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Diaschismic–gothmic equivalence continuum: Difference between revisions

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Swap n and k following the organization of other continua, and rework the intro to better reflect this change. Reserve "m" for 1/n + 1/m = 1 and use "l" for the other relation
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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 ''diaschismic-würschmidt 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 while the denominator represents .
All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}. 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 [[tempering out|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. This is more properly called the ''diaschismic-gothmic equivalence continuum''.  


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.  
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 [[3/1|3]].


{| class="wikitable center-1"
Another 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>''k''</sup> ~ 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.
 
{| class="wikitable center-1 center-2"
|+ Temperaments in the diaschismic-gothic continuum
|+ Temperaments in the diaschismic-gothic continuum
|-
|-
! rowspan="2" | ''k''
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 20: Line 22:
| [[Gothic]]
| [[Gothic]]
| [[134217728/129140163]]
| [[134217728/129140163]]
| {{monzo| 27 -17}}
| {{monzo| 27 -17 }}
|-
|-
| 3
| 3
Line 26: Line 28:
| [[Immunity]]
| [[Immunity]]
| [[1638400/1594323]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2}}
| {{monzo| 16 -13 2 }}
|-
|-
| 2
| 2
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| [[Tetracot]]
| [[Tetracot]]
| [[20000/19683]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
| {{monzo| 5 -9 4 }}
|-
|-
| 3/2
| 3/2
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| [[Fifive]]
| [[Fifive]]
| 9765625/9565938
| 9765625/9565938
| {{monzo| -1 -14 10}}
| {{monzo| -1 -14 10 }}
|-
|-
| 1
| 1
| 3
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[Hanson]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
| {{monzo| -6 -5 6 }}
|-
|-
| 2/3
| 2/3
| 10/3
| 10/3
| [[Gammic]]
| [[Gammic]]
| 95367431640625/95105071448064
| (28 digits)
| {{monzo| -29 -11 20}}
| {{monzo| -29 -11 20 }}
|-
|-
| 3/5
| 3/5
| 17/5
| 17/5
| [[Chlorine]]
| [[Chlorine]]
| [ [[Septendecima|very long]]; equal to ([[25/24]])<sup>17</sup> / [[2/1|2]] ]
| (48 digits)
| {{monzo| -52 -17 34}}
| {{monzo| -52 -17 34 }}
|-
|-
| 1/2
| 1/2
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| [[Vishnu]]
| [[Vishnu]]
| [[6115295232/6103515625]]
| [[6115295232/6103515625]]
| {{monzo| 23 6 -14}}
| {{monzo| 23 6 -14 }}
|-
|-
| 1/3
| 1/3
| 11/3
| 11/3
| [[Majvam]]
| [[Majvam]]
| 2404631929946112/2384185791015625
| (32 digits)
| {{monzo| 40 7 -22}}
| {{monzo| 40 7 -22 }}
|-
|-
| 0
| 0
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| [[Würschmidt]]
| [[Würschmidt]]
| [[393216/390625]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
| {{monzo| 17 1 -8 }}
|-
|-
| -1/2
| -1/2
| 9/2
| 9/2
| [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34&142]
| [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34&142]
| 35184372088832/34332275390625
| (28 digits)
| {{monzo| 45 -2 -18}}
| {{monzo| 45 -2 -18 }}
|-
|-
| -1
| -1
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| [[Mabila_family#Mabila|Mabila]]
| [[Mabila_family#Mabila|Mabila]]
| 268435456/263671875
| 268435456/263671875
| {{monzo| 28 -3 -10}}
| {{monzo| 28 -3 -10 }}
|-
|-
| -2
| -2
| 6
| 6
| [[User:Userminusone/Goldis_comma|Goldis]]
| [[Goldis]]
| 549755813888/533935546875
| 549755813888/533935546875
| {{monzo| 39 -7 -12}}
| {{monzo| 39 -7 -12 }}
|-
|-
| …
| …
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| ∞*
| ∞*
| ∞
| ∞
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[Srutal]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
| {{monzo| 11 -4 -2 }}
|}
|}
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.
: <nowiki>*</nowiki> 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.
We may invert the continuum by setting ''m'' such that 1/''n'' + 1/''m'' = 1. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments with integer ''m'' in the kleismic-würschmidt continuum
|+ Temperaments with integer ''m''
|-
|-
! rowspan="2" | ''m''
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Gothic]]
| [[134217728/129140163]]
| {{monzo| 27 -17 }}
|-
| 1
| [[Srutal]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
|-
| 2
| [[Tetracot]]
| [[20000/19683]]
| {{monzo| 5 -9 4 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Immunity]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2 }}
|}
 
