Kleismic family: Difference between revisions

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Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.

==~~== Other subgroup extensions ==~~==

== 2.3.5.13 ==

~~The main extension of note is a very remarkable~~ extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. ~~Notice 15625~~/~~15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The~~ [[S-expression]]-based comma list of ~~the temperament is~~ {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}.

Kleismic lends itself nicely to an extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup. (For a version of cata with a 1\5 period, see [[thunderclysmic]].)

~~See [[#~~Subgroup ~~extensions]]~~.

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}

Optimal tunings:

* CTE: ~2 = 1\1, ~6/5 = 317.1110

* POTE: ~2 = 1\1, ~6/5 = 317.075611

* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}

* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}

Badness (Sintel): 0.131

== Catakleismic ==

@@ Line 36: / Line 36: @@
 Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
-==== Other subgroup extensions ====
+== 2.3.5.13 ==
-The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}.
+Kleismic lends itself nicely to an extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap| {[[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]],) [[676/675|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup. (For a version of cata with a 1\5 period, see [[thunderclysmic]].)
-See [[#Subgroup extensions]].
+Subgroup: 2.3.5.13
+Comma list: 325/324, 625/624
+Subgroup-val mapping: {{mapping| 1 0 1 0 | 0 6 5 14 }}
+Optimal tunings:
+* CTE: ~2 = 1\1, ~6/5 = 317.1110
+* POTE: ~2 = 1\1, ~6/5 = 317.075611
+* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
+* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
+{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }}
+Badness (Sintel): 0.131
 == Catakleismic ==