Kleismic - Revision history

Overthink: single link (see talk)

2026-03-26T21:30:55Z

single link (see talk)

← Older revision		Revision as of 21:30, 26 March 2026
Line 18:		Line 18:
	'''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 }}).		'''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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup\|2.3.5.13~~]]<nowiki/>~~-[[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.		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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup\|2.3.5.13-subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

	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]] comma [[729/728]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).		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]] comma [[729/728]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

Overthink: link 2.3.5.13 subgroup page; add word (MOS:SEAOFBLUE)

2026-03-26T04:27:27Z

link 2.3.5.13 subgroup page; add word (MOS:SEAOFBLUE)

← Older revision		Revision as of 04:27, 26 March 2026
Line 18:		Line 18:
	'''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 }}).		'''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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.		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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup\|2.3.5.13]]<nowiki/>-[[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

	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]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).		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]] comma [[729/728]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

	Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).		Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).

ArrowHead294 at 18:16, 6 March 2026

2026-03-06T18:16:09Z

← Older revision		Revision as of 18:16, 6 March 2026
Line 10:		Line 10:
	\| Mapping = 1; 6 5 14		\| Mapping = 1; 6 5 14
	\| Generators = 6/5 \| Generators tuning = 317.1 \| Optimization method = CWE		\| Generators = 6/5 \| Generators tuning = 317.1 \| Optimization method = CWE
	\| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]		\| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
	\| Pergen = (P8, P12/6)		\| Pergen = (P8, P12/6)
	\| Color name = Tribiyoti		\| Color name = Tribiyoti
Line 166:		Line 166:
	=== Tuning spectrum ===		=== Tuning spectrum ===
	{\| class="wikitable center-all left-4"		{\| class="wikitable center-all left-4"
			\|-
	! Edo<br>generator		! Edo<br>generator
	! [[Eigenmonzo\|Eigenmonzo<br>(unchanged interval)]]*		! [[Eigenmonzo\|Eigenmonzo<br>(unchanged interval)]]*

FloraC: Cleanup on infobox

2026-02-09T10:06:53Z

Cleanup on infobox

← Older revision		Revision as of 10:06, 9 February 2026
Line 1:		Line 1:
	{{~~interwiki~~		{{Interwiki
			\| en = Kleismic
	\| de = Hanson-Kleismisch		\| de = Hanson-Kleismisch
	~~\| en = Kleismic~~
	}}		}}
	{{Infobox regtemp		{{Infobox regtemp
	\| Title = Kleismic~~; hanson; cata~~		\| Title = Kleismic
	\| Subgroups = 2.3.5, 2.3.5.13		\| Subgroups = 2.3.5, 2.3.5.13
	\| Comma basis = [[15625/15552]] (2.3.5); <br> [[325/324]], [[625/624]] (2.3.5.13)		\| Comma basis = [[15625/15552]] (2.3.5); <br>[[325/324]], [[625/624]] (2.3.5.13)
	\| Edo join 1 = 15 \| Edo join 2 = 19		\| Edo join 1 = 15 \| Edo join 2 = 19
	\| ~~Generator~~ = 6/5 \| ~~Generator~~ tuning = 317.1 \| Optimization method = ~~CTE~~		\| Mapping = 1; 6 5 14
			\| Generators = 6/5 \| Generators tuning = 317.1 \| Optimization method = CWE
	\| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]		\| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]
	~~\| Mapping = 1; 6 5 14~~
	\| Pergen = (P8, P12/6)		\| Pergen = (P8, P12/6)
	\| Color name = Tribiyoti		\| Color name = Tribiyoti
	\| Odd limit 1 = 5 \| Mistuning 1 = 1.35 \| Complexity 1 = 7		\| Odd limit 1 = 5 \| Mistuning 1 = 1.35 \| Complexity 1 = 7
	\| Odd limit 2 = (2.3.5.13) 15 \| Mistuning 2 = 2.35 \| Complexity 2 = 15		\| Odd limit 2 = 2.3.5.13 15 \| Mistuning 2 = 2.35 \| Complexity 2 = 15
	}}		}}
	'''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 }}).		'''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 }}).

