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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: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.
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 }}


=7-limit starling=
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]], [[166edo|166c]], [[185edo|185]], [[243edo|243c]]
Badness: 0.0000699


[[Projection pair]]: 7 125/18
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5


Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
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 lattice basis]]:  
7: 25/24, 81/80
: ~6/5 length = 1.068, ~5/4 length = 1.206
8: 16/15, 648/625
: angle (~6/5, ~5/4) = 100.364°
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==
[[Optimal tuning]]s:
Commas: 126/125, 385/384
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
: [[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 8|, &lt;0 1 0 -2 3|, &lt;0 0 1 3 -4]]
[[Minimax tuning]]:  
Generators: 2, 3, 5
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
Badness: .000677
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


=Thrush=
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
[[Comma|Commas]]: 126/125, 176/175
Associated linear temperament: [[Starling temperaments|myna]]


7 and 9 limit minimax
[[Badness]] (Sintel): 0.308
[|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
[[Projection pair]]: <code>7 125/18</code>
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|]
{{Databox|[[Minkowski blocks]]|
[[Generator|Generators]]: 2, 3, 5
* 7: 25/24, 81/80
EDOs: 12, 15, 31, 46, 58, 89, 135c, 224c
* 8: 16/15, 648/625
Badness: 0.000353
* 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
}}


Projection pairs: 7 125/18 11 3125/288
=== Overview to extensions ===
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]]


Scales: [[thrush12]]
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.


==13-limit==  
== Thrush ==
Commas: 126/125, 176/175, 196/195
{{Main| Starling and thrush }}


Map: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]
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]].  
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf
Badness: 0.000677


==Bluebird==
[[Subgroup]]: 2.3.5.7.11
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|]
[[Comma list]]: 126/125, 176/175
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef
Badness: 0.000915


Projection pairs: 7 125/18 11 3125/288 13 41472/3125
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}


==Nightingale==
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
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|]
Lattice basis:  
[[edo|EDOs]]: [[15edo|15]], [[19edo|19f]], [[21edo|21e]], [[22edo|22f]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]], [[108edo|108ef]]
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
Badness: 0.000837
: angle(~5/4, ~6/5) = 74.6239°


==Veery==  
[[Optimal tuning]]s:
Commas: 91/90, 126/125, 176/175
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
: [[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 }}


Map: [&lt;1 0 0 -1 -5 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 5 -2|]
[[Minimax tuning]]:  
EDOs: 12, 15, 46
* 7- and [[9-odd-limit]]
Badness: 0.000991
: [{{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


=Thrasher=
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
[[Comma|Commas]]: 56/55, 100/99


11-limit minimax
[[Badness]] (Sintel): 0.424
[|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
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
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|]
=== 13-limit ===
[[Generator|Generators]]: 2, 3, 5
Subgroup: 2.3.5.7.11.13
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e
Badness: 0.000480


==13-limit==
Comma list: 126/125, 176/175, 196/195
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|]
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e
Badness: 0.000876


==Mockingbird==  
Optimal tunings:
Commas: 126/125, 100/99, 40/39
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}


Map: [&lt;1 0 0 -1 2 3|, &lt;0 1 0 -2 -2 -1|, &lt;0 0 1 3 2 1|]
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
EDOs: 12f, 15, 27ef
Badness: 0.000859


==Catbird==
Badness (Sintel): 0.633
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|]
=== Bluebird ===
[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[17edo|17c]], [[19edo|19]], [[22edo|22ef]], [[27edo|27e]], [[34edo|34]], [[58edo|58e]]
{{Distinguish| Bluebirds }}
Badness: 0.000905


=Aplonis=
Subgroup: 2.3.5.7.11.13
Commas: 126/125, 540/539


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
Comma list: 126/125, 144/143, 176/175
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce
Badness: 0.000648


==13-limit==
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
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|]
Optimal tunings:  
EDOs: 19, 31, 41, 43, 50, 58, 89f, 125ce, 135ce, 147cf, 166cef, 224cef
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
Badness: 0.000821
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}


=Oxpecker=
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
Commas: 121/120, 126/125


