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{{Technical data page}}
The head of the starling [[family|family]] is starling, which tempers out [[126/125|126/125]], the [[starling_comma|starling comma]] or [[septimal_semicomma|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|77edo]]. Other possible tunings are [[108edo|108edo]] and [[185edo|185edo]], and the nonpatent [[135edo|135edo]] 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)^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.
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 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.


=7-limit starling=
Temperaments discussed elsewhere include
[[Comma|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 }}


[[Projection_pair|Projection pair]]: 7 125/18
[[Badness]]: 0.0699 × 10<sup>-3</sup>


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


Music: [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure A Seed Planted] by Jake Freivald. The melody depends on tempering out 126/125.
{{Databox|[[Minkowski blocks]]|
* 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
}}


==[[Minkowski_blocks|Minkowski blocks]]==
== Undecimal starling ==
7: 25/24, 81/80
[[Subgroup]]: 2.3.5.7.11


8: 16/15, 648/625
[[Comma list]]: 126/125, 385/384


9: 27/25, 128/125
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}


11: 16/15, 15625/15552
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}


12: 128/125, 628/625
[[Badness]]: 0.677 × 10<sup>-3</sup>


15: 128/125, 250/243
== Thrush ==
{{Main| Starling and thrush }}


16: 648/625, 3125/3072
[[Subgroup]]: 2.3.5.7.11


17: 25/24, 20480/19683
[[Comma list]]: 126/125, 176/175


19: 81/80, 3125/3072
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}


27: 128/125, 78732/78125
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]


28: 648/625, 16875/16384
Lattice basis:  
: 5/4 length = 0.8576, 6/5 length = 0.9314
: Angle(5/4, 6/5) = 74.6239 degrees


31: 81/80, 1990656/1953125
[[Minimax tuning]]:  
* 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


34: 15625/15552, 2048/2025
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}


==11-limit==
[[Badness]]: 0.353 × 10<sup>-3</sup>
Commas: 126/125, 385/384


Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
[[Projection pair]]s: 7 125/18 11 3125/288


Generators: 2, 3, 5
[[Associated temperament]]: [[myna]]


EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: .000677
Comma list: 126/125, 176/175, 196/195


=Thrush=
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
[[Comma|Commas]]: 126/125, 176/175


Associated linear temperament: [[Starling_temperaments|myna]]
{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}


7 and 9 limit minimax
Badness: 0.677 × 10<sup>-3</sup>


[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
=== Bluebird ===
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
Subgroup: 2.3.5.7.11.13


[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
Comma list: 126/125, 144/143, 176/175


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}


Angle(5/4, 6/5) = 74.6239 degrees
{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}


Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
Badness: 0.915 × 10<sup>-3</sup>
 
Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
 
[[generator|Generators]]: 2, 3, 5
 
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
 
Badness: 0.000353
 
Projection pairs: 7 125/18 11 3125/288
 
Scales: [[thrush12|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|]
Comma list: 66/65, 126/125, 176/175


EDOs: 12, 15, 46
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}


Badness: 0.000991
{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}


=Thrasher=
Badness: 0.837 × 10<sup>-3</sup>
[[Comma|Commas]]: 56/55, 100/99


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


[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;,
Comma list: 91/90, 126/125, 176/175
|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|Eigenmonzos]]: 2, 8/7, 11/9
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}


Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}


Angle(6/5, 5/4) = 98.8447
Badness: 0.991 × 10<sup>-3</sup>


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


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


[[generator|Generators]]: 2, 3, 5
[[Comma list]]: 56/55, 100/99


EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}


Badness: 0.000480
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]


==13-limit==
Lattice basis:
Commas: 126/125, 100/99, 91/90
: 6/5 length = 0.9089, 5/4 length = 1.2007
: Angle (6/5, 5/4) = 98.8447


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


EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>


Badness: 0.000876
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]


==Mockingbird==
[[Badness]]: 0.480 × 10<sup>-3</sup>
Commas: 126/125, 100/99, 40/39


Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]


EDOs: 12f, 15, 27ef
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.000859
Comma list: 56/55, 91/90, 100/99


==Catbird==
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
Commas: 126/125, 100/99, 78/77


Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>


[[EDO|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]


Badness: 0.000905
Badness: 0.876 × 10<sup>-3</sup>


=Aplonis=
=== Mockingbird ===
Commas: 126/125, 540/539
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
Comma list: 40/39, 56/55, 100/99


EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}


Badness: 0.000648
{{Optimal ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}


==13-limit==
Badness: 0.859 × 10<sup>-3</sup>
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|]
=== Catbird ===
Subgroup: 2.3.5.7.11.13


EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
Comma list: 78/77, 100/99, 126/125


Badness: 0.000821
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}


=Oxpecker=
{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
Commas: 121/120, 126/125


Map: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
Badness: 0.905 × 10<sup>-3</sup>


EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
== Aplonis ==
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000699
[[Comma list]]: 126/125, 540/539


==Woodpecker==
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
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|]
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}


EDOs: 15, 31, 46f, 70f, 101f
[[Badness]]: 0.648 × 10<sup>-3</sup>


Badness: 0.001093
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=Treecreeper=
Comma list: 126/125, 144/143, 196/195
Commas: 126/125, 1232/1215


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


EDOs: 34, 46, 119c, 165c
{{Optimal ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}


Badness: 0.001585
Badness: 0.821 × 10<sup>-3</sup>


==13-limit==
== Treecreeper ==
Commas: 91/90, 126/125, 352/351
[[Subgroup]]: 2.3.5.7.11


Map: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
[[Comma list]]: 126/125, 1232/1215


EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}


Badness: 0.001588
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}


=Cuckoo=
[[Badness]]: 1.585 × 10<sup>-3</sup>
Commas: 126/125, 243/242


Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 31 34 58 65 89 154 185 216
Comma list: 91/90, 126/125, 352/351


