Kleismic: Difference between revisions

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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.

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 ~~in fact~~ 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]], 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.

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

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