Didymus rank-3 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 '''didymus rank-3 family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] rank-3 temperament, with an [[11-limit]] comma we get an 11-limit temperament, and so forth.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-31 13:54:44 UTC</tt>.<br>
: The original revision id was <tt>288922277</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">These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.


[[toc|flat]]
Most of these temperaments were named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100647.html#100648 Yahoo! Tuning Group | ''Higher-rank meantone extensions (muse temperaments?)'']</ref>.


=Euterpe=
Temperaments discussed elsewhere include:
Commas: 81/80, 99/98
* ''[[Urania]]'' (+243/242 or 121/120) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]


Minimax tuning
== Didymus ==
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
[[Subgroup]]: 2.3.5.7
|0 0 0 1 0&gt;, |-1 0 -1/2 2 0&gt;]
Eigenmonzos: 2, 5, 7


Map: [&lt;1 0 -4 0 1|, &lt;0 1 4 0 -2|, &lt;0 0 0 1 2|]
[[Comma list]]: 81/80
EDOs: 5, 12, 26, 31, 88
Badness: 0.000536


=Calliope=
{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}
Commas: 81/80, 45/44


Minimax tuning
[[Optimal tuning]]s:
[|1 0 0 0 0&gt;, |1 0 0 0 1/6&gt;, |0 0 0 0 2/3&gt;,
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}, ~7/4 = 966.0272{{c}}
|1 -1 0 1 1/6&gt;, |0 0 0 0 1&gt;]
: [[error map]]: {{val| +1.391 -3.519 +1.868 -0.018 }}
Eigenmonzos: 2, 7/3, 11
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}, ~7/4 = 966.5654{{c}}
: error map: {{val| 0.000 -5.304 +0.291 -2.261 }}


Map: [&lt;1 0 -4 0 -6|, &lt;0 1 4 0 6|, &lt;0 0 0 1 0|]
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
EDOs: 7, 12, 19, 26, 45
Badness: 0.000530


=Erato=
[[Badness]] (Sintel): 0.419
Commas: 81/80, 126/125


Map: [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0], &lt;0 0 0 0 1|]
== Euterpe ==
EDOs: 12, 19, 31, 81
Euterpe tempers out 99/98 and is related to [[huygens]].  
Badness: 0.000558


=Clio=
[[Subgroup]]: 2.3.5.7.11
Commas: 81/80, 176/175


Minimax tuning
[[Comma list]]: 81/80, 99/98
||1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
|0 0 0 1 0&gt;, |-4 0 2 1 0&gt;]
Eigenmonzos: 2, 5, 7


Map: [&lt;1 0 -4 0 -12|, &lt;0 1 4 0 8|, &lt;0 0 0 1 1|]
{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}
EDOs: 7, 12, 24, 31, 105, 129
Badness: 0.000738


=Polyhymnia=
[[Optimal tuning]]s:
Commas: 81/80, 385/384
* [[WE]]: ~2 = 1201.2743{{c}}, ~3/2 = 696.9375{{c}}, ~7/4 = 969.4564{{c}}
: [[error map]]: {{val| +1.274 -3.743 +1.436 +3.179 -2.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.5797{{c}}, ~7/4 = 969.7942{{c}}
: error map: {{val| 0.000 -5.375 +0.005 +0.968 -4.889 }}


=Thalia=
[[Minimax tuning]]:
Commas: 81/80, 55/54
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7


=Melpomene=
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88, 119be, 150be }}
Commas: 81/80, 56/55


=Urania=
[[Badness]] (Sintel): 0.644
Commas: 81/80, 121/120


Lattice basis: 11/9 length 0.2536 8/7 length 2.807
== Clio ==
Angle(11/9, 8/7) = 90 degrees
Clio tempers out 176/175. It is another temperament related to huygens.  
Map to lattice: [&lt;0 2 8 0 5|, &lt;0 0 0 -1 0|]


