Diaschismic–gothmic equivalence continuum: Difference between revisions

Line 143:

== kleismic-tetracot continuum ==

~~All~~ temperaments in the continuum ~~also satisfy~~ (15625/15552)''k'' ~ 20000/19683, for a value of ''k'' defined such that 1/''r'' + 1/''k'' = 1. Varying ''k'' (for number of kleismas) 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.

We may also describe the set of all [[5-limit]] [[regular temperament|temperaments]] supported by [[34edo|34et]] by expressing the continuum (15625/15552)''k'' ~ 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 kleismas) 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. 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"

|+ Temperaments with half-integer ''k'' in the kleismic-tetracot continuum

|+ Temperaments with half-integer ''k'' in the kleismic-tetracot continuum

|-

! rowspan="2" | ''k''

@@ Line 143: / Line 143: @@
 == kleismic-tetracot continuum ==
-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 number of <u>k</u>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.
+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]] 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"
-|+ Temperaments with half-integer ''k'' in the kleismic-tetracot continuum
+|+ Temperaments with half-integer ''k'' in the<br />kleismic-tetracot continuum
 |-
 ! rowspan="2" | ''k''