Kleismic family - Revision history

Godtone: /* 2.3.5.13 subgroup (cata) */ grammar tweak

2026-10-02T01:38:31Z

2.3.5.13 subgroup (cata): grammar tweak

← Older revision		Revision as of 01:38, 2 October 2026
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	The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap\| S13/S15 {{=}} ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. }} From here, we notice that {{nowrap\| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap\| S10/S12 {{=}} (13/9)/([[6/5\|12/10]])<sup>2</sup>, }} hence explaining the mapping of the entire 2.3.5.13 subgroup, and dividing 3/1 into three 13/9's via [[2197/2187]]. (Note that relatedly, because of tempering out S25 and S26, the tone is trisected via {{nowrap\| 24:25:26:27, }} also implying {{nowrap\| [[27/25]][[~]][[13/12]].) }}		The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap\| S13/S15 {{=}} ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. }} From here, we notice that {{nowrap\| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap\| S10/S12 {{=}} (13/9)/([[6/5\|12/10]])<sup>2</sup>, }} hence explaining the mapping of the entire 2.3.5.13 subgroup, and dividing 3/1 into three 13/9's via [[2197/2187]]. (Note that relatedly, because of tempering out S25 and S26, the tone is trisected via {{nowrap\| 24:25:26:27, }} also implying {{nowrap\| [[27/25]][[~]][[13/12]].) }}

	The accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].		The accuracy of cata as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].

	For a version of cata with a 1\5 period, see [[Thunderclysmic]].		For a version of cata with a 1\5 period, see [[Thunderclysmic]].

Godtone: /* 2.3.5.13 subgroup (cata) */ the threedie is related to the trisection of the tone

2026-10-02T01:37:51Z

2.3.5.13 subgroup (cata): the threedie is related to the trisection of the tone

← Older revision		Revision as of 01:37, 2 October 2026
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	=== 2.3.5.13 subgroup (cata) ===		=== 2.3.5.13 subgroup (cata) ===
	The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap\| S13/S15 {{=}} ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. }} From here, we notice that {{nowrap\| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap\| S10/S12 {{=}} (13/9)/([[6/5\|12/10]])<sup>2</sup>, }} hence explaining the mapping of the entire 2.3.5.13 subgroup. (Note that ~~more trivially~~, because of tempering out S25 and S26, the tone is trisected via 24:25:26:27, also implying {{nowrap\| [[27/25]][[~]][[13/12]].) }}		The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap\| S13/S15 {{=}} ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. }} From here, we notice that {{nowrap\| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap\| S10/S12 {{=}} (13/9)/([[6/5\|12/10]])<sup>2</sup>, }} hence explaining the mapping of the entire 2.3.5.13 subgroup, and dividing 3/1 into three 13/9's via [[2197/2187]]. (Note that relatedly, because of tempering out S25 and S26, the tone is trisected via {{nowrap\| 24:25:26:27, }} also implying {{nowrap\| [[27/25]][[~]][[13/12]].) }}

	The accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].		The accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].

Godtone: /* 2.3.5.13 subgroup (cata) */ restore phrasing but include new info

2026-10-02T01:30:04Z

2.3.5.13 subgroup (cata): restore phrasing but include new info

← Older revision		Revision as of 01:30, 2 October 2026
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	=== 2.3.5.13 subgroup (cata) ===		=== 2.3.5.13 subgroup (cata) ===
	The ~~kleisma has~~ the [[S-expression]] ~~S25<sup>2<~~/~~sup>⋅S26~~, ~~which explains how the tone is trisected, and thus tempering out its constituents~~ [[625/624]] ~~({{S\|25}}~~) ~~and~~ [[676/675]] ({{S\|26}}~~) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences,~~ [[~~325~~/~~324~~]] (~~S25⋅S26) is also tempered out, and two generator steps stand in for ~~~[[13/9]]. ~~Finally~~, ~~since this step~~ is ~~one third of a perfect twelfth,~~ the ~~comma~~ [[~~2197~~/~~2187~~]] ~~(S25⋅S26~~<sup>2</sup>~~) is tempered~~ out~~. The third~~ tone is ~~now~~ 25~~/24~~~26~~/25~~~27~~/26~~, ~~which means~~ 27/25 ~~and~~ [[13/12]] ~~are made equal~~.		The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] {{nowrap\| S13/S15 {{=}} ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. }} From here, we notice that {{nowrap\| (26/15)/(13/9) {{=}} 6/5,}} so all that remains is dividing 13/9 into two 6/5's via the semiparticular {{nowrap\| S10/S12 {{=}} (13/9)/([[6/5\|12/10]])<sup>2</sup>, }} hence explaining the mapping of the entire 2.3.5.13 subgroup. (Note that more trivially, because of tempering out S25 and S26, the tone is trisected via 24:25:26:27, also implying {{nowrap\| [[27/25]][[~]][[13/12]].) }}

