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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-23 17:24:06 UTC</tt>.<br>
: The original revision id was <tt>173026135</tt>.<br>
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<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">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 &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.
The '''starling family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[starling comma]] ([[ratio]]: 126/125, {{monzo|legend=1| 1 2 -3 1 }}), also known as ''septimal semicomma''.  


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.
== Starling ==
{{Main| Starling and thrush }}


===Vital statistics===
The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list 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 [[135edo]] in the nonpatent 135c val {{val| 135 214 '''314''' 379 }}.
[[Comma]] c = 126/125


7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
[&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
[[Subgroup]]: 2.3.5.7
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|]
[[Comma list]]: [[126/125]]
Generators: 2, 3, 5
[[edo|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[185edo|185]], &lt;135 214 314 379|


==11 limit children==
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5


===Thrush===
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
[[Comma|Commas]]: 126/125, 176/175


7 and 9 limit minimax
[[Minkowski lattice basis]]:
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
: ~6/5 length = 1.068, ~5/4 length = 1.206
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
: angle (~6/5, ~5/4) = 100.364°
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
[[Optimal tuning]]s:
Angle(5/4, 6/5) = 74.6239 degrees
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}
: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}


Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
[[Minimax tuning]]:  
[[Generator|Generators]]: 2, 3, 5
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


13-limit
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
Commas: 126/125, 176/175, 196/195


====Bluebird====
[[Badness]] (Sintel): 0.308
Commas 126/125, 176/175, 144/143


====Nightingale====
[[Projection pair]]: <code>7 125/18</code>
Commas: 126/125, 176/175, 66/65


[[edo|EDOs]]: [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]]
{{Databox|[[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
}}


===Thrasher===
=== Overview to extensions ===
[[Comma|Commas]]: 126/125, 100/99
Temperaments discussed elsewhere include
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]


11-limit minimax
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.
[|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
== Thrush ==
Angle(6/5, 5/4) = 98.8447
{{Main| Starling and thrush }}
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|]
Thrush is the main extension of starling into the [[11-limit]]. It may be described as 12 & 15 & 31, and is loosely [[associated temperament|associated]] with [[myna]].
[[Generator|Generators]]: 2, 3, 5


13-limit
[[Subgroup]]: 2.3.5.7.11
Commas: 126/125, 100/99, 91/90


====Mockingbird====
[[Comma list]]: 126/125, 176/175
Commas: 126/125, 100/99, 40/39


====Catbird====
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
Commas: 126/125, 100/99, 78/77


[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[19edo|19]], [[34edo|34]]
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]


===Aplonis===
Lattice basis:
Commas: 126/125, 540/539
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239°


13-limit
[[Optimal tuning]]s:
Commas: 126/125, 144/143, 196/195</pre></div>
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
<h4>Original HTML content:</h4>
: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
<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;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 &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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}
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;
[[Minimax tuning]]:
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;
* 7- and [[9-odd-limit]]
&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 126/125&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;
 
[&amp;lt;1 0 0 0|, &amp;lt;0 1 0 0|, &amp;lt;1/3 2/3 0 1/3|, &amp;lt;0 0 0 1|]&lt;br /&gt;
[[Badness]] (Sintel): 0.424
Eigenmonzos: 2, 8/7, 4/3&lt;br /&gt;
 
&lt;br /&gt;
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
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;
=== 13-limit ===
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
Comma list: 126/125, 176/175, 196/195
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="/185edo"&gt;185&lt;/a&gt;, &amp;lt;135 214 314 379|&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-11 limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11 limit children&lt;/h2&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11 limit children-Thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Thrush&lt;/h3&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
7 and 9 limit minimax&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |0 1 0 0 0&amp;gt;, |1/3 2/3 0 1/3 0&amp;gt;, &lt;br /&gt;
Badness (Sintel): 0.633
|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;
=== Bluebird ===
&lt;br /&gt;
{{Distinguish| Bluebirds }}
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;
Subgroup: 2.3.5.7.11.13
Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 144/143, 176/175
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;
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
&lt;br /&gt;
 
13-limit&lt;br /&gt;
Optimal tunings:
Commas: 126/125, 176/175, 196/195&lt;br /&gt;
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;a name="x-11 limit children-Thrush-Bluebird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Bluebird&lt;/h4&gt;
 
Commas 126/125, 176/175, 144/143&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-11 limit children-Thrush-Nightingale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Nightingale&lt;/h4&gt;
Badness (Sintel): 0.856
Commas: 126/125, 176/175, 66/65&lt;br /&gt;
 
&lt;br /&gt;
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
&lt;a class="wiki_link" href="/edo"&gt;EDOs&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;br /&gt;
 
&lt;br /&gt;
=== Nightingale ===
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-11 limit children-Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Thrasher&lt;/h3&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 100/99&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 66/65, 126/125, 176/175
11-limit minimax&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;, &lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
|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;
Optimal tunings:
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 11/9&lt;br /&gt;
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}
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;
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
Map to lattice: [&amp;lt;0 1 0 -2 -2|, &amp;lt;0 1 1 1 0|]&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.783
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;
=== Veery ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
13-limit&lt;br /&gt;
 
Commas: 126/125, 100/99, 91/90&lt;br /&gt;
Comma list: 91/90, 126/125, 176/175
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc6"&gt;&lt;a name="x-11 limit children-Thrasher-Mockingbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Mockingbird&lt;/h4&gt;
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
Commas: 126/125, 100/99, 40/39&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc7"&gt;&lt;a name="x-11 limit children-Thrasher-Catbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Catbird&lt;/h4&gt;
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
Commas: 126/125, 100/99, 78/77&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
&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="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-11 limit children-Aplonis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Aplonis&lt;/h3&gt;
Badness (Sintel): 0.927
Commas: 126/125, 540/539&lt;br /&gt;
 
&lt;br /&gt;
== Starnova ==
13-limit&lt;br /&gt;
Starnova tempers out [[385/384]]. This temperament used to be known as ''undecimal starling'', but was decanonicalized in 2026. The name ''starnova'' was proposed by [[Flora Canou]] after [[Gene Ward Smith]]'s [[Star]] and [[Nova]] scales.
Commas: 126/125, 144/143, 196/195&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 385/384
 
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
 
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
 
[[Badness]] (Sintel): 0.813
 
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
 
Scales: [[Star and Nova]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 385/384
 
Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
 
Badness (Sintel): 1.07
 
== Thrasher ==
{{See also| Ptolemismic clan }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 100/99
 
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
 
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
 
Lattice basis:
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
: angle (~6/5, ~5/4) = 98.8447°
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
 
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
 
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
 
[[Badness]] (Sintel): 0.577
 
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
 
Optimal tunings:
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
 
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
 
Badness (Sintel): 0.820
 
=== Mockingbird ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 56/55, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
 
Optimal tunings:
* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}
 
Badness (Sintel): 0.804
 
=== Catbird ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
 
Optimal tunings:
* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}
 
{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
 
Badness (Sintel): 0.846
 
== Aplonis ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 540/539
 
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}
: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}
 
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
 
[[Badness]] (Sintel): 0.778
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 196/195
 
Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
 
Badness (Sintel): 0.768
 
== Treecreeper ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 1232/1215
 
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}
 
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
 
[[Badness]] (Sintel): 1.91
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 352/351
 
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}
 
Badness (Sintel): 1.48
 
[[Category:Starling family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]
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