Starling family: Difference between revisions

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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>2010-12-15 20:42:19 UTC</tt>.<br>
: The original revision id was <tt>188547407</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 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|


Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5


===[[Minkowski blocks]]===
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
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


==11 limit children==
[[Minkowski lattice basis]]:
: ~6/5 length = 1.068, ~5/4 length = 1.206
: angle (~6/5, ~5/4) = 100.364°


===Thrush===
[[Optimal tuning]]s:
[[Comma|Commas]]: 126/125, 176/175
* [[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 }}


7 and 9 limit minimax
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;,
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 4/3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7


Lattice basis 5/4 length 0.8576 6/5 length 0.9314
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
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|]
[[Badness]] (Sintel): 0.308
[[Generator|Generators]]: 2, 3, 5
EDOs: 12, 15, 31, 46, 58, 89, 135, 224


13-limit
[[Projection pair]]: <code>7 125/18</code>
Commas: 126/125, 176/175, 196/195


EDOs: 31, 46, 58
{{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
}}


====Bluebird====
=== Overview to extensions ===
Commas 126/125, 176/175, 144/143
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]]


EDOs: 12, 15, 31, 43, 58
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.


====Nightingale====
== Thrush ==
Commas: 126/125, 176/175, 66/65
{{Main| Starling and thrush }}


[[edo|EDOs]]: [[15edo|15]], [[28edo|28]], [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]]
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]].


===Thrasher===
[[Subgroup]]: 2.3.5.7.11
[[Comma|Commas]]: 126/125, 100/99


11-limit minimax
[[Comma list]]: 126/125, 176/175
[|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
{{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
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|]
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
[[Generator|Generators]]: 2, 3, 5
EDOs: 12, 15, 19, 34


13-limit
Lattice basis:
Commas: 126/125, 100/99, 91/90
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239°


EDOs: 12, 15, 18, 34
[[Optimal tuning]]s:  
* [[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 }}


====Mockingbird====
[[Minimax tuning]]:
Commas: 126/125, 100/99, 40/39
* 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


====Catbird====
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
Commas: 126/125, 100/99, 78/77


[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[19edo|19]], [[34edo|34]]
[[Badness]] (Sintel): 0.424


===Aplonis===
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
Commas: 126/125, 540/539


13-limit
=== 13-limit ===
Commas: 126/125, 144/143, 196/195
Subgroup: 2.3.5.7.11.13


===Spreo===
Comma list: 126/125, 176/175, 196/195
Starling can be extended to spreo with essentially no tuning accuracy penalty; [[135edo]] can be used for both of them.


Commas: 126/125, 441/440
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}


11-limit minimax
Optimal tunings:
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1 2/5 0 0 1/5&gt;,
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
|2 -4/5 0 0 3/5&gt;, |0 0 0 0 1&gt;]
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}
Eigenmonzos: 2, 4/3, 11/8


Lattice basis: 5/4 length=0.8576 6/5 length=0.9314
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
Angle(5/4, 6/5) = 74.6239
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|]
Badness (Sintel): 0.633
Generators: 2, 3, 5
Edos: 12, 15, 16, 28, 31, 43, 46, 58, 73, 77, 89,
&lt;135 214 314 379 467|


Scales: [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]</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 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 &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;
 
&lt;br /&gt;
Comma list: 66/65, 126/125, 176/175
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;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
&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;
 
7: 25/24, 81/80&lt;br /&gt;
Optimal tunings:  
8: 16/15, 648/625&lt;br /&gt;
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
9: 27/25, 128/125&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}
11: 16/15, 15625/15552&lt;br /&gt;
 
12: 128/125, 628/625&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
15: 128/125, 250/243&lt;br /&gt;
 
16: 648/625, 3125/3072&lt;br /&gt;
Badness (Sintel): 0.783
17: 25/24, 20480/19683&lt;br /&gt;
 
19: 81/80, 3125/3072&lt;br /&gt;
=== Veery ===
27: 128/125, 78732/78125&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
28: 648/625, 16875/16384&lt;br /&gt;
 
31: 81/80, 1990656/1953125&lt;br /&gt;
Comma list: 91/90, 126/125, 176/175
34: 15625/15552, 2048/2025&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-11 limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11 limit children&lt;/h2&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-11 limit children-Thrush"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Thrush&lt;/h3&gt;
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
&lt;br /&gt;
 
7 and 9 limit minimax&lt;br /&gt;
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
[|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;
Badness (Sintel): 0.927
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;
 
