Sensamagic 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 '''sensamagic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the sensamagic comma, [[245/243]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-17 03:29:34 UTC</tt>.<br>
: The original revision id was <tt>302677848</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]]


=Sensamagic=
For a list of rank-2 temperaments, see [[Sensamagic clan]].
[[Comma]]: 245/243


[[7-limit]] [[minimax]]
== Sensamagic ==
[|1 0 0 0&gt;, |0 0 1/5 2/5&gt;, |0 0 1 0&gt;, |0 0 0 1&gt;]
{{Main| Sensamagic }}
[[Eigenmonzo]]s: 2, 8/7, 5/4


[[9-limit]] minimax
Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].
[|1 0 0 0&gt;, |0 1 0 0&gt;, |0 5/3 2/3 -2/3&gt;, |0 5/3 -1/3 1/3&gt;]
Eigenmonzos: 2, 4/3, 7/5


Lattice basis: 3/2 0.9644 9/7 1.0807
Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.
Angle(3/2, 9/7) = 86.5288
Map to lattice: [&lt;0 1 1 2|, &lt;0 0 2 -1|]


Map: [&lt;1 0 0 0|, &lt;0 1 1 2|, &lt;0 0 2 -1|]
[[Subgroup]]: 2.3.5.7
Generators: 2, 3, 9/7
 
Edos: [[17edo|,17]] [[19edo|19]], [[22edo|22]], [[24edo|24]], [[27edo|27]], [[41edo|41]], [[46edo|46]], [[60edo|60]], [[68edo|68]], [[87edo|87]], [[109edo|109]], [[128edo|128]], [[169edo|169]], [[177edo|177]], [[237edo|237]]
[[Comma list]]: 245/243
[[Badness]]: 0.000129
 
{{Mapping|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}
: mapping generators: ~2, ~3, ~9/7
 
[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
 
Lattice basis:
: 3/2 length = 0.9644, 9/7 length = 1.0807
: Angle (3/2, 9/7) = 86.5288°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9983{{c}}, ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}}
: [[error map]]: {{val| -0.002 +1.785 -0.771 -2.248 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}
: error map: {{val| 0.000 +1.786 -0.769 -2.245 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
 
{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
 
[[Badness]] (Sintel): 0.570


[[Projection pair]]: 5 243/49 to 2.3.7
[[Projection pair]]: 5 243/49 to 2.3.7


===[[Minkowski blocks]]===
{{Databox|[[Minkowski block]]s|
{2, 3, 7} subgroup
2.3.7 subgroup
* 12: 729/686, 64/63
* 17: 64/63, 19683/19208
* 19: 49/48, 177147/175616
* 22: 64/63, 537824/531441
* 24: 64/63, 15059072/14348907
}}
 
=== Overview to extensions ===
The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.
 
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.
 
== Undecimal sensamagic ==
{{Main| Sensamagic }}
 
Undecimal sensamagic tempers out not only [[385/384]], but [[896/891]], making itself a [[strong extension]] of [[parapyth]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 245/243, 385/384
 
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9667{{c}}, ~3/2 = 703.7809{{c}}, ~9/7 = 440.9056{{c}}
: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}
: error map: {{val| 0.000 +1.840 -0.683 -2.154 +0.175 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
 
{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283 }}
 
[[Badness]] (Sintel): 0.868
 
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 245/243, 352/351, 364/363
 
Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9905{{c}}, ~3/2 = 703.7325{{c}}, ~9/7 = 440.9149{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7381{{c}}, ~9/7 = 440.9184{{c}}
 
{{Optimal ET sequence|legend=0| 17, 19f, 22, 41, 46, 63, 87, 237, 283 }}
 
Badness (Sintel): 1.12
 
== Sensawer ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 245/243, 441/440
 
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1654{{c}}, ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}}
: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2917{{c}}, ~9/7 = 441.1849{{c}}
: error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}
 