			The accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].

	For a version of cata with a 1\5 period, see [[Thunderclysmic]].		For a version of cata with a 1\5 period, see [[Thunderclysmic]].

FloraC: Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "

2026-09-30T17:44:17Z

Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "

← Older revision		Revision as of 17:44, 30 September 2026
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	Comma list: 325/324, 625/624		Comma list: 325/324, 625/624

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 1 0 \| 0 6 5 14 }}		{{Mapping\|legend=2\| 1 0 1 0 \| 0 6 5 14 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 169/168, 225/224, 325/324		Comma list: 169/168, 225/224, 325/324

	~~Subgroup-val mapping:~~ {{~~mapping~~\| 1 0 1 -3 0 \| 0 6 5 22 14 }}		{{Mapping\|legend=2\| 1 0 1 -3 0 \| 0 6 5 22 14 }}

	Optimal tunings:		Optimal tunings:

FloraC: Fix various grammatical, formatting, and linking problems

2026-09-30T07:20:36Z

Fix various grammatical, formatting, and linking problems

← Older revision		Revision as of 07:20, 30 September 2026
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	{{Technical data page}}		{{Technical data page}}
	The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5\|classical minor third]]s falls short of the [[3/1\|3rd~~]] [[~~harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots [[9/8]]. As a result, all kleismic temperaments trisect 9/8, for which 1 third tone is 25/24 and two ~~are~~ 27/25.		The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5\|classical minor third]]s falls short of the [[3/1\|3rd harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots the [[9/8]] whole tone. As a result, all kleismic temperaments trisect 9/8, for which one third-tone represents [[25/24]] and two represents [[27/25]].

	== Kleismic a.k.a. hanson ==		== Kleismic a.k.a. hanson ==
	{{Main\| Kleismic }}		{{Main\| Kleismic }}

	The [[generator]] of kleismic is a [[6/5\|classical minor third]], and to get to the interval class of [[5/4\|major thirds]] requires five of these, and so to get to [[3/2\|fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo\|14\53]] is about perfect as a generator, though [[34edo\|9\34]] also makes sense, and [[19edo\|5\19]] and [[15edo\|4\15]] are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].		The [[generator]] of kleismic is a [[6/5\|classical minor third]], and to get to the interval class of [[5/4\|major thirds]] requires five of these, and so to get to [[3/2\|perfect fifths]] requires six. In fact, (6/5)<sup>5</sup> = (5/2)⋅(15625/15552). This 5-limit temperament (virtually a [[microtemperament]]) is sometimes called ''hanson'', and [[53edo\|14\53]] is about perfect as a generator, though [[34edo\|9\34]] also makes sense, and [[19edo\|5\19]] and [[15edo\|4\15]] are possible. Other edo tunings include [[72edo]], [[87edo]] and [[140edo]].

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	=== Overview to extensions ===		=== Overview to extensions ===
	==== Full 7-limit extensions ====		==== Full 7-limit extensions ====
	The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], ~~hemifamity~~, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.		The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], the aberschisma, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.

