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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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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>2010-07-26 00:11:11 UTC</tt>.<br>
: The original revision id was <tt>154007203</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 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-liit scale, one may as well assume valentine tempering as it will do no harm.</pre></div>
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.
<h4>Original HTML 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;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;
Temperaments discussed elsewhere include
&lt;br /&gt;
* ''[[Erato]]'' (+81/80) → [[Didymus rank three family #Erato|Didymus rank-3 family]]
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;
* ''[[Sensigh]]'' (+245/243) → [[Sensamagic family #Sensigh|Sensamagic family]]
&lt;br /&gt;
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
Because no 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-liit scale, one may as well assume valentine tempering as it will do no harm.&lt;/body&gt;&lt;/html&gt;</pre></div>
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank three clan #Cuckoo|Rastmic rank-3 clan]]
 
Considered below are starling, thrush, thrasher, aplonis, and treecreeper.
 
== Starling ==
{{Main| Starling and thrush }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: [[126/125]]
 
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
 
: mapping generators: ~2, ~3, ~5
 
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
 
[[Minkowski lattice basis]]:  
: 6/5 length = 1.068, 5/4 length = 1.206
: Angle (6/5, 5/4) = 100.364 degrees
 
[[Minimax tuning]]:  
* 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
 
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
 
[[Badness]]: 0.0699 × 10<sup>-3</sup>
 
[[Projection pair]]: 7 125/18
 
{{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
}}
 
== Undecimal starling ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 385/384
 
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
 
[[Badness]]: 0.677 × 10<sup>-3</sup>
 
== Thrush ==
{{Main| Starling and thrush }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 176/175
 
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
 
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
 
Lattice basis:
: 5/4 length = 0.8576, 6/5 length = 0.9314
: 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"