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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 clan''' tempers out the sensamagic comma, [[245/243]], a triprime [[comma]] with no factors of 2, {{val| 0 -5 1 2 }} to be exact. Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo]] is the [[optimal patent val]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-09 18:12:05 UTC</tt>.<br>
: The original revision id was <tt>332701132</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|flat]]


The //sensamagic clan// tempers out the [[sensamagic comma]], [[245_243|245/243]], a [[triprime comma]] with no factors of 2. |0 -5 1 2&gt; to be exact. There are a number of [[linear temperament]]s in the [[clan]] (magic, father, sensi, godzilla, superpyth, octacot, rodan, hedgehog, clyde, shrutar, sidi) but they've mostly been discussed elsewhere. Tempering out 245/243 alone leads to a [[rank three temperament]] for which [[283edo]] is the [[optimal patent val]].
== BPS ==
{{Main| BPS }}


=Bohpier=
BPS, for ''Bohlen–Pierce–Stearns'', is the 3.5.7-subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the ''lambda'' temperament, which was named after the [[4L 5s (tritave-equivalent)|lambda scale]].
[[Comma]]s: 245/243, 3125/3087


[[POTE tuning|POTE generator]]: ~27/25 = 146.474
[[Subgroup]]: 3.5.7


Map: [&lt;1 0 0 0|, &lt;0 13 19 23|]
[[Comma list]]: 245/243
[[Wedgie]]: &lt;&lt;13 19 23 0 0 0||
EDOs: [[41edo|41]], [[49edo|49]], [[90edo|90]], [[139edo|139]]
EDTs: [[13edt|13]]
[[Badness]]: 0.0682


==11-limit==
{{Mapping|legend=2| 1 1 2 | 0 -2 1 }}
Commas: 100/99, 245/243, 1344/1331


POTE generator: ~12/11 = 146.545
: sval mapping generators: ~3, ~9/7


Map: [&lt;1 0 0 0 0|, &lt;0 13 19 23 12|]
[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881
EDOs: 41, 90e, 131e
Badness: 0.0339


==13-limit==
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
Commas: 100/99, 144/143, 245/243, 275/273


POTE generator: ~12/11 = 146.603
=== Overview to extensions ===
The full 7-limit extensions' relation to BPS is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are sensi, cohemiripple, hedgehog, and fourfives.  


Map: [&lt;1 0 0 0 0 0|, &lt;0 13 19 23 12 14|]
These temperaments are distributed into different family pages.
EDOs: 41, 90ef, 131ef, 221bdeff
* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
Badness: 0.0249
* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]


==Music==
The others are weak extensions. Father tempers out [[16/15]], splitting the generator in two. Godzilla tempers out [[49/48]] with a hemitwelfth period. Sidi tempers out [[25/24]], splitting the generator in two with a hemitwelfth period. Clyde tempers out [[3136/3125]] with a 1/6-twelfth period. Superpyth tempers out [[64/63]], splitting the generator in six. Magic tempers out [[225/224]] with a 1/5-twelfth period. Octacot tempers out [[2401/2400]], splitting the generator in five. Hemiaug tempers out [[128/125]]. Pentacloud tempers out [[16807/16384]]. These split the generator in seven. Bamity tempers out [[64827/64000]], splitting the generator in nine. Rodan tempers out [[1029/1024]], splitting the generator in ten. Shrutar tempers out [[2048/2025]], splitting the generator in eleven. Finally, escaped tempers out [[65625/65536]], splitting the generator in sixteen.  
By [[Chris Vaisvil]]
http://micro.soonlabel.com/bophier/bophier-1.mp3
http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3


=Sensa=
Discussed elsewhere are
Commas: 245/243, 65625/65536
* [[Father]] (+16/15 or 28/27) → [[Father family #Father|Father family]]
* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[Clyde]]'' (+3136/3125) → [[Kleismic family #Clyde|Kleismic family]]
* [[Superpyth]] (+64/63) → [[Archytas clan #Superpyth|Archytas clan]]
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* ''[[Octacot]]'' (+2401/2400) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Hemiaug]]'' (+128/125) → [[Augmented family #Hemiaug|Augmented family]]
* ''[[Pentacloud]]'' (+16807/16384) → [[Quintile family #Pentacloud|Quintile family]]
* ''[[Bamity]]'' (+64827/64000) → [[Amity family #Bamity|Amity family]]
* [[Rodan]] (+1029/1024) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Shrutar]]'' (+2048/2025) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Escaped]]'' (+65625/65536) → [[Escapade family #Escaped|Escapade family]]


POTE generator: ~28/27 = 55.122
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].