Map: [&lt;1 1 0 0 2|, &lt;0 2 8 0 5|, &lt;0 0 0 1 0|]
[[Subgroup]]: 2.3.5.7.11
EDOs: 7, 14c, 24, 31, 100de, 131bde, 162bde
Badness: 0.000842
[[Spectrum of a temperament|Spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
Scale: [[urania24]]


=Terpsichore=
[[Comma list]]: 81/80, 176/175
Commas: 81/80, 540/539


Map: [&lt;1 0 -4 0 -2|, &lt;0 1 4 0 7|, &lt;0 0 0 1 -2|]
{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}
EDOs: 19, 31, 81, 112b
 
Badness: 0.000850
[[Optimal tuning]]s:
Spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11</pre></div>
* [[WE]]: ~2 = 1200.6376{{c}}, ~3/2 = 697.6207{{c}}, ~7/4 = 969.1442{{c}}
<h4>Original HTML content:</h4>
: [[error map]]: {{val| +0.638 -3.697 +4.169 +1.593 -2.483 }}
<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;Didymus rank three family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3350{{c}}, ~7/4 = 969.0152{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -4.620 +3.026 +0.189 -3.622 }}
&lt;!-- ws:start:WikiTextTocRule:18:&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:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#Euterpe"&gt;Euterpe&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Calliope"&gt;Calliope&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Erato"&gt;Erato&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Clio"&gt;Clio&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Polyhymnia"&gt;Polyhymnia&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Thalia"&gt;Thalia&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Melpomene"&gt;Melpomene&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Urania"&gt;Urania&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Terpsichore"&gt;Terpsichore&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;br /&gt;
[[Minimax tuning]]:  
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Euterpe"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Euterpe&lt;/h1&gt;
* [[11-odd-limit]]
Commas: 81/80, 99/98&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
&lt;br /&gt;
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7
Minimax tuning&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |1 0 1/4 0 0&amp;gt;, |0 0 1 0 0&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129, 136b }}
|0 0 0 1 0&amp;gt;, |-1 0 -1/2 2 0&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 5, 7&lt;br /&gt;
[[Badness]] (Sintel): 0.886
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -4 0 1|, &amp;lt;0 1 4 0 -2|, &amp;lt;0 0 0 1 2|]&lt;br /&gt;
== Terpsichore ==
EDOs: 5, 12, 26, 31, 88&lt;br /&gt;
Terpsichore tempers out 540/539 and is related to [[meanpop]].
Badness: 0.000536&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Calliope"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Calliope&lt;/h1&gt;
 
Commas: 81/80, 45/44&lt;br /&gt;
[[Comma list]]: 81/80, 540/539
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}
[|1 0 0 0 0&amp;gt;, |1 0 0 0 1/6&amp;gt;, |0 0 0 0 2/3&amp;gt;, &lt;br /&gt;
 
|1 -1 0 1 1/6&amp;gt;, |0 0 0 0 1&amp;gt;]&lt;br /&gt;
[[Optimal tuning]]s:  
Eigenmonzos: 2, 7/3, 11&lt;br /&gt;
* [[WE]]: ~2 = 1201.5055{{c}}, ~3/2 = 697.1093{{c}}, ~7/4 = 965.2101{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +1.506 -3.340 +2.123 -0.605 -0.468 }}
Map: [&amp;lt;1 0 -4 0 -6|, &amp;lt;0 1 4 0 6|, &amp;lt;0 0 0 1 0|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.7166{{c}}, ~7/4 = 964.9479{{c}}
EDOs: 7, 12, 19, 26, 45&lt;br /&gt;
: error map: {{val| 0.000 -5.238 +0.553 -3.878 -4.197 }}
Badness: 0.000530&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12e, 14c, 17c, 19, 31, 81, 112b }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Erato"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Erato&lt;/h1&gt;
 
Commas: 81/80, 126/125&lt;br /&gt;
[[Badness]] (Sintel): 1.02
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -4 -13 0|, &amp;lt;0 1 4 10 0], &amp;lt;0 0 0 0 1|]&lt;br /&gt;
[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
EDOs: 12, 19, 31, 81&lt;br /&gt;
 
