Cathartic family - Revision history

FloraC: Migrate info from cathartic article

2026-08-10T14:30:48Z

Migrate info from cathartic article

← Older revision		Revision as of 14:30, 10 August 2026
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	== Cathartic ==		== Cathartic ==
	{{~~Main~~\| ~~Canou~~ }}		[[File:Lattice Orthocanousmic.png\|thumb\|Lattice for cathartic.]]
			[[File:Lattice Orthocanousmic Rearranged.png\|thumb\|Ditto, but re-arranged into hemitwelfths.]]
			{{See also\| Cathartic scales }}

	Cathartic (formerly ''canou'') features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic.		Cathartic (formerly ''canou'') features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic.

	A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], associating it with the [[amicable]] temperament.		A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], associating it with the [[amicable]] temperament.

			9-, 14- and 19-note scales are highly characteristic for the temperament, with the abundance of [[28/27]] subminor seconds making it melodically active.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]].		The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]].

	Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests tempering out [[595/594]] (S34⋅S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. [[311edo]] remains an excellent tuning in all ~~the subgroups~~.		Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests tempering out [[595/594]] (S34⋅S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. These extensions add little additional error, and [[311edo]] remains an excellent tuning in all cases.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

FloraC: /* Undecimal cathartic */ rework into a no-13 temp

2026-08-10T14:17:10Z

Undecimal cathartic: rework into a no-13 temp

← Older revision		Revision as of 14:17, 10 August 2026
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	== Undecimal cathartic ==		== Undecimal cathartic ==
	The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma~~]], which suggests the [[wilschisma]] should also be tempered out in the [[13-limit~~]].		The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]].

	Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests tempering out [[595/594]] (S34⋅S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. [[311edo]] remains an excellent tuning in all the ~~limits~~.		Since the syntonic comma has been split in two, it is natural to map [[19/17]] to the mean of [[9/8]] and [[10/9]], tempering out [[1445/1444]], while the other 11-limit comma, [[42875/42768]] (S34⋅S35<sup>2</sup>), suggests tempering out [[595/594]] (S34⋅S35), [[1156/1155]] (S34), and [[1225/1224]] (S35), which coincides with above. Finally, we can map [[23/20]] to the fourth complement of 22/19 to make an equidistant sequence consisting of 7/6, 22/19, 23/20, and 8/7, tempering out [[760/759]]. [[311edo]] remains an excellent tuning in all the subgroups.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	: error map: {{val\| 0.0000 +0.228 -0.422 -0.604 +0.107 }}		: error map: {{val\| 0.0000 +0.228 -0.422 -0.604 +0.107 }}

	{{Optimal ET sequence\|legend=1\| 94, 99e, 118, 193, 212, 311, 740, 1051d }}		{{Optimal ET sequence\|legend=1\| 24, …, 75e, 94, 99e, 118, 193, 212, 311, 740, 1051d }}

	[[Badness]] (Sintel): 2.45		[[Badness]] (Sintel): 2.45
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	[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5		[[Complexity spectrum]]: 4/3, 9/8, 9/7, 7/6, 5/4, 6/5, 10/9, 11/9, 8/7, 12/11, 11/10, 14/11, 11/8, 7/5

	=== ~~13-limit~~ ===		=== 2.3.5.7.11.17 subgroup ===
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.17

	Comma list: ~~2080~~/~~2079~~, 19712/19683~~, 42875/42768~~		Comma list: 595/594, 1156/1155, 19712/19683

	~~Mapping~~: {{mapping\| 1 0 0 -1 -7 -13 \| 0 1 2 2 7 10 \| 0 0 -4 3 -3 4 }}		Subgroup-val mapping: {{mapping\| 1 0 0 -1 -7 -5 \| 0 1 2 2 7 6 \| 0 0 -4 3 -3 -2 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~0501~~{{c}}, ~3/2 = 702.~~2100~~{{c}}, ~81/70 = 254.~~6345~~{{c}}		* WE: ~2 = 1200.0485{{c}}, ~3/2 = 702.2481{{c}}, ~51/44 = 254.6343{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.~~1889~~{{c}}, ~81/70 = 254.~~6222~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2325{{c}}, ~51/44 = 254.6253{{c}}