Map: [&lt;1 0 1 2 2|, &lt;0 1 1 1 1|, &lt;0 0 -2 -6 -1|]
Badness (Sintel): 0.856
EDOs: 8d, 15, 31, 46, 77, 185e, 262cde
Badness: 0.000699


==Woodpecker==
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
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|]
=== Nightingale ===
EDOs: 15, 31, 46f, 70f, 101f
Subgroup: 2.3.5.7.11.13
Badness: 0.001093


=Treecreeper=
Comma list: 66/65, 126/125, 176/175
Commas: 126/125, 1232/1215


Map: [&lt;1 0 0 -1 -3|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -2|]
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
EDOs: 34, 46, 119c, 165c
Badness: 0.001585


==13-limit==  
Optimal tunings:
Commas: 91/90, 126/125, 352/351
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}


Map: [&lt;1 0 0 -1 -3 2|, &lt;0 1 0 -2 7 4|, &lt;0 0 1 3 -2 -2|]
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
EDOs: 34, 46, 172cd, 218cdf, 264bcdf, 310bcdf
Badness: 0.001588


=Cuckoo=
Badness (Sintel): 0.783
Commas: 126/125, 243/242


Map: [&lt;1 1 0 -3 2|, &lt;0 2 0 -4 5|, &lt;0 0 1 3 0|]
=== Veery ===
EDOs: 31 34 58 65 89 154 185 216
Subgroup: 2.3.5.7.11.13
Badness: 0.000933


==13-limit==
Comma list: 91/90, 126/125, 176/175
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|]
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
EDOs: 31 58 65f 89f 92e 154 185 216f</pre></div>
 
<h4>Original HTML content:</h4>
Optimal tunings:  
<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;
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
&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;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
&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;
 
&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;
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
&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;
 
&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;
Badness (Sintel): 0.927
&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;
 
&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;
== Starnova ==
&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;
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: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;
 
&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;
[[Subgroup]]: 2.3.5.7.11
&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;
 
&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;
[[Comma list]]: 126/125, 385/384
&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;
 
&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;
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
&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;
[[Optimal tuning]]s:  
&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;
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
&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;
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
&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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;/div&gt;
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
&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;
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
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;
[[Badness]] (Sintel): 0.813
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;
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
&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;
Scales: [[Star and Nova]]
&lt;br /&gt;
 
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;
=== 13-limit ===
[&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;
Subgroup: 2.3.5.7.11.13
Eigenmonzos: 2, 8/7, 4/3&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 196/195, 385/384
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;
Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 0 0 -5|, &amp;lt;0 1 0 2|, &amp;lt;0 0 1 2|]&lt;br /&gt;
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
Generators: 2, 3, 5&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
&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;
{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 125/18&lt;br /&gt;
Badness (Sintel): 1.07
&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;
== Thrasher ==
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;
{{See also| Ptolemismic clan }}
&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;
[[Subgroup]]: 2.3.5.7.11
7: 25/24, 81/80&lt;br /&gt;
 
8: 16/15, 648/625&lt;br /&gt;
[[Comma list]]: 56/55, 100/99
9: 27/25, 128/125&lt;br /&gt;
 
11: 16/15, 15625/15552&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
12: 128/125, 628/625&lt;br /&gt;
 
15: 128/125, 250/243&lt;br /&gt;
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
16: 648/625, 3125/3072&lt;br /&gt;
 
17: 25/24, 20480/19683&lt;br /&gt;
Lattice basis:
19: 81/80, 3125/3072&lt;br /&gt;
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
27: 128/125, 78732/78125&lt;br /&gt;
: angle (~6/5, ~5/4) = 98.8447°
28: 648/625, 16875/16384&lt;br /&gt;
 
31: 81/80, 1990656/1953125&lt;br /&gt;
[[Optimal tuning]]s:  
34: 15625/15552, 2048/2025&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: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;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
Commas: 126/125, 385/384&lt;br /&gt;
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
&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;
[[Minimax tuning]]:  
Generators: 2, 3, 5&lt;br /&gt;
* [[11-odd-limit]]
EDOs: 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de, 300de&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 }}]
Badness: .000677&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
&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;
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
&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;
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
&lt;br /&gt;
 