Map: [&lt;1 2 2 4|, &lt;0 -9 7 -26|]
Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
Wedgie: &lt;&lt;9 -7 26 -32 16 80||
EDOs: [[22edo|22]], [[43edo|43]], [[65edo|65]], [[87edo|87]], [[109edo|109]], [[196edo|196]], [[283edo|283]]
Badness: 0.0887


==11-limit==  
== Bohpier ==
Commas: 245/243, 385/384, 4000/3993
{{Main| Bohpier }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''


POTE generator: ~28/27 = 55.126
Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].


Map: [&lt;1 2 2 4 3|, &lt;0 -9 7 -26 10|]
[[Subgroup]]: 2.3.5.7
EDOs: 22, 43, 65, 87, 109, 196, 283
Badness: 0.0358


==13-limit==
[[Comma list]]: 245/243, 3125/3087
Commas: 245/243, 352/351, 385/384, 625/624


POTE generator: ~28/27 = 55.138
{{Mapping|legend=1| 1 0 0 0 | 0 13 19 23 }}


Map: [&lt;1 2 2 4 3 2|, &lt;0 -9 7 -26 10 37|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 146.474
EDOs: 22, 43, 65, 87, 109, 196, 283
Badness: 0.0317


=Salsa=
[[Minimax tuning]]:
Commas: 245/243, 32805/32768
* [[7-odd-limit]]: ~27/25 = {{monzo| 0 0 1/19 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~27/25 = {{monzo| 0 1/13 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


POTE generator: ~128/105 = 351.049
{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}


Map: [&lt;1 1 7 -1|, &lt;0 2 -16 13|]
[[Badness]]: 0.068237
Wedgie: &lt;&lt;2 -16 13 -30 15 75||
EDOs: 17, 24, 41, 106d, 147d, 188cd, 335cd
Badness: 0.08015


==11-limit==
=== 11-limit ===
Commas: 243/242, 245/242, 385/384
Subgroup: 2.3.5.7.11


POTE generator: ~11/9 = 351.014
Comma list: 100/99, 245/243, 1344/1331


Map: [&lt;1 1 7 -1 2|, &lt;0 2 -16 13 5|]
Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}
EDOs: 17, 24, 41, 106d, 147d
Badness: 0.0394


=Pycnic=
Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545
The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has MOS of size 9, 11, 13, 15, 17... which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.


Commas: 245/243, 525/512
Minimax tuning:  
* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


POTE generator: ~45/32 = 567.720
{{Optimal ET sequence|legend=0| 41, 90e, 131e }}


Map: [&lt;1 3 -1 8|, &lt;0 -3 7 -11|]
Badness: 0.033949
Wedgie: &lt;&lt;3 -7 11 -18 9 45||
EDOs: 17, 19, 55c, 74cd, 93cd
Badness: 0.0737


=Cohemiripple=
==== 13-limit ====
Commas: 245/243, 1323/1250
Subgroup: 2.3.5.7.11.13


POTE generator: ~7/5 = 549.944
Comma list: 100/99, 144/143, 196/195, 275/273


Map: [&lt;1 7 11 12|, &lt;0 -10 -16 -17|]
Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}
Wedgie: &lt;&lt;10 16 17 2 -1 -5||
EDOs: 24
Badness: 0.1902


==11-limit==
Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603
Commas: 77/75, 243/242, 245/242


POTE generator: ~7/5 = 549.945
Minimax tuning:
* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
: Unchanged-interval (eigenmonzo) basis: 2.5


Map: [&lt;1 7 11 12 17|, &lt;0 -10 -16 -17 -25|]
{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}
EDOs: 24
Badness: 0.0827


==13-limit==
Badness: 0.024864
Commas: 66/65, 77/75, 147/143, 351/350


POTE generator: ~7/5 = 549.958
=== Triboh ===
Triboh is named after the "[[39edt|Triple Bohlen–Pierce scale]]", which divides each step of the [[13edt|equal-tempered]] [[Bohlen–Pierce]] scale into three equal parts.  


