Starling family: Difference between revisions

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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|.

~~: 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 <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.

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

[[Comma|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]]

[[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

[[Projection_pair|Projection pair]]: 7 125/18

~~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

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=

[[Comma|Commas]]: 126/125, 176/175

Associated linear temperament: [[~~Starling temperaments~~|myna]]

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

[[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]]

Scales: [[thrush12|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]]

[[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: [<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=

[[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

[[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]]

[[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: 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~~</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"><html><head><title>Starling~~ family</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#x7-limit starling">7-limit starling</a></div>

[[Category:bluebird]]

~~<div style="margin-left: 2em;"><a href="#x7-limit starling-Minkowski blocks">Minkowski blocks</a></div>~~

[[Category:catbird]]

~~<div style="margin-left: 2em;"><a href="#x7-limit starling-11-limit">11-limit</a></div>~~

[[Category:family]]

~~<div style="margin-left: 1em;"><a href="#Thrush">Thrush</a></div>~~

[[Category:index]]

~~<div style="margin-left: 2em;"><a href="#Thrush-13-limit">13-limit</a></div>~~

[[Category:list]]

~~<div style="margin-left: 2em;"><a href="#Thrush-Bluebird">Bluebird</a></div>~~

[[Category:listen]]

~~<div style="margin-left: 2em;"><a href="#Thrush-Nightingale">Nightingale</a></div>~~

[[Category:mockingbird]]

~~<div style="margin-left: 2em;"><a href="#Thrush-Veery">Veery</a></div>~~

[[Category:nightingale]]

~~<div style="margin-left: 1em;"><a href="#Thrasher">Thrasher</a></div>~~

[[Category:overview]]

~~<div style="margin-left: 2em;"><a href="#Thrasher-13-limit">13-limit</a></div>~~

[[Category:planar]]

~~<div style="margin-left: 2em;"><a href="#Thrasher-Mockingbird">Mockingbird</a></div>~~

[[Category:sound_example]]

~~<div style="margin-left~~: ~~2em;"><a href="#Thrasher-Catbird">Catbird</a></div>~~

[[Category:starling]]

~~<div style="margin-left: 1em;"><a href="#Aplonis">Aplonis</a></div>~~

[[Category:theory]]

~~<div style="margin-left: 2em;"><a href="#Aplonis-13-limit">13-limit</a></div>~~

[[Category:thrasher]]

~~<div style="margin-left: 1em;"><a href="#Oxpecker">Oxpecker</a></div>~~

[[Category:thrush]]

~~<div style="margin-left: 2em;"><a href="#Oxpecker-Woodpecker">Woodpecker</a></div>~~

~~<div style="margin-left: 1em;"><a href="#Treecreeper">Treecreeper</a></div>~~

~~<div style="margin-left: 2em;"><a href="#Treecreeper-13-limit">13-limit</a></div>~~

~~<div style="margin-left: 1em;"><a href="#Cuckoo">Cuckoo</a></div>~~

~~<div style="margin-left: 2em;"><a href="#Cuckoo-13-limit">13-limit</a></div>~~

~~</div>~~

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 />~~

~~<h1 id="toc0"><a name="x7-limit starling"></a>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 />~~

~~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 />

Music: <a class="wiki_link_ext" href="https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure" rel="nofollow" target="_blank">A Seed Planted</a> by Jake Freivald. The melody depends on tempering out 126/125.<br />

~~<br />~~

<h2 id="toc1"><a name="x7-limit starling-Minkowski blocks"></a><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 />~~

~~<h2 id="toc2"><a name="x7-limit starling-11-limit"></a>11-limit</h2>~~

~~Commas: 126/125, 385/384<br />~~

~~<br />~~

~~Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4~~]]~~<br />~~

~~Generators~~: ~~2, 3, 5<br />~~

~~EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de<br />~~

~~Badness: .000677<br />~~

~~<br />~~

~~<h1 id="toc3"><a name="Thrush"></a>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 />~~

~~<h2 id="toc4"><a name="Thrush-13-limit"></a>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 />~~

~~<h2 id="toc5"><a name="Thrush-Bluebird"></a>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 />~~

~~<h2 id="toc6"><a name="Thrush-Nightingale"></a>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 />~~

~~Badness: 0.000837<br />~~

~~<br />~~

~~<h2 id="toc7"><a name="Thrush-Veery"></a>Veery</h2>~~

~~Commas: 91/90, 126/125, 176/175<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]<br />~~

~~EDOs~~: ~~12, 15, 46<br />~~

~~Badness: 0.000991<br />~~

~~<br />~~

~~<h1 id="toc8"><a name="Thrasher"></a>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 />~~

~~<h2 id="toc9"><a name="Thrasher-13-limit"></a>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 />~~

~~<h2 id="toc10"><a name="Thrasher-Mockingbird"></a>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 />~~

~~<h2 id="toc11"><a name="Thrasher-Catbird"></a>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 />~~

~~Badness: 0.000905<br />~~

~~<br />~~

~~<h1 id="toc12"><a name="Aplonis"></a>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 />~~

