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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>2011-08-21 16:27:06 UTC</tt>.<br>
: The original revision id was <tt>247441541</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">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 }}


===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|
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 }}]


===[[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


[[Optimal tuning]]s:
* [[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 }}


==Thrush==
[[Minimax tuning]]:
[[Comma|Commas]]: 126/125, 176/175
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
Associated linear temperament: [[Starling temperaments|myna]]
: {{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


7 and 9 limit minimax
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
[|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
[[Badness]] (Sintel): 0.308
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|]
[[Projection pair]]: <code>7 125/18</code>
[[Generator|Generators]]: 2, 3, 5
EDOs: 12, 15, 31, 46, 58, 89, 135, 224
Badness: 0.000353


Scales: [[thrush12]]
{{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
}}


===13-limit===
=== Overview to extensions ===
Commas: 126/125, 176/175, 196/195
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]]


Map: [&lt;1 0 0 -1 -5 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 5 5|]
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.  
EDOs: 31, 46, 58
Badness: 0.000677


===Bluebird===
== Thrush ==
Commas 126/125, 176/175, 144/143
{{Main| Starling and thrush }}


Map: [&lt;1 0 0 -1 -5 9|, &lt;0 1 0 -2 -2 4|, &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: 12, 15, 31, 43, 58
Badness: 0.000915


===Nightingale===
[[Subgroup]]: 2.3.5.7.11
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|]
[[Comma list]]: 126/125, 176/175
[[edo|EDOs]]: [[15edo|15]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]]
Badness: 0.000837


==Thrasher==
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
[[Comma|Commas]]: 56/55, 100/99


11-limit minimax
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
[|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
Lattice basis:  
Angle(6/5, 5/4) = 98.8447
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
: angle(~5/4, ~6/5) = 74.6239°


Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
[[Optimal tuning]]s:  
[[Generator|Generators]]: 2, 3, 5
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
EDOs: 12, 15, 19, 34
: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
Badness: 0.000480
* [[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 }}


===13-limit===
[[Minimax tuning]]:
Commas: 126/125, 100/99, 91/90
* 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 2 2|, &lt;0 1 0 -2 -2 4|, &lt;0 0 1 3 2 -2|]
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
EDOs: 12, 15, 18, 34
Badness: 0.000876


===Mockingbird===
[[Badness]] (Sintel): 0.424
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|]
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
EDOs: 15
Badness: 0.000859


===Catbird===
=== 13-limit ===
Commas: 126/125, 100/99, 78/77
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 0 -1 2 0|, &lt;0 1 0 -2 -2 -5|, &lt;0 0 1 3 2 5|]
Comma list: 126/125, 176/175, 196/195
[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[19edo|19]], [[34edo|34]]
Badness: 0.000905


==Aplonis==
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
Commas: 126/125, 540/539


Map: [&lt;1 0 0 -1 4|, &lt;0 1 0 -2 7|, &lt;0 0 1 3 -5|]
Optimal tunings:  
EDOs: 19, 31, 58, 89
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
Badness: 0.000648
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
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|]
Badness (Sintel): 0.633
EDOs: 19, 31, 50, 58
 
Badness: 0.000821</pre></div>
=== Bluebird ===
<h4>Original HTML content:</h4>
{{Distinguish| Bluebirds }}
<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 &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;
Subgroup: 2.3.5.7.11.13
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;
Comma list: 126/125, 144/143, 176/175
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;
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
&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;
 
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 126/125&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}
[&amp;lt;1 0 0 0|, &amp;lt;0 1 0 0|, &amp;lt;1/3 2/3 0 1/3|, &amp;lt;0 0 0 1|]&lt;br /&gt;
 
Eigenmonzos: 2, 8/7, 4/3&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
&lt;br /&gt;
 
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206&lt;br /&gt;
Badness (Sintel): 0.856
Angle(6/5, 5/4) = 100.364 degrees&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
&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;
=== Nightingale ===
Generators: 2, 3, 5&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
 
Badness: 0.0000699&lt;br /&gt;
Comma list: 66/65, 126/125, 176/175
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 7 125/18&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
&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;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--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;/h3&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}
7: 25/24, 81/80&lt;br /&gt;
 
8: 16/15, 648/625&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
9: 27/25, 128/125&lt;br /&gt;
 
11: 16/15, 15625/15552&lt;br /&gt;
Badness (Sintel): 0.783
12: 128/125, 628/625&lt;br /&gt;
 
15: 128/125, 250/243&lt;br /&gt;
=== Veery ===
16: 648/625, 3125/3072&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
17: 25/24, 20480/19683&lt;br /&gt;
 
