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 [[393216/390625|Würschmidt comma (393216/390625)]].
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]].


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.  
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. 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 ''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.


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.
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 just value of ''m'' is 1.4117…, and temperaments near this tend to be the most accurate ones.


{| class="wikitable center-1 center-2"
{| class="wikitable center-1"
|+ Temperaments with integer ''n''
|+ Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -2
| 6
| [[User:Userminusone/Goldis_comma|Goldis]]
| 549755813888/533935546875
| {{monzo| 39 -7 -12}}
|-
| -1
| 5
| [[Mabila_family#Mabila|Mabila]]
| 268435456/263671875
| {{monzo| 28 -3 -10}}
|-
| 0
| 4
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
|-
| 1
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
|-
| 2
| 2
| [[Tetracot]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
|-
| 3
| 1
| [[Immunity]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2}}
|-
| 4
| 0
| [[Gothic]]
| [[134217728/129140163]]
| {{monzo| 27 -17}}
|-
| …
| …
| …
| …
| …
|-
| ∞
| ∞
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
|}
We may invert the continuum by setting ''n'' such that 1/''n'' - 1/''m'' = 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 center-2"
|+ Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
Line 75: Line 143:
| [[15625/15552]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
| {{monzo| -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.
{| class="wikitable center-1"
|+ Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -2
| 6
| [[User:Userminusone/Goldis_comma|Goldis]]
| 549755813888/533935546875
| {{monzo| 39 -7 -12}}
|-
| -1
| 5
| [[Mabila_family#Mabila|Mabila]]
| 268435456/263671875
| {{monzo| 28 -3 -10}}
|-
| 0
| 4
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
|-
| 1
| 3
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
|-
| 2
| 2
| [[Tetracot]]
| [[20000/19683]]
| {{monzo| 5 -9 4}}
|-
| 3
| 1
| [[Immunity]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2}}
|-
| 4
| 0
| [[Gothic]]
| [[134217728/129140163]]
| {{monzo| 27 -17}}
|-
| …
| …
| …
| …
| …
|-
| ∞
| ∞
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
|}
|}


{| class="wikitable"
{| class="wikitable"
|+ Temperaments with fractional ''n'' and ''m''
|+ Temperaments with fractional ''m'' and ''n''
|-
|-
! Temperament !! ''n'' !! ''m'' !! ''k''
! Temperament !! ''m'' !! ''n'' !! ''k''
|-
|-
| [[High_badness_temperaments#Majvam|Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}}
| [[High_badness_temperaments#Majvam|Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}}

Revision as of 17:34, 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. 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 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. The just value of m is 1.4117…, and temperaments near this tend to be the most accurate ones.

Temperaments with integer n
n 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⟩

We may invert the continuum by setting n such that 1/n - 1/m = 1. This may be called the wurschmidt-diaschismic equivalence continuum, or the diaschismic-gothic equivalence continuum, which is more or less the same thing.

Temperaments with integer m
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 [-28 -11 20⟩
3 13/4 Quatracot 1490116119384765625/1479074071160291328 [-33 -16 26⟩
… … … … …
∞ 3 Hanson/Kleismic 15625/15552 [-6 -5 6⟩
Temperaments with fractional m and n
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
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