Starling temperaments - Revision history

Xenllium at 10:17, 14 May 2026

2026-05-14T10:17:22Z

← Older revision		Revision as of 10:17, 14 May 2026
Line 474:		Line 474:
	== Paraguay ==		== Paraguay ==
	: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''		: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''

			Named by [[User:Xenllium\|Xenllium]] in 2026, paraguay tempers out [[12005/11664]] and may be described as the {{nowrap\| 19 & 61 }} temperament. It is a variant of [[parakleismic]], mapping 7th harmonic to 16 generators.

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

Xenllium at 09:54, 14 May 2026

2026-05-14T09:54:08Z

← Older revision		Revision as of 09:54, 14 May 2026
Line 471:		Line 471:

	Badness (Sintel): 1.19		Badness (Sintel): 1.19

			== Paraguay ==
			: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 126/125, 12005/11664

			{{Mapping\|legend=1\| 1 -8 -8 -9 \| 0 13 14 16 }}
			: mapping generators: ~2, ~5/3

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
			: [[error map]]: {{val\| +0.642 +2.110 +3.074 -9.434 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
			: error map: {{val\| 0.000 +1.678 +2.214 -10.508 }}

			{{Optimal ET sequence\|legend=1\| 19, 61, 80d, 99d }}

			[[Badness]] (Sintel): 2.47

			=== 11-limit ===
			Subgroup: 2.3.5.7.11

			Comma list: 56/55, 100/99, 12005/11664

			Mapping: {{mapping\| 1 -8 -8 -9 2 \| 0 13 14 16 2 }}

			Optimal tunings:
			* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}

			{{Optimal ET sequence\|legend=0\| 19, 42e, 61e }}

			Badness (Sintel): 2.49

			==== 13-limit ====
			Subgroup: 2.3.5.7.11.13

			Comma list: 56/55, 91/90, 100/99, 343/338

			Mapping: {{mapping\| 1 -8 -8 -9 2 -14 \| 0 13 14 16 2 24 }}

			Optimal tunings:
			* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}

			{{Optimal ET sequence\|legend=0\| 19, 42ef, 61e }}

			Badness (Sintel): 1.86

			==== Uruguay ====
			Subgroup: 2.3.5.7.11.13

			Comma list: 56/55, 78/77, 100/99, 1183/1152

			Mapping: {{mapping\| 1 -8 -8 -9 2 0 \| 0 13 14 16 2 5 }}

			Optimal tunings:
			* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}

			{{Optimal ET sequence\|legend=0\| 19, 42e }}

			Badness (Sintel): 2.51

	== Bisemidim ==		== Bisemidim ==

Xenllium: /* Xenial */

2026-05-04T12:52:13Z

Xenial

← Older revision		Revision as of 12:52, 4 May 2026
Line 410:		Line 410:
	Subgroup: 2.3.5.7.11.13.17.19.23		Subgroup: 2.3.5.7.11.13.17.19.23

	Comma list: 126/125, 162/161, 169/168, 171/170, 221/220, 231/230~~, 256/255~~		Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230

	Mapping: {{mapping\| 1 -6 -12 -25 22 -14 26 27 2 \| 0 9 17 33 -22 21 -26 -27 3 }}		Mapping: {{mapping\| 1 -6 -12 -25 22 -14 26 27 2 \| 0 9 17 33 -22 21 -26 -27 3 }}

Xenllium at 00:07, 3 May 2026

2026-05-03T00:07:02Z

← Older revision		Revision as of 00:07, 3 May 2026
Line 325:		Line 325:

	== Xenial ==		== Xenial ==
			{{Main\| Xenial }}
	: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''		: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

	Named by [[User:Xenllium\|Xenllium]] in 2026, xenial may be described as the {{nowrap\| 19 & 70 }} temperament, splitting the [[8/3\|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2\|perfect fifth]] above 7 ~~octaves~~, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.		Named by [[User:Xenllium\|Xenllium]] in 2026, xenial may be described as the {{nowrap\| 19 & 70 }} temperament, splitting the [[8/3\|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2\|perfect fifth]] above 7 [[2/1\|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
Line 342:		Line 343:
	: error map: {{val\| 0.000 -1.644 +3.162 -1.021 }}		: error map: {{val\| 0.000 -1.644 +3.162 -1.021 }}

	{{Optimal ET sequence\|legend=1\| 19, 70, 89 }}		{{Optimal ET sequence\|legend=1\| 19, 51cd, 70, 89 }}

	[[Badness]] (Sintel): 2.13		[[Badness]] (Sintel): 2.13
Line 354:		Line 355:

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.~~614~~{{c}}, ~9/5 = 1010.~~872~~{{c}}		* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
	* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.~~192~~{{c}}		* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

	{{Optimal ET sequence\|legend=0\| 19, 70, 89 }}		{{Optimal ET sequence\|legend=0\| 19, 51cd, 70, 89 }}

	Badness (Sintel): 2.31		Badness (Sintel): 2.31
Line 369:		Line 370:

	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.~~856~~{{c}}, ~9/5 = 1011.~~091~~{{c}}		* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
	* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.~~210~~{{c}}		* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

	{{Optimal ET sequence\|legend=0\| 19, 70, 89 }}		{{Optimal ET sequence\|legend=0\| 19, 51cd, 70, 89 }}