Badness: 0.000933
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}


==13-limit==
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}
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|]
Badness: 1.588 × 10<sup>-3</sup>


EDOs: 31 58 65f 89f 92e 154 185 216f
[[Category:Temperament families]]
[[Category:bluebird]]
[[Category:Pages with mostly numerical content]]
[[Category:catbird]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:family]]
[[Category:Starling| ]] <!-- key article -->
[[Category:index]]
[[Category:Rank 3]]
[[Category:list]]
[[Category:listen]]
[[Category:mockingbird]]
[[Category:nightingale]]
[[Category:overview]]
[[Category:planar]]
[[Category:sound_example]]
[[Category:starling]]
[[Category:theory]]
[[Category:thrasher]]
[[Category:thrush]]

Latest revision as of 00:32, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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 ⟨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.

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.

Temperaments discussed elsewhere include

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

Starling

Main article: Starling and thrush

Subgroup: 2.3.5.7

Comma list: 126/125

Mapping: [⟨1 0 0 -1], ⟨0 1 0 -2], ⟨0 0 1 3]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 0 -2], ⟨0 1 1 1]]

Minkowski lattice basis:

6/5 length = 1.068, 5/4 length = 1.206
Angle (6/5, 5/4) = 100.364 degrees

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [1/3 2/3 0 1/3⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c

Badness: 0.0699 × 10-3

Projection pair: 7 125/18

Minkowski blocks
  • 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

Undecimal starling

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384

Mapping: [⟨1 0 0 -1 8], ⟨0 1 0 -2 3], ⟨0 0 1 3 -4]]

Optimal ET sequence: 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de

Badness: 0.677 × 10-3

Thrush

Main article: Starling and thrush

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175

Mapping: [⟨1 0 0 -1 -5], ⟨0 1 0 -2 -2], ⟨0 0 1 3 5]]

Mapping to lattice: [⟨0 1 1 1 3], ⟨0 1 0 -2 -2]]

Lattice basis:

5/4 length = 0.8576, 6/5 length = 0.9314
Angle(5/4, 6/5) = 74.6239 degrees

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [1/3 2/3 0 1/3 0⟩, [0 0 0 1 0⟩, [-10/3 4/3 0 5/3 0⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c

Badness: 0.353 × 10-3

Projection pairs: 7 125/18 11 3125/288

Associated temperament: myna

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195

Mapping: [⟨1 0 0 -1 -5 0], ⟨0 1 0 -2 -2 -5], ⟨0 0 1 3 5 5]]

Optimal ET sequence: 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf

Badness: 0.677 × 10-3

Bluebird

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175

Mapping: [⟨1 0 0 -1 -5 9], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 5 -5]]

Optimal ET sequence: 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff

Badness: 0.915 × 10-3

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

Nightingale

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 126/125, 176/175

Mapping: [⟨1 0 0 -1 -5 -4], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 3 5 4]]

Optimal ET sequence: 12f, 15, 19e, 27eff, 31

Badness: 0.837 × 10-3

Veery

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 176/175

Mapping: [⟨1 0 0 -1 -5 2], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 5 -2]]

Optimal ET sequence: 12, 15, 19e, 27e, 31f, 46

Badness: 0.991 × 10-3

Thrasher

See also: Ptolemismic clan § Thrasher

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99

Mapping: [⟨1 0 0 -1 2], ⟨0 1 0 -2 -2], ⟨0 0 1 3 2]]

Mapping to lattice: [⟨0 1 0 -2 -2], ⟨0 1 1 1 0]]

Lattice basis:

6/5 length = 0.9089, 5/4 length = 1.2007
Angle (6/5, 5/4) = 98.8447

Minimax tuning:

[[1 0 0 0 0⟩, [1 3/4 0 1/4 -3/8⟩, [1 1/2 0 1/2 -1/4⟩, [0 0 0 1 0⟩, [2 -1/2 0 1/2 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7.11/9

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e *

* optimal patent val: 34

Badness: 0.480 × 10-3

Scales: Thrasher chromatic, Thrasher diatonic

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -1 2 2], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 2 -2]]

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e, 69bceef *

* optimal patent val: 34

Badness: 0.876 × 10-3

Mockingbird

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 100/99

Mapping: [⟨1 0 0 -1 2 3], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 3 2 1]]

Optimal ET sequence: 7d, 8d, 12f, 15, 27eff

Badness: 0.859 × 10-3

Catbird

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -1 2 0], ⟨0 1 0 -2 -2 -5], ⟨0 0 1 3 2 5]]

Optimal ET sequence: 7df, 8d, 12f, 19, 27e, 66cdeeef

Badness: 0.905 × 10-3

Aplonis

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539

Mapping: [⟨1 0 0 -1 4], ⟨0 1 0 -2 7], ⟨0 0 1 3 -5]]

Optimal ET sequence: 12e, 19, 27e, 31, 58, 89, 197c, 228c

Badness: 0.648 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -1 4 0], ⟨0 1 0 -2 7 -5], ⟨0 0 1 3 -5 5]]

Optimal ET sequence: 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff

Badness: 0.821 × 10-3

Treecreeper

Subgroup: 2.3.5.7.11

Comma list: 126/125, 1232/1215

Mapping: [⟨1 0 0 -1 -3], ⟨0 1 0 -2 7], ⟨0 0 1 3 -2]]

Optimal ET sequence: 7d, 12e, 19e, 27e, 39d, 46, 119c

Badness: 1.585 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -1 -3 2], ⟨0 1 0 -2 7 4], ⟨0 0 1 3 -2 -2]]

Optimal ET sequence: 7d, 12e, 19e, 27e, 46

Badness: 1.588 × 10-3

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