Starling family: Difference between revisions

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The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list of {2, 3, 5}; hence a [[5-limit]] scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and [[135edo]] in the nonpatent 135c val {{val| 135 214 '''314''' 379 }}.

In starling, (6/5)3 = (126/125)⋅(12/7), and minor thirds and major sixths are low-complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5, 6/5, 6/5, 7/6 versions of the [[diminished seventh chord]], which is very characteristic of it. Since this is a chord of [[meantone]] in wide use in Western {{w|common practice}} harmony long before [[12edo]] established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.

Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.

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[[Minkowski lattice basis]]:

: ~6/5 length = 1.068, ~5/4 length = 1.206

: angle (~6/5, ~5/4) = 100.~~364 degrees~~

: angle (~6/5, ~5/4) = 100.364°

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}

: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}

: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}

[[Minimax tuning]]:

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{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}

[[Badness]] (~~Smith~~): 0.~~0699 × 10-3~~

[[Badness]] (Sintel): 0.308

[[Projection pair]]: <code>7 125/18</code>

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Temperaments discussed elsewhere include

* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]

* ''[[Sensigh]]'' (+245/243) → [[~~Sensamagic~~ family #Sensigh|~~Sensamagic~~ family]]

* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]

* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]

* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]

Considered below are ~~starling~~, ~~thrush~~, thrasher, aplonis, and treecreeper.

Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.

~~== Undecimal starling ==~~

~~[[Subgroup]]: 2.3.5.7.11~~

~~[[Comma list]]: 126/125, 385/384~~

~~{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}~~

~~{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}~~

~~[[Badness]] (Smith): 0.677 × 10-3~~

== Thrush ==

Thrush is the main extension of starling into the [[11-limit]]. It may be described as 12 & 15 & 31, and is loosely [[associated temperament|associated]] with [[myna]].

[[Subgroup]]: 2.3.5.7.11

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Lattice basis:

: ~5/4 length = 0.8576, ~6/5 length = 0.9314

: angle(~5/4, ~6/5) = 74.~~6239 degrees~~

: angle(~5/4, ~6/5) = 74.6239°

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}

: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}

: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}

[[Minimax tuning]]:

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{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}

[[Badness]] (~~Smith~~): 0.~~353 × 10-3~~

[[Badness]] (Sintel): 0.424

[[Projection pair]]s: <code>7 125/18 11 3125/288</code>

[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>

~~[[Associated temperament]]: [[myna]]~~

=== 13-limit ===

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Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}

Optimal tunings:

* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}

{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}

Badness (~~Smith~~): 0.~~677 × 10-3~~

Badness (Sintel): 0.633

=== Bluebird ===

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Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}

{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58~~, 147cf~~, 205cceff, 263ccdeefff }}

Optimal tunings:

* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}

{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}

Badness (~~Smith~~): 0.~~915 × 10-3~~

Badness (Sintel): 0.856

Projection pairs: 7 125/18 11 3125/288 13 41472/3125

Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>

=== Nightingale ===

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Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}

Optimal tunings:

* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}

Badness (~~Smith~~): 0.~~837 × 10-3~~

Badness (Sintel): 0.783

=== Veery ===

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Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}

Optimal tunings:

* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}

Badness (Smith): 0.~~991 × 10~~<~~sup~~>-3</~~sup>~~

Badness (Sintel): 0.927

== Starnova ==

Starnova tempers out [[385/384]]. This temperament used to be known as ''undecimal starling'', but was decanonicalized in 2026. The name ''starnova'' was proposed by [[Flora Canou]] after [[Gene Ward Smith]]'s [[Star]] and [[Nova]] scales.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 126/125, 385/384

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}

: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}

: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}

{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}

[[Badness]] (Sintel): 0.813

[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>

Scales: [[Star and Nova]]

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384

Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}

Optimal tunings:

* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}

Badness (Sintel): 1.07

== Thrasher ==

[[Subgroup]]: 2.3.5.7.11

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Lattice basis:

