Starling family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 361691190 - Original comment: ** |
Wikispaces>Gedankenwelt **Imported revision 488235590 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:Gedankenwelt|Gedankenwelt]] and made on <tt>2014-02-09 04:36:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>488235590</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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]] | <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|. | |||
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]]. | |||
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. | ||
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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]] c = 126/125 | ||
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Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]] | Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]] | ||
==[[Minkowski blocks]]== | ==[[Minkowski blocks]]== | ||
7: 25/24, 81/80 | 7: 25/24, 81/80 | ||
8: 16/15, 648/625 | 8: 16/15, 648/625 | ||
| Line 49: | Line 48: | ||
34: 15625/15552, 2048/2025 | 34: 15625/15552, 2048/2025 | ||
==11-limit== | ==11-limit== | ||
Commas: 126/125, 385/384 | Commas: 126/125, 385/384 | ||
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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>, |0 1 0 0 0>, |1/3 2/3 0 1/3 0>, | [|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>] | |0 0 0 1 0>, |-10/3 4/3 0 5/3 0>] | ||
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3 | [[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3 | ||
| Line 75: | Line 74: | ||
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]] | ||
==13-limit== | ==13-limit== | ||
Commas: 126/125, 176/175, 196/195 | Commas: 126/125, 176/175, 196/195 | ||
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Badness: 0.000677 | Badness: 0.000677 | ||
==Bluebird== | ==Bluebird== | ||
Commas 126/125, 176/175, 144/143 | Commas 126/125, 176/175, 144/143 | ||
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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 | ||
| Line 102: | Line 101: | ||
Badness: 0.000837 | Badness: 0.000837 | ||
==Veery== | ==Veery== | ||
Commas: 91/90, 126/125, 176/175 | Commas: 91/90, 126/125, 176/175 | ||
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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>, |1 3/4 0 1/4 -3/8>, | [|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>, | |1 1/2 0 1/2 -1/4>, |0 0 0 1 0>, | ||
|2 -1/2 0 1/2 1/4>] | |2 -1/2 0 1/2 1/4>] | ||
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9 | [[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9 | ||
| Line 127: | Line 126: | ||
Badness: 0.000480 | Badness: 0.000480 | ||
==13-limit== | ==13-limit== | ||
Commas: 126/125, 100/99, 91/90 | Commas: 126/125, 100/99, 91/90 | ||
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Badness: 0.000876 | Badness: 0.000876 | ||
==Mockingbird== | ==Mockingbird== | ||
Commas: 126/125, 100/99, 40/39 | Commas: 126/125, 100/99, 40/39 | ||
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Badness: 0.000859 | Badness: 0.000859 | ||
==Catbird== | ==Catbird== | ||
Commas: 126/125, 100/99, 78/77 | Commas: 126/125, 100/99, 78/77 | ||
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Badness: 0.000905 | Badness: 0.000905 | ||
=Aplonis= | =Aplonis= | ||
Commas: 126/125, 540/539 | Commas: 126/125, 540/539 | ||
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Badness: 0.000648 | Badness: 0.000648 | ||
==13-limit== | ==13-limit== | ||
Commas: 126/125, 144/143, 196/195 | Commas: 126/125, 144/143, 196/195 | ||
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Badness: 0.000821 | Badness: 0.000821 | ||
=Oxpecker= | =Oxpecker= | ||
Commas: 121/120, 126/125 | Commas: 121/120, 126/125 | ||
Map: [<1 0 1 2 2|, | Map: [<1 0 1 2 2|, <0 1 1 1 1|, <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: [<1 0 1 2 2 2|, | 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 | EDOs: 15, 31, 46f, 70f, 101f | ||
Badness: 0.001093 | Badness: 0.001093 | ||
=Treecreeper= | =Treecreeper= | ||
Commas: 126/125, 1232/1215 | Commas: 126/125, 1232/1215 | ||
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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: [<1 0 0 -1 -3 2|, <0 1 0 -2 7 4|, <0 0 1 3 -2 -2|] | 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 | EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf | ||
Badness: 0.001588</pre></div> | 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> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Starling family</title></head><body><!-- ws:start:WikiTextTocRule: | <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><!