Diaschismic–gothmic equivalence continuum: Difference between revisions
m fix links |
add back a verbal description of the continuum cuz the sentence about inverting n to m reads kind of confusingly/awkwardly otherwise and to provide structural info and better understanding; minor corrections; also "hanson" is an obscure/confusing term, both "cata" and "kleismic" are better, but 5-limit makes more sense as "kleismic" so is preferred, but to keep the hint about "hanson" being an alt name and to have the redirect proper ive left it as the target of the links to kleismic |
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* [[Immunity]] (''n'' = 1) splits its twelfth in two; | * [[Immunity]] (''n'' = 1) splits its twelfth in two; | ||
* [[Tetracot]] (''n'' = 2) splits its fifth in four; | * [[Tetracot]] (''n'' = 2) splits its fifth in four; | ||
* [[Hanson]] (''n'' = 3) splits its twelfth in six; | * [[Hanson|Kleismic]] (''n'' = 3) splits its twelfth in six; | ||
* Etc. | * Etc. | ||
The factor of 2 between ''n'' and the split of the interval class of 3 has to do with the fact that 34et has two [[ring number|rings]] of 17et's. | The factor of 2 between ''n'' and the split of the interval class of 3 has to do with the fact that 34et has two [[ring number|rings]] of 17et's. | ||
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| 1 | | 1 | ||
| 3 | | 3 | ||
| [[Hanson]] | | [[Hanson|Kleismic]] | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6 }} | | {{monzo| -6 -5 6 }} | ||
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|} | |} | ||
We may invert the continuum by setting ''m'' such that 1/''n'' + 1/''m'' = 1. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones. The [[immunity comma]] is both larger and more complex than the diaschisma | We may invert the continuum by setting ''m'' such that 1/''n'' + 1/''m'' = 1. The just value of ''m'' is 1.41414…, and temperaments near this tend to be the most accurate ones. The resulting continuum equates a number of [[immunity comma]]s to the [[gothic comma]], but as the immunity comma is both larger and more complex than the diaschisma, this continuum does not contain as many useful temperaments at simple points which aren't already found by (half-)integer points on the diaschismic-gothmic and kleismic-tetracot continua. | ||
It is worth briefly noting that on this continuum: ''m'' = 0 yields [[gothic]], ''m'' = 1 yields [[diaschismic]] (a.k.a. srutal), ''m'' = 2 yields [[tetracot]] and the simplest non-integer convergent, ''m'' = 3/2, yields [[Hanson|kleismic]], with ''m'' = 1/2 yielding the 34 & 29c temperament which may also be described as the 34 & 107 temperament, which is essentially complementary to the simpler [[immunity]]. | |||
We may also examine temperaments that are structurally nontrivial in that they correspond to non-half-integer fractional ''n'' and ''m'', presented here for potential insight into meanings of their fractional values of ''n'' and ''m'' as they relate to the pergen structures of the temperaments. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ Temperaments with non-half-integer fractional ''n'' and ''m'' | |+ Temperaments with non-half-integer fractional ''n'' and ''m'' | ||
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| 11/3 = 3.{{overline|6}} || 11/8 = 1.375 || [[Majvam]] || {{monzo| 40 7 -22 }} | | 11/3 = 3.{{overline|6}} || 11/8 = 1.375 || [[Majvam]] || {{monzo| 40 7 -22 }} | ||
|} | |} | ||
Note that all of these correspond to half-integer points of either ''n'' or ''k'' ( | Note that all of these correspond to half-integer points of either ''n'' or ''k'' (defined below), hence part of the usefulness of the inversion discussed in the [[#Kleismic-tetracot continuum]] subsection. | ||
== Significance of tetracot == | == Significance of tetracot == | ||
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== Kleismic-tetracot continuum == | == Kleismic-tetracot continuum == | ||
We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of <u>k</u>leismas) results in different temperaments listed in the table below. It converges to [[ | We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)<sup>''k''</sup> ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1 – corresponding to an inversion of the diaschismic-tetracot continuum with respect to tetracot. Varying ''k'' (for number of <u>k</u>leismas) results in different temperaments listed in the table below. It converges to [[Hanson|kleismic]] 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. This also suggests that the kleisma is, loosely speaking, a type of "super-comma" or "meta-comma" for the 5-limit, in its ability to equate so many commas simultaneously into a general purpose comma. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
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| ∞ | | ∞ | ||
| 3 | | 3 | ||
| [[Hanson]] | | [[Hanson|Kleismic]] | ||
| [[15625/15552]] | | [[15625/15552]] | ||
| {{monzo| -6 -5 6 }} | | {{monzo| -6 -5 6 }} | ||