Starling family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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 &lt;135 214 314 379|.
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on <tt>2015-07-20 09:11:17 UTC</tt>.<br>
: The original revision id was <tt>555518345</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]
The head of the starling [[family]] is starling, which tempers out [[126_125|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 &lt;135 214 314 379|.


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)^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 [[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 [[Starling_temperaments|valentine temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.


=7-limit starling=  
=7-limit starling=
[[Comma]] c = 126/125
[[Comma|Comma]] c = 126/125


7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]
[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]
Eigenmonzos: 2, 8/7, 4/3
Eigenmonzos: 2, 8/7, 4/3


Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206
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 degrees
Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]
Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]


Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]
Generators: 2, 3, 5
Generators: 2, 3, 5
[[edo|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
 
[[EDO|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
 
Badness: 0.0000699
Badness: 0.0000699


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


==[[Minkowski blocks]]==  
==[[Minkowski_blocks|Minkowski blocks]]==
7: 25/24, 81/80
7: 25/24, 81/80
8: 16/15, 648/625
8: 16/15, 648/625
9: 27/25, 128/125
9: 27/25, 128/125
11: 16/15, 15625/15552
11: 16/15, 15625/15552
12: 128/125, 628/625
12: 128/125, 628/625
15: 128/125, 250/243
15: 128/125, 250/243
16: 648/625, 3125/3072
16: 648/625, 3125/3072
17: 25/24, 20480/19683
17: 25/24, 20480/19683
19: 81/80, 3125/3072
19: 81/80, 3125/3072
27: 128/125, 78732/78125
27: 128/125, 78732/78125
28: 648/625, 16875/16384
28: 648/625, 16875/16384
31: 81/80, 1990656/1953125
31: 81/80, 1990656/1953125
34: 15625/15552, 2048/2025
34: 15625/15552, 2048/2025


==11-limit==  
==11-limit==
Commas: 126/125, 385/384
Commas: 126/125, 385/384


Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
Generators: 2, 3, 5
Generators: 2, 3, 5
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
Badness: .000677
Badness: .000677


=Thrush=  
=Thrush=
[[Comma|Commas]]: 126/125, 176/175
[[Comma|Commas]]: 126/125, 176/175
Associated linear temperament: [[Starling temperaments|myna]]
 
Associated linear temperament: [[Starling_temperaments|myna]]


7 and 9 limit minimax
7 and 9 limit minimax
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
[|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;]
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
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 degrees
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]


Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
[[Generator|Generators]]: 2, 3, 5
 
[[generator|Generators]]: 2, 3, 5
 
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
Badness: 0.000353
Badness: 0.000353


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


Scales: [[thrush12]]
Scales: [[thrush12|thrush12]]


==13-limit==  
==13-limit==
Commas: 126/125, 176/175, 196/195
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|]
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
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf
Badness: 0.000677
Badness: 0.000677


==Bluebird==  
==Bluebird==
Commas 126/125, 176/175, 144/143
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|]
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
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef
Badness: 0.000915
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
Commas: 126/125, 176/175, 66/65


Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
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]]
 
[[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
Badness: 0.000837


==Veery==  
==Veery==
Commas: 91/90, 126/125, 176/175
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|]
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
EDOs: 12, 15, 46
Badness: 0.000991
Badness: 0.000991


=Thrasher=  
=Thrasher=
[[Comma|Commas]]: 56/55, 100/99
[[Comma|Commas]]: 56/55, 100/99


11-limit minimax
11-limit minimax
[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;,
[|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;,
|1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;,
|2 -1/2 0 1/2 1/4&gt;]
|2 -1/2 0 1/2 1/4&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9


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


Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
[[Generator|Generators]]: 2, 3, 5
 
[[generator|Generators]]: 2, 3, 5
 
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
Badness: 0.000480
Badness: 0.000480


==13-limit==  
==13-limit==
Commas: 126/125, 100/99, 91/90
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|]
Map: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
Badness: 0.000876
Badness: 0.000876