	{{Optimal ET sequence\|legend=0\| 94, ~~118f~~, ~~193f~~, ~~212~~, 217, 311, ~~740, 1051d~~ }}		{{Optimal ET sequence\|legend=0\| 24, …, 75e, 94, 99e, 118, 193, 212g, 217, 311, 1051dg }}

	Badness (Sintel): 2.39		Badness (Sintel): 1.01

	=== 17~~-limit~~ ===		=== 2.3.5.7.11.17.19 subgroup ===
	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.17.19

	Comma list: 595/594, ~~833~~/~~832~~, 1156/1155, ~~19712~~/~~19683~~		Comma list: 595/594, 969/968, 1156/1155, 1216/1215

	~~Mapping~~: {{mapping\| 1 0 0 -1 -7 -13 -5 \| 0 1 2 2 7 10 6 \| 0 0 -4 3 -3 4 -2 }}		Subgroup-val mapping: {{mapping\| 1 0 0 -1 -7 -5 -6 \| 0 1 2 2 7 6 7 \| 0 0 -4 3 -3 -2 -4 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1200.~~0630~~{{c}}, ~3/2 = 702.~~2317~~{{c}}, ~51/44 = 254.~~6224~~{{c}}		* WE: ~2 = 1200.0444{{c}}, ~3/2 = 702.2569{{c}}, ~22/19 = 254.6305{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.~~2055~~{{c}}, ~51/44 = 254.~~6066~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2421{{c}}, ~22/19 = 254.6224{{c}}

	{{Optimal ET sequence\|legend=0\| 94, ~~118f~~, ~~193f~~, ~~212g~~, 217, 311, ~~740g, 1051dg~~ }}		{{Optimal ET sequence\|legend=0\| 24, …, 75e, 94, 99e, 118, 193, 217, 311, 1051dgh }}

	Badness (Sintel): 1.41		Badness (Sintel): 0.641

	=== 19~~-limit~~ ===		=== 2.3.5.7.11.17.19.23 subgroup ===
	Subgroup: 2.3.5.7.11~~.13~~.17.19		Subgroup: 2.3.5.7.11.17.19.23

	Comma list: 595/594, ~~833~~/~~832~~, 969/968, 1156/1155~~, 1216/1215~~		Comma list: 595/594, 760/759, 875/874, 969/968, 1156/1155

	Mapping: {{mapping\| 1 0 0 -1 -7 ~~-13~~ -5 -6 \| 0 1 2 2 7 10 6 7 \| 0 0 -4 3 -3 4 -2 -4 }}		Mapping: {{mapping\| 1 0 0 -1 -7 -5 -6 4 \| 0 1 2 2 7 6 7 1 \| 0 0 -4 3 -3 -2 -4 -5 }}

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = ~~1200~~.~~0624~~{{c}}, ~3/2 = 702.~~2377~~{{c}}, ~22/19 = 254.~~6139~~{{c}}		* WE: ~2 = 1199.9778{{c}}, ~3/2 = 702.2794{{c}}, ~22/19 = 254.6572{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.~~2117~~{{c}}, ~22/19 = 254.~~5983~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2883{{c}}, ~22/19 = 254.6625{{c}}

	{{Optimal ET sequence\|legend=0\| 94, ~~118f~~, ~~193f~~, ~~212gh~~, 217, 311~~, 740g, 1051dgh~~ }}		{{Optimal ET sequence\|legend=0\| 24, 75e, 94, 99e, 118, 193, 217, 311 }}

