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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 }}.
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.
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.


Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to [[Starling temperaments #Valentine|valentine temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
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
[[Comma]] c = 126/125
* ''[[Erato]]'' (+81/80) → [[Didymus rank three 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 three clan #Cuckoo|Rastmic rank-3 clan]]


7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
Considered below are starling, thrush, thrasher, aplonis, and treecreeper.


[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]
== Starling ==
{{Main| Starling and thrush }}


Eigenmonzos: 2, 8/7, 4/3
[[Subgroup]]: 2.3.5.7


Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206
[[Comma list]]: [[126/125]]


Angle(6/5, 5/4) = 100.364 degrees
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}


Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]
: mapping generators: ~2, ~3, ~5


Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]


Generators: 2, 3, 5
[[Minkowski lattice basis]]:  
: 6/5 length = 1.068, 5/4 length = 1.206
: Angle (6/5, 5/4) = 100.364 degrees


[[EDO|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


Badness: 0.0000699
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
 
[[Badness]]: 0.0699 × 10<sup>-3</sup>


[[Projection pair]]: 7 125/18
[[Projection pair]]: 7 125/18


Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
{{Databox|[[Minkowski blocks]]|
 
* 7: 25/24, 81/80
Music: [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by Jake Freivald. The melody depends on tempering out 126/125.
* 8: 16/15, 648/625
 
* 9: 27/25, 128/125
== [[Minkowski blocks]] ==
* 11: 16/15, 15625/15552
7: 25/24, 81/80
* 12: 128/125, 628/625
 
* 15: 128/125, 250/243
8: 16/15, 648/625
* 16: 648/625, 3125/3072
* 17: 25/24, 20480/19683
* 19: 81/80, 3125/3072
* 27: 128/125, 78732/78125
* 28: 648/625, 16875/16384
* 31: 81/80, 1990656/1953125
* 34: 15625/15552, 2048/2025
}}


9: 27/25, 128/125
== Undecimal starling ==
[[Subgroup]]: 2.3.5.7.11


11: 16/15, 15625/15552
[[Comma list]]: 126/125, 385/384


12: 128/125, 628/625
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}


15: 128/125, 250/243
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}


16: 648/625, 3125/3072
[[Badness]]: 0.677 × 10<sup>-3</sup>


17: 25/24, 20480/19683
== Thrush ==
{{Main| Starling and thrush }}


19: 81/80, 3125/3072
[[Subgroup]]: 2.3.5.7.11


27: 128/125, 78732/78125
[[Comma list]]: 126/125, 176/175


28: 648/625, 16875/16384
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}


31: 81/80, 1990656/1953125
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]


34: 15625/15552, 2048/2025
Lattice basis:  
: 5/4 length = 0.8576, 6/5 length = 0.9314
: Angle(5/4, 6/5) = 74.6239 degrees


== 11-limit ==
[[Minimax tuning]]:
Commas: 126/125, 385/384
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}


Generators: 2, 3, 5
[[Badness]]: 0.353 × 10<sup>-3</sup>


EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
[[Projection pair]]s: 7 125/18 11 3125/288


Badness: .000677
[[Associated temperament]]: [[myna]]


= Thrush =
=== 13-limit ===
[[Commas]]: 126/125, 176/175
Subgroup: 2.3.5.7.11.13


Associated linear temperament: [[Starling temperaments #Myna|myna]]
Comma list: 126/125, 176/175, 196/195


7 and 9 limit minimax
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}


[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]


[[Eigenmonzo]]s: 2, 7/6, 4/3
Badness: 0.677 × 10<sup>-3</sup>


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
=== Bluebird ===
Subgroup: 2.3.5.7.11.13


Angle(5/4, 6/5) = 74.6239 degrees
Comma list: 126/125, 144/143, 176/175


Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}


Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}


[[Generator]]s: 2, 3, 5
Badness: 0.915 × 10<sup>-3</sup>
 
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
 
Badness: 0.000353
 
Projection pairs: 7 125/18 11 3125/288
 
Scales: [[thrush12]]
 
== 13-limit ==
Commas: 126/125, 176/175, 196/195
 
Map: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]
 
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf
 
Badness: 0.000677
 
== Bluebird ==
Commas 126/125, 176/175, 144/143
 
Map: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]
 
