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]], characterized by the vanishing of the [[15625/15552|kleisma]] ([[ratio]]: 15625/15552, {{monzo|legend=1| -6 -5 6 }}), which is the difference between a [[3/1|perfect twelfth (3/1)]] and six [[6/5|classical minor thirds (6/5)]], where 6/5 therefore serves as the [[generator]].

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 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]].

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 closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially in the context of 2.3.5.13 cata, 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]], characterized by the vanishing of the [[15625/15552|kleisma]] ([[ratio]]: 15625/15552, {{monzo|legend=1| -6 -5 6 }}), which is the difference between a [[3/1|perfect twelfth (3/1)]] and six [[6/5|classical minor thirds (6/5)]], where 6/5 therefore serves as the [[generator]].
-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 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]].
 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 closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially in the context of 2.3.5.13 cata, 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).