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The '''diaschismic-tetracot 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]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2.
The '''diaschismic-tetracot 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]]. It is equivalent to the ''diaschismic-gothic equivalence continuum'', where the [[gothic comma]] is found as ([[2048/2025]])<sup>2</sup> * [[20000/19683]], at ''n'' = -2.


Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464…, and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable.
Each ''n'' on the continuum is defined by equating ([[2048/2025]])<sup>''n''</sup> with [[20000/19683]]. The just value of ''n'' is 1.41464… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[2048/2025]]), and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of ''n'' in the general proximity of the just value are also often notable.


: 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.
: 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]] 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"
Line 34: Line 34:
| 1/2
| 1/2
| [[Fifive]]
| [[Fifive]]
| [[9765625/9565938]]
| 9765625/9565938
| {{monzo| -1 -14 10}}
| {{monzo| -1 -14 10}}
|-
|-
| 1
| 1
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[Kleismic|Hanson/kleismic]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
| {{monzo| -6 -5 6}}
Line 63: Line 63:
|-
|-
| 2
| 2
| [[würschmidt]]
| [[Würschmidt]]
| [[393216/390625]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
| {{monzo| 17 1 -8}}
Line 94: Line 94:
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.
: <nowiki>*</nowiki> in projective tuning space, ∞ = -∞.


We may invert the continuum by setting ''n'' 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 ''k'' such that 1/''n'' - 1/''k'' = 1, resulting in a continuum for ([[15625/15552]])<sup>k</sup> ~ [[20000/19683]], motivated by many important temperaments of 34edo being defined by commas connected by kleismas.


The just value of ''k'' is 3.41173… = log<sub>2</sub>([[20000/19683]])/log<sub>2</sub>([[15625/15552]]), and temperaments near this tend to be the most accurate ones.
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments with integer ''m''
|+ Tetracot-kleismic continuum
|-
|-
! rowspan="2" | ''m''
! rowspan="2" | ''m''
! rowspan="2" | ''k''
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 107: Line 108:
! 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
| ∞
| ?
| [[Diaschismic_family#Srutal_aka_diaschismic|Srutal]]
| [[Diaschismic family#Srutal archagall|Srutal]]
| [[2048/2025]]
| [[2048/2025]]
| {{monzo| 11 -4 -2}}
| {{monzo| 11 -4 -2}}
|-
|-
| 0
| 2
| 4
| ?
| [[Würschmidt_family#W.C3.BCrschmidt|Wurschmidt]]
| [[Würschmidt]]
| [[393216/390625]]
| [[393216/390625]]
| {{monzo| 17 1 -8}}
| {{monzo| 17 1 -8}}
|-
|-
| 1
| 3
| 7/2
| ?
| [[Vishnuzmic_family#Vishnu|Vishnu]]
| [[Vishnu]]
| [[6115295232/6103515625]]
| [[6115295232/6103515625]]
| {{monzo| 23 6 -14}}
| {{monzo| 23 6 -14}}
|-
|-
| 2
| 4
| 10/3
| ?
| [[Gammic_family|Gammic]]
| [[Gammic_family|Gammic]]
| 95367431640625/95105071448064
| 95367431640625/95105071448064
| {{monzo| -29 -11 20}}
| {{monzo| -29 -11 20}}
|-
|-
| 3
| 5
| 13/4
| ?
| [[Ragismic_microtemperaments#Quatracot|Quatracot]]
| [[Ragismic_microtemperaments#Quatracot|Quatracot]]
| 1490116119384765625/1479074071160291328
| 1490116119384765625/1479074071160291328
Line 162: Line 163:
|-
|-
| ∞
| ∞
| 3
| ?
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[Hanson_and_cata|Hanson/Kleismic]]
| [[15625/15552]]
| [[15625/15552]]
| {{monzo| -6 -5 6}}
| {{monzo| -6 -5 6}}
|}
|}
{| class="wikitable"
|+ Temperaments with fractional ''m'' and ''n''
|-
! Temperament !! ''m'' !! ''n'' !! ''k''
|-
| [[High_badness_temperaments#Majvam|Majvam]] || 1/2 = 0.5 || 1/3 = 0.{{overline|3}} || 11/3 = 3.{{overline|6}}
|-
| [[Chlorine]] || 3/2 = 1.5 || 3/5 = 0.6 || 17/5 = 3.4
|}
[[Category:34edo]]
[[Category:Equivalence continua]]

Latest revision as of 12:38, 26 April 2025

The diaschismic-tetracot equivalence continuum is a continuum of 5-limit temperaments that describes the set of all 5-limit temperaments supported by 34edo. It is equivalent to the diaschismic-gothic equivalence continuum, where the gothic comma is found as (2048/2025)2 * 20000/19683, at n = -2.

Each n on the continuum is defined by equating (2048/2025)n with 20000/19683. The just value of n is 1.41464… = log2(20000/19683)/log2(2048/2025), and temperaments near this tend to be the most accurate. However, due to this continuum being defined through two reasonably-accurate temperaments and due to the strength of 34edo as a 5-limit temperament (supporting many notable tempered equivalences), simple fractional values of n in the general proximity of the just value are also often notable.

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.
Diaschismic-tetracot equivalence continuum
n Temperament Comma
Ratio Monzo
-2 Gothic 134217728/129140163 [27 -17⟩
-1 Immunity 1638400/1594323 [16 -13 2⟩
0 Tetracot 20000/19683 [5 -9 4⟩
1/2 Fifive 9765625/9565938 [-1 -14 10⟩
1 Hanson/kleismic 15625/15552 [-6 -5 6⟩
4/3 Gammic 95367431640625/95105071448064 [-29 -11 20⟩
7/5 Chlorine [very long but equal to (25/24)17 / 2] [-52 -17 34⟩
3/2 Vishnu 6115295232/6103515625 [23 6 -14⟩
5/3 Majvam 2404631929946112/2384185791015625 [40 7 -22⟩
2 Würschmidt 393216/390625 [17 1 -8⟩
5/2 34&142 35184372088832/34332275390625 [45 -2 18⟩
3 Mabila 268435456/263671875 [28 -3 -10⟩
4 Goldis 549755813888/533935546875 [39 -7 -12⟩
… … … …
∞ Srutal 2048/2025 [11 -4 -2⟩
* in projective tuning space, ∞ = -∞.

We may invert the continuum by setting k such that 1/n - 1/k = 1, resulting in a continuum for (15625/15552)k ~ 20000/19683, motivated by many important temperaments of 34edo being defined by commas connected by kleismas.

The just value of k is 3.41173… = log2(20000/19683)/log2(15625/15552), and temperaments near this tend to be the most accurate ones.

Tetracot-kleismic continuum
m n Temperament Comma
Ratio Monzo
-2 ? 34 & 113 152587890625/148769467776 [-7 -19 16⟩
-1 ? Fifive 9765625/9565938 [-1 -14 10⟩
0 ? Tetracot 20000/19683 [5 -9 4⟩
1 ? Srutal 2048/2025 [11 -4 -2⟩
2 ? Würschmidt 393216/390625 [17 1 -8⟩
3 ? Vishnu 6115295232/6103515625 [23 6 -14⟩
4 ? Gammic 95367431640625/95105071448064 [-29 -11 20⟩
5 ? Quatracot 1490116119384765625/1479074071160291328 [-35 -16 26⟩
… … … … …
∞ ? Hanson/Kleismic 15625/15552 [-6 -5 6⟩
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