Kleismic family: Difference between revisions
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{{Technical data page}} | {{Technical data page}} | ||
The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]]. | The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5|classical minor third]]s falls short of the [[3/1|3rd]] [[harmonic]]. | ||
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=== Overview to extensions === | === Overview to extensions === | ||
==== Full 7-limit extensions ==== | |||
The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan. | The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan. | ||
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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. | Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts. | ||
The | ==== Other subgroup extensions ==== | ||
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]]} }}. | |||
See [[#Subgroup extensions]]. | |||
== Catakleismic == | == Catakleismic == | ||
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== Subgroup extensions == | == Subgroup extensions == | ||
For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]]. | |||
=== Kleismic (2.3.5.13) a.k.a. cata === | === Kleismic (2.3.5.13) a.k.a. cata === | ||
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. | |||
Subgroup: 2.3.5.13 | Subgroup: 2.3.5.13 | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}} | * WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}} | ||
* CWE: ~2 = 1200. | * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }} | {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 140, 193, 246 }} | ||
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Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy. | Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy. | ||
Subgroup: 2.3.5.13.37 | Subgroup: 2.3.5.13.37 | ||
Comma list: 325/324, 481/480, 625/624 | Comma list: 325/324, 481/480, 625/624 | ||
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Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}} | * WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}} | ||
* CWE: ~2 = 1200. | * CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}} | ||
{{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }} | {{Optimal ET sequence|legend=0| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }} | ||
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Badness (Sintel): 0.223 | Badness (Sintel): 0.223 | ||
[[Category:Kleismic family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||