Octagar temperaments - Revision history

FloraC: /* Pluto */ expand a little

2026-04-02T11:31:29Z

Pluto: expand a little

← Older revision		Revision as of 11:31, 2 April 2026
Line 23:		Line 23:
	{{Distinguish\| Plutus }}		{{Distinguish\| Plutus }}

	Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group \| ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap\| 41 & 80 }} temperament. It is generated by a flattened [[10/7]], and [[121edo\|59\121]] is about perfect as a tuning.		Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group \| ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap\| 41 & 80 }} temperament. It is generated by a flattened [[~]][[10/7]] [[tritone]], seven of which give the [[interval class]] of [[3/1\|3]], and the [[ploidacot]] for this temperament is gamma-heptacot. [[121edo\|59\121]] is about perfect as a tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: Move pluto here cuz mirkwai clan is a no-2 clan, which doesn't explain its structure very well

2026-04-02T11:03:30Z

Move pluto here cuz mirkwai clan is a no-2 clan, which doesn't explain its structure very well

@@ Line 16: / Line 16: @@
 * ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
 * ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
 * ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
-Considered below are slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
+Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
 == Slithy ==
@@ Line 269: / Line 395: @@
 Badness (Sintel): 1.55
 [[Category:Temperament collections]]
 [[Category:Octagar temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]

FloraC: + intros

2026-03-13T13:13:58Z

+ intros

← Older revision		Revision as of 13:13, 13 March 2026
Line 22:		Line 22:

	== Slithy ==		== Slithy ==
			Slithy tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap\| 27 & 94 }} temperament with a [[ploidacot]] signature of 17-sheared 19-cot. [[94edo]], [[121edo]] and [[215edo]] can all be used as tunings.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Line 117:		Line 119:
	{{See also\| 13th-octave temperaments #Tridecatonic }}		{{See also\| 13th-octave temperaments #Tridecatonic }}

	~~The tridecatonic temperament (~~{{nowrap\| 26 & 39 }}~~) has a period of 1/13 octave~~. It tempers out ~~the~~ tridecalimma, {{monzo\| -11 26 -13 }} in the 5-limit. ~~There are some extensions for 7-limit, including tridecatonic ({{nowrap\| 26 & 39d }}, tempering out 4000/3969),~~ [[~~ragismic microtemperaments #Trideci\|trideci~~]] ~~({{nowrap\| 26 & 39 }}, tempering out 4375/4374), and [[26th~~-~~octave temperaments\|bosonic]] ({{nowrap\| 26 & 130 }}, tempering out 321489/320000)~~.		Tridecatonic has a period of 1/13 octave, and may be described as {{nowrap\| 26 & 39 }}. It tempers out [[devil's tridecalimma]], {{monzo\| -11 26 -13 }} in the 5-limit. Its [[ploidacot]] is 13-ploid monocot.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
Line 219:		Line 221:

	== Dhaivatic ==		== Dhaivatic ==
			Named by [[Xenllium]] in 2023, dhaivatic tempers out the [[rainy comma]] and may be described as the {{nowrap\| 15 & 94 }} temperament. Its [[ploidacot]] is epsilon-21-cot.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: Sorting by badness

2026-03-13T12:46:37Z

Sorting by badness

@@ Line 2: / Line 2: @@
 This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
-Temperaments considered below are slendi and slithy. Discussed elsewhere are:
+Temperaments discussed elsewhere are:
 * ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
 * ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
@@ Line 19: / Line 19: @@
 * ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
-== Slendi ==
+Considered below are slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
 == Slithy ==
@@ Line 215: / Line 165: @@
 Badness (Sintel): 1.56
 == Dhaivatic ==