Badness: 0.000558&lt;br /&gt;
== Polyhymnia ==
&lt;br /&gt;
Polyhymnia tempers out 385/384. It is another temperament related to meanpop.
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Clio"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Clio&lt;/h1&gt;
 
Commas: 81/80, 176/175&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
[[Comma list]]: 81/80, 385/384
||1 0 0 0 0&amp;gt;, |1 0 1/4 0 0&amp;gt;, |0 0 1 0 0&amp;gt;, &lt;br /&gt;
 
|0 0 0 1 0&amp;gt;, |-4 0 2 1 0&amp;gt;]&lt;br /&gt;
{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}
Eigenmonzos: 2, 5, 7&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
Map: [&amp;lt;1 0 -4 0 -12|, &amp;lt;0 1 4 0 8|, &amp;lt;0 0 0 1 1|]&lt;br /&gt;
* [[WE]]: ~2 = 1201.4003{{c}}, ~3/2 = 697.0429{{c}}, ~7/4 = 965.9954{{c}}
EDOs: 7, 12, 24, 31, 105, 129&lt;br /&gt;
: [[error map]]: {{val| +1.400 -3.512 +1.858 -0.030 -0.040 }}
Badness: 0.000738&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.3248{{c}}, ~7/4 = 964.8389{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -5.630 -1.014 -3.987 -5.131 }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Polyhymnia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Polyhymnia&lt;/h1&gt;
 
Commas: 81/80, 385/384&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31, 81, 112b }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Thalia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Thalia&lt;/h1&gt;
[[Badness]] (Sintel): 1.07
Commas: 81/80, 55/54&lt;br /&gt;
 
&lt;br /&gt;
Scales: [[polyhymnia12]]
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Melpomene"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Melpomene&lt;/h1&gt;
 
Commas: 81/80, 56/55&lt;br /&gt;
== Calliope ==
&lt;br /&gt;
Calliope tempers out 45/44 and is related to [[flattone]].
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Urania"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Urania&lt;/h1&gt;
 
Commas: 81/80, 121/120&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
Lattice basis: 11/9 length 0.2536 8/7 length 2.807&lt;br /&gt;
[[Comma list]]: 45/44, 81/80
Angle(11/9, 8/7) = 90 degrees&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 2 8 0 5|, &amp;lt;0 0 0 -1 0|]&lt;br /&gt;
{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 0 0 2|, &amp;lt;0 2 8 0 5|, &amp;lt;0 0 0 1 0|]&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 7, 14c, 24, 31, 100de, 131bde, 162bde&lt;br /&gt;
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}, ~7/4 = 964.5866{{c}}
Badness: 0.000842&lt;br /&gt;
: [[error map]]: {{val| +2.062 -5.348 -8.135 -0.115 +15.951 }}
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}, ~7/4 = 965.3621{{c}}
Scale: &lt;a class="wiki_link" href="/urania24"&gt;urania24&lt;/a&gt;&lt;br /&gt;
: error map: {{val| 0.000 -8.047 -10.680 -3.464 +12.133 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Terpsichore"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Terpsichore&lt;/h1&gt;
[[Minimax tuning]]:
Commas: 81/80, 540/539&lt;br /&gt;
* [[11-odd-limit]]
&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
Map: [&amp;lt;1 0 -4 0 -2|, &amp;lt;0 1 4 0 7|, &amp;lt;0 0 0 1 -2|]&lt;br /&gt;
: [[eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3.11
EDOs: 19, 31, 81, 112b&lt;br /&gt;
 
Badness: 0.000850&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
Spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Badness]] (Sintel): 0.637
 
== Thalia ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 33/32, 55/54
 
{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}, ~7/4 = 955.4695{{c}}
: [[error map]]: {{val| +6.583 +0.504 -2.809 -0.190 -20.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.1206{{c}}, ~7/4 = 955.1393{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -13.687 -44.439 }}
 