==Mockingbird==  
==Mockingbird==
Commas: 126/125, 100/99, 40/39
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|]
Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
EDOs: 12f, 15, 27ef
EDOs: 12f, 15, 27ef
Badness: 0.000859
Badness: 0.000859


==Catbird==  
==Catbird==
Commas: 126/125, 100/99, 78/77
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|]
Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
 
[[EDO|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
 
Badness: 0.000905
Badness: 0.000905


=Aplonis=  
=Aplonis=
Commas: 126/125, 540/539
Commas: 126/125, 540/539


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
Badness: 0.000648
Badness: 0.000648


==13-limit==  
==13-limit==
Commas: 126/125, 144/143, 196/195
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|]
Map: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
Badness: 0.000821
Badness: 0.000821


=Oxpecker=  
=Oxpecker=
Commas: 121/120, 126/125
Commas: 121/120, 126/125


Map: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
Map: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
Badness: 0.000699
Badness: 0.000699


==Woodpecker==  
==Woodpecker==
Commas: 66/65, 121/120, 126/125
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|]
Map: [&lt;1 0 1 2 2 2|, &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]
EDOs: 15, 31, 46f, 70f, 101f
EDOs: 15, 31, 46f, 70f, 101f
Badness: 0.001093
Badness: 0.001093


=Treecreeper=  
=Treecreeper=
Commas: 126/125, 1232/1215
Commas: 126/125, 1232/1215


Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
EDOs: 34, 46, 119c, 165c
EDOs: 34, 46, 119c, 165c
Badness: 0.001585
Badness: 0.001585


==13-limit==  
==13-limit==
Commas: 91/90, 126/125, 352/351
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|]
Map: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
Badness: 0.001588
Badness: 0.001588


=Cuckoo=  
=Cuckoo=
Commas: 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|]
Map: [&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
EDOs: 31 34 58 65 89 154 185 216
Badness: 0.000933
Badness: 0.000933


==13-limit==  
==13-limit==
Commas: 126/125, 196/195, 243/242
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|]
Map: [&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</pre></div>
 