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef
 
Badness: 0.000915


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


== Nightingale ==
=== Nightingale ===
Commas: 126/125, 176/175, 66/65
Subgroup: 2.3.5.7.11.13
 
Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
 
[[EDO|EDOs]]: [[15edo|15]], [[19edo|19f]], [[21edo|21e]], [[22edo|22f]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]], [[108edo|108ef]]
 
Badness: 0.000837
 
== Veery ==
Commas: 91/90, 126/125, 176/175
 
Map: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]


EDOs: 12, 15, 46
Comma list: 66/65, 126/125, 176/175


Badness: 0.000991
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}


= Thrasher =
{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}
[[Comma]]s: 56/55, 100/99


11-limit minimax
Badness: 0.837 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;,
=== Veery ===
|1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;,
Subgroup: 2.3.5.7.11.13
|2 -1/2 0 1/2 1/4&gt;]


[[Eigenmonzo]]s: 2, 8/7, 11/9
Comma list: 91/90, 126/125, 176/175


Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}


Angle(6/5, 5/4) = 98.8447
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}


Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
Badness: 0.991 × 10<sup>-3</sup>


Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
== Thrasher ==
{{See also| Ptolemismic clan #Thrasher }}


[[Generator]]s: 2, 3, 5
[[Subgroup]]: 2.3.5.7.11


EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
[[Comma list]]: 56/55, 100/99


Badness: 0.000480
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}


== 13-limit ==
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
Commas: 126/125, 100/99, 91/90


Map: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
Lattice basis:  
: 6/5 length = 0.9089, 5/4 length = 1.2007
: Angle (6/5, 5/4) = 98.8447


EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
[[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


Badness: 0.000876
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>


== Mockingbird ==
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
Commas: 40/39, 100/99, 126/125


Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
[[Badness]]: 0.480 × 10<sup>-3</sup>


EDOs: 12f, 15, 27ef
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]


Badness: 0.000859
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== Catbird ==
Comma list: 56/55, 91/90, 100/99
Commas: 78/77, 100/99, 126/125


Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}


[[EDO|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>


Badness: 0.000905
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]


= Aplonis =
Badness: 0.876 × 10<sup>-3</sup>
Commas: 126/125, 540/539


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
=== Mockingbird ===
Subgroup: 2.3.5.7.11.13


EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
Comma list: 40/39, 56/55, 100/99


Badness: 0.000648
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}


== 13-limit ==
{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
Commas: 126/125, 144/143, 196/195


Map: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
Badness: 0.859 × 10<sup>-3</sup>


EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
=== Catbird ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000821
Comma list: 78/77, 100/99, 126/125


= Oxpecker =
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
Commas: 121/120, 126/125


Map: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}


EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
Badness: 0.905 × 10<sup>-3</sup>


Badness: 0.000699
== Aplonis ==
[[Subgroup]]: 2.3.5.7.11


== Woodpecker ==
[[Comma list]]: 126/125, 540/539
Commas: 66/65, 121/120, 126/125


Map: [&lt;1 0 1 2 2 2|, &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}


EDOs: 15, 31, 46f, 70f, 101f
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}


Badness: 0.001093
[[Badness]]: 0.648 × 10<sup>-3</sup>


= Treecreeper =
=== 13-limit ===
Commas: 126/125, 1232/1215
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
Comma list: 126/125, 144/143, 196/195


EDOs: 34, 46, 119c, 165c
Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}


Badness: 0.001585
{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}


== 13-limit ==
Badness: 0.821 × 10<sup>-3</sup>
Commas: 91/90, 126/125, 352/351


Map: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
== Treecreeper ==
[[Subgroup]]: 2.3.5.7.11


EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
[[Comma list]]: 126/125, 1232/1215


Badness: 0.001588
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}


= Cuckoo =
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
Commas: 126/125, 243/242


Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]
[[Badness]]: 1.585 × 10<sup>-3</sup>


EDOs: 31, 34, 58, 65, 89, 154, 185, 216
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000933
Comma list: 91/90, 126/125, 352/351


== 13-limit ==
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
Commas: 126/125, 196/195, 243/242


Map: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|]
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}


EDOs: 31, 58, 65f, 89f, 92e, 154, 185, 216f
Badness: 1.588 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Starling]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Thrush]]
[[Category:Starling| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Starling_family"