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87 }}
 
[[Badness]] (Sintel): 0.957
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 352/351
 
Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.9800{{c}}, ~3/2 = 703.4468{{c}}, ~9/7 = 441.3705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4494{{c}}, ~9/7 = 441.3758{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
 
Badness (Sintel): 0.868
 
== Octarod ==
Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 100/99, 245/243
 
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2854{{c}}, ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}}
: [[error map]]: {{val| -0.715 +1.957 -3.915 -0.245 +4.226 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5246{{c}}, ~9/7 = 439.2798{{c}}
: error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}
 
{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*
 
<nowiki/>*[[Optimal patent val]]: [[104edo|104]]
 
[[Badness]] (Sintel): 0.698
 
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]


12: 729/686, 64/63
== Shrusus ==
17: 64/63, 19683/19208
[[Subgroup]]: 2.3.5.7.11
19: 49/48, 177147/175616
22: 64/63, 537824/531441
24: 64/63, 15059072/14348907


==Undecimal sensamagic==
[[Comma list]]: 176/175, 245/243
Commas: 245/243, 385/384


[[11-limit]] minimax
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}
[|1 0 0 0 0&gt;, |21/13 6/13 -1/13 1/13 -3/13&gt;,
|35/13 10/13 7/13 -7/13 -5/13&gt;,
|35/13 10/13 -6/13 6/13 -5/13&gt;,
|42/13 -14/13 -2/13 2/13 7/13&gt;]
Eigenmonzos: 2, 11/9, 7/5


Map: [&lt;1 0 0 0 7|, &lt;0 1 1 2 -2|, &lt;0 0 2 -1 -1|]
[[Optimal tuning]]s:  
Generators: 2, 3, 9/7
* [[WE]]: ~2 = 1198.9114{{c}}, ~3/2 = 705.7294{{c}}, ~9/7 = 441.7137{{c}}
EDOs: [[5edo|5]], 17, 19, 22, 41, 68, 87, [[196edo|196]], [[283edo|283]], 607bd, 694bd
: [[error map]]: {{val| -1.089 +2.686 +1.754 -1.258 -3.259 }}
Badness: 0.000722
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.8402{{c}}, ~9/7 = 442.1064{{c}}
: error map: {{val| 0.000 +3.885 +3.739 +0.748 -1.638 }}


Projection pairs: 5 243/49 11 896/81 to 2.3.7
{{Optimal ET sequence|legend=1| 19e, 22, 27e, 46, 68, 95, 141bc, 163bc }}


==13-limit==
[[Badness]] (Sintel): 1.05
Commas: 245/243, 352/351, 385/384


Map: [&lt;1 0 0 0 7 12|, &lt;0 1 1 2 -2 -5|, &lt;0 0 2 -1 -1 -1|]
=== Shrusic ===
Generators: 2, 3, 9/7
Subgroup: 2.3.5.7.11.13
EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
Badness: 0.00120


=Sensawer=
Comma list: 91/90, 176/175, 245/243
Commas: 245/243, 441/440


Map: [&lt;1 0 0 0 -3|, &lt;0 1 1 2 5|, &lt;0 0 2 -1 -4|]
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}
EDOs: 41, 87, 302d, 389d, 476bd
Badness: 0.000796


==13-limit==
Optimal tunings:
Commas: 196/195, 245/243, 352/351
* WE: ~2 = 1199.7256{{c}}, ~3/2 = 704.9071{{c}}, ~9/7 = 443.1303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9572{{c}}, ~9/7 = 443.2018{{c}}


Map: [&lt;1 0 0 0 -3 2|, &lt;0 1 1 2 5 2|, &lt;0 0 2 -1 -4 -4|]
{{Optimal ET sequence|legend=0| 19e, 22, 27e, 46 }}
EDOs: 41, 46, 87, 522bd
Badness: 0.000928


