Starling family: Difference between revisions

<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.

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.

[[Comma]] c = 126/125

7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp

[&lt;1 0 0 0|, &lt;0 1 0 0|, &lt;1/3 2/3 0 1/3|, &lt;0 0 0 1|]

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|]

Generators: 2, 3, 5

Scales: [[starling7]], [[starling8]], [[starling9]], [[starling11]], [[starling12]], [[starling15]], [[starling16]], [[starling17]], [[starling19]]

==11 limit children==

[[Comma|Commas]]: 126/125, 176/175

[|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

Angle(5/4, 6/5) = 74.6239 degrees

Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]

[[Generator|Generators]]: 2, 3, 5

Commas: 126/125, 176/175, 196/195

Commas 126/125, 176/175, 144/143

Commas: 126/125, 176/175, 66/65

[[edo|EDOs]]: [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]]

===Thrasher===

[[Comma|Commas]]: 126/125, 100/99

[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9

Lattice basis: 6/5 length 0.9089 5/4 length 1.2007

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|]

[[Generator|Generators]]: 2, 3, 5

13-limit

Commas: 126/125, 100/99, 91/90

====Mockingbird====

Commas: 126/125, 100/99, 78/77

[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[19edo|19]], [[34edo|34]]

===Aplonis===

Commas: 126/125, 144/143, 196/195

===Spreo===

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

[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1 2/5 0 0 1/5&gt;,  

|2 -4/5 0 0 3/5&gt;, |0 0 0 0 1&gt;]

Eigenmonzos: 2, 4/3, 11/8

Lattice basis: 5/4 length=0.8576 6/5 length=0.9314

Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]

Generators: 2, 3, 5

Edos: 12, 15, 16, 28, 31, 43, 46, 58, 73, 77, 89,

&lt;135 214 314 379 467|</pre></div>

<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;

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;

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;!-- 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;

7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp&lt;br /&gt;

[&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;

Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206&lt;br /&gt;

Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 1 1 1|]&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;

&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;

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;!-- 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;

&lt;br /&gt;

&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;

&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 176/175&lt;br /&gt;

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;

|0 0 0 1 0&amp;gt;, |-10/3 4/3 0 5/3 0&amp;gt;]&lt;br /&gt;

&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 4/3&lt;br /&gt;

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;

Map to lattice: [&amp;lt;0 1 1 1 3|, &amp;lt;0 1 0 -2 -2|]&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;

&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;

13-limit&lt;br /&gt;

Commas: 126/125, 176/175, 196/195&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;a name="x-11 limit children-Thrush-Bluebird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Bluebird&lt;/h4&gt;

Commas 126/125, 176/175, 144/143&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-11 limit children-Thrush-Nightingale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Nightingale&lt;/h4&gt;

Commas: 126/125, 176/175, 66/65&lt;br /&gt;

&lt;a class="wiki_link" href="/edo"&gt;EDOs&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;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-11 limit children-Thrasher"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Thrasher&lt;/h3&gt;

&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 100/99&lt;br /&gt;

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;

|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;

&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 11/9&lt;br /&gt;

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;

Map to lattice: [&amp;lt;0 1 0 -2 -2|, &amp;lt;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;

&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;

Commas: 126/125, 100/99, 91/90&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc6"&gt;&lt;a name="x-11 limit children-Thrasher-Mockingbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Mockingbird&lt;/h4&gt;

Commas: 126/125, 100/99, 40/39&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc7"&gt;&lt;a name="x-11 limit children-Thrasher-Catbird"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Catbird&lt;/h4&gt;

Commas: 126/125, 100/99, 78/77&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;

&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-11 limit children-Aplonis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Aplonis&lt;/h3&gt;

Commas: 126/125, 540/539&lt;br /&gt;

Commas: 126/125, 144/143, 196/195&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-11 limit children-Spreo"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Spreo&lt;/h3&gt;