Hstraub at 13:28, 29 January 2026

2026-01-29T13:28:05Z

← Older revision		Revision as of 13:28, 29 January 2026
Line 1:		Line 1:
	{{interwiki		{{interwiki
	\| de = Hanson-Kleismisch		\| de = Hanson-Kleismisch
	\| en = ~~kleismic~~		\| en = Kleismic
	}}		}}
	{{Infobox regtemp		{{Infobox regtemp

Hstraub: Link to German

2026-01-29T13:27:36Z

Link to German

← Older revision		Revision as of 13:27, 29 January 2026
Line 1:		Line 1:
			{{interwiki
			\| de = Hanson-Kleismisch
			\| en = kleismic
			}}
	{{Infobox regtemp		{{Infobox regtemp
	\| Title = Kleismic; hanson; cata		\| Title = Kleismic; hanson; cata

FloraC: Move the image to tunings section. Cleanup

2025-12-21T13:42:07Z

Move the image to tunings section. Cleanup

← Older revision		Revision as of 13:42, 21 December 2025
Line 113:		Line 113:

	== Tunings ==		== Tunings ==
			[[File:Kleismic.png\|thumb\|alt=Kleismic.png\|A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.]]

	=== Optimized tunings ===		=== Optimized tunings ===
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| ~~Norm~~-based tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| 5-limit norm-based tunings
	\|-		\|-
	! rowspan="2" \| ~~Weight-skew\Order~~ !! colspan="2" \| Euclidean		! rowspan="2" \| !! colspan="2" \| Euclidean
	\|-		\|-
	! Constrained !! Destretched		! Constrained !! Destretched
	\|-		\|-
	! Tenney		! Tenney
	\| ~~(2.3.5)~~ CTE: ~6/5 = 317.~~0308¢~~ \|\| ~~(2.3.5)~~ POTE: ~6/5 = 317.~~007¢~~		\| CTE: ~6/5 = 317.0308{{c}} \|\| POTE: ~6/5 = 317.007{{c}}
	\|-		\|-
	! Equilateral		! Equilateral
	\| ~~(2.3.5)~~ CEE: ~6/5 = 317.~~1033¢~~		\| CEE: ~6/5 = 317.1033{{c}}<br>(11/61-kleisma)
	(11/61-kleisma)		\|}

			{\| class="wikitable mw-collapsible mw-collapsed"
			\|+ style="font-size: 105%; white-space: nowrap;" \| 2.3.5.13-subgroup norm-based tunings
			\|-
			! rowspan="2" \| !! colspan="2" \| Euclidean
			\|-
			! Constrained !! Destretched
	\|-		\|-
	! Tenney		! Tenney
	\| ~~(2.3.5.13)~~ CTE: ~6/5 = 317.~~1110¢~~ \|\| ~~(2.3.5.13)~~ POTE: ~6/5 = 317.~~0756¢~~		\| CTE: ~6/5 = 317.1110{{c}} \|\| POTE: ~6/5 = 317.0756{{c}}
	\|}		\|}

Line 147:		Line 156:
	\| 13:15:18 (+2 +3) \|\| ~6/5 = 317.0010 \|\| 3''g''<sup>3</sup> − ''g'' − 4 = 0 \|\| Close to 13/51-marveltwin comma		\| 13:15:18 (+2 +3) \|\| ~6/5 = 317.0010 \|\| 3''g''<sup>3</sup> − ''g'' − 4 = 0 \|\| Close to 13/51-marveltwin comma
	\|}		\|}

			=== Other tunings ===
			* [[DKW theory\|DKW]] (2.3.5): ~2 = 1200.0000{{c}}, ~6/5 = 317.1983{{c}}

	=== Tuning spectrum ===		=== Tuning spectrum ===
	{\| class="wikitable center-all left-4"		{\| class="wikitable center-all left-4"
	! ~~EDO~~<br />generator		! Edo<br>generator
	! [[Eigenmonzo\|Eigenmonzo<br />(unchanged interval)]]*		! [[Eigenmonzo\|Eigenmonzo<br>(unchanged interval)]]*
	! Generator (¢)		! Generator (¢)
	! Comments		! Comments
Line 320:		Line 332:
	\| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''		\| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''
	\|}		\|}
	<nowiki />* Besides the octave		<nowiki/>* Besides the octave