<h4>Original HTML content:</h4>
EDOs: 31 58 65f 89f 92e 154 185 216f
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Starling family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:40:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x7-limit starling"&gt;7-limit starling&lt;/a&gt;&lt;/div&gt;
[[Category:bluebird]]
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x7-limit starling-Minkowski blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;/div&gt;
[[Category:catbird]]
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x7-limit starling-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
[[Category:family]]
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Thrush"&gt;Thrush&lt;/a&gt;&lt;/div&gt;
[[Category:index]]
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Category:list]]
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-Bluebird"&gt;Bluebird&lt;/a&gt;&lt;/div&gt;
[[Category:listen]]
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-Nightingale"&gt;Nightingale&lt;/a&gt;&lt;/div&gt;
[[Category:mockingbird]]
&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-Veery"&gt;Veery&lt;/a&gt;&lt;/div&gt;
[[Category:nightingale]]
&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Thrasher"&gt;Thrasher&lt;/a&gt;&lt;/div&gt;
[[Category:overview]]
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrasher-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Category:planar]]
&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrasher-Mockingbird"&gt;Mockingbird&lt;/a&gt;&lt;/div&gt;
[[Category:sound_example]]
&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrasher-Catbird"&gt;Catbird&lt;/a&gt;&lt;/div&gt;
[[Category:starling]]
&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Aplonis"&gt;Aplonis&lt;/a&gt;&lt;/div&gt;
[[Category:theory]]
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Aplonis-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
[[Category:thrasher]]
&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Oxpecker"&gt;Oxpecker&lt;/a&gt;&lt;/div&gt;
[[Category:thrush]]
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Oxpecker-Woodpecker"&gt;Woodpecker&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Treecreeper"&gt;Treecreeper&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Treecreeper-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Cuckoo"&gt;Cuckoo&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Cuckoo-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;The head of the starling &lt;a class="wiki_link" href="/family"&gt;family&lt;/a&gt; is starling, which tempers out &lt;a class="wiki_link" href="/126_125"&gt;126/125&lt;/a&gt;, the &lt;a class="wiki_link" href="/starling%20comma"&gt;starling comma&lt;/a&gt; or &lt;a class="wiki_link" href="/septimal%20semicomma"&gt;septimal semicomma&lt;/a&gt;. 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 &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;. Other possible tunings are &lt;a class="wiki_link" href="/108edo"&gt;108edo&lt;/a&gt; and &lt;a class="wiki_link" href="/185edo"&gt;185edo&lt;/a&gt;, and the nonpatent &lt;a class="wiki_link" href="/135edo"&gt;135edo&lt;/a&gt; val &amp;lt;135 214 314 379|.&lt;br /&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;valentine temperament&lt;/a&gt;, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x7-limit starling"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;7-limit starling&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 126/125&lt;br /&gt;
&lt;br /&gt;
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;
[&amp;lt;1 0 0 0|, &amp;lt;0 1 0 0|, &amp;lt;1/3 2/3 0 1/3|, &amp;lt;0 0 0 1|]&lt;br /&gt;
Eigenmonzos: 2, 8/7, 4/3&lt;br /&gt;
&lt;br /&gt;
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206&lt;br /&gt;
Angle(6/5, 5/4) = 100.364 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]&lt;br /&gt;
Generators: 2, 3, 5&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, &lt;a class="wiki_link" href="/77edo"&gt;77&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108&lt;/a&gt;, &lt;a class="wiki_link" href="/166edo"&gt;166c&lt;/a&gt;, &lt;a class="wiki_link" href="/185edo"&gt;185&lt;/a&gt;, &lt;a class="wiki_link" href="/243edo"&gt;243c&lt;/a&gt;&lt;br /&gt;
Badness: 0.0000699&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 125/18&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/starling7"&gt;starling7&lt;/a&gt;, &lt;a class="wiki_link" href="/starling8"&gt;starling8&lt;/a&gt;, &lt;a class="wiki_link" href="/starling9"&gt;starling9&lt;/a&gt;, &lt;a class="wiki_link" href="/starling11"&gt;starling11&lt;/a&gt;, &lt;a class="wiki_link" href="/starling12"&gt;starling12&lt;/a&gt;, &lt;a class="wiki_link" href="/starling15"&gt;starling15&lt;/a&gt;, &lt;a class="wiki_link" href="/starling16"&gt;starling16&lt;/a&gt;, &lt;a class="wiki_link" href="/starling17"&gt;starling17&lt;/a&gt;, &lt;a class="wiki_link" href="/starling19"&gt;starling19&lt;/a&gt;&lt;br /&gt;