All temperaments in the continuum also satisfy (15625/15552)<sup>''l''</sup> ~ 393216/390625, for a value of ''l'' defined such that 1/''k'' - 1/''l'' = 1; equivalently, we can offset ''l'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma. Varying ''l'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''l'' 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"
|+ Temperaments with integer ''l'' in the kleismic-würschmidt continuum
|-
! rowspan="2" | ''l''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 122: Line 161:
|-
|-
| -4
| -4
| 8/3
| 34 & 113
| 34 & 113
| 152587890625/148769467776
| 152587890625/148769467776
| {{monzo| -7 -19 16}}
| {{monzo| -7 -19 16 }}
|-
|-
| -3
| -3
| 5/2
| [[Fifive]]
| [[Fifive]]
| 9765625/9565938
| 9765625/9565938
| {{monzo| -1 -14 10}}
| {{monzo| -1 -14 10 }}
|-
|-
| -2
| -2
| 2
| [[Tetracot]]
| [[Tetracot]]
| [[20000/19683]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
| {{monzo| 5 -9 4 }}
|-
|-
| -1
| -1
| ∞
| [[Srutal]]
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
| {{monzo| 11 -4 -2 }}
|-
|-
| 0
| 0
| 4
| [[Würschmidt]]
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[393216/390625]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
| {{monzo| 17 1 -8 }}
|-
|-
| 1
| 1
| 7/2
| [[Vishnu]]
| [[Vishnuzmic_family#Vishnu|Vishnu]]
| [[6115295232/6103515625]]
| [[6115295232/6103515625]]
| {{monzo| 23 6 -14}}
| {{monzo| 23 6 -14 }}
|-
|-
| 2
| 2
| 10/3
| [[Gammic]]
| [[Gammic_family|Gammic]]
| (28 digits)
| 95367431640625/95105071448064
| {{monzo| -29 -11 20 }}
| {{monzo| -29 -11 20}}
|-
|-
| 3
| 3
| 13/4
| [[Quatracot]]
| [[Ragismic_microtemperaments#Quatracot|Quatracot]]
| (38 digits)
| 1490116119384765625/1479074071160291328
| {{monzo| -35 -16 26 }}
| {{monzo| -35 -16 26}}
|-
|-
| …
| …
| …
| …
| …
Line 176: Line 206:
|-
|-
| ∞
| ∞
| 3
| [[Hanson]]
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
| {{monzo| -6 -5 6 }}
|}
|}


{| class="wikitable"
{| class="wikitable"
|+ Temperaments with fractional ''m'' and ''n''
|+ Temperaments with fractional ''k'' and ''l''
|-
|-
! Temperament !! ''m'' !! ''n'' !! ''k''
! Temperament !! ''l'' !! ''k''
|-
|-
| [[High_badness_temperaments#Majvam|Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}}
| [[Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}}
|-
|-
| [[Chlorine]] || 3/2 = 1.5 || 3/5 = 0.6 || 17/5 = 3.4
| [[Chlorine]] || 3/2 = 1.5 || 3/5 = 0.6
|-
|-
| [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34&142]
| [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.13.17&reduce=on&weights=tenney&target=&edos=&commas=35184372088832%2F34332275390625%2C+289%2F288%2C+2197%2F2187&submit_comma=submit 34&142]
| -1/3 = -0.{{overline|3}}
| -1/3 = -0.{{overline|3}}
| -1/2 = -0.5
| -1/2 = -0.5
| 9/2 = 4.5
|}
|}


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