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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>2012-09-03 13:16:46 UTC</tt>.<br>
: The original revision id was <tt>361691190</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|.
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''.  


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


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.
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 }}.  


=7-limit starling=
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.
[[Comma]] c = 126/125


7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
[[Subgroup]]: 2.3.5.7
[&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
[[Comma list]]: [[126/125]]
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|]
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
Generators: 2, 3, 5
: mapping generators: ~2, ~3, ~5
[[edo|EDOs]]: 12, 15, 16, 19, 27, 31, 34, 43, 46, 50, 58, 65, [[77edo|77]], [[108edo|108]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
Badness: 0.0000699


[[Projection pair]]: 7 125/18
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]


Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
[[Minkowski lattice basis]]:
: ~6/5 length = 1.068, ~5/4 length = 1.206
: angle (~6/5, ~5/4) = 100.364°


==[[Minkowski blocks]]==
[[Optimal tuning]]s:  
7: 25/24, 81/80
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
8: 16/15, 648/625
: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}
9: 27/25, 128/125
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}
11: 16/15, 15625/15552
: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}
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==
[[Minimax tuning]]:
Commas: 126/125, 385/384
* 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


Map: [&lt;1 0 0 -1 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
Generators: 2, 3, 5
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
Badness: .000677


=Thrush=
[[Badness]] (Sintel): 0.308
[[Comma|Commas]]: 126/125, 176/175
Associated linear temperament: [[Starling temperaments|myna]]


7 and 9 limit minimax
[[Projection pair]]: <code>7 125/18</code>
[|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
{{Databox|[[Minkowski blocks]]|
Angle(5/4, 6/5) = 74.6239 degrees
* 7: 25/24, 81/80
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]
* 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
}}


Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
=== Overview to extensions ===
[[Generator|Generators]]: 2, 3, 5
Temperaments discussed elsewhere include
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
Badness: 0.000353
* ''[[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]]


Projection pairs: 7 125/18 11 3125/288
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.


Scales: [[thrush12]]
== Thrush ==
{{Main| Starling and thrush }}


==13-limit==
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]].
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|]
[[Subgroup]]: 2.3.5.7.11
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf
Badness: 0.000677


==Bluebird==
[[Comma list]]: 126/125, 176/175
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|]
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef
Badness: 0.000915


Projection pairs: 7 125/18 11 3125/288 13 41472/3125
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]


==Nightingale==
Lattice basis:
Commas: 126/125, 176/175, 66/65
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239°


Map: [&lt;1 0 0 -1 -5 -4|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 5 4|]
[[Optimal tuning]]s:  
[[edo|EDOs]]: [[15edo|15]], [[19edo|19f]], [[21edo|21e]], [[22edo|22f]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]], [[108edo|108ef]]
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
Badness: 0.000837
: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}
: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}


==Veery==
[[Minimax tuning]]:
Commas: 91/90, 126/125, 176/175
* 7- and [[9-odd-limit]]
: [{{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 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


Map: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
EDOs: 12, 15, 46
Badness: 0.000991


=Thrasher=
[[Badness]] (Sintel): 0.424
[[Comma|Commas]]: 56/55, 100/99


11-limit minimax
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
[|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
=== 13-limit ===
Angle(6/5, 5/4) = 98.8447
Subgroup: 2.3.5.7.11.13
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|]
Comma list: 126/125, 176/175, 196/195
[[Generator|Generators]]: 2, 3, 5
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
Badness: 0.000480


==13-limit==
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
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|]
Optimal tunings:  
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
Badness: 0.000876
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}


==Mockingbird==
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
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|]
Badness (Sintel): 0.633
EDOs: 12f, 15, 27ef
Badness: 0.000859


==Catbird==
=== Bluebird ===
Commas: 126/125, 100/99, 78/77
{{Distinguish| Bluebirds }}


Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
Subgroup: 2.3.5.7.11.13
[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
Badness: 0.000905