Map: [&lt;1 7 11 12 17 14|, &lt;0 -10 -16 -17 -25 -19|]
Subgroup: 2.3.5.7.11
EDOs: 24
Badness: 0.0499


Comma list: 245/243, 1331/1323, 3125/3087


Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}


Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828


</pre></div>
{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
<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 clan&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&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:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Bohpier"&gt;Bohpier&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt; | &lt;a href="#Sensa"&gt;Sensa&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Salsa"&gt;Salsa&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt; | &lt;a href="#Pycnic"&gt;Pycnic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#Cohemiripple"&gt;Cohemiripple&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;
Badness: 0.162592
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;br /&gt;
 
The &lt;em&gt;sensamagic clan&lt;/em&gt; tempers out the &lt;a class="wiki_link" href="/sensamagic%20comma"&gt;sensamagic comma&lt;/a&gt;, &lt;a class="wiki_link" href="/245_243"&gt;245/243&lt;/a&gt;, a &lt;a class="wiki_link" href="/triprime%20comma"&gt;triprime comma&lt;/a&gt; with no factors of 2. |0 -5 1 2&amp;gt; to be exact. There are a number of &lt;a class="wiki_link" href="/linear%20temperament"&gt;linear temperament&lt;/a&gt;s in the &lt;a class="wiki_link" href="/clan"&gt;clan&lt;/a&gt; (magic, father, sensi, godzilla, superpyth, octacot, rodan, hedgehog, clyde, shrutar, sidi) but they've mostly been discussed elsewhere. Tempering out 245/243 alone leads to a &lt;a class="wiki_link" href="/rank%20three%20temperament"&gt;rank three temperament&lt;/a&gt; for which &lt;a class="wiki_link" href="/283edo"&gt;283edo&lt;/a&gt; is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt;.&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Bohpier"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Bohpier&lt;/h1&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 245/243, 3125/3087&lt;br /&gt;
Comma list: 245/243, 275/273, 847/845, 1331/1323
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~27/25 = 146.474&lt;br /&gt;
Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 0|, &amp;lt;0 13 19 23|]&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822
&lt;a class="wiki_link" href="/Wedgie"&gt;Wedgie&lt;/a&gt;: &amp;lt;&amp;lt;13 19 23 0 0 0||&lt;br /&gt;
 
EDOs: &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49&lt;/a&gt;, &lt;a class="wiki_link" href="/90edo"&gt;90&lt;/a&gt;, &lt;a class="wiki_link" href="/139edo"&gt;139&lt;/a&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}
EDTs: &lt;a class="wiki_link" href="/13edt"&gt;13&lt;/a&gt;&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0682&lt;br /&gt;
Badness: 0.082158
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Bohpier-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
== Salsa ==
Commas: 100/99, 245/243, 1344/1331&lt;br /&gt;
{{See also| Schismatic family }}
&lt;br /&gt;
 
POTE generator: ~12/11 = 146.545&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 0 0|, &amp;lt;0 13 19 23 12|]&lt;br /&gt;
[[Comma list]]: 245/243, 32805/32768
EDOs: 41, 90e, 131e&lt;br /&gt;
 
Badness: 0.0339&lt;br /&gt;
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bohpier-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049
Commas: 100/99, 144/143, 245/243, 275/273&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
POTE generator: ~12/11 = 146.603&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]]: 0.080152
Map: [&amp;lt;1 0 0 0 0 0|, &amp;lt;0 13 19 23 12 14|]&lt;br /&gt;
 