-- ws:start:WikiTextTocRule:40:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#x7-limit starling">7-limit starling</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#x7-limit starling-Minkowski blocks">Minkowski blocks</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#x7-limit starling-11-limit">11-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 1em;"><a href="#Thrush">Thrush</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 2em;"><a href="#Thrush-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Thrush-Bluebird">Bluebird</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Thrush-Nightingale">Nightingale</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Thrush-Veery">Veery</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Thrasher">Thrasher</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Thrasher-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#Thrasher-Mockingbird">Mockingbird</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#Thrasher-Catbird">Catbird</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#Aplonis">Aplonis</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 2em;"><a href="#Aplonis-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 1em;"><a href="#Oxpecker">Oxpecker</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 2em;"><a href="#Oxpecker-Woodpecker">Woodpecker</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 1em;"><a href="#Treecreeper">Treecreeper</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 2em;"><a href="#Treecreeper-13-limit">13-limit</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 1em;"><a href="#Cuckoo">Cuckoo</a></div> | ||
< | <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 2em;"><a href="#Cuckoo-13-limit">13-limit</a></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>. | <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --></div> | ||
<!-- ws:end:WikiTextTocRule:61 -->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 /> | <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 /> | 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 /> | ||
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<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> | <!-- 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 /> | <a class="wiki_link" href="/Comma">Comma</a> c = 126/125<br /> | ||
<br /> | <br /> | ||
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp<br /> | 7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp<br /> | ||
| Line 237: | Line 250: | ||
<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> | <!-- 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 /> | 7: 25/24, 81/80<br /> | ||
8: 16/15, 648/625<br /> | 8: 16/15, 648/625<br /> | ||
9: 27/25, 128/125<br /> | 9: 27/25, 128/125<br /> | ||
| Line 252: | Line 265: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x7-limit starling-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x7-limit starling-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h2> | ||
Commas: 126/125, 385/384<br /> | Commas: 126/125, 385/384<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]<br /> | Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]<br /> | ||
| Line 260: | Line 273: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Thrush"></a><!-- ws:end:WikiTextHeadingRule:6 -->Thrush</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Thrush"></a><!-- ws:end:WikiTextHeadingRule:6 -->Thrush</h1> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 126/125, 176/175<br /> | <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 /> | Associated linear temperament: <a class="wiki_link" href="/Starling%20temperaments">myna</a><br /> | ||
<br /> | <br /> | ||
7 and 9 limit minimax<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 /> | [|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 /> | |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 /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/6, 4/3<br /> | ||
| Line 277: | Line 290: | ||
Badness: 0.000353<br /> | Badness: 0.000353<br /> | ||
<br /> | <br /> | ||
Projection pairs: 7 125/18 11 3125/288 <br /> | Projection pairs: 7 125/18 11 3125/288<br /> | ||
<br /> | <br /> | ||
Scales: <a class="wiki_link" href="/thrush12">thrush12</a><br /> | Scales: <a class="wiki_link" href="/thrush12">thrush12</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Thrush-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Thrush-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> | ||
Commas: 126/125, 176/175, 196/195<br /> | Commas: 126/125, 176/175, 196/195<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]<br /> | ||
| Line 289: | Line 302: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Thrush-Bluebird"></a><!-- ws:end:WikiTextHeadingRule:10 -->Bluebird</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Thrush-Bluebird"></a><!-- ws:end:WikiTextHeadingRule:10 -->Bluebird</h2> | ||
Commas 126/125, 176/175, 144/143<br /> | Commas 126/125, 176/175, 144/143<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -5|]<br /> | ||
| Line 298: | Line 311: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Thrush-Nightingale"></a><!-- ws:end:WikiTextHeadingRule:12 -->Nightingale</h2> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Thrush-Nightingale"></a><!-- ws:end:WikiTextHeadingRule:12 -->Nightingale</h2> | ||
Commas: 126/125, 176/175, 66/65<br /> | Commas: 126/125, 176/175, 66/65<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]<br /> | ||
| Line 305: | Line 318: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Thrush-Veery"></a><!-- ws:end:WikiTextHeadingRule:14 -->Veery</h2> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Thrush-Veery"></a><!-- ws:end:WikiTextHeadingRule:14 -->Veery</h2> | ||
Commas: 91/90, 126/125, 176/175<br /> | Commas: 91/90, 126/125, 176/175<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]<br /> | ||
