Starling family

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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 <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.

===Vital statistics===
[[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]], [[185edo|185]], <135 214 314 379|
Badness: 0.0000699

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

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, 135, 224
Badness: 0.000353

===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
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, 58
Badness: 0.000915

===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]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]]
Badness: 0.000837

==Thrasher==
[[Comma|Commas]]: 126/125, 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, 34
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, 34
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: 15
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]], [[19edo|19]], [[34edo|34]]
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, 89
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, 50, 58
Badness: 0.000821

Original HTML content:

<html><head><title>Starling family</title></head><body>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 <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;h3&gt; --><h3 id="toc0"><a name="x--Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vital statistics</h3>
<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="/185edo">185</a>, &lt;135 214 314 379|<br />
Badness: 0.0000699<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;h3&gt; --><h3 id="toc1"><a name="x--Minkowski blocks"></a><!-- ws:end:WikiTextHeadingRule:2 --><a class="wiki_link" href="/Minkowski%20blocks">Minkowski blocks</a></h3>
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 />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Thrush"></a><!-- ws:end:WikiTextHeadingRule:4 -->Thrush</h2>
<a class="wiki_link" href="/Comma">Commas</a>: 126/125, 176/175<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, 135, 224<br />
Badness: 0.000353<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Thrush-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h3>
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<br />
Badness: 0.000677<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Thrush-Bluebird"></a><!-- ws:end:WikiTextHeadingRule:8 -->Bluebird</h3>
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, 58<br />
Badness: 0.000915<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Thrush-Nightingale"></a><!-- ws:end:WikiTextHeadingRule:10 -->Nightingale</h3>
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>, <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><br />
Badness: 0.000837<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Thrasher"></a><!-- ws:end:WikiTextHeadingRule:12 -->Thrasher</h2>
<a class="wiki_link" href="/Comma">Commas</a>: 126/125, 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, 34<br />
Badness: 0.000480<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Thrasher-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3>
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, 34<br />
Badness: 0.000876<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Thrasher-Mockingbird"></a><!-- ws:end:WikiTextHeadingRule:16 -->Mockingbird</h3>
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: 15<br />
Badness: 0.000859<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Thrasher-Catbird"></a><!-- ws:end:WikiTextHeadingRule:18 -->Catbird</h3>
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="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a><br />
Badness: 0.000905<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-Aplonis"></a><!-- ws:end:WikiTextHeadingRule:20 -->Aplonis</h2>
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, 89<br />
Badness: 0.000648<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="x-Aplonis-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h3>
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, 50, 58<br />
Badness: 0.000821</body></html>