=Aplonis=
Comma list: 126/125, 144/143, 176/175
Commas: 126/125, 540/539


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
Badness: 0.000648


==13-limit==
Optimal tunings:
Commas: 126/125, 144/143, 196/195
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}


Map: [&lt;1 0 0 -1 4 0|, &lt;0 1 0 -2 7 -5|, &lt;0 0 1 3 -5 5|]
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
Badness: 0.000821


=Oxpecker=
Badness (Sintel): 0.856
Commas: 121/120, 126/125


Map: [&lt;1 0 1 2 2|,  &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
Badness: 0.000699


==Woodpecker==
=== Nightingale ===
Commas: 66/65, 121/120, 126/125
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 1 2 2 2|, &lt;0 1 1 1 1 1|, &lt;0 0 -2 -6 -1 -1|]
Comma list: 66/65, 126/125, 176/175
EDOs: 15, 31, 46f, 70f, 101f
Badness: 0.001093


=Treecreeper=
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
Commas: 126/125, 1232/1215


Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
Optimal tunings:  
EDOs: 34, 46, 119c, 165c
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
Badness: 0.001585
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
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|]
Badness (Sintel): 0.783
EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
 
Badness: 0.001588</pre></div>
=== Veery ===
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13
<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:36:&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:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x7-limit starling"&gt;7-limit starling&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&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;
Comma list: 91/90, 126/125, 176/175
&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&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;
 
&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Thrush"&gt;Thrush&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-Bluebird"&gt;Bluebird&lt;/a&gt;&lt;/div&gt;
Optimal tunings:  
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-Nightingale"&gt;Nightingale&lt;/a&gt;&lt;/div&gt;
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrush-Veery"&gt;Veery&lt;/a&gt;&lt;/div&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Thrasher"&gt;Thrasher&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrasher-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrasher-Mockingbird"&gt;Mockingbird&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Thrasher-Catbird"&gt;Catbird&lt;/a&gt;&lt;/div&gt;
Badness (Sintel): 0.927
&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Aplonis"&gt;Aplonis&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Aplonis-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
== Starnova ==
&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Oxpecker"&gt;Oxpecker&lt;/a&gt;&lt;/div&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.
&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&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:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Treecreeper"&gt;Treecreeper&lt;/a&gt;&lt;/div&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&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:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;/div&gt;
[[Comma list]]: 126/125, 385/384
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;br /&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;
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
&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;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
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;
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
&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;
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
&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| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
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): 0.813
Eigenmonzos: 2, 8/7, 4/3&lt;br /&gt;
 
&lt;br /&gt;
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</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;
Scales: [[Star and Nova]]
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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;
Comma list: 126/125, 196/195, 385/384
Badness: 0.0000699&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 125/18&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
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;
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
&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;
 
7: 25/24, 81/80&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
8: 16/15, 648/625&lt;br /&gt;
 
9: 27/25, 128/125&lt;br /&gt;
Badness (Sintel): 1.07
11: 16/15, 15625/15552&lt;br /&gt;
 
12: 128/125, 628/625&lt;br /&gt;
== Thrasher ==
15: 128/125, 250/243&lt;br /&gt;
{{See also| Ptolemismic clan }}
16: 648/625, 3125/3072&lt;br /&gt;
 
17: 25/24, 20480/19683&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
19: 81/80, 3125/3072&lt;br /&gt;
 
27: 128/125, 78732/78125&lt;br /&gt;
[[Comma list]]: 56/55, 100/99
28: 648/625, 16875/16384&lt;br /&gt;
 
31: 81/80, 1990656/1953125&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
34: 15625/15552, 2048/2025&lt;br /&gt;
 
&lt;br /&gt;
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
&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;
Lattice basis:
&lt;br /&gt;
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
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;
: angle (~6/5, ~5/4) = 98.8447°
Generators: 2, 3, 5&lt;br /&gt;
 