&lt;br /&gt;
== Starnova ==
Lattice basis 5/4 length 0.8576 6/5 length 0.9314&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.
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;
[[Subgroup]]: 2.3.5.7.11
&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;
[[Comma list]]: 126/125, 385/384
&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, 135, 224&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
&lt;br /&gt;
 
13-limit&lt;br /&gt;
[[Optimal tuning]]s:  
Commas: 126/125, 176/175, 196/195&lt;br /&gt;
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
EDOs: 31, 46, 58&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-11 limit children-Thrush-Bluebird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Bluebird&lt;/h4&gt;
 
Commas 126/125, 176/175, 144/143&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
&lt;br /&gt;
 
EDOs: 12, 15, 31, 43, 58&lt;br /&gt;
[[Badness]] (Sintel): 0.813
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc5"&gt;&lt;a name="x-11 limit children-Thrush-Nightingale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Nightingale&lt;/h4&gt;
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
Commas: 126/125, 176/175, 66/65&lt;br /&gt;
 
&lt;br /&gt;
Scales: [[Star and Nova]]
&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;
 
&lt;br /&gt;
=== 13-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-11 limit children-Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Thrasher&lt;/h3&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 100/99&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 196/195, 385/384
11-limit minimax&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |1 3/4 0 1/4 -3/8&amp;gt;, &lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
|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;
Optimal tunings:  
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 11/9&lt;br /&gt;
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
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 ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
Map to lattice: [&amp;lt;0 1 0 -2 -2|, &amp;lt;0 1 1 1 0|]&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.07
Map: [&amp;lt;1 0 0 -1 2|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 3 2|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
== Thrasher ==
EDOs: 12, 15, 19, 34&lt;br /&gt;
{{See also| Ptolemismic clan }}
&lt;br /&gt;
 
13-limit&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Commas: 126/125, 100/99, 91/90&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 56/55, 100/99
EDOs: 12, 15, 18, 34&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc7"&gt;&lt;a name="x-11 limit children-Thrasher-Mockingbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Mockingbird&lt;/h4&gt;
 
Commas: 126/125, 100/99, 40/39&lt;br /&gt;
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc8"&gt;&lt;a name="x-11 limit children-Thrasher-Catbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Catbird&lt;/h4&gt;
Lattice basis:
Commas: 126/125, 100/99, 78/77&lt;br /&gt;
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
&lt;br /&gt;
: angle (~6/5, ~5/4) = 98.8447°
&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;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-11 limit children-Aplonis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Aplonis&lt;/h3&gt;
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
Commas: 126/125, 540/539&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}}
13-limit&lt;br /&gt;
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
Commas: 126/125, 144/143, 196/195&lt;br /&gt;
 
&lt;br /&gt;
[[Minimax tuning]]:  
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-11 limit children-Spreo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Spreo&lt;/h3&gt;
* [[11-odd-limit]]
Starling can be extended to spreo with essentially no tuning accuracy penalty; &lt;a class="wiki_link" href="/135edo"&gt;135edo&lt;/a&gt; can be used for both of them.&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 }}]
&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
Commas: 126/125, 441/440&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
11-limit minimax&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |0 1 0 0 0&amp;gt;, |1 2/5 0 0 1/5&amp;gt;, &lt;br /&gt;
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
|2 -4/5 0 0 3/5&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 4/3, 11/8&lt;br /&gt;
[[Badness]] (Sintel): 0.577
&lt;br /&gt;
 
Lattice basis: 5/4 length=0.8576 6/5 length=0.9314&lt;br /&gt;
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
Angle(5/4, 6/5) = 74.6239&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
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;
 
Generators: 2, 3, 5&lt;br /&gt;
Comma list: 56/55, 91/90, 100/99
Edos: 12, 15, 16, 28, 31, 43, 46, 58, 73, 77, 89, &lt;br /&gt;
 
&amp;lt;135 214 314 379 467|&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
&lt;br /&gt;
 
Scales: &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;/body&gt;&lt;/html&gt;</pre></div>
Optimal tunings:  
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
 
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
 
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:Starling family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]

Latest revision as of 12:23, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The starling family of rank-3 temperaments tempers out the starling comma (ratio: 126/125, monzo: [1 2 -3 1⟩), also known as septimal semicomma.

Starling

Main article: Starling and thrush

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 ⟨135 214 314 379].

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.