Music: &lt;a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure" rel="nofollow" target="_blank"&gt;A Seed Planted&lt;/a&gt; by Jake Freivald. The melody depends on tempering out 126/125.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x7-limit starling-Minkowski blocks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/Minkowski%20blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;/h2&gt;
7: 25/24, 81/80&lt;br /&gt;
8: 16/15, 648/625&lt;br /&gt;
9: 27/25, 128/125&lt;br /&gt;
11: 16/15, 15625/15552&lt;br /&gt;
12: 128/125, 628/625&lt;br /&gt;
15: 128/125, 250/243&lt;br /&gt;
16: 648/625, 3125/3072&lt;br /&gt;
17: 25/24, 20480/19683&lt;br /&gt;
19: 81/80, 3125/3072&lt;br /&gt;
27: 128/125, 78732/78125&lt;br /&gt;
28: 648/625, 16875/16384&lt;br /&gt;
31: 81/80, 1990656/1953125&lt;br /&gt;
34: 15625/15552, 2048/2025&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x7-limit starling-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Commas: 126/125, 385/384&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 8|, &amp;lt;0 1 0 -2 3|, &amp;lt;0 0 1 3 -4]]&lt;br /&gt;
Generators: 2, 3, 5&lt;br /&gt;
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de&lt;br /&gt;
Badness: .000677&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Thrush&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;
Associated linear temperament: &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;myna&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
7 and 9 limit minimax&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |0 1 0 0 0&amp;gt;, |1/3 2/3 0 1/3 0&amp;gt;,&lt;br /&gt;
|0 0 0 1 0&amp;gt;, |-10/3 4/3 0 5/3 0&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
&lt;br /&gt;
Lattice basis 5/4 length 0.8576 6/5 length 0.9314&lt;br /&gt;
Angle(5/4, 6/5) = 74.6239 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -5|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 5|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c&lt;br /&gt;
Badness: 0.000353&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 125/18 11 3125/288&lt;br /&gt;
&lt;br /&gt;
Scales: &lt;a class="wiki_link" href="/thrush12"&gt;thrush12&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Thrush-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h2&gt;
Commas: 126/125, 176/175, 196/195&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -5 0|, &amp;lt;0 1 0 -2 -2 -5|, &amp;lt;0 0 1 3 5 5|]&lt;br /&gt;
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf&lt;br /&gt;
Badness: 0.000677&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Thrush-Bluebird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Bluebird&lt;/h2&gt;
Commas 126/125, 176/175, 144/143&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -5 9|, &amp;lt;0 1 0 -2 -2 4|, &amp;lt;0 0 1 3 5 -5|]&lt;br /&gt;
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef&lt;br /&gt;
Badness: 0.000915&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 7 125/18 11 3125/288 13 41472/3125&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Thrush-Nightingale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Nightingale&lt;/h2&gt;
Commas: 126/125, 176/175, 66/65&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -5 -4|, &amp;lt;0 1 0 -2 -2 -1|, &amp;lt;0 0 1 3 5 4|]&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19f&lt;/a&gt;, &lt;a class="wiki_link" href="/21edo"&gt;21e&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22f&lt;/a&gt;, &lt;a class="wiki_link" href="/28edo"&gt;28&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/89edo"&gt;89&lt;/a&gt;, &lt;a class="wiki_link" href="/108edo"&gt;108ef&lt;/a&gt;&lt;br /&gt;
Badness: 0.000837&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Thrush-Veery"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Veery&lt;/h2&gt;
Commas: 91/90, 126/125, 176/175&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -5 2|, &amp;lt;0 1 0 -2 -2 4|, &amp;lt;0 0 1 3 5 -2|]&lt;br /&gt;
EDOs: 12, 15, 46&lt;br /&gt;
Badness: 0.000991&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Thrasher&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 56/55, 100/99&lt;br /&gt;
&lt;br /&gt;
11-limit minimax&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;,&lt;br /&gt;
|1 1/2 0 1/2 -1/4&amp;gt;, |0 0 0 1 0&amp;gt;,&lt;br /&gt;
