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Didymus rank-3 family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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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>2011-02-18 00:32:17 UTC</tt>.<br>
: The original revision id was <tt>202996868</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.


==Euterpe==
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>.
Commas: 81/80, 99/98


Minimax tuning
Temperaments discussed elsewhere include:
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
* ''[[Urania]]'' (+243/242 or 121/120) → [[Rastmic rank-3 clan #Urania|Rastmic rank-3 clan]]
|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|]
== Didymus ==
EDOs: 31, 74, 88, 105
[[Subgroup]]: 2.3.5.7
Badness: 0.000536


==Calliope==
[[Comma list]]: 81/80
Commas: 81/80, 45/44


Minimax tuning
{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}
[|1 0 0 0 0&gt;, |1 0 0 0 1/6&gt;, |0 0 0 0 2/3&gt;,
|1 -1 0 1 1/6&gt;, |0 0 0 0 1&gt;]
Eigenmonzos: 2, 7/3, 11


Map: [&lt;1 0 -4 0 -6|, &lt;0 1 4 0 6|, &lt;0 0 0 1 0|]
[[Optimal tuning]]s:  
EDOs: 19, 26, 45
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}, ~7/4 = 966.0272{{c}}
Badness: 0.000530
: [[error map]]: {{val| +1.391 -3.519 +1.868 -0.018 }}
* [[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 }}


==Erato==
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}
Commas: 81/80, 126/125


Map: [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0], &lt;0 0 0 0 1|]
[[Badness]] (Sintel): 0.419
EDOs: 12, 19, 31, 81
Badness: 0.000558


==Clio==
== Euterpe ==
Commas: 81/80, 176/175
Euterpe tempers out 99/98 and is related to [[huygens]].


Minimax tuning
[[Subgroup]]: 2.3.5.7.11
||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|]
[[Comma list]]: 81/80, 99/98
EDOs: 31, 74, 98, 105, 129
Badness: 0.000738


==Polyhymnia==
{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}
Commas: 81/80, 385/384


==Thalia==
[[Optimal tuning]]s:
Commas: 81/80, 55/54
* [[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 }}


==Melpomene==
[[Minimax tuning]]:
Commas: 81/80, 56/55
* [[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


==Urania==
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88, 119be, 150be }}
Commas: 81/80, 121/120


==Terpsichore==
[[Badness]] (Sintel): 0.644
Commas: 81/80, 540/539</pre></div>
 
<h4>Original HTML content:</h4>
== Clio ==
<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;
Clio tempers out 176/175. It is another temperament related to huygens.
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Euterpe"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Euterpe&lt;/h2&gt;
[[Subgroup]]: 2.3.5.7.11
Commas: 81/80, 99/98&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 81/80, 176/175
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;
{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}
|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;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1200.6376{{c}}, ~3/2 = 697.6207{{c}}, ~7/4 = 969.1442{{c}}
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;
: [[error map]]: {{val| +0.638 -3.697 +4.169 +1.593 -2.483 }}
EDOs: 31, 74, 88, 105&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3350{{c}}, ~7/4 = 969.0152{{c}}
Badness: 0.000536&lt;br /&gt;
: error map: {{val| 0.000 -4.620 +3.026 +0.189 -3.622 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Calliope"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Calliope&lt;/h2&gt;
[[Minimax tuning]]:
Commas: 81/80, 45/44&lt;br /&gt;
* [[11-odd-limit]]
&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 }}]
Minimax tuning&lt;br /&gt;
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7
[|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 ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129, 136b }}
Eigenmonzos: 2, 7/3, 11&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 0.886
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;
 
EDOs: 19, 26, 45&lt;br /&gt;
== Terpsichore ==
Badness: 0.000530&lt;br /&gt;
Terpsichore tempers out 540/539 and is related to [[meanpop]].
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Erato"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Erato&lt;/h2&gt;
[[Subgroup]]: 2.3.5.7.11
Commas: 81/80, 126/125&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 81/80, 540/539
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;
 
EDOs: 12, 19, 31, 81&lt;br /&gt;
{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}
Badness: 0.000558&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Clio"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Clio&lt;/h2&gt;
* [[WE]]: ~2 = 1201.5055{{c}}, ~3/2 = 697.1093{{c}}, ~7/4 = 965.2101{{c}}
Commas: 81/80, 176/175&lt;br /&gt;
: [[error map]]: {{val| +1.506 -3.340 +2.123 -0.605 -0.468 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.7166{{c}}, ~7/4 = 964.9479{{c}}
Minimax tuning&lt;br /&gt;
: error map: {{val| 0.000 -5.238 +0.553 -3.878 -4.197 }}
||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;
{{Optimal ET sequence|legend=1| 12e, 14c, 17c, 19, 31, 81, 112b }}
Eigenmonzos: 2, 5, 7&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 1.02
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;
 
EDOs: 31, 74, 98, 105, 129&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
Badness: 0.000738&lt;br /&gt;
 
&lt;br /&gt;
== Polyhymnia ==
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Polyhymnia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Polyhymnia&lt;/h2&gt;
Polyhymnia tempers out 385/384. It is another temperament related to meanpop.
Commas: 81/80, 385/384&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x-Thalia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Thalia&lt;/h2&gt;
 
Commas: 81/80, 55/54&lt;br /&gt;
[[Comma list]]: 81/80, 385/384
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x-Melpomene"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Melpomene&lt;/h2&gt;
{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}
Commas: 81/80, 56/55&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Urania"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Urania&lt;/h2&gt;
* [[WE]]: ~2 = 1201.4003{{c}}, ~3/2 = 697.0429{{c}}, ~7/4 = 965.9954{{c}}
Commas: 81/80, 121/120&lt;br /&gt;
: [[error map]]: {{val| +1.400 -3.512 +1.858 -0.030 -0.040 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.3248{{c}}, ~7/4 = 964.8389{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x-Terpsichore"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Terpsichore&lt;/h2&gt;
: error map: {{val| 0.000 -5.630 -1.014 -3.987 -5.131 }}
Commas: 81/80, 540/539&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Optimal ET sequence|legend=1| 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|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}, ~7/4 = 964.5866{{c}}
: [[error map]]: {{val| +2.062 -5.348 -8.135 -0.115 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}, ~7/4 = 965.3621{{c}}
: error map: {{val| 0.000 -8.047 -10.680 -3.464 +12.133 }}
 
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [{{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 }}]
: [[eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3.11
 
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
 
[[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]]
Retrieved from "https://en.xen.wiki/w/Didymus_rank-3_family"