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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 }}.


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== Starling  ==
== Starling  ==
[[Comma]] c = 126/125
Subgroup: 2.3.5.7


7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
[[Comma list]]: [[126/125]]


[<1 0 0 0|, <0 1 0 0|, <1/3 2/3 0 1/3|, <0 0 0 1|]
[[Mapping]]: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]


Eigenmonzos: 2, 8/7, 4/3
Mapping 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


Map to lattice: [<0 1 0 -2|, <0 1 1 1|]
Map to lattice: [<0 1 0 -2|, <0 1 1 1|]


Map: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]
Minkowski lattice basis:  
: 6/5 length = 1.068, 5/4 length = 1.206
: Angle (6/5, 5/4) = 100.364 degrees


Generators: 2, 3, 5
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3c sharp
: [<1 0 0 0|, <0 1 0 0|, <1/3 2/3 0 1/3|, <0 0 0 1|]
: Eigenmonzos: 2, 8/7, 4/3


[[EDO|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
{{Val list|legend=1| 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, 77, 108, 166c, 185, 243c }}


Badness: 0.0000699
[[Badness]]: 0.0699 × 10<sup>-3</sup>


[[Projection pair]]: 7 125/18
[[Projection pair]]: 7 125/18
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Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]


Music: [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by Jake Freivald. The melody depends on tempering out 126/125.
; Music
* [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by [[Jake Freivald]]. The melody depends on tempering out 126/125.
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">[[Minkowski blocks]]</div>
<div class="mw-collapsible-content">


=== [[Minkowski blocks]]  ===
* 7: 25/24, 81/80
7: 25/24, 81/80
* 8: 16/15, 648/625
* 9: 27/25, 128/125
* 11: 16/15, 15625/15552
* 12: 128/125, 628/625
* 15: 128/125, 250/243
* 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


8: 16/15, 648/625
</div></div>


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]]: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]


15: 128/125, 250/243
Mapping generators: ~2, ~3, ~5


16: 648/625, 3125/3072
{{Val list|legend=1| 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de }}


17: 25/24, 20480/19683
[[Badness]]: 0.677 × 10<sup>-3</sup>


19: 81/80, 3125/3072
== Thrush ==
Subgroup: 2.3.5.7.11


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


28: 648/625, 16875/16384
[[Mapping]]: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]


31: 81/80, 1990656/1953125
Mapping generators: ~2, ~3, ~5


34: 15625/15552, 2048/2025
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
: [|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;, |0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
: [[Eigenmonzo]]s: 2, 7/6, 4/3


=== 11-limit  ===
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
Commas: 126/125, 385/384


Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
Lattice basis:  
: 5/4 length = 0.8576, 6/5 length = 0.9314
: Angle(5/4, 6/5) = 74.6239 degrees


Generators: 2, 3, 5
{{Val list|legend=1| 12, 15, 31, 46, 58, 89, 135c, 224c }}


EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
[[Badness]]: 0.353 × 10<sup>-3</sup>


Badness: .000677
[[Projection pair]]s: 7 125/18 11 3125/288


== Thrush  ==
[[Associated temperament]]: [[Starling temperaments #Myna|myna]]
[[Commas]]: 126/125, 176/175


Associated linear temperament: [[Starling temperaments #Myna|myna]]
Scales: [[thrush12]]


7 and 9 limit minimax
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
Comma list: 126/125, 176/175, 196/195
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]


[[Eigenmonzo]]s: 2, 7/6, 4/3
Mapping: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
Vals: {{Val list| 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf }}


Angle(5/4, 6/5) = 74.6239 degrees
Badness: 0.677 × 10<sup>-3</sup>


Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
=== Bluebird ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
Comma list: 126/125, 176/175, 144/143


[[Generator]]s: 2, 3, 5
Mapping: [&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, 46, 58, 89, 135c, 224c
Vals: {{Val list| 12, 15, 31, 43, 48c, 58, 147cf, 205cef }}


Badness: 0.000353
Badness: 0.915 × 10<sup>-3</sup>


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


Scales: [[thrush12]]
=== Nightingale ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 126/125, 176/175, 66/65
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|]
Mapping: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]


EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf
Vals: {{Val list| 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef }}


Badness: 0.000677
Badness: 0.837 × 10<sup>-3</sup>


=== Bluebird  ===
=== Veery ===
Commas 126/125, 176/175, 144/143
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]
Comma list: 91/90, 126/125, 176/175


EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef
Mapping: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]


Badness: 0.000915
Vals: {{Val list| 12, 15, 46 }}


Projection pairs: 7 125/18 11 3125/288 13 41472/3125
Badness: 0.991 × 10<sup>-3</sup>


=== Nightingale  ===
== Thrasher ==
Commas: 126/125, 176/175, 66/65
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
[[Comma list]]: 56/55, 100/99


