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Hemimage 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>
The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-30 06:09:55 UTC</tt>.<br>
: The original revision id was <tt>190263826</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">===Vital statistics===
[[Comma]]: hemimage 10976/10935


7-limit minimax
== Hemimage ==
[|1 0 0 0&gt;, |5/8 1/8 -1/8 3/8&gt;,
Hemimage is generated by a [[3/2|perfect fifth]] and a [[~]][[28/27]] subminor second. Two of the subminor seconds make ~[[15/14]], and three make ~[[10/9]]. The ubiquitous presence of this little interval gives rise to the highly intense [[hemimage bleeding chord]]s.
|5/8 -7/8 7/8 3/8&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 8/7, 6/5


9-limit minimax
[[Subgroup]]: 2.3.5.7
[|1 0 0 0&gt;, |5/9 2/9 -1/9 1/3&gt;,
|10/9 -14/9 7/9 2/3&gt;, |0 0 0 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 8/7


Map: [&lt;1 0 2 -1|, &lt;0 1 2 3|, &lt;0 0 -3 -1|]
[[Comma list]]: 10976/10935
[[Generator|Generators]]: 2, 3, 27/14
[[edo|Edos]]: [[99edo|99]], [[118edo|118]], [[140edo|140]], [[316edo|316]], [[415edo|415]], [[654edo|654]]


==11-limit children==
{{Mapping|legend=1| 1 0 -1 -2 | 0 1 2 3 | 0 0 3 1 }}
: mapping generators: ~2, ~3, ~28/27


===Uni===
[[Optimal tuning]]s:
[[Comma|Commas]]: 540/539, 896/891
* [[WE]]: ~2 = 1199.8555{{c}}, ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}
: [[error map]]: {{val| -0.145 +0.507 +0.155 -0.683 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6137{{c}}, ~28/27 = 60.4761{{c}}
: error map: {{val| 0.000 +0.659 +0.342 -0.509 }}


[[Minimax tuning]]
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |16/11 6/11 1/11 0 -3/11&gt;,
* [[7-odd-limit]]
|0 0 1 0 0&gt;, |19/11 14/11 6/11 0 -7/11&gt;,
: {{monzo list| 1 0 0 0 | 5/8 1/8 -1/8 3/8 | 5/8 -7/8 7/8 3/8 | 0 0 0 1 }}
|32/11 -10/11 2/11 0 5/11&gt;]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 5/4
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 5/9 2/9 -1/9 1/3 | 10/9 -14/9 7/9 2/3 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


Map: [&lt;1 0 2 -1 6|, &lt;0 1 2 3 -1|, &lt;0 0 -3 -1 -1|]
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
[[Generator|Generators]]: 2, 3, 27/14
[[edo|Edos]]: [[19edo|19]], [[22edo|22]], [[41edo|41]], [[58edo|58]], [[80edo|80]], [[121edo|121]]


===Turan===
[[Badness]] (Sintel): 0.878
[[Comma|Commas]]: 10976/10935, 441/440


[[Minimax tuning]]
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7
[|1 0 0 0 0&gt;, |19/12 -2/3 1/12 0 1/4&gt;,
|19/6 -10/3 7/6 0 1/2&gt;,
|37/12 -8/3 7/12 0 3/4&gt;,
|19/6 -10/3 1/6 0 3/2&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9, 11/9


Map: [&lt;1 0 2 -1 -7|, &lt;0 1 2 3 6|, &lt;0 0 -3 -1 1|]
== Uni ==
[[Generator|Generators]]: 2, 3, 27/14
[[Subgroup]]: 2.3.5.7.11
[[edo|Edos]]: [[41edo|41]], [[58edo|58]], [[118edo|118]], [[159edo|159]], [[217edo|217]], [[335edo|335]]</pre></div>
 
<h4>Original HTML content:</h4>
[[Comma list]]: 540/539, 896/891
<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;Hemimage family&lt;/title&gt;&lt;/head&gt;&lt;body&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;: hemimage 10976/10935&lt;br /&gt;
{{Mapping|legend=1| 1 0 -1 -2 5 | 0 1 2 3 -1 | 0 0 3 1 1 }}
&lt;br /&gt;
 