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de&lt;br /&gt;
[[Optimal tuning]]s:  
Badness: .000677&lt;br /&gt;
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
&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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
Associated linear temperament: &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;myna&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
[[Minimax tuning]]:  
7 and 9 limit minimax&lt;br /&gt;
* [[11-odd-limit]]
[|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;
: [{{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 }}]
|0 0 0 1 0&amp;gt;, |-10/3 4/3 0 5/3 0&amp;gt;]&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
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;
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.577
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;
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c&lt;br /&gt;
 
Badness: 0.000353&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Projection pairs: 7 125/18 11 3125/288 &lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 91/90, 100/99
Scales: &lt;a class="wiki_link" href="/thrush12"&gt;thrush12&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
&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;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
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;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf&lt;br /&gt;
 
Badness: 0.000677&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
&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;
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
Commas 126/125, 176/175, 144/143&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.820
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;
=== Mockingbird ===
Badness: 0.000915&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Projection pairs: 7 125/18 11 3125/288 13 41472/3125&lt;br /&gt;
Comma list: 40/39, 56/55, 100/99
&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;
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
Commas: 126/125, 176/175, 66/65&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
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;
* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
&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;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}
Badness: 0.000837&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}
&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;
Badness (Sintel): 0.804
&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;
=== Catbird ===
EDOs: 12, 15, 46&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.000991&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 78/77, 100/99
&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;
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
&lt;br /&gt;
 
11-limit minimax&lt;br /&gt;
Optimal tunings:
[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;, &lt;br /&gt;
* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
|1 1/2 0 1/2 -1/4&amp;gt;, |0 0 0 1 0&amp;gt;, &lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}
|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;
{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
&lt;br /&gt;
 
Lattice basis: 6/5 length 0.9089 5/4 length 1.2007&lt;br /&gt;
Badness (Sintel): 0.846
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;
== Aplonis ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
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;
[[Comma list]]: 126/125, 540/539
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e&lt;br /&gt;
 
Badness: 0.000480&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
&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;
[[Optimal tuning]]s:
Commas: 126/125, 100/99, 91/90&lt;br /&gt;
* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}
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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e&lt;br /&gt;
: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}
Badness: 0.000876&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
&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;
[[Badness]] (Sintel): 0.778
&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;
=== 13-limit ===
EDOs: 12f, 15, 27ef&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.000859&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 144/143, 196/195
&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;
Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
&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;
Optimal tunings:
&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;
* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
Badness: 0.000905&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}
&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;
{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
Commas: 126/125, 540/539&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.768
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;
== Treecreeper ==
Badness: 0.000648&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
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[[Comma list]]: 126/125, 1232/1215
Commas: 126/125, 144/143, 196/195&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
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;
[[Optimal tuning]]s:
Badness: 0.000821&lt;br /&gt;
* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
Commas: 121/120, 126/125&lt;br /&gt;
: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}
&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;
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
EDOs: 8d, 15, 31, 46, 77, 185e, 262cde&lt;br /&gt;
 
Badness: 0.000699&lt;br /&gt;
[[Badness]] (Sintel): 1.91
&lt;br /&gt;
 
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=== 13-limit ===
Commas: 66/65, 121/120, 126/125&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
Comma list: 91/90, 126/125, 352/351
EDOs: 15, 31, 46f, 70f, 101f &lt;br /&gt;
 
Badness: 0.001093&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
&lt;br /&gt;
 
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Optimal tunings:  
Commas: 126/125, 1232/1215&lt;br /&gt;
* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}
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;
{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}
Badness: 0.001585&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.48
&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;
[[Category:Temperament families]]
&lt;br /&gt;
[[Category:Starling family| ]] <!-- main article -->
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;
[[Category:Rank 3]]
EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf&lt;br /&gt;
Badness: 0.001588&lt;/body&gt;&lt;/html&gt;</pre></div>
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