	Badness (Sintel): ~~1.03~~		Badness (Sintel): 0.703

	~~=== 23-limit ===~~
	~~Subgroup: 2.3.5.7.11.13.17.19.23~~

	~~Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155~~

	~~Mapping: {{mapping\| 1 0 0 -1 -7 -13 -5 -6 4 \| 0 1 2 2 7 10 6 7 1 \| 0 0 -4 3 -3 4 -2 -4 -5 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.0004{{c}}, ~3/2 = 702.2361{{c}}, ~22/19 = 254.6225{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~22/19 = 254.6223{{c}}

	~~{{Optimal ET sequence\|legend=~~0~~\| 94, 193f, 212gh, 217, 311 }}~~

	~~Badness (Sintel): 1~~.09

	== Canta ==		== Canta ==

FloraC: FloraC moved page Canou family to Cathartic family: Rename (see talk)

2026-08-10T12:40:54Z

FloraC moved page Canou family to Cathartic family: Rename (see talk)

← Older revision	Revision as of 12:40, 10 August 2026
(No difference)

FloraC: Initiate the rename (see talk)

2026-08-10T12:40:31Z

Initiate the rename (see talk)

← Older revision		Revision as of 12:40, 10 August 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''~~canou~~ family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the [[~~canousma~~]] ({{monzo\|legend=1\| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s.		The '''canthartic family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the [[cathartic comma]] ({{monzo\|legend=1\| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969), a 7-limit comma measuring about 6.9 [[cent]]s.

	== ~~Canou~~ ==		== Cathartic ==
	{{Main\| Canou }}		{{Main\| Canou }}

	~~The~~ canou ~~temperament~~ features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic.		Cathartic (formerly ''canou'') features a [[period]] of an [[octave]] and [[generator]]s of [[3/2]] and [[81/70]]. The ~81/70 generator is about 255 cents wide, three of which make [[14/9]], and four make [[9/5]]. It therefore splits the large septimal diesis, [[49/48]], into three equal parts, guaranteeing the existence of two [[interseptimal interval]]s related to the 35th harmonic.

	A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], associating it with the [[amicable]] temperament.		A basic tuning option would be [[99edo]], although [[80edo]] is even simpler and distinctive. More intricate tunings are provided by [[311edo]] and [[410edo]], whereas the [[optimal patent val]] goes up to [[1131edo]], associating it with the [[amicable]] temperament.
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	[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5		[[Complexity spectrum]]: 4/3, 9/7, 9/8, 7/6, 6/5, 10/9, 5/4, 8/7, 7/5

	== Undecimal ~~canou~~ ==		== Undecimal cathartic ==
	The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]], which suggests the [[wilschisma]] should also be tempered out in the [[13-limit]].		The fifth is in the range where a stack of four (i.e. a major third) can serve as ~[[19/15]] and a stack of five (i.e. a major seventh) can serve as ~[[19/10]], tempering out [[1216/1215]]. Moreover, the last generator of ~81/70 is sharpened to slightly overshoot [[22/19]], so it only makes sense to temper out their difference, [[1540/1539]]. The implied 11-limit comma is the [[symbiotic comma]], which suggests the [[wilschisma]] should also be tempered out in the [[13-limit]].

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	Badness (Sintel): 4.47		Badness (Sintel): 4.47

	== ~~Semicanou~~ ==		== Semicathart ==
	~~Semicanou~~ adds 9801/9800, the [[kalisma]], to the comma list, and may be described as {{nowrap\| 80 & 94 & 118 }}. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that {{nowrap\| 99/70 {{=}} (81/70)⋅(11/9) }}.		Semicathart (formerly ''semicanou'') adds 9801/9800, the [[kalisma]], to the comma list, and may be described as {{nowrap\| 80 & 94 & 118 }}. It splits the octave into two equal parts, each representing 99/70~140/99. This takes advantage of the fact that {{nowrap\| 99/70 {{=}} (81/70)⋅(11/9) }}.