{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e, 33cdee, 52cdeee }}
 
[[Badness]] (Sintel): 0.898
 
== Melpomene ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 81/80
 
{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.7942{{c}}, ~3/2 = 698.5204{{c}}, ~7/4 = 963.2674{{c}}
: [[error map]]: {{val| -1.206 -4.640 +7.768 -7.970 +11.839 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.9964{{c}}, ~7/4 = 962.5723{{c}}
: error map: {{val| 0.000 -2.959 +9.672 -6.254 +15.269 }}
 
{{Optimal ET sequence|legend=1| 12, 17c, 19, 24, 31e, 36 }}
 
[[Badness]] (Sintel): 0.929
 
== Erato ==
Erato uses the same mapping for the 7-limit as septimal meantone, but adds an independent generator for prime 11.  
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 81/80, 126/125
 
{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}, ~11/8 = 547.5886{{c}}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 -0.022 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}, ~11/8 = 548.5396{{c}}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 -2.778 }}
 
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81, 112b }}
 
[[Badness]] (Sintel): 0.670
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 81/80, 105/104, 126/125
 
Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}
 
Optimal tunings:  
* WE: ~2 = 1201.6213{{c}}, ~3/2 = 696.9287{{c}}, ~11/8 = 546.4190{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1573{{c}}, ~11/8 = 547.5865{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 19, 31, 50, 69de, 81, 119bdde }}
 
Badness (Sintel): 0.823
 
== References ==
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Didymus rank-3 family| ]] <!-- main article -->
[[Category:Didymus| ]] <!-- key article -->
[[Category:Rank 3]]

Latest revision as of 13:50, 27 August 2025

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 didymus rank-3 family of rank-3 temperaments tempers out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit rank-3 temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Most of these temperaments were named by Gene Ward Smith in 2010[1].

Temperaments discussed elsewhere include:

Didymus

Subgroup: 2.3.5.7

Comma list: 81/80

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

Optimal tunings:

  • WE: ~2 = 1201.3906 ¢, ~3/2 = 697.0455 ¢, ~7/4 = 966.0272 ¢
error map: ⟨+1.391 -3.519 +1.868 -0.018]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6512 ¢, ~7/4 = 966.5654 ¢
error map: ⟨0.000 -5.304 +0.291 -2.261]

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness (Sintel): 0.419

Euterpe

Euterpe tempers out 99/98 and is related to huygens.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98

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

Optimal tunings:

  • WE: ~2 = 1201.2743 ¢, ~3/2 = 696.9375 ¢, ~7/4 = 969.4564 ¢
error map: ⟨+1.274 -3.743 +1.436 +3.179 -2.457]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.5797 ¢, ~7/4 = 969.7942 ¢
error map: ⟨0.000 -5.375 +0.005 +0.968 -4.889]

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-1 0 -1/2 2 0⟩]
eigenmonzo (unchanged-interval) basis: 2.5.7

Optimal ET sequence: 12, 17c, 19e, 26, 31, 88, 119be, 150be

Badness (Sintel): 0.644

Clio

Clio tempers out 176/175. It is another temperament related to huygens.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175

Mapping: [⟨1 0 -4 0 -12], ⟨0 1 4 0 8], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.6376 ¢, ~3/2 = 697.6207 ¢, ~7/4 = 969.1442 ¢
error map: ⟨+0.638 -3.697 +4.169 +1.593 -2.483]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3350 ¢, ~7/4 = 969.0152 ¢
error map: ⟨0.000 -4.620 +3.026 +0.189 -3.622]

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-4 0 2 1 0⟩]
eigenmonzo (unchanged-interval) basis: 2.5.7

Optimal ET sequence: 7, 12, 19e, 24, 31, 105, 129, 136b

Badness (Sintel): 0.886

Terpsichore

Terpsichore tempers out 540/539 and is related to meanpop.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 540/539

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

Optimal tunings:

  • WE: ~2 = 1201.5055 ¢, ~3/2 = 697.1093 ¢, ~7/4 = 965.2101 ¢
error map: ⟨+1.506 -3.340 +2.123 -0.605 -0.468]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.7166 ¢, ~7/4 = 964.9479 ¢
error map: ⟨0.000 -5.238 +0.553 -3.878 -4.197]

