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Diaschismic–gothmic equivalence continuum: Difference between revisions

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switched to tetracot version of the continuum due to making the inversion relation positive-signed
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The '''diaschismic-gothic equivalence continuum''', also referrable as the ''diaschismic-würschmidt equivalence continuum'', ''diaschismic-kleismic equivalence continuum'', or ''kleismic-würschmidt 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-gothic equivalence continuum''', also referrable as the ''diaschismic-tetracot equivalence continuum'', ''kleismic-tetracot equivalence continuum'', ''diaschismic-würschmidt equivalence continuum'', ''diaschismic-kleismic equivalence continuum'', or ''kleismic-würschmidt 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]].


All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|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 [[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.  
All temperaments in the continuum satisfy (2048/2025)<sup>''n''</sup> ~ {{monzo| 27 -17 }}, equating a number of [[2048/2025|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 [[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.  
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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]].
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]].


Another reasonable way of defining this continuum equates a number of diaschismas 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 labeling may also be called the ''diaschismic-würschmidt equivalence continuum''. The just value of ''k'' is 0.5853…, and temperaments near this tend to be the most accurate.  
Another reasonable way of defining this continuum equates a number of diaschismas with the [[20000/19683|tetracot comma (20000/19683)]], so that (2048/2025)<sup>''r''</sup> ~ 20000/19683. As a result, ''r'' = ''n'' - 2, and this labeling may also be called the ''diaschismic-tetracot equivalence continuum''. The just value of ''r'' is 1.4146…, and temperaments near this tend to be the most accurate.  


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments with integer ''n'' and ''k''
|+ Temperaments with integer ''n'' and ''r''
|-
|-
! rowspan="2" | ''k''
! rowspan="2" | ''r''
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
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! Monzo
! Monzo
|-
|-
| 4
| -2
| 0
| 0
| [[Gothic]]
| [[Gothic]]
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| {{monzo| 27 -17 }}
| {{monzo| 27 -17 }}
|-
|-
| 3
| -1
| 1
| 1
| [[Immunity]]
| [[Immunity]]
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| {{monzo| 16 -13 2 }}
| {{monzo| 16 -13 2 }}
|-
|-
| 2
| 0
| 2
| 2
| [[Tetracot]]
| [[Tetracot]]
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| {{monzo| -6 -5 6 }}
| {{monzo| -6 -5 6 }}
|-
|-
| 0
| 2
| 4
| 4
| [[Würschmidt]]
| [[Würschmidt]]
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| {{monzo| 17 1 -8 }}
| {{monzo| 17 1 -8 }}
|-
|-
| -1
| 3
| 5
| 5
| [[Mabila_family#Mabila|Mabila]]
| [[Mabila_family#Mabila|Mabila]]
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| {{monzo| 28 -3 -10 }}
| {{monzo| 28 -3 -10 }}
|-
|-
| -2
| 4
| 6
| 6
| [[Goldis]]
| [[Goldis]]
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: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.


All temperaments in the continuum also satisfy (15625/15552)<sup>''s''</sup> ~ 393216/390625, for a value of ''s'' defined such that 1/''k'' - 1/''s'' = 1; equivalently, we can offset ''s'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''s'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''s'' 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>''s''</sup> ~ 20000/19683, for a value of ''s'' defined such that 1/''r'' + 1/''s'' = 1; equivalently, we can offset ''s'' by 1, and equate a number of [[15625/15552|kleismas (15625/15552)]] with the diaschisma, giving rise to the name ''diaschismic-kleismic equivalence continuum''. Varying ''s'' results in different temperaments listed in the table below. It converges to [[hanson]] as ''s'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas. The just value of ''s'' is 3.4117…, and temperaments near this tend to be the most accurate.


{| class="wikitable center-1"
{| class="wikitable center-1"
|+ Temperaments with integer ''s'' in the kleismic-würschmidt continuum
|+ Temperaments with integer ''s'' in the kleismic-tetracot continuum
|-
|-
! rowspan="2" | ''s''
! rowspan="2" | ''s''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
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! Monzo
! Monzo
|-
|-
| -4
| -2
| 8/3
| 34 & 113
| 34 & 113
| 152587890625/148769467776
| 152587890625/148769467776
| {{monzo| -7 -19 16 }}
| {{monzo| -7 -19 16 }}
|-
|-
| -3
| -1
| 5/2
| [[Fifive]]
| [[Fifive]]
| 9765625/9565938
| 9765625/9565938
| {{monzo| -1 -14 10 }}
| {{monzo| -1 -14 10 }}
|-
|-
| -2
| 0
| 2
| [[Tetracot]]
| [[Tetracot]]
| [[20000/19683]]
| [[20000/19683]]
| {{monzo| 5 -9 4 }}
| {{monzo| 5 -9 4 }}
|-
|-
| -1
| 1
| ∞
| [[Srutal]]
| [[Srutal]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| {{monzo| 11 -4 -2 }}
|-
|-
| 0
| 2
| 4
| [[Würschmidt]]
| [[Würschmidt]]
| [[393216/390625]]
| [[393216/390625]]
| {{monzo| 17 1 -8 }}
| {{monzo| 17 1 -8 }}
|-
|-
| 1
| 3
| 7/2
| [[Vishnu]]
| [[Vishnu]]
| [[6115295232/6103515625]]
| [[6115295232/6103515625]]
| {{monzo| 23 6 -14 }}
| {{monzo| 23 6 -14 }}
|-
|-
| 2
| 4
| 10/3
| [[Gammic]]
| [[Gammic]]
| (28 digits)
| (28 digits)
| {{monzo| -29 -11 20 }}
| {{monzo| -29 -11 20 }}
|-
|-
| 3
| 5
| 13/4
| [[Quatracot]]
| [[Quatracot]]
| (38 digits)
| (38 digits)
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|-
|-
| ∞
| ∞
| 3
| [[Hanson]]
| [[Hanson]]
| [[15625/15552]]
| [[15625/15552]]
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{| class="wikitable"
{| class="wikitable"
|+ Temperaments with fractional ''k'' and ''s''
|+ Temperaments with fractional ''r'' and ''s''
|-
|-
! ''s'' !! ''k'' !! ''n'' !! Temperament !! Comma
! ''s'' !! ''r'' !! ''n'' !! Temperament !! Comma
|-
|-
| 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}} || [[Majvam]] || {{monzo| 40 7 -22 }}
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || 11/3 = 3.{{overline|6}} || [[Majvam]] || {{monzo| 40 7 -22 }}
|-
|-
| 3/2 = 1.5 || 3/5 = 0.6 || 17/5 = 3.4 || [[Chlorine]] || {{monzo| -52 -17 34 }}
| 7/2 = 3.5 || 7/5 = 1.4 || 17/5 = 3.4 || [[Chlorine]] || [[Septendecima|{{monzo| -52 -17 34 }}]]
|-
|-
| -1/3 = -0.{{overline|3}} || -1/2 = -0.5 || 9/2 = 4.5 || [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] || {{monzo| 45 -2 -18 }}
| 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || 9/2 = 4.5 || [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] || {{monzo| 45 -2 -18 }}
|}
|}


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