Starling temperaments: Difference between revisions
Wikispaces>genewardsmith **Imported revision 187269421 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 190334190 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2010-12-10 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-31 01:10:30 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>190334190</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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[|1 0 0 0>, |0 1 0 0>, |5/11 13/11 0 0>, |4/11 17/11 0 0>] | [|1 0 0 0>, |0 1 0 0>, |5/11 13/11 0 0>, |4/11 17/11 0 0>] | ||
[[Eigenmonzo|Eigenmonzos]]: 2, 3 | [[Eigenmonzo|Eigenmonzos]]: 2, 3 | ||
[[POTE tuning|POTE generator]]: 154.579 | |||
Map: [<1 3 4 5|, <0 -11 -13 -17|] | Map: [<1 3 4 5|, <0 -11 -13 -17|] | ||
[[Generator|Generators]]: 2, 49/45 | [[Generator|Generators]]: 2, 49/45 | ||
EDOs: 7, 8, 31, 101, 132, 163 | |||
====11-limit==== | ====11-limit==== | ||
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[[Eigenmonzo|Eigenmonzos]]: 2, 11/9 | [[Eigenmonzo|Eigenmonzos]]: 2, 11/9 | ||
[[POTE tuning|POTE generator]]: 154.645 | |||
Algebraic generator: [[Algebraic number|positive root]] of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly. | Algebraic generator: [[Algebraic number|positive root]] of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly. | ||
Map: [<1 3 4 5 5|, <0 -11 -13 -17 -12|] | Map: [<1 3 4 5 5|, <0 -11 -13 -17 -12|] | ||
[[Generator|Generators]]: 2, 11/10</pre></div> | [[Generator|Generators]]: 2, 11/10 | ||
EDOs: 7, 8, 31, 101, 194</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>Starling temperaments</title></head><body>This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before <a class="wiki_link" href="/12edo">12edo</a> 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 /> | <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"><html><head><title>Starling temperaments</title></head><body>This page discusses some of the temperaments tempering out 126/125, the starling comma or septimal semicomma. Since (6/5)^3 = 126/125 * 12/7, these temperaments tend to have a relatively small complexity for 6/5. They also possess 6/5-6/5-6/5-7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before <a class="wiki_link" href="/12edo">12edo</a> 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 /> | ||
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[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]<br /> | [|1 0 0 0&gt;, |0 1 0 0&gt;, |5/11 13/11 0 0&gt;, |4/11 17/11 0 0&gt;]<br /> | ||
<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br /> | ||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 154.579<br /> | |||
<br /> | <br /> | ||
Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]<br /> | Map: [&lt;1 3 4 5|, &lt;0 -11 -13 -17|]<br /> | ||
<a class="wiki_link" href="/Generator">Generators</a>: 2, 49/45<br /> | <a class="wiki_link" href="/Generator">Generators</a>: 2, 49/45<br /> | ||
EDOs: 7, 8, 31, 101, 132, 163<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h4&gt; --><h4 id="toc5"><a name="x--Nusecond temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h4> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h4&gt; --><h4 id="toc5"><a name="x--Nusecond temperament-11-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->11-limit</h4> | ||
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<a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/9<br /> | <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 11/9<br /> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 154.645<br /> | |||
Algebraic generator: <a class="wiki_link" href="/Algebraic%20number">positive root</a> of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.<br /> | Algebraic generator: <a class="wiki_link" href="/Algebraic%20number">positive root</a> of 15x^2-10x-7, or (5+sqrt(130))/15, at 154.6652 cents. The recurrence converges very quickly.<br /> | ||
<br /> | <br /> | ||
Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]<br /> | Map: [&lt;1 3 4 5 5|, &lt;0 -11 -13 -17 -12|]<br /> | ||
<a class="wiki_link" href="/Generator">Generators</a>: 2, 11/10</body></html></pre></div> | <a class="wiki_link" href="/Generator">Generators</a>: 2, 11/10<br /> | ||
EDOs: 7, 8, 31, 101, 194</body></html></pre></div> | |||