Starling family

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[[toc|flat]]

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 <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 [[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=
[[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
[[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

[[Projection pair]]: 7 125/18

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

==[[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

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

==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|]
[[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

=Thrasher=
[[Comma|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>]
[[Eigenmonzo|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|]
[[Generator|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|]
[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|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: 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

Original HTML content:

<html><head><title>Starling family</title></head><body><!-- ws:start:WikiTextTocRule:28:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#x7-limit starling">7-limit starling</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#Thrush">Thrush</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Thrasher">Thrasher</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Aplonis">Aplonis</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --> | <a href="#Oxpecker">Oxpecker</a><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: -->
<!-- ws:end:WikiTextTocRule:43 --><br />
The head of the starling <a class="wiki_link" href="/family">family</a> is starling, which tempers out <a class="wiki_link" href="/126_125">126/125</a>, the <a class="wiki_link" href="/starling%20comma">starling comma</a> or <a class="wiki_link" href="/septimal%20semicomma">septimal semicomma</a>. 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 <a class="wiki_link" href="/77edo">77edo</a>.  Other possible tunings are <a class="wiki_link" href="/108edo">108edo</a> and <a class="wiki_link" href="/185edo">185edo</a>, and the nonpatent <a class="wiki_link" href="/135edo">135edo</a> val &lt;135 214 314 379|.<br />
<br />
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.<br />
<br />
Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to <a class="wiki_link" href="/Starling%20temperaments">valentine temperament</a>, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x7-limit starling"></a><!-- ws:end:WikiTextHeadingRule:0 -->7-limit starling</h1>
<a class="wiki_link" href="/Comma">Comma</a> c = 126/125<br />
<br />
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp<br />
[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]<br />
Eigenmonzos: 2, 8/7, 4/3<br />
<br />
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206<br />
Angle(6/5, 5/4) = 100.364 degrees<br />
Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]<br />
<br />
Map: [&lt;1 0 0 -5|, &lt;0 1 0 2|, &lt;0 0 1 2|]<br />
Generators: 2, 3, 5<br />
<a class="wiki_link" href="/edo">EDOs</a>: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/108edo">108</a>, <a class="wiki_link" href="/166edo">166c</a>, <a class="wiki_link" href="/185edo">185</a>, <a class="wiki_link" href="/243edo">243c</a><br />
Badness: 0.0000699<br />
<br />
<a class="wiki_link" href="/Projection%20pair">Projection pair</a>: 7 125/18<br />
<br />
Scales: <a class="wiki_link" href="/starling7">starling7</a>, <a class="wiki_link" href="/starling8">starling8</a>, <a class="wiki_link" href="/starling9">starling9</a>, <a class="wiki_link" href="/starling11">starling11</a>, <a class="wiki_link" href="/starling12">starling12</a>, <a class="wiki_link" href="/starling15">starling15</a>, <a class="wiki_link" href="/starling16">starling16</a>, <a class="wiki_link" href="/starling17">starling17</a>, <a class="wiki_link" href="/starling19">starling19</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x7-limit starling-Minkowski blocks"></a><!-- ws:end:WikiTextHeadingRule:2 --><a class="wiki_link" href="/Minkowski%20blocks">Minkowski blocks</a></h2>
7: 25/24, 81/80<br />
8: 16/15, 648/625<br />
9: 27/25, 128/125<br />
11: 16/15, 15625/15552<br />
12: 128/125, 628/625<br />
15: 128/125, 250/243<br />
16: 648/625, 3125/3072<br />
17: 25/24, 20480/19683<br />
19: 81/80, 3125/3072<br />
27: 128/125, 78732/78125<br />
28: 648/625, 16875/16384<br />
31: 81/80, 1990656/1953125<br />
34: 15625/15552, 2048/2025<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Thrush"></a><!-- ws:end:WikiTextHeadingRule:4 -->Thrush</h1>
<a class="wiki_link" href="/Comma">Commas</a>: 126/125, 176/175<br />