: ~6/5 length = 0.9089, ~5/4 length = 1.2007

: angle (~6/5, ~5/4) = 98.~~8447~~

: angle (~6/5, ~5/4) = 98.8447°

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}

: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}

: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}

[[Minimax tuning]]:

* [[11-odd-limit]]

: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]

: [[~~Eigenmonzo~~ basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9

{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *

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<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]

[[Badness]] (~~Smith~~): 0.~~480 × 10-3~~

[[Badness]] (Sintel): 0.577

Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]

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Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}

Optimal tunings:

* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}

{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *

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<nowiki>*</nowiki> optimal patent val: [[34edo|34]]

Badness (~~Smith~~): 0.~~876 × 10-3~~

Badness (Sintel): 0.820

=== Mockingbird ===

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Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}

Optimal tunings:

* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}

Badness (~~Smith~~): 0.~~859 × 10-3~~

Badness (Sintel): 0.804

=== Catbird ===

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Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}

Optimal tunings:

* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}

{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}

Badness (~~Smith~~): 0.~~905 × 10-3~~

Badness (Sintel): 0.846

== Aplonis ==

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}

: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}

: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}

{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}

[[Badness]] (~~Smith~~): 0.~~648 × 10-3~~

[[Badness]] (Sintel): 0.778

=== 13-limit ===

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Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}

Optimal tunings:

* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}

{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}

Badness (~~Smith~~): 0.~~821 × 10-3~~

Badness (Sintel): 0.768

== Treecreeper ==

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[[Optimal tuning]]s:

* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}

: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}

: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}

{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}

[[Badness]] (~~Smith~~): 1.~~585 × 10-3~~

[[Badness]] (Sintel): 1.91

=== 13-limit ===

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Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}

Optimal tunings:

* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}

Badness (~~Smith~~): 1.~~588 × 10-3~~

Badness (Sintel): 1.48

[[Category:Temperament families]]

[[Category:Starling family| ]]

[[Category:Rank 3]]