EDOs: 41, 90ef, 131ef, 221bdeff&lt;br /&gt;
=== 11-limit ===
Badness: 0.0249&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bohpier-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Music&lt;/h2&gt;
Comma list: 243/242, 245/242, 385/384
By &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextUrlRule:175:http://micro.soonlabel.com/bophier/bophier-1.mp3 --&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/bophier/bophier-1.mp3" rel="nofollow"&gt;http://micro.soonlabel.com/bophier/bophier-1.mp3&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:175 --&gt;&lt;br /&gt;
Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
&lt;!-- ws:start:WikiTextUrlRule:176:http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3 --&gt;&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3" rel="nofollow"&gt;http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:176 --&gt;&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.014
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Sensa"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Sensa&lt;/h1&gt;
 
Commas: 245/243, 65625/65536&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 55.122&lt;br /&gt;
Badness: 0.039444
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 4|, &amp;lt;0 -9 7 -26|]&lt;br /&gt;
=== 13-limit ===
Wedgie: &amp;lt;&amp;lt;9 -7 26 -32 16 80||&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43&lt;/a&gt;, &lt;a class="wiki_link" href="/65edo"&gt;65&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="/196edo"&gt;196&lt;/a&gt;, &lt;a class="wiki_link" href="/283edo"&gt;283&lt;/a&gt;&lt;br /&gt;
 
Badness: 0.0887&lt;br /&gt;
Comma list: 105/104, 144/143, 243/242, 245/242
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Sensa-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
Commas: 245/243, 385/384, 4000/3993&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025
POTE generator: ~28/27 = 55.126&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}
Map: [&amp;lt;1 2 2 4 3|, &amp;lt;0 -9 7 -26 10|]&lt;br /&gt;
 
EDOs: 22, 43, 65, 87, 109, 196, 283&lt;br /&gt;
Badness: 0.030793
Badness: 0.0358&lt;br /&gt;
 
&lt;br /&gt;
== Pycnic ==
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Sensa-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
Commas: 245/243, 352/351, 385/384, 625/624&lt;br /&gt;
 
&lt;br /&gt;
The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has [[mos]] of size 9, 11, 13, 15, 17… which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.
POTE generator: ~28/27 = 55.138&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Map: [&amp;lt;1 2 2 4 3 2|, &amp;lt;0 -9 7 -26 10 37|]&lt;br /&gt;
 
EDOs: 22, 43, 65, 87, 109, 196, 283&lt;br /&gt;
[[Comma list]]: 245/243, 525/512
Badness: 0.0317&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 3 -1 8 | 0 -3 7 -11 }}
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Salsa"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Salsa&lt;/h1&gt;
 
Commas: 245/243, 32805/32768&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~45/32 = 567.720
&lt;br /&gt;
 
POTE generator: ~128/105 = 351.049&lt;br /&gt;
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 7 -1|, &amp;lt;0 2 -16 13|]&lt;br /&gt;
[[Badness]]: 0.073735
Wedgie: &amp;lt;&amp;lt;2 -16 13 -30 15 75||&lt;br /&gt;
 
EDOs: 17, 24, 41, 106d, 147d, 188cd, 335cd&lt;br /&gt;
== Superthird ==
Badness: 0.08015&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Salsa-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
[[Subgroup]]: 2.3.5.7
Commas: 243/242, 245/242, 385/384&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 245/243, 78125/76832
POTE generator: ~11/9 = 351.014&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
Map: [&amp;lt;1 1 7 -1 2|, &amp;lt;0 2 -16 13 5|]&lt;br /&gt;
 
EDOs: 17, 24, 41, 106d, 147d&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076
Badness: 0.0394&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Pycnic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Pycnic&lt;/h1&gt;
 
The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has MOS of size 9, 11, 13, 15, 17... which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.&lt;br /&gt;
[[Badness]]: 0.139379
&lt;br /&gt;
 
Commas: 245/243, 525/512&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~45/32 = 567.720&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 100/99, 245/243, 78125/76832
Map: [&amp;lt;1 3 -1 8|, &amp;lt;0 -3 7 -11|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 -7 11 -18 9 45||&lt;br /&gt;
Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}
EDOs: 17, 19, 55c, 74cd, 93cd&lt;br /&gt;
 