FloraC: - POTE tunings

2026-03-13T12:44:55Z

- POTE tunings

← Older revision		Revision as of 12:44, 13 March 2026
Line 84:		Line 84:
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7026{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7026{{c}}
	: error map: {{val\| 0.000 +1.394 -1.021 +2.415 }}		: error map: {{val\| 0.000 +1.394 -1.021 +2.415 }}
	~~<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~40/21 = 1110.718{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 27, 67c, 94, 121, 215, 336d }}		{{Optimal ET sequence\|legend=1\| 27, 67c, 94, 121, 215, 336d }}
Line 100:		Line 99:
	* WE: ~2 = 1199.5884{{c}}, ~40/21 = 1110.3276{{c}}		* WE: ~2 = 1199.5884{{c}}, ~40/21 = 1110.3276{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.6969{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.6969{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~40/21 = 1110.709{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 27e, 67ce, 94, 121, 215, 336de }}		{{Optimal ET sequence\|legend=0\| 27e, 67ce, 94, 121, 215, 336de }}
Line 116:		Line 114:
	* WE: ~2 = 1199.5032{{c}}, ~40/21 = 1110.2545{{c}}		* WE: ~2 = 1199.5032{{c}}, ~40/21 = 1110.2545{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7037{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7037{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~40/21 = 1110.7140{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 27e, 67ce, 94, 121, 215, 336def }}		{{Optimal ET sequence\|legend=0\| 27e, 67ce, 94, 121, 215, 336def }}
Line 132:		Line 129:
	* WE: ~2 = 1199.5051{{c}}, ~40/21 = 1110.2555{{c}}		* WE: ~2 = 1199.5051{{c}}, ~40/21 = 1110.2555{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7033{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7033{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~40/21 = 1110.714{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 27eg, 67ceg, 94, 121, 215, 336defg }}		{{Optimal ET sequence\|legend=0\| 27eg, 67ceg, 94, 121, 215, 336defg }}
Line 148:		Line 144:
	* WE: ~2 = 1199.6184{{c}}, ~19/10 = 1110.3583{{c}}		* WE: ~2 = 1199.6184{{c}}, ~19/10 = 1110.3583{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.7036{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.7036{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~19/10 = 1110.712{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 27eg, 67cegh, 94, 121, 215 }}		{{Optimal ET sequence\|legend=0\| 27eg, 67cegh, 94, 121, 215 }}
Line 164:		Line 159:
	* WE: ~2 = 1199.6651{{c}}, ~19/10 = 1110.3928{{c}}		* WE: ~2 = 1199.6651{{c}}, ~19/10 = 1110.3928{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.6965{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.6965{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~19/10 = 1110.703{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 27egi, 94, 121i, 215 }}		{{Optimal ET sequence\|legend=0\| 27egi, 94, 121i, 215 }}
Line 187:		Line 181:
	* [[CWE]]: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2352{{c}}		* [[CWE]]: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2352{{c}}
	: error map: {{val\| 0.000 -0.720 +0.772 +9.332 }}		: error map: {{val\| 0.000 -0.720 +0.772 +9.332 }}
	~~<!-- * [[POTE]]: ~21/20 = 92.3077{{c}}, ~3/2 = 701.9605{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 26, 39d, 65d }}		{{Optimal ET sequence\|legend=1\| 26, 39d, 65d }}
Line 203:		Line 196:
	* WE: ~21/20 = 92.2333{{c}}, ~3/2 = 701.1371{{c}}		* WE: ~21/20 = 92.2333{{c}}, ~3/2 = 701.1371{{c}}
	* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2751{{c}}		* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2751{{c}}
	~~<!-- * POTE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.7028{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 26, 39d, 65d }}		{{Optimal ET sequence\|legend=0\| 26, 39d, 65d }}
Line 219:		Line 211:
	* WE: ~21/20 = 92.2248{{c}}, ~3/2 = 700.7272{{c}}		* WE: ~21/20 = 92.2248{{c}}, ~3/2 = 700.7272{{c}}
	* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 700.8621{{c}}		* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 700.8621{{c}}
	~~<!-- * POTE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.3573{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 26, 39df, 65d }}		{{Optimal ET sequence\|legend=0\| 26, 39df, 65d }}
Line 238:		Line 229:
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 319.1980{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 319.1980{{c}}
	: error map: {{val\| 0.000 +1.202 -2.354 +0.798 }}		: error map: {{val\| 0.000 +1.202 -2.354 +0.798 }}
	~~<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 319.198{{c}} -->~~