|2 -1/2 0 1/2 1/4&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 11/9&lt;br /&gt;
&lt;br /&gt;
Lattice basis: 6/5 length 0.9089 5/4 length 1.2007&lt;br /&gt;
Angle(6/5, 5/4) = 98.8447&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 0 -2 -2|, &amp;lt;0 1 1 1 0|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 2|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 2|]&lt;br /&gt;
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e&lt;br /&gt;
Badness: 0.000480&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Thrasher-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;13-limit&lt;/h2&gt;
Commas: 126/125, 100/99, 91/90&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 2 2|, &amp;lt;0 1 0 -2 -2 4|, &amp;lt;0 0 1 3 2 -2|]&lt;br /&gt;
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e&lt;br /&gt;
Badness: 0.000876&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Thrasher-Mockingbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Mockingbird&lt;/h2&gt;
Commas: 126/125, 100/99, 40/39&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 2 3|, &amp;lt;0 1 0 -2 -2 -1|, &amp;lt;0 0 1 3 2 1|]&lt;br /&gt;
EDOs: 12f, 15, 27ef&lt;br /&gt;
Badness: 0.000859&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Thrasher-Catbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Catbird&lt;/h2&gt;
Commas: 126/125, 100/99, 78/77&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 2 0|, &amp;lt;0 1 0 -2 -2 -5|, &amp;lt;0 0 1 3 2 5|]&lt;br /&gt;
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/17edo"&gt;17c&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22ef&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27e&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58e&lt;/a&gt;&lt;br /&gt;
Badness: 0.000905&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Aplonis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Aplonis&lt;/h1&gt;
Commas: 126/125, 540/539&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 4|, &amp;lt;0 1 0 -2 7|, &amp;lt;0 0 1 3 -5|]&lt;br /&gt;
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce&lt;br /&gt;
Badness: 0.000648&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Aplonis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
Commas: 126/125, 144/143, 196/195&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 4 0|, &amp;lt;0 1 0 -2 7 -5|, &amp;lt;0 0 1 3 -5 5|]&lt;br /&gt;
EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef&lt;br /&gt;
Badness: 0.000821&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Oxpecker"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Oxpecker&lt;/h1&gt;
Commas: 121/120, 126/125&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 2 2|, &amp;lt;0 1 1 1 1|, &amp;lt;0 0 -2 -6 -1|]&lt;br /&gt;
EDOs: 8d, 15, 31, 46, 77, 185e, 262cde&lt;br /&gt;
Badness: 0.000699&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Oxpecker-Woodpecker"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Woodpecker&lt;/h2&gt;
Commas: 66/65, 121/120, 126/125&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 2 2 2|, &amp;lt;0 1 1 1 1 1|, &amp;lt;0 0 -2 -6 -1 -1|]&lt;br /&gt;
EDOs: 15, 31, 46f, 70f, 101f&lt;br /&gt;
Badness: 0.001093&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Treecreeper"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Treecreeper&lt;/h1&gt;
Commas: 126/125, 1232/1215&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -3|, &amp;lt;0 1 0 -2 7|, &amp;lt;0 0 1 3 -2|]&lt;br /&gt;
EDOs: 34, 46, 119c, 165c&lt;br /&gt;
Badness: 0.001585&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Treecreeper-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;13-limit&lt;/h2&gt;
Commas: 91/90, 126/125, 352/351&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 -1 -3 2|, &amp;lt;0 1 0 -2 7 4|, &amp;lt;0 0 1 3 -2 -2|]&lt;br /&gt;
EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf&lt;br /&gt;
Badness: 0.001588&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;a name="Cuckoo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Cuckoo&lt;/h1&gt;
Commas: 126/125, 243/242&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 1 0 -3 2|, &amp;lt;0 2 0 -4 5|, &amp;lt;0 0 1 3 0|]&lt;br /&gt;
EDOs: 31 34 58 65 89 154 185 216&lt;br /&gt;
Badness: 0.000933&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Cuckoo-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;13-limit&lt;/h2&gt;
Commas: 126/125, 196/195, 243/242&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 1 0 -3 2 -5|, &amp;lt;0 2 0 -4 5 -10|, &amp;lt;0 0 1 3 0 5|]&lt;br /&gt;
EDOs: 31 58 65f 89f 92e 154 185 216f&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

