Starling family: Difference between revisions

Revision as of 16:12, 28 September 2010

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2010-09-28 16:12:43 UTC.

The original revision id was 166170881.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

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.

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.

===Vital statistics===
[[Comma]] c = 126/125
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206
Angle(6/5, 5/4) = 100.364 degrees
Map to lattice: [<0 1 0 -2|, <0 1 1 1|]
[[edo|EDOs]]: [[77edo|77]], [[108edo|108]], [[185edo|185]]

==11 limit children==

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

7 and 9 limit minimax
[|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>]
[[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 to lattice: [<0 1 1 1 3|, <0 1 0 -2 -2|]

Map: [<1 0 0 -1 -5|, <0 1 0 -2 -2|, <0 0 1 3 5|]
[[Generator|Generators]]: 2, 3, 5

13-limit
Commas: 126/125, 176/175, 196/195

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

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

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

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

11-limit minimax
[|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>]
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 11/9

Lattice basis: 6/5 length 0.9089 5/4 length 1.2007
Angle(6/5, 5/4) = 98.8447
Map to lattice: [<0 1 0 -2 -2|, <0 1 1 1 0|]

Map: [<1 0 0 -1 2|, <0 1 0 -2 -2|, <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, 40/39

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

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

===Aplonis===
Commas: 126/125, 540/539

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

Original HTML content:

<html><head><title>Starling family</title></head><body>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 <a class="wiki_link" href="/77edo">77edo</a>.  Other possible tunings are <a class="wiki_link" href="/108edo">108edo</a> and <a class="wiki_link" href="/185edo">185edo</a>. <br />
<br />
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.<br />
<br />
Because no appreciable tuning accuracy is lost by including 1029/1024 along with 126/125 in the comma list, which leads to <a class="wiki_link" href="/Starling%20temperaments">valentine temperament</a>, there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vital statistics</h3>
<a class="wiki_link" href="/Comma">Comma</a> c = 126/125<br />
7- and 9-limit minimax: 3 and 7 just, 5 1/3c sharp<br />
Minkowski lattice basis: 6/5 length 1.068, 5/4 length 1.206<br />
Angle(6/5, 5/4) = 100.364 degrees<br />
Map to lattice: [&lt;0 1 0 -2|, &lt;0 1 1 1|]<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/108edo">108</a>, <a class="wiki_link" href="/185edo">185</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-11 limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->11 limit children</h2>
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-11 limit children-Thrush"></a><!-- ws:end:WikiTextHeadingRule:4 -->Thrush</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 126/125, 176/175<br />
<br />
7 and 9 limit minimax<br />
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |1/3 2/3 0 1/3 0&gt;, <br />
|0 0 0 1 0&gt;, |-10/3 4/3 0 5/3 0&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/6, 4/3<br />
<br />
Lattice basis 5/4 length 0.8576 6/5 length 0.9314<br />
Angle(5/4, 6/5) = 74.6239 degrees<br />
Map to lattice: [&lt;0 1 1 1 3|, &lt;0 1 0 -2 -2|]<br />
<br />
Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
<br />
13-limit<br />
Commas: 126/125, 176/175, 196/195<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h4&gt; --><h4 id="toc3"><a name="x-11 limit children-Thrush-Bluebird"></a><!-- ws:end:WikiTextHeadingRule:6 -->Bluebird</h4>
Commas 126/125, 176/175, 144/143<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h4&gt; --><h4 id="toc4"><a name="x-11 limit children-Thrush-Nightingale"></a><!-- ws:end:WikiTextHeadingRule:8 -->Nightingale</h4>
Commas: 126/125, 176/175, 66/65<br />
<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/46edo">46</a>, <a class="wiki_link" href="/58edo">58</a>, <a class="wiki_link" href="/89edo">89</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-11 limit children-Thrasher"></a><!-- ws:end:WikiTextHeadingRule:10 -->Thrasher</h3>
<a class="wiki_link" href="/Comma">Commas</a>: 126/125, 100/99<br />
<br />
11-limit minimax<br />
[|1 0 0 0 0&gt;, |1 3/4 0 1/4 -3/8&gt;, <br />
|1 1/2 0 1/2 -1/4&gt;, |0 0 0 1 0&gt;, <br />
|2 -1/2 0 1/2 1/4&gt;]<br />
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 8/7, 11/9<br />
<br />
Lattice basis: 6/5 length 0.9089 5/4 length 1.2007<br />
Angle(6/5, 5/4) = 98.8447<br />
Map to lattice: [&lt;0 1 0 -2 -2|, &lt;0 1 1 1 0|]<br />
<br />
Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]<br />
<a class="wiki_link" href="/Generator">Generators</a>: 2, 3, 5<br />
<br />
13-limit<br />
Commas: 126/125, 100/99, 91/90<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h4&gt; --><h4 id="toc6"><a name="x-11 limit children-Thrasher-Mockingbird"></a><!-- ws:end:WikiTextHeadingRule:12 -->Mockingbird</h4>
Commas: 126/125, 100/99, 40/39<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h4&gt; --><h4 id="toc7"><a name="x-11 limit children-Thrasher-Catbird"></a><!-- ws:end:WikiTextHeadingRule:14 -->Catbird</h4>
Commas: 126/125, 100/99, 78/77<br />
<br />
<a class="wiki_link" href="/edo">EDOs</a>: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/15edo">15</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/34edo">34</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-11 limit children-Aplonis"></a><!-- ws:end:WikiTextHeadingRule:16 -->Aplonis</h3>
Commas: 126/125, 540/539<br />
<br />
13-limit<br />
Commas: 126/125, 144/143, 196/195</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-09-26 09:13:07 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-28 16:12:43 UTC</tt>.<br>
-: The original revision id was <tt>165446689</tt>.<br>
+: The original revision id was <tt>166170881</tt>.<br>
-: The revision comment was: <tt>links added</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>
@@ Line 36: / Line 36: @@
 Map: [&lt;1 0 0 -1 -5|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 5|]
 [[Generator|Generators]]: 2, 3, 5
+-limit
+Commas: 126/125, 176/175, 196/195
+====Bluebird====
+Commas 126/125, 176/175, 144/143
+====Nightingale====
+Commas: 126/125, 176/175, 66/65
 [[edo|EDOs]]: [[31edo|31]], [[46edo|46]], [[58edo|58]], [[89edo|89]]
@@ Line 53: / Line 63: @@
 Map: [&lt;1 0 0 -1 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 3 2|]
 [[Generator|Generators]]: 2, 3, 5
-[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[19edo|19]], [[34edo|34]]</pre></div>
+-limit
+Commas: 126/125, 100/99, 91/90
+====Mockingbird====
+Commas: 126/125, 100/99, 40/39
+====Catbird====
+Commas: 126/125, 100/99, 78/77
+[[edo|EDOs]]: [[12edo|12]], [[15edo|15]], [[19edo|19]], [[34edo|34]]
+===Aplonis===
+Commas: 126/125, 540/539
+-limit
+Commas: 126/125, 144/143, 196/195</pre></div>
 <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;
@@ Line 85: / Line 111: @@
 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;
+&lt;br /&gt;
+-limit&lt;br /&gt;
+Commas: 126/125, 176/175, 196/195&lt;br /&gt;
+&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;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;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;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;
+&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;
 &lt;br /&gt;
@@ Line 102: / Line 138: @@
 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;
-&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;/body&gt;&lt;/html&gt;</pre></div>
+&lt;br /&gt;
+-limit&lt;br /&gt;
+Commas: 126/125, 100/99, 91/90&lt;br /&gt;
+&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;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;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;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;
+&lt;br /&gt;
+-limit&lt;br /&gt;
+Commas: 126/125, 144/143, 196/195&lt;/body&gt;&lt;/html&gt;</pre></div>

Starling family: Difference between revisions

Revision as of 16:12, 28 September 2010

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

Navigation menu

Search