	The other comma necessary to define it is 14641/14580, the [[~~semicanousma~~]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by about one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9, 11/9, 11/9, and 11/10.		The other comma necessary to define it is 14641/14580, the [[semicathartisma]], which is the difference between [[121/120]] and [[243/242]]. By flattening the 11th harmonic by about one cent, it identifies [[20/11]] by three [[11/9]]'s stacked, so an octave can be divided into 11/9, 11/9, 11/9, and 11/10.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	[[Category:~~Canou~~ family\| ]] <!-- main article -->		[[Category:Cathartic family\| ]] <!-- main article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

FloraC: Switch to Sintel's badness, WE & CWE tunings. - all 13-limit semicanou extensions (not very notable). - redundant category

2026-06-09T06:42:16Z

Switch to Sintel's badness, WE & CWE tunings. - all 13-limit semicanou extensions (not very notable). - redundant category

Show changes

FloraC: Text replacement - "rank-3 temperament" to "rank-3 temperament"

2026-06-08T08:36:18Z

Text replacement - "rank-3 temperament" to "rank-3 temperament"

← Older revision		Revision as of 08:36, 8 June 2026
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	{{Technical data page}}		{{Technical data page}}
	The '''canou family''' of [[rank-3 ~~temperament\|rank-3]] [[regular temperament\|~~temperament]]s [[tempering out\|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo\| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.		The '''canou family''' of [[rank-3 temperament]]s [[tempering out\|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo\| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.

	== Canou ==		== Canou ==

FloraC: Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

2025-11-27T12:22:26Z

Text replacement - "Category:Temperament families↵Category:Pages with mostly numerical content" to "Category:Temperament families"

← Older revision		Revision as of 12:22, 27 November 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
	~~[[Category:Pages with mostly numerical content]]~~
	[[Category:Canou family\| ]] <!-- main article -->		[[Category:Canou family\| ]] <!-- main article -->
	[[Category:Canou\| ]] <!-- key article -->		[[Category:Canou\| ]] <!-- key article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

Lériendil: Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

2025-06-24T00:35:25Z

Text replacement - "Category:Temperament families" to "Category:Temperament families Category:Pages with mostly numerical content"

← Older revision		Revision as of 00:35, 24 June 2025
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	[[Category:Temperament families]]		[[Category:Temperament families]]
			[[Category:Pages with mostly numerical content]]
	[[Category:Canou family\| ]] <!-- main article -->		[[Category:Canou family\| ]] <!-- main article -->
	[[Category:Canou\| ]] <!-- key article -->		[[Category:Canou\| ]] <!-- key article -->
	[[Category:Rank 3]]		[[Category:Rank 3]]

Lériendil: Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "

2025-06-21T06:58:23Z

Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "

← Older revision		Revision as of 06:58, 21 June 2025
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	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit]]: 3 +c/14, 5 and 7 just		* [[7-odd-limit]]: 3 +c/14, 5 and 7 just
	: [[eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5.7		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5.7
	* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just		* [[9-odd-limit]]: 3 just, 5 and 7 -c/7 to 3 +c/14, 5 and 7 just
	: [[eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5		: [[eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7/5

	{{Optimal ET sequence\|legend=1\| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}		{{Optimal ET sequence\|legend=1\| 19, 56d, 61d, 75, 80, 94, 99, 212, 292, 311, 410, 1131, 1541b, 1659b }}

Lériendil: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:47:08Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

← Older revision		Revision as of 13:47, 13 April 2025
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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	The '''canou family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperament]]s [[tempering out\|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo\| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.		The '''canou family''' of [[rank-3 temperament\|rank-3]] [[regular temperament\|temperament]]s [[tempering out\|tempers out]] the [[canousma]], 4802000/4782969 ({{monzo\| 4 -14 3 4 }}), a 7-limit comma measuring about 6.9 [[cent]]s.