	[[6144/6125]], the porwell comma, gives [[#Hemikleismic\|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde\|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic\|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths\|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh\|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic\|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds\|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.		[[6144/6125]], the porwell comma, gives [[#Hemikleismic\|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde\|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic\|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths\|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh\|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic\|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds\|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
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	=== 2.3.5.13 subgroup (cata) ===		=== 2.3.5.13 subgroup (cata) ===
	The kleisma has the [[S-expression]] ~~S25²S26~~ which explains how the tone is trisected, thus tempering out its ~~constituyents S25 =~~ [[625/624]] and ~~S26 =~~ [[676/675]] is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] ~~= S25S26 it~~ is also tempered out~~. Cata finds the third-tritave [[³√3]] to be~~ [[13/9]] ~~as its~~ comma [[2197/2187]] ~~also has the expression S25*S26², and the half-tritave [[√3]] to be [[26~~/~~15]], whose difference~~ is ~~the basis S26~~.		The kleisma has the [[S-expression]] S25<sup>2</sup>⋅S26, which explains how the tone is trisected, and thus tempering out its constituents [[625/624]] ({{S\|25}}) and [[676/675]] ({{S\|26}}) is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] (S25⋅S26) is also tempered out, and two generator steps stand in for ~[[13/9]]. Finally, since this step is one third of a perfect twelfth, the comma [[2197/2187]] (S25⋅S26<sup>2</sup>) is tempered out. The third tone is now 25/24~26/25~27/26, which means 27/25 and [[13/12]] are made equal.

	The third tone is now 25/24 ~ 26/25 ~ 27/26, which means 27/25 ~ [[13/12]] are made equal.

	For a version of cata with a 1\5 period, see [[Thunderclysmic]].		For a version of cata with a 1\5 period, see [[Thunderclysmic]].
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	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}[[error map]]: {{val\|+0.121 +0.690 -0.655 -1.022}}		* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}[[error map]]: {{val\|+0.000 +0.597 -0.854 -1.240}}		: error map: {{val\| +0.121 +0.690 -0.655 -1.022 }}
			* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
			: error map: {{val\| 0.000 +0.597 -0.854 -1.240 }}

	{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 140, 193, 246 }}		{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 140, 193, 246 }}
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	{{Main\| Catakleismic }}		{{Main\| Catakleismic }}

	Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap\| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/~~24 - 27~~/~~25 - 9~~/~~8 - 7~~/6 are made equidistant.		Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap\| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/24–27/25–9/8–7/6 are made equidistant.

	=== 7-limit ===		=== 7-limit ===
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	==== 2.3.5.7.13 subgroup ====		==== 2.3.5.7.13 subgroup ====
	Catakleismic extends easily with [[prime interval\|prime]] [[13/1\|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression ~~S25S26S27~~ and the ragisma S25/S27, thus ~~S27 =~~ [[729/728]] is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25 ~ 13/12 is added.		Catakleismic extends easily with [[prime interval\|prime]] [[13/1\|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25⋅S26⋅S27 and the ragisma S25/S27, thus [[729/728]] ({{S\|27}}) is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25~13/12 is added.

	Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions - cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.		Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions – cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.

	Subgroup: 2.3.5.7.13		Subgroup: 2.3.5.7.13

FloraC: Restore subsection of the overview; - no-11 countercata (no viable alternative 11-limit extension)

2026-09-30T07:08:16Z

Restore subsection of the overview; - no-11 countercata (no viable alternative 11-limit extension)

@@ Line 35: / Line 35: @@
 Temperaments involving larger splits include [[#Sqrtphi|sqrtphi]], [[#Quartkeenlig|quartkeenlig]], [[#Novemkleismic|novemkleismic]]. Those split the kleismic structure into five to nine parts.
 === 2.3.5.13 subgroup (cata) ===
@@ Line 523: / Line 526: @@
 [[Badness]] (Sintel): 1.32
 === 11-limit ===

Eufalesio: Added and redescribed many things

2026-09-29T19:43:20Z

Added and redescribed many things

← Older revision		Revision as of 19:43, 29 September 2026
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	{{Technical data page}}		{{Technical data page}}
	The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5\|classical minor third]]s falls short of the [[3/1\|3rd]] [[harmonic]].		The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5\|classical minor third]]s falls short of the [[3/1\|3rd]] [[harmonic]], or equivalently the amount by which a stack of three [[25/24]] overshoots [[9/8]]. As a result, all kleismic temperaments trisect 9/8, for which 1 third tone is 25/24 and two are 27/25.