	{{Optimal ET sequence\|legend=1\| 15, 64c, 79, 94, 203, 297c }}		{{Optimal ET sequence\|legend=1\| 15, 64c, 79, 94, 203, 297c }}
Line 254:		Line 244:
	* WE: ~2 = 1200.0983{{c}}, ~6/5 = 319.2229{{c}}		* WE: ~2 = 1200.0983{{c}}, ~6/5 = 319.2229{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1985{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1985{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 319.197{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 15, 64c, 79, 94, 203, 297c }}		{{Optimal ET sequence\|legend=0\| 15, 64c, 79, 94, 203, 297c }}
Line 270:		Line 259:
	* WE: ~2 = 1200.0732{{c}}, ~6/5 = 319.1840{{c}}		* WE: ~2 = 1200.0732{{c}}, ~6/5 = 319.1840{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1658{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1658{{c}}
	~~<!-- * POTE: ~2 = 1200.000{{c}}, ~6/5 = 319.165{{c}} -->~~

	{{Optimal ET sequence\|legend=0\| 15, 79, 94 }}		{{Optimal ET sequence\|legend=0\| 15, 79, 94 }}

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-03-13T12:39:38Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Review data for slendi. + intro

2026-03-13T11:20:48Z

Review data for slendi. + intro

← Older revision		Revision as of 11:20, 13 March 2026
Line 20:		Line 20:

	== Slendi ==		== Slendi ==
			Slendi may be described as the {{nowrap\| 41 & 42 }} temperament. It has a generator of [[49/48]], seventeen of which give [[4/3]]. Its [[ploidacot]] is omega-17-cot.

			The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Line 25:		Line 29:

	{{Mapping\|legend=1\| 1 2 3 3 \| 0 -17 -28 -8 }}		{{Mapping\|legend=1\| 1 2 3 3 \| 0 -17 -28 -8 }}
			: mapping generators: ~2, ~49/48

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = 1200.~~252~~{{c}}, ~49/48 = 29.~~187~~{{c}}		* [[WE]]: ~2 = 1200.2515{{c}}, ~49/48 = 29.1870{{c}}
	* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~49/48 = 29.~~172~~{{c}}		: [[error map]]: {{val\| +0.252 +2.368 -2.796 -1.568 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 29.1715{{c}}
			: error map: {{val\| 0.000 +2.129 -3.116 -2.198 }}

	{{Optimal ET sequence\|legend=1\| 41, 206~~, 247bc, 288bcd, 329bcd~~ }}		{{Optimal ET sequence\|legend=1\| 1c, 40c, 41, 206 }}

	[[Badness]] (Sintel): 3.~~616~~		[[Badness]] (Sintel): 3.62

	=== 11-limit ===		=== 11-limit ===
Line 42:		Line 49:

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.~~803~~{{c}}, ~49/48 = 29.~~175~~{{c}}		* WE: ~2 = 1199.8027{{c}}, ~49/48 = 29.1751{{c}}
	* CWE: ~2 = 1200.~~000~~{{c}}, ~49/48 = 29.~~188~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1875{{c}}

	{{Optimal ET sequence\|legend=0\| 41~~, 206e~~ }}		{{Optimal ET sequence\|legend=0\| 1ce, 40c, 41 }}

	Badness (Sintel): 2.~~090~~		Badness (Sintel): 2.09

	=== 13-limit ===		=== 13-limit ===
Line 57:		Line 64:

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.~~091~~{{c}}, ~40/39 = 29.~~103~~{{c}}		* WE: ~2 = 1199.0915{{c}}, ~49/48 = 29.1033{{c}}
	* CWE: ~2 = 1200.~~000~~{{c}}, ~40/39 = 29.~~158~~{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1576{{c}}