=Octarod=
Badness (Sintel): 1.05
Commas: 100/99, 245/243


Map: [&lt;1 0 0 0 2|, &lt;0 1 1 2 0|, &lt;0 0 2 -1 4|]
== Bisector ==
EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e
[[Subgroup]]: 2.3.5.7.11
Badness: 0.000581


=Shrusus=
[[Comma list]]: 121/120, 245/243
Commas: 176/175, 245/243


Map: [&lt;1 0 0 0 -4|, &lt;0 1 1 2 4|, &lt;0 0 2 -1 3|]
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}
EDOs: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
: mapping generators: ~77/54, ~3, ~9/7
Badness: 0.000877


==Shrusic==
[[Optimal tuning]]s:
Commas: 91/90, 176/175, 245/243
* [[WE]]: ~2 = 600.3096{{c}}, ~3/2 = 703.4512{{c}}, ~9/7 = 441.3336{{c}}
: [[error map]]: {{val| +0.619 +2.115 +0.424 -2.019 -4.985 }}
* [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 703.5671{{c}}, ~9/7 = 441.2436{{c}}
: error map: {{val| 0.000 +1.612 -0.259 -2.935 -6.507 }}


Map: [&lt;1 0 0 0 -4 1|, &lt;0 1 1 2 4 1|, &lt;0 0 2 -1 3 3|]
{{Optimal ET sequence|legend=1| 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e }}
EDOs: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
Badness: 0.001125