	== Kleismic a.k.a. hanson ==		== Kleismic a.k.a. hanson ==
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	=== 2.3.5.13 subgroup (cata) ===		=== 2.3.5.13 subgroup (cata) ===
	The ~~structure of~~ the ~~temperament as dividing 3/1 into 6 equal parts can be deduced completely from its~~ [[S-expression]]~~-based comma list of {{nowrap\| {[[325/324\|S10/S12~~ = ~~S25⋅S26]], (~~[[625/624~~\|S25~~]],) [[676/675~~\|S13/S15 = S26~~]]~~} }}~~. ~~Specifically~~, ~~dividing 3~~/~~1 into two halves of ~26/15~~ is ~~equivalent to dividing 4/3 into two halves of ~15/13, hence~~ the [[~~semiparticular~~]] ~~S13/S15 = (~~[[4/~~3\|16/12~~]]~~)/(~~[[15/13]]~~)<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5~~, ~~so all that remains is dividing 13/9 into two 6/5's via~~ the ~~semiparticular S10/S12 = (13/9)/(~~[[6/~~5\|12/10~~]]~~)<sup>2</sup>~~, ~~hence explaining~~ the ~~mapping of the entire 2.3.5.13 subgroup~~.		The kleisma has the [[S-expression]] S25²S26 which explains how the tone is trisected, thus tempering out its constituyents S25 = [[625/624]] and S26 = [[676/675]] is a readily natural subgroup extension. As a result of coincidental S-expression equivalences, [[325/324]] = S25S26 it is also tempered out. Cata finds the third-tritave [[³√3]] to be [[13/9]] as its comma [[2197/2187]] also has the expression S25*S26², and the half-tritave [[√3]] to be [[26/15]], whose difference is the basis S26.

	The ~~accuracy of it as providing a slightly flat 5~~/~~4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32~~/25~~]][[~~~~~]][[41~~/~~32]]~~, ~~tempering out~~ [[~~1025~~/~~1024~~]].		The third tone is now 25/24 ~ 26/25 ~ 27/26, which means 27/25 ~ [[13/12]] are made equal.

	For a version of cata with a 1\5 period, see [[Thunderclysmic]].		For a version of cata with a 1\5 period, see [[Thunderclysmic]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13
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	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}		* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}[[error map]]: {{val\|+0.121 +0.690 -0.655 -1.022}}
	* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}[[error map]]: {{val\|+0.000 +0.597 -0.854 -1.240}}

	{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 140, 193, 246 }}		{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 140, 193, 246 }}
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	{{Main\| Catakleismic }}		{{Main\| Catakleismic }}

	Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap\| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings.		Catakleismic tempers out 225/224, the [[marvel comma]], and 4375/4374, the [[ragisma]], and may be described as the {{nowrap\| 53 & 72 }} temperament. [[125edo]] and especially [[197edo]] make for excellent tunings. In here, the JI intervals 25/24 - 27/25 - 9/8 - 7/6 are made equidistant.

	~~Catakleismic extends easily with [[prime interval\|prime]] [[13~~/~~1\|13]]. The [[S~~-~~expression]]~~-~~based comma list of this extension is {[[169~~/~~168\|S13]], [[225/224\|S15 = S25⋅S26⋅S27]], [[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]], [[676/675\|S26 = S13/S15]], [[729~~/~~728\|S27]])}~~.

	=== 7-limit ===		=== 7-limit ===
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	==== 2.3.5.7.13 subgroup ====		==== 2.3.5.7.13 subgroup ====
			Catakleismic extends easily with [[prime interval\|prime]] [[13/1\|13]] as seen before with cata, which has the S-expression list {S25, S26}. The marvel comma has an equivalent S-expression S25S26S27 and the ragisma S25/S27, thus S27 = [[729/728]] is another constituent that can be naturally tempered out. As a result, the third tone becomes 25/24~26/25~27/26~28/27, and the equivalence 27/25 ~ 13/12 is added.

			Add-13 catakleismic is less accurate than cata or catakleismic alone as the error of 7 and 13 go in separate directions - cata wants a sharper 6/5 but catakleismic wants a flatter one. Nontheless, its structural properties make it quite notable.

