Kleismic: Difference between revisions

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{{Redirect|Hanson and cata|the rank-3 temperament family|Kleismic rank three family}}

'''~~Hanson~~''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] of the [[kleismic family]], characterized by the vanishing of the [[15625/15552~~|kleisma~~]]. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]]. This naturally gives us hemitwelfths at only 3 generator steps, which can be interpreted as [[26/15]] (and thus hemifourths as [[15/13]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''.

'''Kleismic''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] of the [[kleismic family]], characterized by the vanishing of the kleisma ( [[15625/15552]]). In the [[5-limit]], it is also known as '''hanson'''. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].

7-~~limit extensions include~~ [[~~keemun~~]], [[~~catalan~~]], [[catakleismic]], [[countercata]], and [[~~metakleismic~~]].

However, 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 would be illogical not to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering 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 S26 = [[676/675]].

Extensions to prime 7 include [[catakleismic]], [[countercata]], [[metakleismic]], [[keemun]], and [[catalan]]. Of these, catakleismic can be considered 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 [[225/224]] and [[4375/4374]].

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

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[[Category:Kleismic| ]]

[[Category:Kleismic family]]

~~[[Category:Kleismic]]~~

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 {{Redirect|Hanson and cata|the rank-3 temperament family|Kleismic rank three family}}
-'''Hanson''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] of the [[kleismic family]], characterized by the vanishing of the [[15625/15552|kleisma]]. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]]. This naturally gives us hemitwelfths at only 3 generator steps, which can be interpreted as [[26/15]] (and thus hemifourths as [[15/13]]), resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]], sometimes known as '''cata'''.
+'''Kleismic''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] of the [[kleismic family]], characterized by the vanishing of the kleisma ( [[15625/15552]]). In the [[5-limit]], it is also known as '''hanson'''. It is [[generator|generated]] by a [[6/5|classical minor third (6/5)]], six of which make a [[3/1|twelfth (3/1)]].
--limit extensions include [[keemun]], [[catalan]], [[catakleismic]], [[countercata]], and [[metakleismic]].
+However, 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 would be illogical not to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering 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 S26 = [[676/675]].
+Extensions to prime 7 include [[catakleismic]], [[countercata]], [[metakleismic]], [[keemun]], and [[catalan]]. Of these, catakleismic can be considered 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 [[225/224]] and [[4375/4374]].
 For technical data, see [[Kleismic family #Hanson]].
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 [[Category:Kleismic| ]] <!-- main article -->
 [[Category:Kleismic family]]
-[[Category:Kleismic]]