7 and 9 limit minimax&lt;br /&gt;
[[Badness]] (Sintel): 0.577
[|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;
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Lattice basis 5/4 length 0.8576 6/5 length 0.9314&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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;
Comma list: 56/55, 91/90, 100/99
&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;
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
&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;
Optimal tunings:  
Badness: 0.000353&lt;br /&gt;
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}
Projection pairs: 7 125/18 11 3125/288&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
Scales: &lt;a class="wiki_link" href="/thrush12"&gt;thrush12&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
&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;
Badness (Sintel): 0.820
&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;
=== Mockingbird ===
EDOs: 31, 46, 58, 89f, 104c, 135c, 193cf, 239cf, 328cf&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.000677&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 40/39, 56/55, 100/99
&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;
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
&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;
Optimal tunings:
EDOs: 12, 15, 31, 43, 48c, 58, 147cf, 205cef&lt;br /&gt;
* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
Badness: 0.000915&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}
&lt;br /&gt;
 
Projection pairs: 7 125/18 11 3125/288 13 41472/3125&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}
&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;
Badness (Sintel): 0.804
Commas: 126/125, 176/175, 66/65&lt;br /&gt;
 
&lt;br /&gt;
=== Catbird ===
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;
Subgroup: 2.3.5.7.11.13
&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;
Comma list: 56/55, 78/77, 100/99
&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;
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
Commas: 91/90, 126/125, 176/175&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
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;
* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
EDOs: 12, 15, 46&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}
Badness: 0.000991&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
&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;
Badness (Sintel): 0.846
&lt;br /&gt;
 
11-limit minimax&lt;br /&gt;
== Aplonis ==
[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;,&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
|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;
[[Comma list]]: 126/125, 540/539
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 11/9&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
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 tuning]]s:
Map to lattice: [&amp;lt;0 1 0 -2 -2|, &amp;lt;0 1 1 1 0|]&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|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 2|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}
EDOs: 12, 15, 19, 27e, 34, 46e, 58e, 61e, 73e&lt;br /&gt;
 
Badness: 0.000480&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
&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;
[[Badness]] (Sintel): 0.778
Commas: 126/125, 100/99, 91/90&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
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;
Subgroup: 2.3.5.7.11.13
EDOs: 12, 15, 18, 25e, 27e, 34, 46e, 51ce, 61e&lt;br /&gt;
 
Badness: 0.000876&lt;br /&gt;
Comma list: 126/125, 144/143, 196/195
&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;
Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
Commas: 126/125, 100/99, 40/39&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
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;
* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
EDOs: 12f, 15, 27ef&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}
Badness: 0.000859&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
&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;
Badness (Sintel): 0.768
&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;
== Treecreeper ==
&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;
[[Subgroup]]: 2.3.5.7.11
Badness: 0.000905&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 126/125, 1232/1215
&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;
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
&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;
[[Optimal tuning]]s:  
EDOs: 19, 31, 58, 67, 89, 197c, 286ce, 375ce&lt;br /&gt;
* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
Badness: 0.000648&lt;br /&gt;
: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
&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;
: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}
Commas: 126/125, 144/143, 196/195&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
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]] (Sintel): 1.91
Badness: 0.000821&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
&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;
Subgroup: 2.3.5.7.11.13
Commas: 121/120, 126/125&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 126/125, 352/351
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;
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
Badness: 0.000699&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&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;
* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
Commas: 66/65, 121/120, 126/125&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}
&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;
{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}
EDOs: 15, 31, 46f, 70f, 101f&lt;br /&gt;
 
Badness: 0.001093&lt;br /&gt;
Badness (Sintel): 1.48
&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;
[[Category:Temperament families]]
Commas: 126/125, 1232/1215&lt;br /&gt;
[[Category:Starling family| ]] <!-- main article -->
&lt;br /&gt;
[[Category:Rank 3]]
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>
Retrieved from "https://en.xen.wiki/w/Starling_family"