~~<h2 id="toc13"><a name="Aplonis-13-limit"></a>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 />~~

~~<h1 id="toc14"><a name="Oxpecker"></a>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: 8d, 15, 31, 46, 77, 185e, 262cde<br />~~

~~Badness: 0.000699<br />~~

~~<br />~~

~~<h2 id="toc15"><a name="Oxpecker-Woodpecker"></a>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<br />~~

~~<br />~~

~~<h1 id="toc16"><a name="Treecreeper"></a>Treecreeper</h1>~~

~~Commas: 126/125, 1232/1215<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]<br />~~

~~EDOs: 34, 46, 119c, 165c<br />~~

~~Badness: 0.001585<br />~~

~~<br />~~

~~<h2 id="toc17"><a name="Treecreeper-13-limit"></a>13-limit</h2>~~

~~Commas: 91/90, 126/125, 352/351<br />~~

~~<br />~~

~~Map:~~ [~~&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]<br />~~

~~EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf<br />~~

~~Badness: 0.001588<br />~~

~~<br />~~

~~<h1 id="toc18"><a name="Cuckoo"></a>Cuckoo</h1>~~

~~Commas: 126/125, 243/242<br />~~

~~<br />~~

~~Map~~: ~~[&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|~~]~~<br />~~

~~EDOs: 31 34 58 65 89 154 185 216<br />~~

~~Badness: 0.000933<br />~~

~~<br />~~

~~<h2 id="toc19"><a name="Cuckoo-13-limit"></a>13-limit</h2>~~

~~Commas: 126/125, 196/195, 243/242<br />~~

~~<br />~~

~~Map: [&lt;1 1 0 -3 2 -5|, &lt;0 2 0 -4 5 -10|, &lt;0 0 1 3 0 5|~~]~~<br />~~

~~EDOs: 31 58 65f 89f 92e 154 185 216f</body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+__FORCETOC__
-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.
-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=
-[[Comma]] c = 126/125
+[[Comma|Comma]] c = 126/125
 - 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|]
 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: [&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|]
 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
-[[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]]==
 : 25/24, 81/80
 : 16/15, 648/625
 : 27/25, 128/125
 : 16/15, 15625/15552
 : 128/125, 628/625
 : 128/125, 250/243
 : 648/625, 3125/3072
 : 25/24, 20480/19683
 : 81/80, 3125/3072
 : 128/125, 78732/78125
 : 648/625, 16875/16384
 : 81/80, 1990656/1953125
 : 15625/15552, 2048/2025
 ==11-limit==
 Commas: 126/125, 385/384
 Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
 Generators: 2, 3, 5
 EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
 Badness: .000677
 =Thrush=
 [[Comma|Commas]]: 126/125, 176/175
-Associated linear temperament: [[Starling temperaments|myna]]
+Associated linear temperament: [[Starling_temperaments|myna]]
 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;,
 |0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
 [[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: [&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|]
-[[Generator|Generators]]: 2, 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]]
+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
 ==Nightingale==
 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|]
-[[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
 ==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|]
 EDOs: 12, 15, 46
 Badness: 0.000991
 =Thrasher=
 [[Comma|Commas]]: 56/55, 100/99
 -limit minimax
 [|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|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: [&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|]
-[[Generator|Generators]]: 2, 3, 5
+[[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: [&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
 Badness: 0.000876
 ==Mockingbird==
 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|]
 EDOs: 12f, 15, 27ef
 Badness: 0.000859
 ==Catbird==
 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|]
-[[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
 =Aplonis=
 Commas: 126/125, 540/539
 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
 Badness: 0.000648
 ==13-limit==
 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|]
 EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
 Badness: 0.000821
 =Oxpecker=
 Commas: 121/120, 126/125
 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
 Badness: 0.000699
 ==Woodpecker==
 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|]
 EDOs: 15, 31, 46f, 70f, 101f
 Badness: 0.001093
 =Treecreeper=
 Commas: 126/125, 1232/1215
 Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
 EDOs: 34, 46, 119c, 165c
 Badness: 0.001585
 ==13-limit==
 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|]
 EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
 Badness: 0.001588
 =Cuckoo=
 Commas: 126/125, 243/242
 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
 Badness: 0.000933
 ==13-limit==
 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|]
-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;
-- 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;
-: 25/24, 81/80&lt;br /&gt;
-: 16/15, 648/625&lt;br /&gt;
-: 27/25, 128/125&lt;br /&gt;
-: 16/15, 15625/15552&lt;br /&gt;
-: 128/125, 628/625&lt;br /&gt;
-: 128/125, 250/243&lt;br /&gt;
-: 648/625, 3125/3072&lt;br /&gt;
-: 25/24, 20480/19683&lt;br /&gt;
-: 81/80, 3125/3072&lt;br /&gt;
-: 128/125, 78732/78125&lt;br /&gt;
-: 648/625, 16875/16384&lt;br /&gt;
-: 81/80, 1990656/1953125&lt;br /&gt;
-: 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;
-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;
--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>