	Subgroup: 2.3.5.7.13		Subgroup: 2.3.5.7.13

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	[[Badness]] (Sintel): 1.32		[[Badness]] (Sintel): 1.32

			==== 2.3.5.7.13 subgroup ====
			Subgroup: 2.3.5.7.13

			Comma list: 169/168, 225/224, 325/324

			Subgroup-val mapping: {{mapping\| 1 0 1 11 0 \| 0 6 5 -31 14 }}

			Optimal tunings:
			* WE: ~2 = 1199.981{{c}}, ~6/5 = 317.124{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.129{{c}}

			{{Optimal ET sequence\|legend=0\|TBA}}

			Badness (Sintel): 0.675

	=== 11-limit ===		=== 11-limit ===

Godtone: /* Subgroup extensions */ out of place

2026-09-29T15:58:38Z

Subgroup extensions: out of place

← Older revision		Revision as of 15:58, 29 September 2026
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	Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.		Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.

	~~==== Subgroup extensions ====~~
	~~Kleismic lends itself to a natural extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. This is discussed immediately below.~~

	=== 2.3.5.13 subgroup (cata) ===		=== 2.3.5.13 subgroup (cata) ===

FloraC: Restore categories

2026-09-29T07:25:09Z

Restore categories

← Older revision		Revision as of 07:25, 29 September 2026
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	Badness (Sintel): 1.61		Badness (Sintel): 1.61

			[[Category:Kleismic family\| ]] <!-- main article -->
			[[Category:Kleismic]]
			[[Category:Temperament families]]
			[[Category:Catalogs of rank-2 temperaments]]

FloraC: Move cata under overview (least surprise principle). Cata isn't officially deprecated. - CTE & POTE tunings (covered in the main page)

2026-09-29T07:24:06Z

Move cata under overview (least surprise principle). *Cata* isn't officially deprecated. - CTE & POTE tunings (covered in the main page)

← Older revision		Revision as of 07:24, 29 September 2026
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	[[Badness]] (Sintel): 0.310		[[Badness]] (Sintel): 0.310
	=== 2.3.5.13 ===
	Kleismic lends itself to a natural extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5\|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup. ~~This temperament has historically been called cata. Note that the~~ accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension to prime 41 via [[32/25]][[~]][[41/32]] ~~(via~~ [[1025/1024]]). (For a version of cata with a 1\5 period, see [[~~thunderclysmic~~]].)		=== Overview to extensions ===
			==== Full 7-limit extensions ====
			The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.

			[[6144/6125]], the porwell comma, gives [[#Hemikleismic\|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde\|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic\|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths\|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh\|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic\|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds\|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

			Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.

			==== Subgroup extensions ====
			Kleismic lends itself to a natural extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. This is discussed immediately below.

			=== 2.3.5.13 subgroup (cata) ===
			The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5\|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.

			The accuracy of it as providing a slightly flat 5/4 in ideal tunings lends a possible (but complex) extension for prime 41 via [[32/25]][[~]][[41/32]], tempering out [[1025/1024]].

			For a version of cata with a 1\5 period, see [[Thunderclysmic]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13
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	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~2 = ~~1\1, ~6/5 = 317~~.~~1110~~		* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
	* POTE: ~2 = 1\1, ~6/5 = 317.~~075611~~
	* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{c}}
	* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}

	{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 140, 193, 246 }}		{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 140, 193, 246 }}

	Badness (Sintel): 0.131		Badness (Sintel): 0.131

	~~=== Overview to extensions ===~~
	The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.

	[[6144/6125]], the porwell comma, gives [[#Hemikleismic\|hemikleismic]]. [[245/243]], sensamagic, gives [[#Clyde\|clyde]]. [[1029/1024]], the gamelisma, gives [[#Tritikleismic\|tritikleismic]]. [[10976/10935]], hemimage, gives [[#Marfifths\|marfifths]]. [[1728/1715]], the orwellismia, gives [[#Kleiboh\|kleiboh]]. [[2401/2400]], the breedsma, gives [[#Quadritikleismic\|quadritikleismic]]. [[2460375/2458624]], the breeze comma, gives [[#Marthirds\|marthirds]]. Hemikleismic splits the 6/5 in half to get a neutral second generator of ~35/32, and clyde similarly splits the 5/3 in half to get a ~9/7 generator. Marfifths splits the 12/5 into three. Kleiboh splits the 24/5 into three. Marthirds splits the 12/5 into four. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

	~~Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.~~

	== Catakleismic ==		== Catakleismic ==