=Sensigh=
[[Badness]] (Sintel): 1.31
Commas: 91/90, 126/125, 169/168


Map: [&lt;1 6 8 11 0 10|, &lt;0 -7 -9 -13 0 -10|, &lt;0 0 0 0 1 0|]
[[Category:Temperament families]]
EDOs: 19, 27, 46, 111df, 157df
[[Category:Sensamagic family| ]] <!-- main article -->
Badness: 0.000939
[[Category:Rank 3]]
</pre></div>
<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;Sensamagic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sensamagic"&gt;Sensamagic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Sensamagic--Minkowski blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sensamagic-Undecimal sensamagic"&gt;Undecimal sensamagic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sensamagic-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sensawer"&gt;Sensawer&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Sensawer-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Octarod"&gt;Octarod&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Shrusus"&gt;Shrusus&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Shrusus-Shrusic"&gt;Shrusic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sensigh"&gt;Sensigh&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Sensamagic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Sensamagic&lt;/h1&gt;
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;: 245/243&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; &lt;a class="wiki_link" href="/minimax"&gt;minimax&lt;/a&gt;&lt;br /&gt;
[|1 0 0 0&amp;gt;, |0 0 1/5 2/5&amp;gt;, |0 0 1 0&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzo&lt;/a&gt;s: 2, 8/7, 5/4&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/9-limit"&gt;9-limit&lt;/a&gt; minimax&lt;br /&gt;
[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |0 5/3 2/3 -2/3&amp;gt;, |0 5/3 -1/3 1/3&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 4/3, 7/5 &lt;br /&gt;
&lt;br /&gt;
Lattice basis: 3/2 0.9644 9/7 1.0807&lt;br /&gt;
Angle(3/2, 9/7) = 86.5288&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 1 2|, &amp;lt;0 0 2 -1|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0|, &amp;lt;0 1 1 2|, &amp;lt;0 0 2 -1|]&lt;br /&gt;
Generators: 2, 3, 9/7&lt;br /&gt;
Edos: &lt;a class="wiki_link" href="/17edo"&gt;,17&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="/24edo"&gt;24&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&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="/60edo"&gt;60&lt;/a&gt;, &lt;a class="wiki_link" href="/68edo"&gt;68&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87&lt;/a&gt;, &lt;a class="wiki_link" href="/109edo"&gt;109&lt;/a&gt;, &lt;a class="wiki_link" href="/128edo"&gt;128&lt;/a&gt;, &lt;a class="wiki_link" href="/169edo"&gt;169&lt;/a&gt;, &lt;a class="wiki_link" href="/177edo"&gt;177&lt;/a&gt;, &lt;a class="wiki_link" href="/237edo"&gt;237&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.000129&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Projection%20pair"&gt;Projection pair&lt;/a&gt;: 5 243/49 to 2.3.7&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="Sensamagic--Minkowski blocks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/Minkowski%20blocks"&gt;Minkowski blocks&lt;/a&gt;&lt;/h3&gt;
{2, 3, 7} subgroup&lt;br /&gt;
&lt;br /&gt;
12: 729/686, 64/63&lt;br /&gt;
17: 64/63, 19683/19208&lt;br /&gt;
19: 49/48, 177147/175616&lt;br /&gt;
22: 64/63, 537824/531441&lt;br /&gt;
24: 64/63, 15059072/14348907&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Sensamagic-Undecimal sensamagic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Undecimal sensamagic&lt;/h2&gt;
Commas: 245/243, 385/384&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; minimax&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |21/13 6/13 -1/13 1/13 -3/13&amp;gt;, &lt;br /&gt;
|35/13 10/13 7/13 -7/13 -5/13&amp;gt;, &lt;br /&gt;
|35/13 10/13 -6/13 6/13 -5/13&amp;gt;, &lt;br /&gt;
|42/13 -14/13 -2/13 2/13 7/13&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 11/9, 7/5&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 7|, &amp;lt;0 1 1 2 -2|, &amp;lt;0 0 2 -1 -1|]&lt;br /&gt;
Generators: 2, 3, 9/7&lt;br /&gt;
EDOs: &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, 17, 19, 22, 41, 68, 87, &lt;a class="wiki_link" href="/196edo"&gt;196&lt;/a&gt;, &lt;a class="wiki_link" href="/283edo"&gt;283&lt;/a&gt;, 607bd, 694bd&lt;br /&gt;
Badness: 0.000722&lt;br /&gt;
&lt;br /&gt;
Projection pairs: 5 243/49 11 896/81 to 2.3.7&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Sensamagic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Commas: 245/243, 352/351, 385/384&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 7 12|, &amp;lt;0 1 1 2 -2 -5|, &amp;lt;0 0 2 -1 -1 -1|]&lt;br /&gt;
Generators: 2, 3, 9/7&lt;br /&gt;
EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd&lt;br /&gt;
Badness: 0.00120&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Sensawer"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Sensawer&lt;/h1&gt;
Commas: 245/243, 441/440&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 -3|, &amp;lt;0 1 1 2 5|, &amp;lt;0 0 2 -1 -4|]&lt;br /&gt;
EDOs: 41, 87, 302d, 389d, 476bd&lt;br /&gt;
Badness: 0.000796&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Sensawer-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
Commas: 196/195, 245/243, 352/351&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 -3 2|, &amp;lt;0 1 1 2 5 2|, &amp;lt;0 0 2 -1 -4 -4|]&lt;br /&gt;
EDOs: 41, 46, 87, 522bd&lt;br /&gt;
Badness: 0.000928&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Octarod"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Octarod&lt;/h1&gt;
Commas: 100/99, 245/243&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 2|, &amp;lt;0 1 1 2 0|, &amp;lt;0 0 2 -1 4|]&lt;br /&gt;
EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e &lt;br /&gt;
Badness: 0.000581&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Shrusus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Shrusus&lt;/h1&gt;
Commas: 176/175, 245/243&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 -4|, &amp;lt;0 1 1 2 4|, &amp;lt;0 0 2 -1 3|]&lt;br /&gt;
EDOs: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc&lt;br /&gt;
Badness: 0.000877&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Shrusus-Shrusic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Shrusic&lt;/h2&gt;
Commas: 91/90, 176/175, 245/243&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 0 0 -4 1|, &amp;lt;0 1 1 2 4 1|, &amp;lt;0 0 2 -1 3 3|]&lt;br /&gt;
EDOs: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf&lt;br /&gt;
Badness: 0.001125&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Sensigh"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Sensigh&lt;/h1&gt;
Commas: 91/90, 126/125, 169/168&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 6 8 11 0 10|, &amp;lt;0 -7 -9 -13 0 -10|, &amp;lt;0 0 0 0 1 0|]&lt;br /&gt;
EDOs: 19, 27, 46, 111df, 157df&lt;br /&gt;
Badness: 0.000939&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 10:14, 11 April 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 sensamagic family of rank-3 temperaments tempers out the sensamagic comma, 245/243.

