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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>2010-11-05 18:24:05 UTC</tt>.<br>
: The original revision id was <tt>176950455</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">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 }}
Eigenmonzos: 2, 8/7, 5/4


9-limit minimax
[[Subgroup]]: 2.3.5.7
[|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
[[Comma list]]: 245/243
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|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}
Generators: 2, 3, 9/7
Edos: 17, 19, 22, 24, 27, 41, 46, 60, 68, 87, 109, 128, 169, 177, 237


===[[Minkowski blocks]]===
: mapping generators: ~2, ~3, ~9/7
{2, 3, 7} subgroup


12: 729/686, 64/63
[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
17: 64/63, 19683/19208
19: 49/48, 177147/175616
22: 64/63, 537824/531441
24: 64/63, 15059072/14348907


==Unidecimal sensamagic==
Lattice basis:
Commas: 245/243, 385/384
: 3/2 length = 0.9644, 9/7 length = 1.0807
: Angle (3/2, 9/7) = 86.5288°


11-limit minimax
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.7424, ~9/7 = 440.9020
[|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|]
[[Minimax tuning]]:  
Generators: 2, 3, 9/7</pre></div>
* [[7-odd-limit]]
<h4>Original HTML content:</h4>
: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
<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;Comma: 245/243&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
&lt;br /&gt;
* [[9-odd-limit]]
7-limit minimax&lt;br /&gt;
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
[|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;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
Eigenmonzos: 2, 8/7, 5/4&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
9-limit 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;
[[Badness]]: 0.129 × 10<sup>-3</sup>
Eigenmonzos: 2, 4/3, 7/5 &lt;br /&gt;
 
&lt;br /&gt;
[[Projection pair]]: 5 243/49 to 2.3.7
Lattice basis: 3/2 0.9644 9/7 1.0807&lt;br /&gt;
 
Angle(3/2, 9/7) = 86.5288&lt;br /&gt;
{{Databox|[[Minkowski block]]s|
Map to lattice: [&amp;lt;0 1 1 2|, &amp;lt;0 0 2 -1|]&lt;br /&gt;
2.3.7 subgroup
&lt;br /&gt;
* 12: 729/686, 64/63
Map: [&amp;lt;1 0 0 0|, &amp;lt;0 1 1 2|, &amp;lt;0 0 2 -1|]&lt;br /&gt;
* 17: 64/63, 19683/19208
Generators: 2, 3, 9/7&lt;br /&gt;
* 19: 49/48, 177147/175616
Edos: 17, 19, 22, 24, 27, 41, 46, 60, 68, 87, 109, 128, 169, 177, 237&lt;br /&gt;
* 22: 64/63, 537824/531441
&lt;br /&gt;
* 24: 64/63, 15059072/14348907
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Minkowski blocks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&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;
=== Overview to extensions ===
12: 729/686, 64/63&lt;br /&gt;
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]). Considered below are undecimal sensamagic, sensawer, octarod, shrusus, bisector and sensigh.
17: 64/63, 19683/19208&lt;br /&gt;
 
19: 49/48, 177147/175616&lt;br /&gt;
== Undecimal sensamagic ==
22: 64/63, 537824/531441&lt;br /&gt;
{{Main| Sensamagic }}
24: 64/63, 15059072/14348907&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Unidecimal sensamagic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Unidecimal sensamagic&lt;/h2&gt;
 
Commas: 245/243, 385/384&lt;br /&gt;
[[Comma list]]: 245/243, 385/384
&lt;br /&gt;
 
11-limit minimax&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}
[|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;
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8004, ~9/7 = 440.9178
|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;
[[Minimax tuning]]:
Eigenmonzos: 2, 11/9, 7/5&lt;br /&gt;
* [[11-odd-limit]]
&lt;br /&gt;
: [{{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 }}]
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;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
Generators: 2, 3, 9/7&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{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
 
{{Optimal ET sequence|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
 
[[Badness]]: 1.089 × 10<sup>-3</sup>
 
== Sensigh ==
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168
 
Mapping: {{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|legend=1| 19, 27, 46, 111df, 157df }}
 
Badness: 0.939 × 10<sup>-3</sup>
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 126/125, 154/153, 169/168
 
Mapping: {{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|legend=1| 19, 27, 46 }}
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Sensamagic| ]] <!-- key article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Sensamagic_family"