Kleismic: Difference between revisions

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Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As 9/8 = (25/24)(26/25)(27/26), it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the marveltwin comma (S25 × S26 = [[325/324]]), and the tunbarsma (S25 = [[625/624]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''. As the chain of generators naturally gives us hemitwelfths at only 3 generator steps, this also corresponds directly to an interpretation of these as [[26/15]] (and thus hemifourths as [[15/13]]) by tempering out S26 = [[676/675]].

Extensions with prime 7 include [[catakleismic]] (which adds [[225/224]], finding 7 at 22 generators up), [[countercata]] (which adds [[5120/5103]], finding 7 at 31 generators down), [[metakleismic]] (which adds [[179200/177147]], finding 7 at 56 generators up), [[keemun]] (which adds [[49/48]], finding 7 at 3 generators up), anakleismic (which adds [[2240/2187]], finding 7 at 37 generators up), and [[catalan]] (which adds [[64/63]], finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical, as it makes ~~a natural~~ further equivalence of 25/24~26/25~27/26 to [[28/27]] and can be defined in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata ~~exists~~ closer to the ~~truly~~ optimal range of kleismic (between [[53edo]] and [[87edo]]) ~~and tempers out~~ [[~~4096~~/~~4095~~]] ~~where~~ [[65/64]] ~~and therefore~~ [[64/63]] ~~are close to just~~. Catakleismic and countercata merge in [[53edo]] ~~because one~~ finds 7 at 22 ~~gens~~ up ~~and~~ the ~~other~~ finds it at 31 ~~gens~~ down ~~and~~ 22+31=53~~, making it a good tuning for accessing harmonies of 7~~.

Extensions with prime 7 include [[catakleismic]] (which adds [[225/224]], finding 7 at 22 generators up), [[countercata]] (which adds [[5120/5103]], finding 7 at 31 generators down), [[metakleismic]] (which adds [[179200/177147]], finding 7 at 56 generators up), [[keemun]] (which adds [[49/48]], finding 7 at 3 generators up), anakleismic (which adds [[2240/2187]], finding 7 at 37 generators up), and [[catalan]] (which adds [[64/63]], finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical extension, as it makes an intuitive further equivalence of [[25/24]]~[[26/25]]~[[27/26]] to [[28/27]] (by tempering out [[square-particular]]s [[625/624|S25]], [[676/675|S26]], and [[729/728|S27]]), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), identifying [[64/63]] with [[65/64]] by tempering out [[4096/4095]]. Catakleismic and countercata merge in [[53edo]], as the former finds 7 at 22 generators up while the latter finds it at 31 generators down (22 + 31 = 53).

Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]~~, which~~ works well since ~6/5 ~~should be tuned sharp of~~ just~~, bringing it closer to~~ 77/64~~, which is just at very close to [[15edo]]'s minor third of 320c~~.

Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[Just_intonation|just]] 6/5 and 77/64.

For technical data, see [[Kleismic family #Hanson]].

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 Another useful interpretation of the kleisma as a comma is that it makes the classical chromatic semitone, [[25/24]], into a third-tone by equating three of this interval to [[9/8]]. As 9/8 = (25/24)(26/25)(27/26), it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the marveltwin comma (S25 × S26 = [[325/324]]), and the tunbarsma (S25 = [[625/624]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''. As the chain of generators naturally gives us hemitwelfths at only 3 generator steps, this also corresponds directly to an interpretation of these as [[26/15]] (and thus hemifourths as [[15/13]]) by tempering out S26 = [[676/675]].
-Extensions with prime 7 include [[catakleismic]] (which adds [[225/224]], finding 7 at 22 generators up), [[countercata]] (which adds [[5120/5103]], finding 7 at 31 generators down), [[metakleismic]] (which adds [[179200/177147]], finding 7 at 56 generators up), [[keemun]] (which adds [[49/48]], finding 7 at 3 generators up), anakleismic (which adds [[2240/2187]], finding 7 at 37 generators up), and [[catalan]] (which adds [[64/63]], finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical, as it makes a natural further equivalence of 25/24~26/25~27/26 to [[28/27]] and can be defined in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata exists closer to the truly optimal range of kleismic (between [[53edo]] and [[87edo]]) and tempers out [[4096/4095]] where [[65/64]] and therefore [[64/63]] are close to just. Catakleismic and countercata merge in [[53edo]] because one finds 7 at 22 gens up and the other finds it at 31 gens down and 22+31=53, making it a good tuning for accessing harmonies of 7.
+Extensions with prime 7 include [[catakleismic]] (which adds [[225/224]], finding 7 at 22 generators up), [[countercata]] (which adds [[5120/5103]], finding 7 at 31 generators down), [[metakleismic]] (which adds [[179200/177147]], finding 7 at 56 generators up), [[keemun]] (which adds [[49/48]], finding 7 at 3 generators up), anakleismic (which adds [[2240/2187]], finding 7 at 37 generators up), and [[catalan]] (which adds [[64/63]], finding 7 at 12 generators down). Of these, catakleismic can perhaps be considered the canonical extension, as it makes an intuitive further equivalence of [[25/24]]~[[26/25]]~[[27/26]] to [[28/27]] (by tempering out [[square-particular]]s [[625/624|S25]], [[676/675|S26]], and [[729/728|S27]]), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), identifying [[64/63]] with [[65/64]] by tempering out [[4096/4095]]. Catakleismic and countercata merge in [[53edo]], as the former finds 7 at 22 generators up while the latter finds it at 31 generators down (22 + 31 = 53).
-Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]], which works well since ~6/5 should be tuned sharp of just, bringing it closer to 77/64, which is just at very close to [[15edo]]'s minor third of 320c.
+Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[Just_intonation|just]] 6/5 and 77/64.
 For technical data, see [[Kleismic family #Hanson]].