User:Godtone/diaschismic-tetracot equivalence continuum: Difference between revisions

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m just value of m & exact forms for just values
Godtone (talk | contribs)
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motivation & inverted continuum
Line 94: Line 94:
: <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 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.
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:34edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Revision as of 21:18, 21 July 2024

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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