Subgroup: 2.3.5.7

Comma list: 126/125

Mapping: [⟨1 0 0 -1], ⟨0 1 0 -2], ⟨0 0 1 3]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 0 -2], ⟨0 1 1 1]]

Minkowski lattice basis:

~6/5 length = 1.068, ~5/4 length = 1.206
angle (~6/5, ~5/4) = 100.364°

Optimal tunings:

  • WE: ~2 = 1199.7931 ¢, ~3/2 = 701.1306 ¢, ~5/4 = 390.0298 ¢
error map: ⟨-0.207 -1.031 +3.302 -1.618]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0811 ¢, ~5/4 = 389.8833 ¢
error map: ⟨0.000 -0.874 +3.570 -1.338]

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [1/3 2/3 0 1/3⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c

Badness (Sintel): 0.308

Projection pair: 7 125/18

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

Overview to extensions

Temperaments discussed elsewhere include

Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.

Thrush

Main article: Starling and thrush

Thrush is the main extension of starling into the 11-limit. It may be described as 12 & 15 & 31, and is loosely associated with myna.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175

Mapping: [⟨1 0 0 -1 -5], ⟨0 1 0 -2 -2], ⟨0 0 1 3 5]]

Mapping to lattice: [⟨0 1 1 1 3], ⟨0 1 0 -2 -2]]

Lattice basis:

~5/4 length = 0.8576, ~6/5 length = 0.9314
angle(~5/4, ~6/5) = 74.6239°

Optimal tunings:

  • WE: ~2 = 1199.4742 ¢, ~3/2 = 701.6379 ¢, ~5/4 = 391.0192 ¢
error map: ⟨-0.526 -0.843 +3.654 -0.621 -1.075]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6298 ¢, ~5/4 = 390.8558 ¢
error map: ⟨0.000 -0.325 +4.542 +0.482 -0.298]

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [1/3 2/3 0 1/3 0⟩, [0 0 0 1 0⟩, [-10/3 4/3 0 5/3 0⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7

Optimal ET sequence: 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c

Badness (Sintel): 0.424

Projection pairs: 7 125/18 11 3125/288

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195

Mapping: [⟨1 0 0 -1 -5 0], ⟨0 1 0 -2 -2 -5], ⟨0 0 1 3 5 5]]

Optimal tunings:

  • WE: ~2 = 1199.2727 ¢, ~3/2 = 702.2858 ¢, ~5/4 = 391.3567 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.4562 ¢, ~5/4 = 391.2080 ¢

Optimal ET sequence: 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf

Badness (Sintel): 0.633

Bluebird

Not to be confused with Bluebirds.

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175

Mapping: [⟨1 0 0 -1 -5 9], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 5 -5]]

Optimal tunings:

  • WE: ~2 = 1198.7153 ¢, ~3/2 = 701.647 ¢, ~5/4 = 391.9180 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6263 ¢, ~5/4 = 391.7786 ¢

Optimal ET sequence: 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff

Badness (Sintel): 0.856

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: [⟨1 0 0 -1 -5 -4], ⟨0 1 0 -2 -2 -1], ⟨0 0 1 3 5 4]]

Optimal tunings:

  • WE: ~2 = 1200.1476 ¢, ~3/2 = 699.2886 ¢, ~5/4 = 388.2054 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 699.2840 ¢, ~5/4 = 388.2437 ¢

Optimal ET sequence: 12f, 15, 19e, 27eff, 31

Badness (Sintel): 0.783

Veery

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 176/175

Mapping: [⟨1 0 0 -1 -5 2], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 5 -2]]

Optimal tunings:

  • WE: ~2 = 1199.7998 ¢, ~3/2 = 704.4117 ¢, ~5/4 = 391.7571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.3959 ¢, ~5/4 = 391.6910 ¢

Optimal ET sequence: 12, 15, 19e, 27e, 31f, 46

Badness (Sintel): 0.927

Starnova

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.

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384

Mapping: [⟨1 0 0 -1 8], ⟨0 1 0 -2 3], ⟨0 0 1 3 -4]]

Optimal tunings:

  • WE: ~2 = 1200.6133 ¢, ~3/2 = 700.4314 ¢, ~5/4 = 388.3236 ¢
error map: ⟨+0.613 -0.910 +3.237 -2.878 -1.478]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4020 ¢, ~5/4 = 388.3631 ¢
error map: ⟨0.000 -1.553 +2.049 -4.541 -3.564]

Optimal ET sequence: 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de

Badness (Sintel): 0.813

Projection pairs: 7 125/18 11 6912/625

Scales: Star and Nova

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384

Mapping: [⟨1 0 0 -1 8 0], ⟨0 1 0 -2 3 -5], ⟨0 0 1 3 -4 5]]

Optimal tunings:

  • WE: ~2 = 1200.5083 ¢, ~3/2 = 700.7452 ¢, ~5/4 = 388.4655 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.5708 ¢, ~5/4 = 388.4346 ¢

