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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The head of the '''starling family''' is starling, which tempers out [[126/125]], the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and the nonpatent [[135edo]] val {{val| 135 214 314 379 }} (135c).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-26 05:51:07 UTC</tt>.<br>
: The original revision id was <tt>165435135</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, 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 marvel tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]].  


In starling, (6/5)^3 = 126/125 * 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.
In starling, (6/5)<sup>3</sup> = 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.


Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to [[Starling temperaments|valentine temperament]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.


===Vital statistics===
Temperaments discussed elsewhere include
Comma c = 126/125
* ''[[Erato]]'' (+81/80) → [[Didymus rank three family #Erato|Didymus rank-3 family]]
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic family]]
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
Angle(6/5, 5/4) = 100.364 degrees
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank three clan #Cuckoo|Rastmic rank-3 clan]]
Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]
EDOs 77, 108, 185


==11 limit children==
Considered below are starling, thrush, thrasher, aplonis, and treecreeper.


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


7 and 9 limit minimax
[[Subgroup]]: 2.3.5.7
[|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;]
Eigenmonzos: 2, 7/6, 4/3


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
[[Comma list]]: [[126/125]]
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|]
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
Generators: 2, 3, 5
Edos: 31, 46, 58, 89


===Thrasher===
: mapping generators: ~2, ~3, ~5
Commas: 126/125, 100/99


11-limit minimax
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
[|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;]
Eigenmonzos: 2, 8/7, 11/9


Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
[[Minkowski lattice basis]]:
Angle(6/5, 5/4) = 98.8447
: 6/5 length = 1.068, 5/4 length = 1.206
Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]
: Angle (6/5, 5/4) = 100.364 degrees


Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
[[Minimax tuning]]:  
Generators: 2, 3, 5
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
Edos: 12, 15, 19, 34</pre></div>
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
<h4>Original HTML content:</h4>
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Starling family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The head of the starling family is starling, which tempers out 126/125, the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent marvel 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;. &lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
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]]: 0.0699 × 10<sup>-3</sup>
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]]: 7 125/18
&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;
 
Comma c = 126/125&lt;br /&gt;
{{Databox|[[Minkowski blocks]]|
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;
* 7: 25/24, 81/80
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206&lt;br /&gt;
* 8: 16/15, 648/625
Angle(6/5, 5/4) = 100.364 degrees&lt;br /&gt;
* 9: 27/25, 128/125
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&lt;br /&gt;
* 11: 16/15, 15625/15552
EDOs 77, 108, 185&lt;br /&gt;
* 12: 128/125, 628/625
&lt;br /&gt;
* 15: 128/125, 250/243
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-11 limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11 limit children&lt;/h2&gt;
* 16: 648/625, 3125/3072
&lt;br /&gt;
* 17: 25/24, 20480/19683
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11 limit children-Thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Thrush&lt;/h3&gt;
* 19: 81/80, 3125/3072
Commas: 126/125, 176/175&lt;br /&gt;
* 27: 128/125, 78732/78125
&lt;br /&gt;
* 28: 648/625, 16875/16384
7 and 9 limit minimax&lt;br /&gt;
* 31: 81/80, 1990656/1953125
[|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;
* 34: 15625/15552, 2048/2025
|0 0 0 1 0&amp;gt;, |-10/3 4/3 0 5/3 0&amp;gt;]&lt;br /&gt;
}}
Eigenmonzos: 2, 7/6, 4/3&lt;br /&gt;
 
&lt;br /&gt;
== Undecimal starling ==
Lattice basis 5/4 length 0.8576 6/5 length 0.9314&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
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]]: 126/125, 385/384
&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|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
Generators: 2, 3, 5&lt;br /&gt;
 
Edos: 31, 46, 58, 89&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-11 limit children-Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Thrasher&lt;/h3&gt;
[[Badness]]: 0.677 × 10<sup>-3</sup>
Commas: 126/125, 100/99&lt;br /&gt;
 
&lt;br /&gt;
== Thrush ==
11-limit minimax&lt;br /&gt;
{{Main| Starling and thrush }}
[|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;
 
Eigenmonzos: 2, 8/7, 11/9&lt;br /&gt;
[[Comma list]]: 126/125, 176/175
&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 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
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 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
&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:  
Generators: 2, 3, 5&lt;br /&gt;
: 5/4 length = 0.8576, 6/5 length = 0.9314
Edos: 12, 15, 19, 34&lt;/body&gt;&lt;/html&gt;</pre></div>
: Angle(5/4, 6/5) = 74.6239 degrees
 
[[Minimax tuning]]:
* 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
 
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
 
[[Badness]]: 0.353 × 10<sup>-3</sup>
 
[[Projection pair]]s: 7 125/18 11 3125/288
 
[[Associated temperament]]: [[myna]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195
 
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
 
{{Optimal ET sequence|legend=1| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
 
Badness: 0.677 × 10<sup>-3</sup>
 
=== Bluebird ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 176/175
 
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
 
{{Optimal ET sequence|legend=1| 12, 15, 27e, 31, 43, 58, 147cf, 205cceff, 263ccdeefff }}
 
Badness: 0.915 × 10<sup>-3</sup>
 
Projection pairs: 7 125/18 11 3125/288 13 41472/3125
 
=== Nightingale ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 126/125, 176/175
 
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
 
{{Optimal ET sequence|legend=1| 12f, 15, 19e, 27eff, 31 }}
 
Badness: 0.837 × 10<sup>-3</sup>
 
=== Veery ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 176/175
 
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
 
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31f, 46 }}
 
Badness: 0.991 × 10<sup>-3</sup>
 
== Thrasher ==
{{See also| Ptolemismic clan #Thrasher }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 100/99
 
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
 
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
 
Lattice basis:  
: 6/5 length = 0.9089, 5/4 length = 1.2007
: Angle (6/5, 5/4) = 98.8447
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
 
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} <nowiki>*</nowiki>
 
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
 
[[Badness]]: 0.480 × 10<sup>-3</sup>
 
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
 
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e, 69bceef }} <nowiki>*</nowiki>
 
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
 
Badness: 0.876 × 10<sup>-3</sup>
 
=== 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 ET sequence|legend=1| 7d, 8d, 12f, 15, 27eff }}
 
Badness: 0.859 × 10<sup>-3</sup>
 
=== Catbird ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 78/77, 100/99, 126/125
 
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
 
{{Optimal ET sequence|legend=1| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
 
Badness: 0.905 × 10<sup>-3</sup>
 
== 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 ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
 
[[Badness]]: 0.648 × 10<sup>-3</sup>
 
=== 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 ET sequence|legend=1| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
 
Badness: 0.821 × 10<sup>-3</sup>
 
== 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 ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
 
[[Badness]]: 1.585 × 10<sup>-3</sup>
 
=== 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 ET sequence|legend=1| 7d, 12e, 19e, 27e, 46 }}
 
Badness: 1.588 × 10<sup>-3</sup>
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Starling family| ]] <!-- main article -->
[[Category:Starling| ]] <!-- key article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Starling_family"