Kleismic: Difference between revisions

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: ''"Kleismic" redirects here. For the temperament families, see [[Kleismic family]] and [[Kleismic rank three family]].''

'''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]]~~.

'''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)]], where six of them 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]].

@@ Line 14: / Line 14: @@
 : ''"Kleismic" redirects here. For the temperament families, see [[Kleismic family]] and [[Kleismic rank three family]].''
-'''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]].
+'''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)]], where six of them 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]].