Kleismic: Difference between revisions

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'''Kleismic''', known in the [[5-limit]] as '''hanson''' or simply ''kleismic'', 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 9/8 = 27/24 = 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 ~~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]]~~.

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 = 27/24 = 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 {{nowrap| tunbarsma (S25 {{=}} (25/24)/(26/25) {{=}} [[625/624]]) }} and the {{nowrap| marveltwin comma (S25 × S26 {{=}} (25/24)/(27/26) {{=}} [[325/324]] {{=}} S10/S12) }} respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. {{nowrap| From S25 × S26 and }} S25 we can see that {{nowrap| S26 {{=}} (26/25)/(27/26) {{=}} [[676/675]] {{=}} S13/S15 {{=}} ([[4/3|16/12]])/([[15/13]])2 }} is also tempered out, meaning 4/3 is split into two 15/13's and thus {{nowrap| 3/1 (from 22/(4/3)) }} is split into two {{nowrap| 26/15's (from 2/(15/13)) }}. {{nowrap| From 325/324 {{=}} S10/S12 {{=}} ([[13/9]])/([[6/5|12/10]])2 }} we can see that 13/9 is split into two 6/5's, so that it's equated with 36/25; the consequence of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5 because of {{nowrap| (6/5)2 × 6/5 [[~]] 13/9 × 6/5 {{=}} 26/15 }} being half of 3/1 as discussed.

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|S27]] in addition to S25 and S26), 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).

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 '''Kleismic''', known in the [[5-limit]] as '''hanson''' or simply ''kleismic'', 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 9/8 = 27/24 = 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 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]].
+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 = 27/24 = 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 {{nowrap| tunbarsma (S25 {{=}} (25/24)/(26/25) {{=}} [[625/624]]) }} and the {{nowrap| marveltwin comma (S25 × S26 {{=}} (25/24)/(27/26) {{=}} [[325/324]] {{=}} S10/S12) }} respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. {{nowrap| From S25 × S26 and }} S25 we can see that {{nowrap| S26 {{=}} (26/25)/(27/26) {{=}} [[676/675]] {{=}} S13/S15 {{=}} ([[4/3|16/12]])/([[15/13]])<sup>2</sup> }} is also tempered out, meaning 4/3 is split into two 15/13's and thus {{nowrap| 3/1 (from 2<sup>2</sup>/(4/3)) }} is split into two {{nowrap| 26/15's (from 2/(15/13)) }}. {{nowrap| From 325/324 {{=}} S10/S12 {{=}} ([[13/9]])/([[6/5|12/10]])<sup>2</sup> }} we can see that 13/9 is split into two 6/5's, so that it's equated with 36/25; the consequence of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5 because of {{nowrap| (6/5)<sup>2</sup> × 6/5 [[~]] 13/9 × 6/5 {{=}} 26/15 }} being half of 3/1 as discussed.
 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|S27]] in addition to S25 and S26), 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).