Kleismic: Difference between revisions

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'''Kleismic''', alternatively called '''hanson''' in the [[5-limit]], is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which stacked are equated to the [[3/1|perfect twelfth (3/1)]], and thereby characterized by the vanishing of the [[15625/15552|kleisma]] ([[ratio]]: 15625/15552, {{monzo|legend=1| -6 -5 6 }}).

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 {{nowrap| 9/8 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S|25}}) and the marveltwin comma [[325/324]] ([[S-expression|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

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 {{nowrap| 9/8 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S|25}}) and the marveltwin comma [[325/324]] ([[S-expression|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup|2.3.5.13]]<nowiki/>-[[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

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 the [[square superparticular]] [[729/728]] ({{S|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

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 the [[square superparticular]] comma [[729/728]] ({{S|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).

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 '''Kleismic''', alternatively called '''hanson''' in the [[5-limit]], is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[kleismic family]], [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which stacked are equated to the [[3/1|perfect twelfth (3/1)]], and thereby characterized by the vanishing of the [[15625/15552|kleisma]] ([[ratio]]: 15625/15552, {{monzo|legend=1| -6 -5 6 }}).
-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 {{nowrap| 9/8 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S|25}}) and the marveltwin comma [[325/324]] ([[S-expression|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.
+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 {{nowrap| 9/8 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S|25}}) and the marveltwin comma [[325/324]] ([[S-expression|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup|2.3.5.13]]<nowiki/>-[[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.
-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 the [[square superparticular]] [[729/728]] ({{S|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).
+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 the [[square superparticular]] comma [[729/728]] ({{S|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).