Kleismic family: Difference between revisions
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=== Overview to extensions === | === Overview to 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. [[875/864]], the keemic comma, gives keemun. [[ | 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]] 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. | ||
[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator. | [[6144/6125]], the porwell comma, gives [[#Hemikleismic|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator. | ||
Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts. | |||
The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]]. | |||
== Catakleismic == | == Catakleismic == | ||
{{Main| Catakleismic }} | {{Main| Catakleismic }} | ||
Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. | |||
Catakleismic extends easily with [[prime interval|prime]] [[13/1|13]]. The [[S-expression]]-based comma list of this extension is {[[169/168|S13]], [[225/224|S15 = S25⋅S26⋅S27]], [[325/324|S10/S12 = S25⋅S26]], ([[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}. | |||
=== 7-limit === | === 7-limit === | ||
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==== 2.3.5.7.13 subgroup ==== | ==== 2.3.5.7.13 subgroup ==== | ||
Subgroup: 2.3.5.7.13 | Subgroup: 2.3.5.7.13 | ||