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]]s tempers out [[245/243]]. For a list of rank-2 temperaments, see [[Sensamagic clan]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-12-28 03:13:14 UTC</tt>.<br>
: The original revision id was <tt>288626852</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=
== Sensamagic ==
[[Comma]]: 245/243
{{Main| Sensamagic }}


[[7-limit]] [[minimax]]
[[Subgroup]]: 2.3.5.7
[|1 0 0 0&gt;, |0 0 1/5 2/5&gt;, |0 0 1 0&gt;, |0 0 0 1&gt;]
[[Eigenmonzo]]s: 2, 8/7, 5/4


[[9-limit]] minimax
[[Comma list]]: 245/243
[|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
{{Mapping|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}
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|]
: mapping generators: ~2, ~3, ~9/7
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]]
[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
[[Badness]]: 0.000129
 
Lattice basis:
: 3/2 length = 0.9644, 9/7 length = 1.0807
: Angle (3/2, 9/7) = 86.5288°
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020
 
[[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]]: 0.129 × 10<sup>-3</sup>


[[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 ===
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
 
== Undecimal sensamagic ==
{{Main| Sensamagic }}
 
[[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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178
 
[[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, 607bd, 694bd }}
 
[[Badness]]: 0.722 × 10<sup>-3</sup>
 
[[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 ET sequence|legend=1| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}
 
Badness: 1.20 × 10<sup>-3</sup>
 
== 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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359
 
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}
 
[[Badness]]: 0.796 × 10<sup>-3</sup>
 
=== 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 ET sequence|legend=1| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
 
Badness: 0.928 × 10<sup>-3</sup>
 
== Octarod ==
[[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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050
 
{{Optimal ET sequence|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
 
[[Badness]]: 0.581 × 10<sup>-3</sup>
 
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]
 
== Shrusus ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 176/175, 245/243
 
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147
 
{{Optimal ET sequence|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
 
[[Badness]]: 0.877 × 10<sup>-3</sup>
 
=== Shrusic ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 176/175, 245/243
 
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}
 
{{Optimal ET sequence|legend=1| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
 
Badness: 1.125 × 10<sup>-3</sup>
 
== Bisector ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 121/120, 245/243
 
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}
 
: mapping generators: ~77/54, ~3, ~9/7
 
[[Optimal tuning]] ([[POTE]]): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060


12: 729/686, 64/63
{{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
17: 64/63, 19683/19208
19: 49/48, 177147/175616
22: 64/63, 537824/531441
24: 64/63, 15059072/14348907


==Undecimal sensamagic==
[[Badness]]: 1.089 × 10<sup>-3</sup>
Commas: 245/243, 385/384


[[11-limit]] minimax
== Sensigh ==
[|1 0 0 0 0&gt;, |21/13 6/13 -1/13 1/13 -3/13&gt;,
Subgroup: 2.3.5.7.11.13
|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|]
Comma list: 91/90, 126/125, 169/168
Generators: 2, 3, 9/7
EDOs: [[5edo|5]], 17, 19, 22, 41, 68, 87, [[196edo|196]], [[283edo|283]], 607bd, 694bd
Badness: 0.000722


Projection pairs: 5 243/49 11 896/81 to 2.3.7
Mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 | 0 0 0 0 1 0 }}


==13-limit==
: mapping generators: ~2, ~9/7, ~11
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|]
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df, 157df }}
Generators: 2, 3, 9/7
EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
Badness: 0.00120


=Sensawer=
Badness: 0.939 × 10<sup>-3</sup>
Commas: 245/243, 441/440


Map: [&lt;1 0 0 0 -3|, &lt;0 1 1 2 5|, &lt;0 0 2 -1 -4|]
=== 17-limit ===
EDOs: 41, 87, 302d, 389d, 476bd
Subgroup: 2.3.5.7.11.13.17
Badness: 0.000796


==13-limit==
Comma list: 91/90, 126/125, 154/153, 169/168
Commas: 196/195, 245/243, 352/351


Map: [&lt;1 0 0 0 -3 2|, &lt;0 1 1 2 5 2|, &lt;0 0 2 -1 -4 -4|]
Mapping: {{mapping| 1 6 8 11 0 10 0 | 0 -7 -9 -13 0 -10 1 | 0 0 0 0 1 0 1 }}
EDOs: 41, 46, 87, 522bd
Badness: 0.000928


=Octarod=
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
Commas: 100/99, 245/243


Map: [&lt;1 0 0 0 2|, &lt;0 1 1 2 0|, &lt;0 0 2 -1 4|]
[[Category:Temperament families]]
EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e
[[Category:Pages with mostly numerical content]]
Badness: 0.000581
[[Category:Sensamagic family| ]] <!-- main article -->
</pre></div>
[[Category:Sensamagic| ]] <!-- key article -->
<h4>Original HTML content:</h4>
[[Category:Rank 3]]
<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:14:&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:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sensamagic"&gt;Sensamagic&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&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:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&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:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Sensawer"&gt;Sensawer&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&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:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Octarod"&gt;Octarod&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:22 --&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;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 00:40, 24 June 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 sensamagic family of rank-3 temperaments tempers out 245/243. For a list of rank-2 temperaments, see Sensamagic clan.

Sensamagic

Main article: Sensamagic

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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020

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: 0.129 × 10-3

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

Temperaments discussed elsewhere include supernatural (→ Keemic family). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.

Undecimal sensamagic

Main article: Sensamagic

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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178

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, 607bd, 694bd

Badness: 0.722 × 10-3

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 ET sequence: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd

Badness: 1.20 × 10-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.1900, ~9/7 = 441.1359

Optimal ET sequence: 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd

Badness: 0.796 × 10-3

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

Badness: 0.928 × 10-3

Octarod

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 tuning (POTE): ~2 = 1\1, ~3/2 = 705.0464, ~9/7 = 439.5050

Optimal ET sequence: 8d, 14c, 19, 22, 27e, 41, 104, 131e

Badness: 0.581 × 10-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 706.3702, ~9/7 = 442.1147

Optimal ET sequence: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc

Badness: 0.877 × 10-3

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 ET sequence: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf

Badness: 1.125 × 10-3

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 tuning (POTE): ~77/54 = 1\2, ~3/2 = 703.0884, ~9/7 = 441.1060

Optimal ET sequence: 22, 46, 68, 82e, 106de, 114, 128e

Badness: 1.089 × 10-3

Sensigh

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168

Mapping: [⟨1 6 8 11 0 10], ⟨0 -7 -9 -13 0 -10], ⟨0 0 0 0 1 0]]

mapping generators: ~2, ~9/7, ~11

Optimal ET sequence: 19, 27, 46, 111df, 157df

Badness: 0.939 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168

Mapping: [⟨1 6 8 11 0 10 0], ⟨0 -7 -9 -13 0 -10 1], ⟨0 0 0 0 1 0 1]]

Optimal ET sequence: 19, 27, 46

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