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|.

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 temperament, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.

7-limit starling

Comma c = 126/125

7- and 9-limit minimax: 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

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: [<1 0 0 -5|, <0 1 0 2|, <0 0 1 2|]

Generators: 2, 3, 5

EDOs: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, 77, 108, 166c, 185, 243c

Badness: 0.0000699

Projection pair: 7 125/18

Scales: starling7, starling8, starling9, starling11, starling12, starling15, starling16, starling17, starling19

Music: A Seed Planted by Jake Freivald. The melody depends on tempering out 126/125.

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

11-limit

Commas: 126/125, 385/384

Map: [<1 0 0 -1 8|, <0 1 0 -2 3|, <0 0 1 3 -4]]

Generators: 2, 3, 5

EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de

Badness: .000677

Thrush

Commas: 126/125, 176/175

Associated linear temperament: myna

7 and 9 limit minimax

[|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>]

Eigenmonzos: 2, 7/6, 4/3

Lattice basis 5/4 length 0.8576 6/5 length 0.9314

Angle(5/4, 6/5) = 74.6239 degrees

Map to lattice: [<0 1 1 1 3|, <0 1 0 -2 -2|]

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

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

13-limit

Commas: 126/125, 176/175, 196/195

Map: [<1 0 0 -1 -5 0|, <0 1 0 -2 -2 -5|, <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: [<1 0 0 -1 -5 9|, <0 1 0 -2 -2 4|, <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

Nightingale

Commas: 126/125, 176/175, 66/65

Map: [<1 0 0 -1 -5 -4|, <0 1 0 -2 -2 -1|, <0 0 1 3 5 4|]

EDOs: 15, 19f, 21e, 22f, 28, 31, 46, 58, 89, 108ef

Badness: 0.000837

Veery

Commas: 91/90, 126/125, 176/175

Map: [<1 0 0 -1 -5 2|, <0 1 0 -2 -2 4|, <0 0 1 3 5 -2|]

EDOs: 12, 15, 46

Badness: 0.000991

Thrasher

Commas: 56/55, 100/99

11-limit minimax

[|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>]

Eigenmonzos: 2, 8/7, 11/9

Lattice basis: 6/5 length 0.9089 5/4 length 1.2007

Angle(6/5, 5/4) = 98.8447

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

Map: [<1 0 0 -1 2|, <0 1 0 -2 -2|, <0 0 1 3 2|]

Generators: 2, 3, 5

EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e

Badness: 0.000480

13-limit

Commas: 126/125, 100/99, 91/90

Map: [<1 0 0 -1 2 2|, <0 1 0 -2 -2 4|, <0 0 1 3 2 -2|]

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

Badness: 0.000876

Mockingbird

Commas: 126/125, 100/99, 40/39

Map: [<1 0 0 -1 2 3|, <0 1 0 -2 -2 -1|, <0 0 1 3 2 1|]

EDOs: 12f, 15, 27ef

Badness: 0.000859

Catbird

Commas: 126/125, 100/99, 78/77

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

EDOs: 12, 15, 17c, 19, 22ef, 27e, 34, 58e

Badness: 0.000905

Aplonis

Commas: 126/125, 540/539

Map: [<1 0 0 -1 4|, <0 1 0 -2 7|, <0 0 1 3 -5|]

EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce

Badness: 0.000648

13-limit

Commas: 126/125, 144/143, 196/195

Map: [<1 0 0 -1 4 0|, <0 1 0 -2 7 -5|, <0 0 1 3 -5 5|]

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

Badness: 0.000821

Oxpecker

Commas: 121/120, 126/125

Map: [<1 0 1 2 2|, <0 1 1 1 1|, <0 0 -2 -6 -1|]

EDOs: 8d, 15, 31, 46, 77, 185e, 262cde

Badness: 0.000699

Woodpecker

Commas: 66/65, 121/120, 126/125

Map: [<1 0 1 2 2 2|, <0 1 1 1 1 1|, <0 0 -2 -6 -1 -1|]

EDOs: 15, 31, 46f, 70f, 101f

Badness: 0.001093

Treecreeper

Commas: 126/125, 1232/1215

Map: [<1 0 0 -1 -3|, <0 1 0 -2 7|, <0 0 1 3 -2|]

EDOs: 34, 46, 119c, 165c

Badness: 0.001585

13-limit

Commas: 91/90, 126/125, 352/351

Map: [<1 0 0 -1 -3 2|, <0 1 0 -2 7 4|, <0 0 1 3 -2 -2|]

EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf

Badness: 0.001588

Cuckoo

Commas: 126/125, 243/242

Map: [<1 1 0 -3 2|, <0 2 0 -4 5|, <0 0 1 3 0|]

EDOs: 31 34 58 65 89 154 185 216

Badness: 0.000933

13-limit

Commas: 126/125, 196/195, 243/242

Map: [<1 1 0 -3 2 -5|, <0 2 0 -4 5 -10|, <0 0 1 3 0 5|]

EDOs: 31 58 65f 89f 92e 154 185 216f

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