For a list of rank-2 temperaments, see Sensamagic clan.

Sensamagic

Main article: Sensamagic

Sensamagic is generated by a perfect fifth and a wide supermajor third of ~9/7, two of which make ~5/3. Among the good edo tunings are 87edo and 128edo, as well as the optimal patent val 283edo.

Another notable tuning is given by TE, CTE and POTE, all coinciding at 703.7424 ¢, 440.9020 ¢ with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from CWE is practically unnoticeable.

Subgroup: 2.3.5.7

Comma list: 245/243

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

mapping generators: ~2, ~3, ~9/7

Mapping to lattice: [⟨0 1 1 2], ⟨0 0 2 -1]]

Lattice basis:

3/2 length = 0.9644, 9/7 length = 1.0807
Angle (3/2, 9/7) = 86.5288°

Optimal tunings:

  • WE: ~2 = 1199.9983 ¢, ~3/2 = 703.7414 ¢, ~9/7 = 440.9014 ¢
error map: ⟨-0.002 +1.785 -0.771 -2.248]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7411 ¢, ~9/7 = 440.9017 ¢
error map: ⟨0.000 +1.786 -0.769 -2.245]

Minimax tuning:

[[1 0 0 0⟩, [0 0 1/5 2/5⟩, [0 0 1 0⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7
[[1 0 0 0⟩, [0 1 0 0⟩, [0 5/3 2/3 -2/3⟩, [0 5/3 -1/3 1/3⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283

Badness (Sintel): 0.570

Projection pair: 5 243/49 to 2.3.7

Minkowski blocks

2.3.7 subgroup

  • 12: 729/686, 64/63
  • 17: 64/63, 19683/19208
  • 19: 49/48, 177147/175616
  • 22: 64/63, 537824/531441
  • 24: 64/63, 15059072/14348907

Overview to extensions

The second comma in the comma list defines which 11-limit family member we are looking at. Undecimal sensamagic adds 385/384, sensawer adds 441/440, octarod adds 100/99, shrusus adds 176/175. These temperaments use the same generators as sensamagic. Bisector adds 121/120 with a half-octave period.

Temperaments discussed elsewhere include supernatural (→ Keemic family) and sensigh (→ Sengic family). The rest are considered below.

Undecimal sensamagic

Main article: Sensamagic

Undecimal sensamagic tempers out not only 385/384, but 896/891, making itself a strong extension of parapyth.

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384

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

Optimal tunings:

  • WE: ~2 = 1199.9667 ¢, ~3/2 = 703.7809 ¢, ~9/7 = 440.9056 ¢
error map: ⟨-0.033 +1.793 -0.755 -2.236 +0.048]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7948 ¢, ~9/7 = 440.9180 ¢
error map: ⟨0.000 +1.840 -0.683 -2.154 +0.175]

Minimax tuning:

[[1 0 0 0 0⟩, [21/13 6/13 -1/13 1/13 -3/13⟩, [35/13 10/13 7/13 -7/13 -5/13⟩, [35/13 10/13 -6/13 6/13 -5/13⟩, [42/13 -14/13 -2/13 2/13 7/13⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/9

Optimal ET sequence: 17, 19, 22, 41, 68, 87, 196, 283

Badness (Sintel): 0.868

Projection pairs: 5 243/49 11 896/81 to 2.3.7

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 1 2 -2 -5], ⟨0 0 2 -1 -1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9905 ¢, ~3/2 = 703.7325 ¢, ~9/7 = 440.9149 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.7381 ¢, ~9/7 = 440.9184 ¢