	Badness (Sintel): 1.98		Badness (Sintel): 1.98

			=== 17-limit ===
			Subgroup: 2.3.5.7.11.13.17

			Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

			Mapping: {{mapping\| 1 -6 -12 -25 22 -14 26 \| 0 9 17 33 -22 21 -26 }}

			Optimal tunings:
			* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

			{{Optimal ET sequence\|legend=0\| 19, 51cd, 70, 89 }}

			Badness (Sintel): 2.06

			=== 19-limit ===
			Subgroup: 2.3.5.7.11.13.17.19

			Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

			Mapping: {{mapping\| 1 -6 -12 -25 22 -14 26 27 \| 0 9 17 33 -22 21 -26 -27 }}

			Optimal tunings:
			* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

			{{Optimal ET sequence\|legend=0\| 19, 51cdh, 70, 89 }}

			Badness (Sintel): 2.03

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

			Comma list: 126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255

			Mapping: {{mapping\| 1 -6 -12 -25 22 -14 26 27 2 \| 0 9 17 33 -22 21 -26 -27 3 }}

			Optimal tunings:
			* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
			* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

			{{Optimal ET sequence\|legend=0\| 19, 51cdh, 70, 89 }}

			Badness (Sintel): 1.93

	== Kumonga ==		== Kumonga ==

Xenllium at 06:59, 2 May 2026

2026-05-02T06:59:12Z

@@ Line 327: / Line 327: @@
 : ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
-Xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 octaves, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name (named by [[User:Xenllium|Xenllium]] in 2026).
+Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 octaves, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.
 [[Subgroup]]: 2.3.5.7
@@ Line 338: / Line 338: @@
 [[Optimal tuning]]s:
 * [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
-: [[error map]]: {{val| +0.0095, -1.6335, +3.1760, -1.0091 }}
+: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
-: error map: {{val| 0.0000 -1.6444, +3.1618, -1.0205 }}
+: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}
 {{Optimal ET sequence|legend=1| 19, 70, 89 }}
 [[Badness]] (Sintel): 2.13
 == Kumonga ==

Xenllium at 04:52, 2 May 2026

2026-05-02T04:52:26Z

Show changes

FloraC: Status update

2026-04-15T18:57:56Z

Status update

← Older revision		Revision as of 18:57, 15 April 2026
Line 14:		Line 14:
	* [[Muggles]] (+525/512) → [[Magic family #Muggles\|Magic family]]		* [[Muggles]] (+525/512) → [[Magic family #Muggles\|Magic family]]
	* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine\|Gamelismic clan]]		* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine\|Gamelismic clan]]
	* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic\|Diaschismic family]]		* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic\|Diaschismic family]]
	* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia\|Tetracot family]]		* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia\|Tetracot family]]
	* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn\|Unicorn family]]		* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn\|Unicorn family]]
	* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack\|Trisedodge family]] / [[Cloudy clan #Coblack\|cloudy clan]]		* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack\|Trisedodge family]] / [[Cloudy clan #Coblack\|cloudy clan]]

Xenllium at 03:05, 23 February 2026

2026-02-23T03:05:32Z

← Older revision		Revision as of 03:05, 23 February 2026
Line 712:		Line 712:

	== Cobalt ==		== Cobalt ==
			: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''

	Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap\| 27 & 81 }}.		Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap\| 27 & 81 }}.

FloraC: + descriptions to certain entries

2026-02-21T12:38:46Z

+ descriptions to certain entries

← Older revision		Revision as of 12:38, 21 February 2026
Line 274:		Line 274:
	== Vines ==		== Vines ==
	: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''		: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''

			Vines may be described as the {{nowrap\| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2\|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
Line 422:		Line 424:

	== Bisemidim ==		== Bisemidim ==
			Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap\| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2\|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.

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

Line 472:		Line 476:
	: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''		: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''

	Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap\| 31 & 73 }}. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.		Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap\| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

	It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.		It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.
Line 643:		Line 647:

	== Supersensi ==		== Supersensi ==
	~~Supersensi~~ may be described as {{nowrap\| 8d & 43 }}. It has ~~supermajor~~ third ~~as a~~ generator ~~like~~ [[sensi]], ~~but~~ the ~~no-fives comma~~ [[~~17496/16807~~]] ~~rather than [[245/243]] tempered out~~.		Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap\| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
Line 710:		Line 714:
	Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap\| 27 & 81 }}.		Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap\| 27 & 81 }}.

	~~The name of the cobalt temperament comes from~~ the 27th element.		Cobalt was named by [[Xenllium]] in 2022 after the 27th element.

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

FloraC: - typo

2026-02-19T17:23:03Z

- typo

← Older revision		Revision as of 17:23, 19 February 2026
Line 26:		Line 26:
	* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee\|Mabila family]]		* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee\|Mabila family]]

	Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and ~~cobald~~, sorted by increasing [[badness]].		Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].

	Since {{nowrap\|(6/5)<sup>3</sup> {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.		Since {{nowrap\|(6/5)<sup>3</sup> {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

← Older revision		Revision as of 18:57, 15 April 2026
Line 14:		Line 14:
	* [[Muggles]] (+525/512) → [[Magic family #Muggles\|Magic family]]		* [[Muggles]] (+525/512) → [[Magic family #Muggles\|Magic family]]
	* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine\|Gamelismic clan]]		* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine\|Gamelismic clan]]
	* ''[[Diaschismic]]'' (+2048/2025) → [[Diaschismic family #Septimal diaschismic\|Diaschismic family]]		* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic\|Diaschismic family]]
	* ''[[Wollemia]]'' (+2240/2187) → [[Tetracot family #Wollemia\|Tetracot family]]		* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia\|Tetracot family]]
	* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn\|Unicorn family]]		* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn\|Unicorn family]]
	* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack\|Trisedodge family]] / [[Cloudy clan #Coblack\|cloudy clan]]		* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack\|Trisedodge family]] / [[Cloudy clan #Coblack\|cloudy clan]]