	{{Optimal ET sequence\|legend=0\| 41~~, 124ef, 165ef~~ }}		{{Optimal ET sequence\|legend=0\| 1ce, 41 }}

	Badness (Sintel): 1.~~803~~		Badness (Sintel): 1.80

	== Slithy ==		== Slithy ==

FloraC: Units & misc. cleanup. - redundant category

2026-03-13T10:41:43Z

Units & misc. cleanup. - redundant category

Show changes

Xenllium at 00:43, 4 March 2026

2026-03-04T00:43:12Z

← Older revision		Revision as of 00:43, 4 March 2026
Line 3:		Line 3:

	Temperaments considered below are slendi and slithy. Discussed elsewhere are:		Temperaments considered below are slendi and slithy. Discussed elsewhere are:
			* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida\|Dicot family]]
	* ''[[Ripple]]'' (+36/35) → [[Ripple family #Septimal ripple\|Ripple family]]		* ''[[Ripple]]'' (+36/35) → [[Ripple family #Septimal ripple\|Ripple family]]
	* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus\|Porcupine family]]		* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus\|Porcupine family]]

Xenllium at 22:37, 10 February 2026

2026-02-10T22:37:09Z

← Older revision		Revision as of 22:37, 10 February 2026
Line 25:		Line 25:
	{{Mapping\|legend=1\| 1 2 3 3 \| 0 -17 -28 -8 }}		{{Mapping\|legend=1\| 1 2 3 3 \| 0 -17 -28 -8 }}

	[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1200.~~000~~, ~49/48 = 29.~~181~~		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.252¢, ~49/48 = 29.187¢
			* [[CWE]]: ~2 = 1200.000¢, ~49/48 = 29.172¢

	{{Optimal ET sequence\|legend=1\| 41, 206, 247bc, 288bcd, 329bcd }}		{{Optimal ET sequence\|legend=1\| 41, 206, 247bc, 288bcd, 329bcd }}

	[[Badness]]: 0.~~142903~~		[[Badness]] (Sintel): 3.616

	=== 11-limit ===		=== 11-limit ===
Line 38:		Line 40:
	Mapping: {{mapping\| 1 2 3 3 4 \| 0 -17 -28 -8 -22 }}		Mapping: {{mapping\| 1 2 3 3 4 \| 0 -17 -28 -8 -22 }}

	Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~, ~49/48 = 29.~~180~~		Optimal tunings:
			* WE: ~2 = 1199.803¢, ~49/48 = 29.175¢
			* CWE: ~2 = 1200.000¢, ~49/48 = 29.188¢

	{{Optimal ET sequence\|legend=0\| 41, 206e }}		{{Optimal ET sequence\|legend=0\| 41, 206e }}

	Badness: 0.~~063217~~		Badness (Sintel): 2.090

	=== 13-limit ===		=== 13-limit ===
Line 49:		Line 53:
	Comma list: 100/99, 144/143, 245/242, 343/338		Comma list: 100/99, 144/143, 245/242, 343/338

	Mapping: {{mapping\| 1 2 3 3 4 4 \| 0 -17 -28 -8 -22 -1 }}		Mapping: {{mapping\| 1 2 3 3 4 4 \| 0 -17 -28 -8 -22 -12 }}

	Optimal ~~tuning (POTE)~~: ~2 = 1200.~~000~~, ~49/48 = 29.~~125~~		Optimal tunings:
			* WE: ~2 = 1199.091¢, ~40/39 = 29.103¢
			* CWE: ~2 = 1200.000¢, ~40/39 = 29.158¢

	{{Optimal ET sequence\|legend=0\| 41, 124ef, 165ef }}		{{Optimal ET sequence\|legend=0\| 41, 124ef, 165ef }}

	Badness: 0.~~043644~~		Badness (Sintel): 1.803

	== Slithy ==		== Slithy ==