Keenanismic 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 '''keenanismic family''' of [[rank-4 temperament|rank-4]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[keenanisma]], the 11-limit comma 385/384, {{monzo| -7 -1 1 1 1 }} of 4.503 [[cent]]s.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-01-21 13:23:17 UTC</tt>.<br>
: The original revision id was <tt>400169386</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">[[toc|flat]]
The keenanismic family of rank four temperaments tempers out the [[385_384|keenanisma]], the 11-limit comma 385/384 = |-7 -1 1 1 1&gt; of 4.503 cents.


=Keenanismic=  
== Keenanismic ==
Comma: 385/384
[[Subgroup]]: 2.3.5.7.11


Minimax tuning
[[Comma list]]: 385/384
[|1 0 0 0 0&gt;, |0 1 0 0 0&gt;, |7/3 1/3 2/3 -1/3 -1/3&gt;,
|7/3 1/3 -1/3 2/3 -1/3&gt;, |7/3 1/3 -1/3 -1/3 2/3&gt;]
Eigenmonzos: 2, 3, 7/5, 11/5


Map: [&lt;1 0 0 0 7|, &lt;0 1 0 0 1|, &lt;0 0 1 0 -1|, &lt;0 0 0 1 -1|]
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}
Transpose: [2 3 5 7 385/35]
: Mapping generators: ~2, ~3, ~5, ~7
EDOs: [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[68edo|53]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]]
10^9*Badness: 15.159


=Martwin=  
[[Optimal tuning]]s:
Commas: 385/384, 325/324
* [[WE]]: ~2 = 1200.4104{{c}}, ~3/2 = 701.6916{{c}}, ~5/4 = 385.1761{{c}}, ~7/4 = 967.5421{{c}}
: [[error map]]: {{val| +0.410 +0.147 -0.317 -0.463 -0.703 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6933{{c}}, ~5/4 = 385.1491{{c}}, ~7/4 = 967.3004{{c}}
: error map: {{val| 0.000 -0.262 -1.165 -1.525 -2.074 }}


Minimax tuning
[[Minimax tuning]]:
||1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;, |2/3 4/3 1/3 0 0 -1/3&gt;,
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 0 1 0 0 0 }}, ~5 = {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, ~7 = {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}
|17/3 -5/3 1/3 0 -1 2/3&gt;, |2/3 4/3 -2/3 0 1 -1/3&gt;, |2/3 4/3 -2/3 0 0 2/3&gt;]
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5
Eigenmonzos: 2, 3, 11/5, 13/5


Map: [&lt;1 0 0 0 7 2|, &lt;0 1 0 0 1 4|, &lt;0 0 1 0 -1 -2|, &lt;0 0 0 1 -1 0|]
{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
Transpose: [2 3 5 7 385/35 324/25]
EDOs: [[7edo|7]], [[15edo|15]], [[19edo|19]], [[26edo|26]], [[34edo|34]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[72edo|72]], [[87edo|87]], [[140edo|140]], [[159edo|159]], [[212edo|212]] and[[299edo|299]]
10^6*Badness: 2.206


=Ancient=
[[Badness]] (Sintel): 0.277
Commas: 385/384, 625/624


Map: [&lt;1 0 0 0 7 -4|, &lt;0 1 0 0 1 -1|, &lt;0 0 1 0 -1 4|, &lt;0 0 0 1 -1 0|]
== Martwin ==
Transpose: [2 3 5 7 385/35 625/48]
[[Subgroup]]: 2.3.5.7.11.13
EDOs: [[6edo|6]], [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[50edo|50]], [[53edo|53]], [[72edo|72]], [[87edo|87]], [[103edo|103]], [[140edo|140]], [[159edo|159]], [[190edo|190]]
 
10^6*Badness: 2.573</pre></div>
[[Comma list]]: 325/324, 385/384
<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;Keenanismic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Keenanismic"&gt;Keenanismic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Martwin"&gt;Martwin&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#Ancient"&gt;Ancient&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 0 0 1 4 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 0 }}
&lt;!-- ws:end:WikiTextTocRule:10 --&gt;The keenanismic family of rank four temperaments tempers out the &lt;a class="wiki_link" href="/385_384"&gt;keenanisma&lt;/a&gt;, the 11-limit comma 385/384 = |-7 -1 1 1 1&amp;gt; of 4.503 cents.&lt;br /&gt;
 