[[EDO|EDOs]]: [[15edo|15]], [[19edo|19f]], [[21edo|21e]], [[22edo|22f]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]], [[108edo|108ef]]
[[Mapping]]: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]


Badness: 0.000837
Mapping generators: ~2, ~3, ~5


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


EDOs: 12, 15, 46
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;, |1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;, |2 -1/2 0 1/2 1/4&gt;]
: [[Eigenmonzo]]s: 2, 8/7, 11/9


Badness: 0.000991
{{Val list|legend=1| 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e }}


== Thrasher  ==
[[Badness]]: 0.480 × 10<sup>-3</sup>
[[Comma]]s: 56/55, 100/99


11-limit minimax
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;,
Comma list: 126/125, 100/99, 91/90
|1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;,
|2 -1/2 0 1/2 1/4&gt;]


[[Eigenmonzo]]s: 2, 8/7, 11/9
Mapping: [&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
Vals: {{Val list| 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e }}


Angle(6/5, 5/4) = 98.8447
Badness: 0.876 × 10<sup>-3</sup>


Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
=== Mockingbird ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
Comma list: 40/39, 100/99, 126/125


[[Generator]]s: 2, 3, 5
Mapping: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]


EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
Vals: {{Val list| 12f, 15, 27ef }}


Badness: 0.000480
Badness: 0.859 × 10<sup>-3</sup>


=== 13-limit  ===
=== Catbird ===
Commas: 126/125, 100/99, 91/90
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
Comma list: 78/77, 100/99, 126/125


EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
Mapping: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]


Badness: 0.000876
Vals: {{Val list| 12, 15, 17c, 19, 22ef], 27e, 34, 58e }}


=== Mockingbird  ===
Badness: 0.905 × 10<sup>-3</sup>
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|]
== Aplonis ==
Subgroup: 2.3.5.7.11


EDOs: 12f, 15, 27ef
[[Comma list]]: 126/125, 540/539


Badness: 0.000859
[[Mapping]]: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]


=== Catbird  ===
{{Val list|legend=1| 19, 31, 58, 67, 89, 197c, 286ce, 375ce }}
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|]
[[Badness]]: 0.648 × 10<sup>-3</sup>


[[EDO|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000905
Comma list: 126/125, 144/143, 196/195


== Aplonis  ==
Mapping: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
Commas: 126/125, 540/539


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
Vals: {{Val list| 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef }}


EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
Badness: 0.821 × 10<sup>-3</sup>


Badness: 0.000648
== Oxpecker ==
Subgroup: 2.3.5.7.11


=== 13-limit  ===
[[Comma list]]: 121/120, 126/125
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|]
[[Mapping]]: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]


EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
{{Val list|legend=1| 8d, 15, 31, 46, 77, 185e, 262cde }}


Badness: 0.000821
[[Badness]]: 0.699 × 10<sup>-3</sup>


== Oxpecker  ==
=== Woodpecker ===
Commas: 121/120, 126/125
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
Comma list: 66/65, 121/120, 126/125


EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
Mapping: [&lt;1 0 1 2 2 2|, &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]


Badness: 0.000699
Vals: {{Val list| 15, 31, 46f, 70f, 101f }}


=== Woodpecker  ===
Badness: 1.093 × 10<sup>-3</sup>
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|]
== Treecreeper ==
Subgroup: 2.3.5.7.11


EDOs: 15, 31, 46f, 70f, 101f
[[Comma list]]: 126/125, 1232/1215


Badness: 0.001093
[[Mapping]]: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]


== Treecreeper  ==
{{Val list|legend=1| 34, 46, 119c, 165c }}
Commas: 126/125, 1232/1215


Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
[[Badness]]: 1.585 × 10<sup>-3</sup>


EDOs: 34, 46, 119c, 165c
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.001585
Comma list: 91/90, 126/125, 352/351


=== 13-limit  ===
Mapping: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
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|]
Vals: {{Val list| 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf }}


EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
Badness: 1.588 × 10<sup>-3</sup>


Badness: 0.001588
== Cuckoo ==
Subgroup: 2.3.5.7.11


== Cuckoo  ==
[[Comma list]]: 126/125, 243/242
Commas: 126/125, 243/242


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


EDOs: 31, 34, 58, 65, 89, 154, 185, 216
{{Val list|legend=1| 31, 34, 58, 65, 89, 154, 185, 216 }}


Badness: 0.000933
[[Badness]]: 0.933 × 10<sup>-3</sup>


=== 13-limit  ===
=== 13-limit  ===
Commas: 126/125, 196/195, 243/242
Subgroup: 2.3.5.7.11.13
 
Comma list: 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|]
Mapping: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|]


EDOs: 31, 58, 65f, 89f, 92e, 154, 185, 216f
Vals: {{Val list| 31, 58, 65f, 89f, 92e, 154, 185, 216f }}


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Starling]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Thrush]]
[[Category:Thrush]]
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Starling_family"