19: 81/80, 3125/3072&lt;br /&gt;
Comma list: 91/90, 126/125, 176/175
27: 128/125, 78732/78125&lt;br /&gt;
 
28: 648/625, 16875/16384&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
31: 81/80, 1990656/1953125&lt;br /&gt;
 
34: 15625/15552, 2048/2025&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Thrush&lt;/h2&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
Associated linear temperament: &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;myna&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.927
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;
== Starnova ==
|0 0 0 1 0&amp;gt;, |-10/3 4/3 0 5/3 0&amp;gt;]&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.
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
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;
[[Comma list]]: 126/125, 385/384
Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
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;
[[Optimal tuning]]s:  
EDOs: 12, 15, 31, 46, 58, 89, 135, 224&lt;br /&gt;
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
Badness: 0.000353&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}}
Scales: &lt;a class="wiki_link" href="/thrush12"&gt;thrush12&lt;/a&gt;&lt;br /&gt;
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Thrush-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h3&gt;
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
Commas: 126/125, 176/175, 196/195&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.813
Map: [&amp;lt;1 0 0 -1 -5 0|, &amp;lt;0 1 0 -2 -2 -5|, &amp;lt;0 0 1 3 5 5|]&lt;br /&gt;
 
EDOs: 31, 46, 58&lt;br /&gt;
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
Badness: 0.000677&lt;br /&gt;
 
&lt;br /&gt;
Scales: [[Star and Nova]]
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Thrush-Bluebird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Bluebird&lt;/h3&gt;
 
Commas 126/125, 176/175, 144/143&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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, 58&lt;br /&gt;
Comma list: 126/125, 196/195, 385/384
Badness: 0.000915&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;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Thrush-Nightingale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Nightingale&lt;/h3&gt;
 
Commas: 126/125, 176/175, 66/65&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
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;
* 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;: &lt;a class="wiki_link" href="/15edo"&gt;15&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;br /&gt;
 
Badness: 0.000837&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Thrasher&lt;/h2&gt;
Badness (Sintel): 1.07
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 56/55, 100/99&lt;br /&gt;
 
&lt;br /&gt;
== Thrasher ==
11-limit minimax&lt;br /&gt;
{{See also| Ptolemismic clan }}
[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;, &lt;br /&gt;
 
|1 1/2 0 1/2 -1/4&amp;gt;, |0 0 0 1 0&amp;gt;, &lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
|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;
[[Comma list]]: 56/55, 100/99
&lt;br /&gt;
 
Lattice basis: 6/5 length 0.9089 5/4 length 1.2007&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
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;
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 -1 2|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 2|]&lt;br /&gt;
Lattice basis:  
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
EDOs: 12, 15, 19, 34&lt;br /&gt;
: angle (~6/5, ~5/4) = 98.8447°
Badness: 0.000480&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Thrasher-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h3&gt;
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
Commas: 126/125, 100/99, 91/90&lt;br /&gt;
: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
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;
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
EDOs: 12, 15, 18, 34&lt;br /&gt;
 
Badness: 0.000876&lt;br /&gt;
[[Minimax tuning]]:  
&lt;br /&gt;
* [[11-odd-limit]]
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Thrasher-Mockingbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Mockingbird&lt;/h3&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 }}]
Commas: 126/125, 100/99, 40/39&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
&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;
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
EDOs: 15&lt;br /&gt;
 
Badness: 0.000859&lt;br /&gt;
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Thrasher-Catbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Catbird&lt;/h3&gt;
[[Badness]] (Sintel): 0.577
Commas: 126/125, 100/99, 78/77&lt;br /&gt;
 
&lt;br /&gt;
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
Map: [&amp;lt;1 0 0 -1 2 0|, &amp;lt;0 1 0 -2 -2 -5|, &amp;lt;0 0 1 3 2 5|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;&lt;br /&gt;
=== 13-limit ===
Badness: 0.000905&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x-Aplonis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Aplonis&lt;/h2&gt;
Comma list: 56/55, 91/90, 100/99
Commas: 126/125, 540/539&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 }}
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, 89&lt;br /&gt;
Optimal tunings:  
Badness: 0.000648&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}}
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="x-Aplonis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h3&gt;
 
Commas: 126/125, 144/143, 196/195&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 -1 4 0|, &amp;lt;0 1 0 -2 7 -5|, &amp;lt;0 0 1 3 -5 5|]&lt;br /&gt;
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
EDOs: 19, 31, 50, 58&lt;br /&gt;
 
Badness: 0.000821&lt;/body&gt;&lt;/html&gt;</pre></div>
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:Temperament families]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Starling_family"