| Line 312: | Line 325: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Thrasher"></a><!-- ws:end:WikiTextHeadingRule:16 -->Thrasher</h1> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Thrasher"></a><!-- ws:end:WikiTextHeadingRule:16 -->Thrasher</h1> | ||
<a class="wiki_link" href="/Comma">Commas</a>: 56/55, 100/99<br /> | <a class="wiki_link" href="/Comma">Commas</a>: 56/55, 100/99<br /> | ||
<br /> | <br /> | ||
11-limit minimax<br /> | 11-limit minimax<br /> | ||
[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;, <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 /> | |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 /> | |2 -1/2 0 1/2 1/4&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7, 11/9<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7, 11/9<br /> | ||
| Line 330: | Line 343: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Thrasher-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Thrasher-13-limit"></a><!-- ws:end:WikiTextHeadingRule:18 -->13-limit</h2> | ||
Commas: 126/125, 100/99, 91/90<br /> | Commas: 126/125, 100/99, 91/90<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]<br /> | ||
| Line 337: | Line 350: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Thrasher-Mockingbird"></a><!-- ws:end:WikiTextHeadingRule:20 -->Mockingbird</h2> | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Thrasher-Mockingbird"></a><!-- ws:end:WikiTextHeadingRule:20 -->Mockingbird</h2> | ||
Commas: 126/125, 100/99, 40/39<br /> | Commas: 126/125, 100/99, 40/39<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]<br /> | ||
| Line 344: | Line 357: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Thrasher-Catbird"></a><!-- ws:end:WikiTextHeadingRule:22 -->Catbird</h2> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Thrasher-Catbird"></a><!-- ws:end:WikiTextHeadingRule:22 -->Catbird</h2> | ||
Commas: 126/125, 100/99, 78/77<br /> | Commas: 126/125, 100/99, 78/77<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]<br /> | ||
| Line 351: | Line 364: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Aplonis"></a><!-- ws:end:WikiTextHeadingRule:24 -->Aplonis</h1> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc12"><a name="Aplonis"></a><!-- ws:end:WikiTextHeadingRule:24 -->Aplonis</h1> | ||
Commas: 126/125, 540/539<br /> | Commas: 126/125, 540/539<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]<br /> | Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]<br /> | ||
| Line 358: | Line 371: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Aplonis-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Aplonis-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2> | ||
Commas: 126/125, 144/143, 196/195<br /> | Commas: 126/125, 144/143, 196/195<br /> | ||
<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 /> | Map: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]<br /> | ||
| Line 365: | Line 378: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Oxpecker"></a><!-- ws:end:WikiTextHeadingRule:28 -->Oxpecker</h1> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Oxpecker"></a><!-- ws:end:WikiTextHeadingRule:28 -->Oxpecker</h1> | ||
Commas: 121/120, 126/125<br /> | Commas: 121/120, 126/125<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 1 2 2|, | 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 /> | EDOs: 8d, 15, 31, 46, 77, 185e, 262cde<br /> | ||
Badness: 0.000699<br /> | Badness: 0.000699<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Oxpecker-Woodpecker"></a><!-- ws:end:WikiTextHeadingRule:30 -->Woodpecker</h2> | <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Oxpecker-Woodpecker"></a><!-- ws:end:WikiTextHeadingRule:30 -->Woodpecker</h2> | ||
Commas: 66/65, 121/120, 126/125<br /> | Commas: 66/65, 121/120, 126/125<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 1 2 2 2|, | 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 /> | EDOs: 15, 31, 46f, 70f, 101f<br /> | ||
Badness: 0.001093<br /> | Badness: 0.001093<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Treecreeper"></a><!-- ws:end:WikiTextHeadingRule:32 -->Treecreeper</h1> | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Treecreeper"></a><!-- ws:end:WikiTextHeadingRule:32 -->Treecreeper</h1> | ||
Commas: 126/125, 1232/1215<br /> | Commas: 126/125, 1232/1215<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]<br /> | Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]<br /> | ||
| Line 386: | Line 399: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Treecreeper-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2> | <!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Treecreeper-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2> | ||
Commas: 91/90, 126/125, 352/351<br /> | Commas: 91/90, 126/125, 352/351<br /> | ||
<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 /> | 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 /> | EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf<br /> | ||
Badness: 0.001588</body></html></pre></div> | Badness: 0.001588<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="Cuckoo"></a><!-- ws:end:WikiTextHeadingRule:36 -->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 /> | |||
<!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Cuckoo-13-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->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> | |||