&lt;br /&gt;
Lattice basis:
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Keenanismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Keenanismic&lt;/h1&gt;
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247
Comma: 385/384&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
Minimax tuning&lt;br /&gt;
* [[WE]]: ~2 = 1200.3494{{c}}, ~3/2 = 702.2131{{c}}, ~5/4 = 384.8666{{c}}, ~7/4 = 967.8487{{c}}
[|1 0 0 0 0&amp;gt;, |0 1 0 0 0&amp;gt;, |7/3 1/3 2/3 -1/3 -1/3&amp;gt;,&lt;br /&gt;
: [[error map]]: {{val| +0.349 +0.608 -0.748 -0.278 -0.422 -0.710 }}
|7/3 1/3 -1/3 2/3 -1/3&amp;gt;, |7/3 1/3 -1/3 -1/3 2/3&amp;gt;]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1659{{c}}, ~5/4 = 384.8721{{c}}, ~7/4 = 967.6109{{c}}
Eigenmonzos: 2, 3, 7/5, 11/5&lt;br /&gt;
: error map: {{val| 0.000 +0.211 -1.442 -1.215 -1.635 -1.608 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 0 7|, &amp;lt;0 1 0 0 1|, &amp;lt;0 0 1 0 -1|, &amp;lt;0 0 0 1 -1|]&lt;br /&gt;
[[Minimax tuning]]:  
Transpose: [2 3 5 7 385/35]&lt;br /&gt;
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~2 = {{val| 1 0 0 0 0 0 }}, ~3 = {{val| 0 1 0 0 0 0 }}, ~5 = {{val| 2/3 4/3 1/3 0 0 -1/3 }}, ~7 = {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}
EDOs: &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="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&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="/190edo"&gt;190&lt;/a&gt;, &lt;a class="wiki_link" href="/212edo"&gt;212&lt;/a&gt; and &lt;a class="wiki_link" href="/284edo"&gt;284&lt;/a&gt;&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5
10^9*Badness: 15.159&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Martwin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Martwin&lt;/h1&gt;
 
Commas: 385/384, 325/324&lt;br /&gt;
[[Badness]] (Sintel): 0.579
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
== Ancient ==
||1 0 0 0 0 0&amp;gt;, |0 1 0 0 0 0&amp;gt;, |2/3 4/3 1/3 0 0 -1/3&amp;gt;,&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11.13
|17/3 -5/3 1/3 0 -1 2/3&amp;gt;, |2/3 4/3 -2/3 0 1 -1/3&amp;gt;, |2/3 4/3 -2/3 0 0 2/3&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 3, 11/5, 13/5&lt;br /&gt;
[[Comma list]]: 385/384, 625/624
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 0 7 2|, &amp;lt;0 1 0 0 1 4|, &amp;lt;0 0 1 0 -1 -2|, &amp;lt;0 0 0 1 -1 0|]&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}
Transpose: [2 3 5 7 385/35 324/25]&lt;br /&gt;
 
EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&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="/26edo"&gt;26&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/212edo"&gt;212&lt;/a&gt; and&lt;a class="wiki_link" href="/299edo"&gt;299&lt;/a&gt;&lt;br /&gt;
[[Optimal tuning]]s:
10^6*Badness: 2.206&lt;br /&gt;
* [[WE]]: ~2 = 1200.4130{{c}}, ~3/2 = 701.6863{{c}}, ~5/4 = 385.2319{{c}}, ~7/4 = 967.5188{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +0.413 +0.144 -0.256 -0.481 -0.730 -0.047 }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Ancient"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Ancient&lt;/h1&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6838{{c}}, ~5/4 = 385.2497{{c}}, ~7/4 = 967.2556{{c}}
Commas: 385/384, 625/624&lt;br /&gt;
: error map: {{val| 0.000 -0.271 -1.064 -1.570 -2.139 -1.213 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 0 7 -4|, &amp;lt;0 1 0 0 1 -1|, &amp;lt;0 0 1 0 -1 4|, &amp;lt;0 0 0 1 -1 0|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
Transpose: [2 3 5 7 385/35 625/48]&lt;br /&gt;
 
