Kleismic: Difference between revisions

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

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

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 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]].

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

	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 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 = (25/24)(26/25)(27/26), 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]].