Kleismic family - Revision history

FloraC: /* Catakleismic */ move music to the dedicated article

2026-02-16T09:24:34Z

Catakleismic: move music to the dedicated article

← Older revision		Revision as of 09:24, 16 February 2026
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	Badness (Sintel): 0.410		Badness (Sintel): 0.410

	~~Music:~~
	* [https://youtu.be/6Kxq5uxu-c8 Improv in CWE Catakleismic11] - [[Budjarn Lambeth]] (2026)

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

BudjarnLambeth: /* 2.3.5.7.13 subgroup */ add music

2026-02-16T05:43:22Z

2.3.5.7.13 subgroup: add music

← Older revision		Revision as of 05:43, 16 February 2026
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	Badness (Sintel): 0.410		Badness (Sintel): 0.410

			Music:
			* [https://youtu.be/6Kxq5uxu-c8 Improv in CWE Catakleismic11] - [[Budjarn Lambeth]] (2026)

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

BudjarnLambeth: /* 2.3.5.13.37 subgroup */

2026-02-03T05:52:24Z

2.3.5.13.37 subgroup

← Older revision		Revision as of 05:52, 3 February 2026
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	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}		* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
	* CWE: ~2 = 1200.~~9000~~{{c}}, ~6/5 = 317.0452{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0452{{c}}

	{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}		{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}

BudjarnLambeth: /* Kleismic (2.3.5.13) a.k.a. cata */ incorrect cwe

2026-02-03T05:52:12Z

Kleismic (2.3.5.13) a.k.a. cata: incorrect cwe

← Older revision		Revision as of 05:52, 3 February 2026
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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}}
	* CWE: ~2 = 1200.~~1210~~{{c}}, ~6/5 = 317.0920{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0920{{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 }}
Line 1,262:		Line 1,262:
	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}		* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
	* CWE: ~2 = 1200.~~000~~{{c}}, ~6/5 = 317.0452{{c}}		* CWE: ~2 = 1200.9000{{c}}, ~6/5 = 317.0452{{c}}

	{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}		{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}

Xenllium at 13:00, 10 January 2026

2026-01-10T13:00:14Z

← Older revision		Revision as of 13:00, 10 January 2026
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	=== Overview to extensions ===		=== 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]] 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]], 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.		[[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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	=== Diatessic ===		=== Diatessic ===
	Diatessic may be described as {{nowrap\| 121 & 140 }}} and is closely related to the Diatess tuning (generator: 505.727281 cents).		Diatessic may be described as {{nowrap\| 121 & 140 }} and is closely related to the Diatess tuning (generator: 505.727281 cents).

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

FloraC: Expand

2026-01-06T13:55:34Z

Expand

← Older revision		Revision as of 13:55, 6 January 2026
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	=== Overview to extensions ===		=== 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. [[875/864]], the keemic comma, gives keemun. [[~~4375~~/~~4374~~]], ~~the ragisma~~, gives ~~catakleismic~~. [[~~5120~~/~~5103~~]], ~~hemifamity~~, gives countercata~~. Keemun~~, ~~catakleismic~~ and ~~countercata~~ all have octave period and use the minor third as a generator; catakleismic and ~~countercata~~ define the 7/4 more complexly but more accurately than keemun.		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]] 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. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. 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. 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.

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

			The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].

	== Catakleismic ==		== Catakleismic ==
	{{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 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 ====
	The [[S-expression]]-based comma list of this temperament is {[[169/168\|S13]], [[225/224\|S15 = S25S26S27]], [[325/324\|S10/S12 = S25*S26]], ([[625/624\|S25]], [[676/675\|S26 = S13/S15]], [[729/728\|S27]])}.

	Subgroup: 2.3.5.7.13		Subgroup: 2.3.5.7.13

FloraC: Improve intros to certain entries

2026-01-06T13:00:05Z

Improve intros to certain entries

← Older revision		Revision as of 13:00, 6 January 2026
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	[[Badness]] (Sintel): 1.43		[[Badness]] (Sintel): 1.43

	Music:		; Music
	* ''[https://www.youtube.com/watch?v=vdjhC9i5KF4 Four Short Experiments in Octave Stretched 42edo (~~Dec~~ 2024)~~]''~~ by [[Budjarn Lambeth]]		* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024) by [[Budjarn Lambeth]]

	=== 11-limit ===		=== 11-limit ===
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	== Marfifths ==		== Marfifths ==
	~~The ''~~marfifths~~'' temperament (19 & 140)~~ tempers out the [[hemimage comma]], ~~10976/10935~~. It ~~splits the interval~~ of a ~~major thirteenth (~10/3) into three marvelous~~ fifth ([[112/75]]) ~~intervals~~, ~~and uses it for a~~ generator.		Named by [[Xenllium]] in 2021, marfifths tempers out the 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap\| 19 & 140 }} temperament. It is generated by a marvel fourth of [[75/56]] (or a marvel fifth of [[112/75]]), three of which minus an octave make the hanson generator of ~6/5. Its [[ploidacot]] is zeta-18-cot.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	=== Diatessic ===		=== Diatessic ===
	~~The ''diatessic'' temperament (~~121 & 140) is closely related to the ~~'''diatess~~ tuning~~'''~~ (generator: 505.727281 cents).		Diatessic may be described as {{nowrap\| 121 & 140 }}} and is closely related to the Diatess tuning (generator: 505.727281 cents).