Badness: 0.0737&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Cohemiripple"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Cohemiripple&lt;/h1&gt;
{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}
Commas: 245/243, 1323/1250&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.070917
POTE generator: ~7/5 = 549.944&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 7 11 12|, &amp;lt;0 -10 -16 -17|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Wedgie: &amp;lt;&amp;lt;10 16 17 2 -1 -5||&lt;br /&gt;
 
EDOs: 24&lt;br /&gt;
Comma list: 100/99, 144/143, 196/195, 1375/1352
Badness: 0.1902&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Cohemiripple-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
 
Commas: 77/75, 243/242, 245/242&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119
&lt;br /&gt;
 
POTE generator: ~7/5 = 549.945&lt;br /&gt;
{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 7 11 12 17|, &amp;lt;0 -10 -16 -17 -25|]&lt;br /&gt;
Badness: 0.052835
EDOs: 24&lt;br /&gt;
 
Badness: 0.0827&lt;br /&gt;
== Superenneadecal ==
&lt;br /&gt;
Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Cohemiripple-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h2&gt;
 
Commas: 66/65, 77/75, 147/143, 351/350&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
POTE generator: ~7/5 = 549.958&lt;br /&gt;
[[Comma list]]: 245/243, 395136/390625
&lt;br /&gt;
 
Map: [&amp;lt;1 7 11 12 17 14|, &amp;lt;0 -10 -16 -17 -25 -19|]&lt;br /&gt;
{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
EDOs: 24&lt;br /&gt;
 
Badness: 0.0499&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166
 
{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
 
[[Badness]]: 0.132311
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 2560/2541, 3773/3750
 
Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
 
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667
 
{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
 
Badness: 0.101496
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 832/825, 1001/1000
 
Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
 
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801
 
{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e }}
 
Badness: 0.053197
 
== Magus ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
 
Magus temperament tempers out [[50331648/48828125]] (salegu) in the 5-limit. This temperament can be described as {{nowrap| 46 & 49 }} temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension [[starling temperaments #Amigo|amigo]] ({{nowrap|43 & 46}}) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.
 
Magus has a generator of a sharp ~5/4 (so that ~[[25/16]] is twice as sharp so that it makes sense to equate with [[11/7]] by tempering [[176/175]]), so that three reaches [[128/125]] short of the octave (where 128/125 is tuned narrow); this is significant because magus reaches [[3/2]] as ([[25/16]])/([[128/125]])<sup>3</sup>, that is, {{nowrap|2 + 3 × 3 {{=}} 11}} generators. Therefore, it implies that [[25/24]] is split into three [[128/125]]'s. Therefore, in the 5-limit, magus can be thought of as a higher-complexity and sharper analogue of [[würschmidt]] (which reaches [[3/2]] as (25/16)/(128/125)<sup>2</sup> implying 25/24 is split into two 128/125's thus having a guaranteed neutral third), which itself is a higher-complexity and sharper analogue of [[magic]] (which equates 25/24 with 128/125 by flattening 5). For more details on these connections see [[Würschmidt comma]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 28672/28125
 
{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 391.465
 
{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
 
[[Badness]]: 0.108417
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 245/243, 1331/1323
 
Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.503
 
{{Optimal ET sequence|legend=0| 46, 95, 141bc }}
 
Badness: 0.045108
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 176/175, 245/243, 1331/1323
 
Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.366
 
{{Optimal ET sequence|legend=0| 46, 233bcff, 279bccff }}
 
Badness: 0.043024
 
== Leapweek ==
: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 2097152/2066715
 
{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 704.536
 
{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
 
[[Badness]]: 0.140577
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 1331/1323
 
Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554
 
{{Optimal ET sequence|legend=0| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
 
Badness: 0.050679
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 245/243, 352/351, 364/363
 
Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571
 
{{Optimal ET sequence|legend=0| 17, 29c, 46, 63, 109 }}
 
Badness: 0.032727
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
 
Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540
 
{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}
 
Badness: 0.026243
 
==== Leapweeker ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 136/135, 169/168, 221/220, 245/243, 364/363
 
Mapping: {{mapping| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}
 
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537
 
{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}
 
Badness: 0.026774
 
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Sensamagic_clan"