7-limit minimax&lt;br /&gt;
[[Optimal tuning]]s:
[|1 0 0 0&amp;gt;, |5/8 1/8 -1/8 3/8&amp;gt;, &lt;br /&gt;
* [[WE]]: ~2 = 1199.1497{{c}}, ~3/2 = 703.3014{{c}}, ~28/27 = 60.0617{{c}}
|5/8 -7/8 7/8 3/8&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
: [[error map]]: {{val| -0.859 +0.496 -0.376 +0.290 +2.041 }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 8/7, 6/5&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6475{{c}}, ~28/27 = 59.9675{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +1.693 +0.884 +2.084 +5.002 }}
9-limit minimax&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |5/9 2/9 -1/9 1/3&amp;gt;, &lt;br /&gt;
[[Minimax tuning]]:  
|10/9 -14/9 7/9 2/3&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
* [[11-odd-limit]]
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 8/7&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}]
&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9
Map: [&amp;lt;1 0 2 -1|, &amp;lt;0 1 2 3|, &amp;lt;0 0 -3 -1|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 27/14&lt;br /&gt;
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/316edo"&gt;316&lt;/a&gt;, &lt;a class="wiki_link" href="/415edo"&gt;415&lt;/a&gt;, &lt;a class="wiki_link" href="/654edo"&gt;654&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.844
&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;
=== 13-limit  ===
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11-limit children-Uni"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Uni&lt;/h3&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 540/539, 896/891&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 352/351, 364/363, 540/539
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |16/11 6/11 1/11 0 -3/11&amp;gt;, &lt;br /&gt;
Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}
|0 0 1 0 0&amp;gt;, |19/11 14/11 6/11 0 -7/11&amp;gt;, &lt;br /&gt;
 
|32/11 -10/11 2/11 0 5/11&amp;gt;]&lt;br /&gt;
Optimal tunings:
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 5/4&lt;br /&gt;
* WE: ~2 = 1199.0867{{c}}, ~3/2 = 703.4048{{c}}, ~28/27 = 60.0034{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9457{{c}}, ~28/27 = 59.8167{{c}}
Map: [&amp;lt;1 0 2 -1 6|, &amp;lt;0 1 2 3 -1|, &amp;lt;0 0 -3 -1 -1|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 27/14&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/80edo"&gt;80&lt;/a&gt;, &lt;a class="wiki_link" href="/121edo"&gt;121&lt;/a&gt;&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.993
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-11-limit children-Turan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Turan&lt;/h3&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 10976/10935, 441/440&lt;br /&gt;
=== Tin ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |19/12 -2/3 1/12 0 1/4&amp;gt;, &lt;br /&gt;
Comma list: 144/143, 196/195, 729/728
|19/6 -10/3 7/6 0 1/2&amp;gt;, &lt;br /&gt;
 
|37/12 -8/3 7/12 0 3/4&amp;gt;, &lt;br /&gt;
Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}
|19/6 -10/3 1/6 0 3/2&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9, 11/9&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1198.8801{{c}}, ~3/2 = 702.5101{{c}}, ~28/27 = 60.9480{{c}}
Map: [&amp;lt;1 0 2 -1 -7|, &amp;lt;0 1 2 3 6|, &amp;lt;0 0 -3 -1 1|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8952{{c}}, ~28/27 = 60.9129{{c}}
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 27/14&lt;br /&gt;
 
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/217edo"&gt;217&lt;/a&gt;, &lt;a class="wiki_link" href="/335edo"&gt;335&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
 
Badness (Sintel): 1.00
 
== Turan ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 10976/10935
 
{{Mapping|legend=1| 1 0 -1 -2 -6 | 0 1 2 3 6 | 0 0 3 1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0289{{c}}, ~3/2 = 702.1366{{c}}, ~28/27 = 60.8231{{c}}
: [[error map]]: {{val| +0.029 +0.211 +0.458 -1.564 +0.679 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1290{{c}}, ~28/27 = 60.8257{{c}}
: error map: {{val| 0.000 +0.174 +0.421 -1.613 +0.630 }}
 
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9
 
{{Optimal ET sequence|legend=1| 17c, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335 }}
 
[[Badness]] (Sintel): 1.13
 
[[Category:Hemimage family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]
Retrieved from "https://en.xen.wiki/w/Hemimage_family"