Optimal ET sequence: 17, 19f, 22, 41, 46, 63, 87, 237, 283

Badness (Sintel): 1.12

Sensawer

Subgroup: 2.3.5.7.11

Comma list: 245/243, 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 1 2 5], ⟨0 0 2 -1 -4]]

Optimal tunings:

  • WE: ~2 = 1200.1654 ¢, ~3/2 = 703.2870 ¢, ~9/7 = 441.1967 ¢
error map: ⟨-0.033 +1.793 -0.755 -2.236 +0.048]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2917 ¢, ~9/7 = 441.1849 ¢
error map: ⟨0.000 +1.840 -0.683 -2.1554 +0.175]

Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87

Badness (Sintel): 0.957

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 1 2 5 2], ⟨0 0 2 -1 -4 -4]]

Optimal tunings:

  • WE: ~2 = 1199.9800 ¢, ~3/2 = 703.4468 ¢, ~9/7 = 441.3705 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4494 ¢, ~9/7 = 441.3758 ¢

Optimal ET sequence: 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd

Badness (Sintel): 0.868

Octarod

Octarod tempers out 100/99 and the interval class of 11 is found as a stack of four ~9/7's. The name octarod was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from octacot and rodan; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243

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

Optimal tunings:

  • WE: ~2 = 1199.2854 ¢, ~3/2 = 704.6266 ¢, ~9/7 = 439.2433 ¢
error map: ⟨-0.715 +1.957 -3.915 -0.245 +4.226]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5246 ¢, ~9/7 = 439.2798 ¢
error map: ⟨0.000 +2.570 -3.230 +0.944 +5.801]

Optimal ET sequence: 14c, 19, 22, 27e, 41, 90e, 131e*

*Optimal patent val: 104

Badness (Sintel): 0.698

Scales: octarod1, octarod2, octarod3, octarod4, octarod5

Shrusus

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243

Mapping: [⟨1 0 0 0 -4], ⟨0 1 1 2 4], ⟨0 0 2 -1 3]]

Optimal tunings:

  • WE: ~2 = 1198.9114 ¢, ~3/2 = 705.7294 ¢, ~9/7 = 441.7137 ¢
error map: ⟨-1.089 +2.686 +1.754 -1.258 -3.259]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.8402 ¢, ~9/7 = 442.1064 ¢
error map: ⟨0.000 +3.885 +3.739 +0.748 -1.638]

Optimal ET sequence: 19e, 22, 27e, 46, 68, 95, 141bc, 163bc

Badness (Sintel): 1.05

Shrusic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243

Mapping: [⟨1 0 0 0 -4 1], ⟨0 1 1 2 4 1], ⟨0 0 2 -1 3 3]]

Optimal tunings:

  • WE: ~2 = 1199.7256 ¢, ~3/2 = 704.9071 ¢, ~9/7 = 443.1303 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9572 ¢, ~9/7 = 443.2018 ¢

Optimal ET sequence: 19e, 22, 27e, 46

Badness (Sintel): 1.05

Bisector

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243

Mapping: [⟨2 0 0 0 3], ⟨0 1 1 2 1], ⟨0 0 2 -1 1]]

mapping generators: ~77/54, ~3, ~9/7

Optimal tunings:

  • WE: ~2 = 600.3096 ¢, ~3/2 = 703.4512 ¢, ~9/7 = 441.3336 ¢
error map: ⟨+0.619 +2.115 +0.424 -2.019 -4.985]
  • CWE: ~2 = 600.0000 ¢, ~3/2 = 703.5671 ¢, ~9/7 = 441.2436 ¢
error map: ⟨0.000 +1.612 -0.259 -2.935 -6.507]

Optimal ET sequence: 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e

Badness (Sintel): 1.31

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