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 The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list of {2, 3, 5}; hence a [[5-limit]] scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and [[135edo]] in the nonpatent 135c val {{val| 135 214 '''314''' 379 }}.
-In starling, (6/5)<sup>3</sup> = (126/125)⋅(12/7), and minor thirds and major sixths are low-complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5, 6/5, 6/5, 7/6 versions of the [[diminished seventh chord]], which is very characteristic of it. Since this is a chord of [[meantone]] in wide use in Western {{w|common practice}} harmony long before [[12edo]] established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.
 Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
@@ Line 23: / Line 21: @@
 [[Minkowski lattice basis]]:
 : ~6/5 length = 1.068, ~5/4 length = 1.206
-: angle (~6/5, ~5/4) = 100.364 degrees
+: angle (~6/5, ~5/4) = 100.364°
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
+: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}
+: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}
 [[Minimax tuning]]:
@@ Line 32: / Line 36: @@
 {{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
-[[Badness]] (Smith): 0.0699 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.308
 [[Projection pair]]: <code>7 125/18</code>
@@ Line 55: / Line 59: @@
 Temperaments discussed elsewhere include
 * ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
-* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic family]]
+* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
 * ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
 * ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]
-Considered below are starling, thrush, thrasher, aplonis, and treecreeper.
+Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.
-== Undecimal starling ==
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 126/125, 385/384
-{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
-{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
-[[Badness]] (Smith): 0.677 × 10<sup>-3</sup>
 == Thrush ==
 {{Main| Starling and thrush }}
+Thrush is the main extension of starling into the [[11-limit]]. It may be described as 12 & 15 & 31, and is loosely [[associated temperament|associated]] with [[myna]].
 [[Subgroup]]: 2.3.5.7.11
@@ Line 85: / Line 80: @@
 Lattice basis:
 : ~5/4 length = 0.8576, ~6/5 length = 0.9314
-: angle(~5/4, ~6/5) = 74.6239 degrees
+: angle(~5/4, ~6/5) = 74.6239°
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
+: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}
+: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}
 [[Minimax tuning]]:
@@ Line 94: / Line 95: @@
 {{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
-[[Badness]] (Smith): 0.353 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.424
-[[Projection pair]]s: <code>7 125/18 11 3125/288</code>
+[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
-[[Associated temperament]]: [[myna]]
 === 13-limit ===
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 Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
+Optimal tunings:
+* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}
 {{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
-Badness (Smith): 0.677 × 10<sup>-3</sup>
+Badness (Sintel): 0.633
 === Bluebird ===
@@ Line 120: / Line 123: @@
 Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
-{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
+Optimal tunings:
+* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}
+{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
-Badness (Smith): 0.915 × 10<sup>-3</sup>
+Badness (Sintel): 0.856
-Projection pairs: 7 125/18 11 3125/288 13 41472/3125
+Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
 === Nightingale ===
@@ Line 132: / Line 139: @@
 Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
+Optimal tunings:
+* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}
 {{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
-Badness (Smith): 0.837 × 10<sup>-3</sup>
+Badness (Sintel): 0.783
 === Veery ===
@@ Line 143: / Line 154: @@
 Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
+Optimal tunings:
+* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
 {{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
-Badness (Smith): 0.991 × 10<sup>-3</sup>
+Badness (Sintel): 0.927
+== Starnova ==
+Starnova tempers out [[385/384]]. This temperament used to be known as ''undecimal starling'', but was decanonicalized in 2026. The name ''starnova'' was proposed by [[Flora Canou]] after [[Gene Ward Smith]]'s [[Star]] and [[Nova]] scales.
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 126/125, 385/384
+{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
+: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
+: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
+{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
+[[Badness]] (Sintel): 0.813
+[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
+Scales: [[Star and Nova]]
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 196/195, 385/384
+Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
+Optimal tunings:
+* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
+{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
+Badness (Sintel): 1.07
 == Thrasher ==
-{{See also| Ptolemismic clan #Thrasher }}
+{{See also| Ptolemismic clan }}
 [[Subgroup]]: 2.3.5.7.11
@@ Line 161: / Line 214: @@
 Lattice basis:
 : ~6/5 length = 0.9089, ~5/4 length = 1.2007
-: angle (~6/5, ~5/4) = 98.8447
+: angle (~6/5, ~5/4) = 98.8447°
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
+: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
+: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
-: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
 {{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
@@ Line 172: / Line 231: @@
 <nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
-[[Badness]] (Smith): 0.480 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.577
 Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
@@ Line 182: / Line 241: @@
 Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
+Optimal tunings:
+* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}
 {{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
@@ Line 187: / Line 250: @@
 <nowiki>*</nowiki> optimal patent val: [[34edo|34]]
-Badness (Smith): 0.876 × 10<sup>-3</sup>
+Badness (Sintel): 0.820
 === Mockingbird ===
@@ Line 195: / Line 258: @@
 Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
+Optimal tunings:
+* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}
 {{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}
-Badness (Smith): 0.859 × 10<sup>-3</sup>
+Badness (Sintel): 0.804
 === Catbird ===
@@ Line 206: / Line 273: @@
 Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
+Optimal tunings:
+* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}
 {{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
-Badness (Smith): 0.905 × 10<sup>-3</sup>
+Badness (Sintel): 0.846
 == Aplonis ==
@@ Line 217: / Line 288: @@
 {{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}
+: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
+: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}
 {{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
-[[Badness]] (Smith): 0.648 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.778
 === 13-limit ===
@@ Line 228: / Line 305: @@
 Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
+Optimal tunings:
+* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}
 {{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
-Badness (Smith): 0.821 × 10<sup>-3</sup>
+Badness (Sintel): 0.768
 == Treecreeper ==
@@ Line 239: / Line 320: @@
 {{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
+: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
+: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}
 {{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
-[[Badness]] (Smith): 1.585 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.91
 === 13-limit ===
@@ Line 250: / Line 337: @@
 Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
+Optimal tunings:
+* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}
 {{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}
-Badness (Smith): 1.588 × 10<sup>-3</sup>
+Badness (Sintel): 1.48
 [[Category:Temperament families]]
 [[Category:Starling family| ]] <!-- main article -->
 [[Category:Rank 3]]