/Goldis_comma|Goldis]] || 1/3 = 0.{{overline|3}} || -1/2 = -0.5
|-
|-
| [[Logarithmic_approximants#A_small_34edo_comma|Selenic]] || 41/17 || 41/24
| [[High_badness_temperaments#Majvam|Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}}
|-
|-
| [[Chlorine]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}}
| [[Chlorine]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || 7/3 = 2.{{overline|3}}
|}
|}


[[Category:34edo]]
[[Category:34edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Diaschismic–gothmic_equivalence_continuum"