Associated linear temperament: <a class="wiki_link" href="/Starling%20temperaments">myna</a><br />
<br />
7 and 9 limit minimax<br />
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;, <br />
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/6, 4/3<br />
<br />
Lattice basis 5/4 length 0.8576 6/5 length 0.9314<br />
Angle(5/4, 6/5) = 74.6239 degrees<br />
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]<br />
<br />
Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c<br />
Badness: 0.000353<br />
<br />
Projection pairs: 7 125/18 11 3125/288 <br />
<br />
Scales: <a class="wiki_link" href="/thrush12">thrush12</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Thrush-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h2>
Commas: 126/125, 176/175, 196/195<br />
<br />
Map: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]<br />
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf<br />
Badness: 0.000677<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Thrush-Bluebird"></a><!-- ws:end:WikiTextHeadingRule:8 -->Bluebird</h2>
Commas 126/125, 176/175, 144/143<br />
<br />
Map: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]<br />
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef<br />
Badness: 0.000915<br />
<br />
Projection pairs: 7 125/18 11 3125/288 13 41472/3125<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Thrush-Nightingale"></a><!-- ws:end:WikiTextHeadingRule:10 -->Nightingale</h2>
Commas: 126/125, 176/175, 66/65<br />
<br />
Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/15edo">15</a>, [[19edo}19f]], <a class="wiki_link" href="/21edo">21e</a>, <a class="wiki_link" href="/22edo">22f</a>, <a class="wiki_link" href="/28edo">28</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/89edo">89</a>, <a class="wiki_link" href="/108edo">108ef</a><br />
Badness: 0.000837<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Thrasher"></a><!-- ws:end:WikiTextHeadingRule:12 -->Thrasher</h1>
<a class="wiki_link" href="/Comma">Commas</a>: 56/55, 100/99<br />
<br />
11-limit minimax<br />
[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;, <br />
|1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;, <br />
|2 -1/2 0 1/2 1/4&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7, 11/9<br />
<br />
Lattice basis: 6/5 length 0.9089 5/4 length 1.2007<br />
Angle(6/5, 5/4) = 98.8447<br />
Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]<br />
<br />
Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e<br />
Badness: 0.000480<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Thrasher-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h2>
Commas: 126/125, 100/99, 91/90<br />
<br />
Map: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]<br />
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e<br />
Badness: 0.000876<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Thrasher-Mockingbird"></a><!-- ws:end:WikiTextHeadingRule:16 -->Mockingbird</h2>
Commas: 126/125, 100/99, 40/39<br />
<br />
Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]<br />
EDOs: 12f, 15, 27ef<br />
Badness: 0.000859<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Thrasher-Catbird"></a><!-- ws:end:WikiTextHeadingRule:18 -->Catbird</h2>
Commas: 126/125, 100/99, 78/77<br />
<br />
Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22ef</a>, <a class="wiki_link" href="/27edo">27e</a>, <a class="wiki_link" href="/34edo">34</a>, <a class="wiki_link" href="/58edo">58e</a><br />
Badness: 0.000905<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Aplonis"></a><!-- ws:end:WikiTextHeadingRule:20 -->Aplonis</h1>
Commas: 126/125, 540/539<br />
<br />
Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]<br />
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce<br />
Badness: 0.000648<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Aplonis-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h2>
Commas: 126/125, 144/143, 196/195<br />
<br />
Map: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]<br />
EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef<br />
Badness: 0.000821<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Oxpecker"></a><!-- ws:end:WikiTextHeadingRule:24 -->Oxpecker</h1>
Commas: 121/120, 126/125<br />
<br />
Map: [&lt;1 0 1 2 2|,  &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]<br />
EDOs: 15, 31, 46, 77, 185e, 262cde<br />
Badness: 0.000699<br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Oxpecker-Woodpecker"></a><!-- ws:end:WikiTextHeadingRule:26 -->Woodpecker</h2>
Commas: 66/65, 121/120, 126/125<br />
<br />
Map: [&lt;1 0 1 2 2 2|,  &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]<br />
EDOs: 15, 31, 46f, 70f, 101f <br />
Badness: 0.001093</body></html>