Starling family: Difference between revisions

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The head of the '''starling family''' is starling, which tempers out [[126/125]], the starling comma or septimal semicomma. Starling has a normal list basis 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 the nonpatent [[135edo]] val {{val| 135 214 314 379 }} (135c).

In starling, (~~6/5)3 =~~ 126/125 ~~× 12/7~~, ~~and minor thirds/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 temperament in wide use in Western 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~~.

The '''starling family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[starling comma]] ([[ratio]]: 126/125, {{monzo|legend=1| 1 2 -3 1 }}), also known as ''septimal semicomma''.

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.

== Starling ==

~~Temperaments discussed elsewhere include~~

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 }}.

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

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

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.

~~== Starling ==~~

~~{{Main| Starling and thrush }}~~

[[Subgroup]]: 2.3.5.7

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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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}}

== ~~Undecimal starling~~ ==

=== Overview to extensions ===

[[~~Subgroup~~]]~~: 2.~~3~~.5.7.11~~

Temperaments discussed elsewhere include

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

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

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

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

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

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

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

~~[[Badness]] (Smith): 0~~.~~677 × 10-3~~

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

== 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>~~

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

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

=== 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}}

~~Badness (Smith):~~ 0~~.915 × 10-3~~

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

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

Badness (Sintel): 0.856

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]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The head of the '''starling family''' is starling, which tempers out [[126/125]], the starling comma or septimal semicomma. Starling has a normal list basis 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 the nonpatent [[135edo]] val {{val| 135 214 314 379 }} (135c).
-In starling, (6/5)<sup>3</sup> = 126/125 × 12/7, and minor thirds/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 temperament in wide use in Western 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.
+The '''starling family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[starling comma]] ([[ratio]]: 126/125, {{monzo|legend=1| 1 2 -3 1 }}), also known as ''septimal semicomma''.
-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.
+== Starling ==
+{{Main| Starling and thrush }}
-Temperaments discussed elsewhere include
+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 }}.
-* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
-* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic 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.
+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.
-== Starling ==
-{{Main| Starling and thrush }}
 [[Subgroup]]: 2.3.5.7
@@ Line 28: / 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 37: / 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 57: / Line 56: @@
 }}
-== Undecimal starling ==
+=== Overview to extensions ===
-[[Subgroup]]: 2.3.5.7.11
+Temperaments discussed elsewhere include
+* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
-[[Comma list]]: 126/125, 385/384
+* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
+* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
-{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
+* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]
-{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
-[[Badness]] (Smith): 0.677 × 10<sup>-3</sup>
+Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.
 == 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 81: / 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 90: / 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>
-[[Associated temperament]]: [[myna]]
+[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
 === 13-limit ===
@@ Line 102: / Line 105: @@
 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 116: / 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}}
-Badness (Smith): 0.915 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
-Projection pairs: 7 125/18 11 3125/288 13 41472/3125
+Badness (Sintel): 0.856
+Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
 === Nightingale ===
@@ Line 128: / 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 139: / 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 157: / 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 168: / 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 178: / 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 183: / Line 250: @@
 <nowiki>*</nowiki> optimal patent val: [[34edo|34]]
-Badness (Smith): 0.876 × 10<sup>-3</sup>
+Badness (Sintel): 0.820
 === Mockingbird ===
@@ Line 191: / 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 202: / 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 213: / 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 224: / 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 235: / 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 246: / 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]]