EDOs: &lt;a class="wiki_link" href="/6edo"&gt;6&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="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/103edo"&gt;103&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140&lt;/a&gt;, &lt;a class="wiki_link" href="/159edo"&gt;159&lt;/a&gt;, &lt;a class="wiki_link" href="/190edo"&gt;190&lt;/a&gt;&lt;br /&gt;
[[Badness]] (Sintel): 0.675
10^6*Badness: 2.573&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Zaxa ==
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 352/351, 385/384
 
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 -2 | 0 0 1 0 -1 -1 | 0 0 0 1 -1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1117{{c}}, ~3/2 = 702.6588{{c}}, ~5/4 = 385.6821{{c}}, ~7/4 = 968.0061{{c}}
: [[error map]]: {{val| +0.112 +0.815 -0.408 -0.596 -1.901 +1.137 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.6029{{c}}, ~5/4 = 385.6441{{c}}, ~7/4 = 967.9030{{c}}
: error map: {{val| 0.000 +0.648 -0.670 -0.923 -2.262 +0.720 }}
 
{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
 
[[Badness]] (Sintel): 0.880
 
[[Category:Temperament families]]
[[Category:Keenanismic family| ]] <!-- main article -->
[[Category:Rank 4]]

Latest revision as of 18:42, 4 March 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 keenanismic family of rank-4 temperaments tempers out the keenanisma, the 11-limit comma 385/384, [-7 -1 1 1 1⟩ of 4.503 cents.

Keenanismic

Subgroup: 2.3.5.7.11

Comma list: 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 1], ⟨0 0 1 0 -1], ⟨0 0 0 1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1200.4104 ¢, ~3/2 = 701.6916 ¢, ~5/4 = 385.1761 ¢, ~7/4 = 967.5421 ¢
error map: ⟨+0.410 +0.147 -0.317 -0.463 -0.703]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6933 ¢, ~5/4 = 385.1491 ¢, ~7/4 = 967.3004 ¢
error map: ⟨0.000 -0.262 -1.165 -1.525 -2.074]

Minimax tuning:

  • 11-odd-limit: ~2 = [1 0 0 0 0⟩, ~3 = [0 1 0 0 0⟩, ~5 = [7/3 1/3 2/3 -1/3 -1/3⟩, ~7 = [7/3 1/3 -1/3 2/3 -1/3⟩
Unchanged-interval (eigenmonzo) basis: 2.3.7/5.11/5

Optimal ET sequence: 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

Badness (Sintel): 0.277

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

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

Lattice basis:

4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

Optimal tunings:

  • WE: ~2 = 1200.3494 ¢, ~3/2 = 702.2131 ¢, ~5/4 = 384.8666 ¢, ~7/4 = 967.8487 ¢
error map: ⟨+0.349 +0.608 -0.748 -0.278 -0.422 -0.710]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1659 ¢, ~5/4 = 384.8721 ¢, ~7/4 = 967.6109 ¢
error map: ⟨0.000 +0.211 -1.442 -1.215 -1.635 -1.608]

Minimax tuning:

  • 13- and 15-odd-limit: ~2 = ⟨1 0 0 0 0 0], ~3 = ⟨0 1 0 0 0 0], ~5 = ⟨2/3 4/3 1/3 0 0 -1/3], ~7 = ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6]
unchanged-interval (eigenmonzo) basis: 2.3.11/7.13/5

Optimal ET sequence: 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff

Badness (Sintel): 0.579

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Optimal tunings:

  • WE: ~2 = 1200.4130 ¢, ~3/2 = 701.6863 ¢, ~5/4 = 385.2319 ¢, ~7/4 = 967.5188 ¢
error map: ⟨+0.413 +0.144 -0.256 -0.481 -0.730 -0.047]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6838 ¢, ~5/4 = 385.2497 ¢, ~7/4 = 967.2556 ¢
error map: ⟨0.000 -0.271 -1.064 -1.570 -2.139 -1.213]

Optimal ET sequence: 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def

Badness (Sintel): 0.675

Zaxa

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 -2], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1200.1117 ¢, ~3/2 = 702.6588 ¢, ~5/4 = 385.6821 ¢, ~7/4 = 968.0061 ¢
error map: ⟨+0.112 +0.815 -0.408 -0.596 -1.901 +1.137]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 702.6029 ¢, ~5/4 = 385.6441 ¢, ~7/4 = 967.9030 ¢
error map: ⟨0.000 +0.648 -0.670 -0.923 -2.262 +0.720]

Optimal ET sequence: 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e

Badness (Sintel): 0.880

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