	~~=== Other tunings ===~~
	* [[DKW theory\|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983

	== Scales ==		== Scales ==
Line 330:		Line 339:
	* [[Cata15]] ([[4L 11s]])		* [[Cata15]] ([[4L 11s]])
	* [[Cata19]] ([[15L 4s]])		* [[Cata19]] ([[15L 4s]])

	~~== Images ==~~
	~~[[File:Kleismic.png\|alt=Kleismic.png\|600x560px]]~~

	A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.

	== Music ==		== Music ==

FloraC: - extravagant arithmetic expressions, false causalities, and misplaced nowraps

2025-12-21T13:34:18Z

- extravagant arithmetic expressions, false causalities, and misplaced nowraps

← Older revision		Revision as of 13:34, 21 December 2025
Line 14:		Line 14:
	'''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 }}).		'''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~~.		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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

	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 well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).		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]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

	Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[~~Just_intonation~~\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).		Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).

	For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].		For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].

FloraC: /* Interval chain */ CTE -> CWE. - enormously complex ratios

2025-12-21T13:14:34Z

Interval chain: CTE -> CWE. - enormously complex ratios

← Older revision		Revision as of 13:14, 21 December 2025
Line 26:		Line 26:

	{\| class="wikitable sortable center-1 right-2"		{\| class="wikitable sortable center-1 right-2"
	! &#~~35;~~		! #
	! Cents*		! Cents*
	! class="unsortable" \| Approximate ratios		! class="unsortable" \| Approximate ratios
Line 36:		Line 36:
	\| 1		\| 1
	\| 317.1		\| 317.1
	\| 6/5~~, 65/54~~		\| 6/5
	\|-		\|-
	\| 2		\| 2
Line 43:		Line 43:
	\|-		\|-
	\| 3		\| 3
	\| ~~950~~.3		\| 951.3
	\| 26/15~~, 45/26~~		\| 26/15
	\|-		\|-
	\| 4		\| 4
Line 51:		Line 51:
	\|-		\|-
	\| 5		\| 5
	\| 385.6		\| 385.5
	\| '''5/4'''~~, 81/65~~		\| '''5/4'''
	\|-		\|-
	\| 6		\| 6
	\| 702.7		\| 702.6
	\| '''3/2'''		\| '''3/2'''
	\|-		\|-
	\| 7		\| 7
	\| 1019.8		\| 1019.6
	\| 9/5~~, 65/36~~		\| 9/5
	\|-		\|-
	\| 8		\| 8
	\| 136.9		\| 136.7
	\| 13/12, 27/25		\| 13/12, 27/25
	\|-		\|-
	\| 9		\| 9
	\| ~~454~~.0		\| 453.8
	\| 13/10		\| 13/10
	\|-		\|-
	\| 10		\| 10
	\| ~~771~~.1		\| 770.9
	\| 25/16, 39/25~~, 81/52~~		\| 25/16, 39/25
	\|-		\|-
	\| 11		\| 11
	\| 1088.2		\| 1088.0
	\| '''15/8'''		\| '''15/8'''
	\|-		\|-
	\| 12		\| 12
	\| 205.3		\| 205.1
	\| '''9/8'''		\| '''9/8'''
	\|-		\|-
	\| 13		\| 13
	\| 522.4		\| 522.2
	\| 27/20~~, 65/48~~		\| 27/20
	\|-		\|-
	\| 14		\| 14
	\| 839.6		\| 839.3
	\| '''13/8'''~~, 81/50~~		\| '''13/8'''
	\|-		\|-
	\| 15		\| 15
	\| 1156.7		\| 1156.4
	\| 39/20		\| 39/20
	\|-		\|-
	\| 16		\| 16
	\| 273.8		\| 273.5
	\| 75/64		\| 75/64
	\|-		\|-
	\| 17		\| 17
	\| 590.9		\| 590.6
	\| 45/32		\| 45/32
	\|-		\|-
	\| 18		\| 18
	\| ~~908~~.0		\| 907.7
	\| 27/16		\| 27/16
	\|-		\|-
	\| 19		\| 19
	\| 25.1		\| 24.7
	\| 65/64, 81/80		\| 65/64, 81/80
	\|}		\|}
	<nowiki />* In 2.3.5.13-subgroup [[~~CTE~~ tuning]]		<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced

