Kleismic: Difference between revisions

Line 12:

| Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34

}}

'''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 }}).

'''Kleismic''', alternatively called '''hanson''' in the [[5-limit]], 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 {{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]] {{=}} [[semiparticular|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 {{=}} [[semiparticular|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.

@@ Line 12: / Line 12: @@
 | Odd limit 2 = (2.3.5.13) 15 | Mistuning 2 = 2.35 | Complexity 2 = 34
 }}
-'''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 }}).
+'''Kleismic''', alternatively called '''hanson''' in the [[5-limit]], 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 {{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]] {{=}} [[semiparticular|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 {{=}} [[semiparticular|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.