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	=== Marf ===		=== Marf ===
	~~The ''marf'' temperament (~~19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.		Marf may be described as {{nowrap\| 19 & 121 }}. It has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	== Marthirds ==		== Marthirds ==
	~~The ''~~marthirds~~'' temperament (19 & 193)~~ tempers out the breeze comma ~~(laquadru~~-~~atruyo~~ comma), [[~~2460375~~/~~2458624~~]]~~. It splits~~ the ~~interval~~ of ~~minor tenth (~12~~/5~~) into four marvelous major third (~~[[~~56/45~~]]~~) intervals, and uses it for a generator~~.		Named by [[Xenllium]] in 2021, marthirds tempers out 2460375/2458624, the [[breeze comma]], and may be described as the {{nowrap\| 19 & 193 }} temperament. It is generated by a marvel-comma-flat classical major third, [[56/45]], four of which minus an octave make the hanson generator of [[6/5]]. Its [[ploidacot]] is zeta-24-cot.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	{{Main\| Sqrtphi }}		{{Main\| Sqrtphi }}

	The just value of sqrt (φ) is 416.545 cents.		Sqrtphi tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap\| 49 & 72 }} temperament. The just value of sqrt(φ) is 416.545 cents, and this temperament gives a close approximation of it.

			Note that in the data below, the generator is given as its [[octave complement]], which stands in for [[~]][[11/7]] from the [[11-limit]] onwards. Five generators octave reduced make the hanson generator of ~[[6/5]]. The [[ploidacot]] for this temperament is 19-sheared 30-cot.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Quartkeenlig ==		== Quartkeenlig ==
	~~Quartkeenlig~~ uses a generator ~~in the 11-limit~~ that is 33/32~36/35 tempered together, and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching\|stretched 23edo]].		Named by [[Eliora]] in 2022, quartkeenlig uses a generator that is a quartertone of [[33/32]][[~]][[36/35]] tempered together in the [[11-limit]], and is called so because it tempers out the [[quartisma]] by virtue of five 33/32's being with [[7/6]], keenanisma, [[385/384]], tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). As six quartertones make the hanson generator of ~[[6/5]], its [[ploidacot]] is alpha-36-cot. It can also be viewed as a regular temperament interpretation of [[23edo and octave stretching\|stretched 23edo]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	== Subgroup extensions ==		== Subgroup extensions ==
	=== Kleismic (2.3.5.13) a.k.a. cata ===		=== Kleismic (2.3.5.13) a.k.a. cata ===
	Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])([[625/624]]) and 325/324 = (625/624)([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324\|S10/S12 = ~~S25*S26~~]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].		Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

FloraC: Sorting

2026-01-06T10:33:16Z

Sorting

Show changes

FloraC: - CTE & POTE tunings

2026-01-06T10:31:16Z

- CTE & POTE tunings

Show changes

FloraC: Switch to Sintel's badness, WE & CWE tunings (3/3)

2026-01-06T10:28:27Z

Switch to Sintel's badness, WE & CWE tunings (3/3)

← Older revision		Revision as of 10:28, 6 January 2026
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	Optimal tunings:		Optimal tunings:
	* CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}		* WE: ~2 = 1200.1210{{c}}, ~6/5 = 317.1076{{c}}
	* POTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0756{{c}}		* CWE: ~2 = 1200.1210{{c}}, ~6/5 = 317.0920{{c}}
			<!-- * CTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.1110{{c}}
			* POTE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0756{{c}} -->

	{{Optimal ET sequence\|legend=1\| 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

	==== 2.3.5.13.37~~.41~~ subgroup ====		==== 2.3.5.13.37 subgroup ====
	Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.		Hanson can be extended even further to the 2.3.5.13.37.41 subgroup while maintaining a rather low complexity and high accuracy.

	Subgroup: 2.3.5.13.37.41		Subgroup: 2.3.5.13.37.41

	Comma list: 325/324, 625/624, [[481/480]], ~~[[1600~~/~~1599]]~~		Comma list: 325/324, 481/480, 625/624

			Subgroup-val mapping: {{mapping\| 1 0 1 0 6 \| 0 6 5 14 -3 }}

			Optimal tunings:
			* WE: ~2 = 1200.2924{{c}}, ~6/5 = 317.0998{{c}}
			* CWE: ~2 = 1200.000{{c}}, ~6/5 = 317.0452{{c}}

			{{Optimal ET sequence\|legend=0\| 15, 19, 34, 53, 299l, 352fl, 405fl, 458fl, 511cfll, 564cffll }}

			Badness (Sintel): 0.167

			==== 2.3.5.13.37.41 subgroup ====
			Subgroup: 2.3.5.13.37.41

			Comma list: 325/324, 481/480, 625/624, 1025/1024

	Subgroup-val mapping: {{mapping\| 1 0 1 0 6 8 \| 0 6 5 14 -3 -10 }}		Subgroup-val mapping: {{mapping\| 1 0 1 0 6 8 \| 0 6 5 14 -3 -10 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~165~~{{c}}, ~6/5 = 317.~~113~~{{c}}		* WE: ~2 = 1200.1651{{c}}, ~6/5 = 317.1126{{c}}
	* CWE: ~2 = 1200.~~000~~{{c}}, ~6/5 = 317.~~075~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.0748{{c}}

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

	Badness (Sintel): 0.223		Badness (Sintel): 0.223