	== Tunings ==		== Tunings ==

Overthink: update scale sizes

2025-12-21T02:00:26Z

update scale sizes

← Older revision		Revision as of 02:00, 21 December 2025
Line 9:		Line 9:
	\| Pergen = (P8, P12/6)		\| Pergen = (P8, P12/6)
	\| Color name = Tribiyoti		\| Color name = Tribiyoti
	\| Odd limit 1 = 5 \| Mistuning 1 = 1.35 \| Complexity 1 = 15		\| Odd limit 1 = 5 \| Mistuning 1 = 1.35 \| Complexity 1 = 7
	\| Odd limit 2 = (2.3.5.13) 15 \| Mistuning 2 = 2.35 \| Complexity 2 = 34		\| Odd limit 2 = (2.3.5.13) 15 \| Mistuning 2 = 2.35 \| Complexity 2 = 15
	}}		}}
	'''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 }}).		'''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 }}).
Line 20:		Line 20:
	Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[Just_intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).		Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[Just_intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).

	For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].		For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].

	== Interval chain ==		== Interval chain ==

← Older revision		Revision as of 21:30, 26 March 2026
Line 18:		Line 18:
	'''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 }}).		'''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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup\|2.3.5.13~~]]<nowiki/>~~-[[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.		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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup\|2.3.5.13-subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

	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]] comma [[729/728]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).		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]] comma [[729/728]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

← Older revision		Revision as of 04:27, 26 March 2026
Line 18:		Line 18:
	'''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 }}).		'''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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.		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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the [[2.3.5.13 subgroup\|2.3.5.13]]<nowiki/>-[[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

	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]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).		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]] comma [[729/728]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

	Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).		Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).

← Older revision		Revision as of 13:34, 21 December 2025
Line 14:		Line 14:
	'''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 }}).		'''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~~.		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 {{=}} (27/26)⋅(26/25)⋅(25/24) }}, it is natural to equate 25/24 to [[26/25]] and [[27/26]] as well, thereby tempering out the tunbarsma [[625/624]] ({{S\|25}}) and the marveltwin comma [[325/324]] ([[S-expression\|S25⋅S26]]) respectively, and resulting in a low-complexity but high-accuracy [[extension]] to the 2.3.5.13 [[subgroup]] sometimes known as '''cata'''. From there we can see that [[676/675]] ({{S\|26}}) is also tempered out, meaning [[4/3]] is split into two [[15/13]]'s and that 3/1 is split into two [[26/15]]'s. From {{nowrap\| 325/324 {{=}} (13/9)/(6/5)<sup>2</sup> }} we can see that [[13/9]] is split into two 6/5's, so that it is equated with [[36/25]] (giving rise to the other S-expression of 325/324, [[semiparticular\|S10/S12]]); the implication of this is that the chain of generators naturally gives us hemitwelfths at 3 generator steps of a slightly sharpened ~6/5.

	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 well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).		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]] ({{S\|27}}) in addition to 625/624 and 676/675), and can be defined independently in the [[7-limit]] by tempering out [[225/224]] and [[4375/4374]]. However, countercata is well-tuned closer to the optimal range of kleismic (between [[53edo]] and [[87edo]]), especially that of 2.3.5.13 cata, and naturally emerges in that context, 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).

	Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[~~Just_intonation~~\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).		Most of these extensions can also incorporate prime 11 (and thereby reach the full 13-limit) by tempering out [[385/384]], equating the ~6/5 generator to [[77/64]]. This works well since the optimal tunings of cata's ~6/5 are usually intermediate between [[just intonation\|just]] 6/5 (just flat of [[19edo]]) and 77/64 (just sharp of [[15edo]]).

	For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].		For technical data, see [[Kleismic family #Kleismic a.k.a. hanson]].