Diaschismic–gothmic equivalence continuum: Difference between revisions
m simplify starting sentence |
m change s to k for kleisma and credit Lerendiel for the important discovery about the commas connected by a chain of kleisma |
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| … | | … | ||
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| ∞ | | ∞ | ||
| ∞ | | ∞ | ||
| [[Srutal]] | | [[Srutal]] | ||
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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>'' | All temperaments in the continuum also satisfy (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1. Varying ''k'' (for <sup>(number of)</sup> ''k''leismas) results in different temperaments listed in the table below. It converges to [[hanson]] as ''k'' approaches infinity, and is motivated by the fact that many important temperaments of 34edo follow a chain of commas connected by kleismas as discovered by [[User:Lériendil|Lériendil]]. The just value of ''k'' is 3.4117…, and temperaments near this tend to be the most accurate. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ Temperaments with integer '' | |+ Temperaments with integer ''k'' in the kleismic-tetracot continuum | ||
|- | |- | ||
! rowspan="2" | '' | ! rowspan="2" | ''k'' | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ Temperaments with fractional ''r'' and '' | |+ Temperaments with fractional ''r'' and ''k'' | ||
|- | |- | ||
! '' | ! ''k'' !! ''r'' !! ''n'' !! Temperament !! Comma | ||
|- | |- | ||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || 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 }} | ||