Optimal ET sequence: 15f, 19, 27, 31, 46, 77, 96d

Badness (Sintel): 1.07

Thrasher

See also: Ptolemismic clan

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99

Mapping: [⟨1 0 0 -1 2], ⟨0 1 0 -2 -2], ⟨0 0 1 3 2]]

Mapping to lattice: [⟨0 1 0 -2 -2], ⟨0 1 1 1 0]]

Lattice basis:

~6/5 length = 0.9089, ~5/4 length = 1.2007
angle (~6/5, ~5/4) = 98.8447°

Optimal tunings:

  • WE: ~2 = 1196.9183 ¢, ~3/2 = 706.1793 ¢, ~5/4 = 394.5223 ¢
error map: ⟨-3.082 +1.143 +2.045 -6.863 +13.041]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.2523 ¢, ~5/4 = 392.7970 ¢
error map: ⟨0.000 +4.297 +6.483 -2.940 +21.771]

Minimax tuning:

[[1 0 0 0 0⟩, [1 3/4 0 1/4 -3/8⟩, [1 1/2 0 1/2 -1/4⟩, [0 0 0 1 0⟩, [2 -1/2 0 1/2 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7.11/9

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e *

* optimal patent val: 34

Badness (Sintel): 0.577

Scales: Thrasher chromatic, Thrasher diatonic

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99

Mapping: [⟨1 0 0 -1 2 2], ⟨0 1 0 -2 -2 4], ⟨0 0 1 3 2 -2]]

Optimal tunings:

  • WE: ~2 = 1197.4723 ¢, ~3/2 = 707.3847 ¢, ~5/4 = 394.2040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.1440 ¢, ~5/4 = 392.7909 ¢

Optimal ET sequence: 7d, 8d, 12, 15, 19, 27e, 69bceef *

* optimal patent val: 34

Badness (Sintel): 0.820

Mockingbird

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 100/99

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 ¢, ~3/2 = 710.0779 ¢, ~5/4 = 400.7221 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.2922 ¢, ~5/4 = 398.0867 ¢

Optimal ET sequence: 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: [⟨1 0 0 -1 2 0], ⟨0 1 0 -2 -2 -5], ⟨0 0 1 3 2 5]]

Optimal tunings:

  • WE: ~2 = 1196.8488 ¢, ~3/2 = 704.8749 ¢, ~5/4 = 394.9837 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9205 ¢, ~5/4 = 393.2290 ¢

Optimal ET sequence: 7df, 8d, 12f, 19, 27e, 66cdeeef

Badness (Sintel): 0.846

Aplonis

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539

Mapping: [⟨1 0 0 -1 4], ⟨0 1 0 -2 7], ⟨0 0 1 3 -5]]

Optimal tunings:

  • WE: ~2 = 1199.6006 ¢, ~3/2 = 700.5746 ¢, ~5/4 = 390.2558 ¢
error map: ⟨-0.399 -1.780 +3.143 -0.406 +1.026]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.4342 ¢, ~5/4 = 389.9796 ¢
error map: ⟨0.000 -1.521 +3.666 +0.244 +1.824]

Optimal ET sequence: 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: [⟨1 0 0 -1 4 0], ⟨0 1 0 -2 7 -5], ⟨0 0 1 3 -5 5]]

Optimal tunings:

  • WE: ~2 = 1199.0609 ¢, ~3/2 = 701.2333 ¢, ~5/4 = 390.8385 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.0203 ¢, ~5/4 = 390.2413 ¢

Optimal ET sequence: 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: [⟨1 0 0 -1 -3], ⟨0 1 0 -2 7], ⟨0 0 1 3 -2]]

Optimal tunings:

  • WE: ~2 = 1198.8957 ¢, ~3/2 = 704.9055 ¢, ~5/4 = 392.8725 ¢
error map: ⟨-1.104 +1.846 +4.350 -3.332 -2.725]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.0045 ¢, ~5/4 = 392.3159 ¢
error map: ⟨0.000 +3.050 +6.002 -1.887 -0.918]

Optimal ET sequence: 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: [⟨1 0 0 -1 -3 2], ⟨0 1 0 -2 7 4], ⟨0 0 1 3 -2 -2]]

Optimal tunings:

  • WE: ~2 = 1199.7878 ¢, ~3/2 = 704.6703 ¢, ~5/4 = 391.3513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.7011 ¢, ~5/4 = 391.2846 ¢

Optimal ET sequence: 7d, 12e, 19e, 27e, 46

Badness (Sintel): 1.48

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