Optimal ET sequence: 12e, 14c, 17c, 19, 31, 81, 112b

Badness (Sintel): 1.02

Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11

Polyhymnia

Polyhymnia tempers out 385/384. It is another temperament related to meanpop.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384

Mapping: [⟨1 0 -4 0 11], ⟨0 1 4 0 -3], ⟨0 0 0 1 -1]]

Optimal tunings:

  • WE: ~2 = 1201.4003 ¢, ~3/2 = 697.0429 ¢, ~7/4 = 965.9954 ¢
error map: ⟨+1.400 -3.512 +1.858 -0.030 -0.040]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.3248 ¢, ~7/4 = 964.8389 ¢
error map: ⟨0.000 -5.630 -1.014 -3.987 -5.131]

Optimal ET sequence: 7, 12e, 19, 24, 26, 31, 81, 112b

Badness (Sintel): 1.07

Scales: polyhymnia12

Calliope

Calliope tempers out 45/44 and is related to flattone.

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80

Mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6], ⟨0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1202.0621 ¢, ~3/2 = 694.5448 ¢, ~7/4 = 964.5866 ¢
error map: ⟨+2.062 -5.348 -8.135 -0.115 +15.951]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.9085 ¢, ~7/4 = 965.3621 ¢
error map: ⟨0.000 -8.047 -10.680 -3.464 +12.133]

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 0 0 1/6⟩, [0 0 0 0 2/3⟩, [1 -1 0 1 1/6⟩, [0 0 0 0 1⟩]
Eigenmonzo (unchanged-interval) basis: 2.7/3.11

Optimal ET sequence: 7d, 12, 19, 26, 45

Badness (Sintel): 0.637

Thalia

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54

Mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1], ⟨0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1206.5832 ¢, ~3/2 = 695.8763 ¢, ~7/4 = 955.4695 ¢
error map: ⟨+6.583 +0.504 -2.809 -0.190 -20.862]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 693.1206 ¢, ~7/4 = 955.1393 ¢
error map: ⟨0.000 -8.834 -13.831 -13.687 -44.439]

Optimal ET sequence: 5, 7, 12e, 14c, 19e, 33cdee, 52cdeee

Badness (Sintel): 0.898

Melpomene

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80

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

Optimal tunings:

  • WE: ~2 = 1198.7942 ¢, ~3/2 = 698.5204 ¢, ~7/4 = 963.2674 ¢
error map: ⟨-1.206 -4.640 +7.768 -7.970 +11.839]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 698.9964 ¢, ~7/4 = 962.5723 ¢
error map: ⟨0.000 -2.959 +9.672 -6.254 +15.269]

Optimal ET sequence: 12, 17c, 19, 24, 31e, 36

Badness (Sintel): 0.929

Erato

Erato uses the same mapping for the 7-limit as septimal meantone, but adds an independent generator for prime 11.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13 0], ⟨0 1 4 10 0], ⟨0 0 0 0 1]]

Optimal tunings:

  • WE: ~2 = 1201.2358 ¢, ~3/2 = 697.2122 ¢, ~11/8 = 547.5886 ¢
error map: ⟨+1.236 -3.507 +2.535 -0.412 -0.022]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.6562 ¢, ~11/8 = 548.5396 ¢
error map: ⟨0.000 -5.299 +0.311 -2.264 -2.778]

Optimal ET sequence: 12, 19, 31, 50, 81, 112b

Badness (Sintel): 0.670

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125

Mapping: [⟨1 0 -4 -13 0 -20], ⟨0 1 4 10 0 15], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1201.6213 ¢, ~3/2 = 696.9287 ¢, ~11/8 = 546.4190 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 696.1573 ¢, ~11/8 = 547.5865 ¢

Optimal ET sequence: 12f, 19, 31, 50, 69de, 81, 119bdde

Badness (Sintel): 0.823

References

  1. ↑ Yahoo! Tuning Group | Higher-rank meantone extensions (muse temperaments?)
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