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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 '''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>2012-05-18 16:45:21 UTC</tt>.<br>
: The original revision id was <tt>337176058</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: 1220703125/1162261467


POTE generator: ~27/25 = 146.476
[[Subgroup]]: 3.5.7


Map: [&lt;1 0 0|, &lt;0 13 19|]
[[Comma list]]: 245/243
EDOs: 8, 41, 131, 172, 213c
Badness: 0.8605


==7-limit==
{{Mapping|legend=2| 1 1 2 | 0 -2 1 }}
[[Comma]]s: 245/243, 3125/3087


[[POTE tuning|POTE generator]]: ~27/25 = 146.474
: sval mapping generators: ~3, ~9/7


Map: [&lt;1 0 0 0|, &lt;0 13 19 23|]
[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881
[[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==
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
Commas: 100/99, 245/243, 1344/1331


POTE generator: ~12/11 = 146.545
=== 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|, &lt;0 13 19 23 12|]
These temperaments are distributed into different family pages.
EDOs: 41, 90e, 131e
* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
Badness: 0.0339
* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]


==13-limit==
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.
Commas: 100/99, 144/143, 245/243, 275/273


POTE generator: ~12/11 = 146.603
Discussed elsewhere are
* [[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]]


Map: [&lt;1 0 0 0 0 0|, &lt;0 13 19 23 12 14|]
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].
EDOs: 41, 90ef, 131ef, 221bdeff
Badness: 0.0249


==Music==
Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
By [[Chris Vaisvil]]
http://micro.soonlabel.com/bophier/bophier-1.mp3
http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3


=Sensa=  
== Bohpier ==
Commas: 245/243, 65625/65536
{{Main| Bohpier }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''


POTE generator: ~28/27 = 55.122
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|, &lt;0 -9 7 -26|]
[[Subgroup]]: 2.3.5.7
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==
[[Comma list]]: 245/243, 3125/3087
Commas: 245/243, 385/384, 4000/3993


POTE generator: ~28/27 = 55.126
{{Mapping|legend=1| 1 0 0 0 | 0 13 19 23 }}


Map: [&lt;1 2 2 4 3|, &lt;0 -9 7 -26 10|]
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 146.474
EDOs: 22, 43, 65, 87, 109, 196, 283
Badness: 0.0358


==13-limit==
[[Minimax tuning]]:
Commas: 245/243, 352/351, 385/384, 625/624
* [[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: ~28/27 = 55.138
{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}


Map: [&lt;1 2 2 4 3 2|, &lt;0 -9 7 -26 10 37|]
[[Badness]]: 0.068237
EDOs: 22, 43, 65, 87, 109, 196, 283
Badness: 0.0317


=Salsa=
=== 11-limit ===
Commas: 245/243, 32805/32768
Subgroup: 2.3.5.7.11


POTE generator: ~128/105 = 351.049
Comma list: 100/99, 245/243, 1344/1331


Map: [&lt;1 1 7 -1|, &lt;0 2 -16 13|]
Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}
Wedgie: &lt;&lt;2 -16 13 -30 15 75||
EDOs: 17, 24, 41, 106d, 147d, 188cd, 335cd
Badness: 0.08015


==11-limit==
Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545
Commas: 243/242, 245/242, 385/384


POTE generator: ~11/9 = 351.014
Minimax tuning:
* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9


Map: [&lt;1 1 7 -1 2|, &lt;0 2 -16 13 5|]
{{Optimal ET sequence|legend=0| 41, 90e, 131e }}
EDOs: 17, 24, 41, 106d, 147d
Badness: 0.0394


==13-limit==
Badness: 0.033949
Commas: 105/104, 144/143, 243/242, 245/242


POTE generator: ~11/9 = 351.025
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 1 7 -1 2 4|, &lt;0 2 -16 13 5 -1|]
Comma list: 100/99, 144/143, 196/195, 275/273
EDOs: 17, 24, 41, 106df, 147df
Badness: 0.0310


=Pycnic=
Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}
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
Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603


POTE generator: ~45/32 = 567.720
Minimax tuning:
* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
: Unchanged-interval (eigenmonzo) basis: 2.5


Map: [&lt;1 3 -1 8|, &lt;0 -3 7 -11|]
{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}
Wedgie: &lt;&lt;3 -7 11 -18 9 45||
EDOs: 17, 19, 55c, 74cd, 93cd
Badness: 0.0737


=Cohemiripple=
Badness: 0.024864
Commas: 245/243, 1323/1250


POTE generator: ~7/5 = 549.944
=== 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|, &lt;0 -10 -16 -17|]
Subgroup: 2.3.5.7.11
Wedgie: &lt;&lt;10 16 17 2 -1 -5||
EDOs: 24
Badness: 0.1902


==11-limit==
Comma list: 245/243, 1331/1323, 3125/3087
Commas: 77/75, 243/242, 245/242


POTE generator: ~7/5 = 549.945
Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}


Map: [&lt;1 7 11 12 17|, &lt;0 -10 -16 -17 -25|]
Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828
EDOs: 24
Badness: 0.0827


==13-limit==
{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
Commas: 66/65, 77/75, 147/143, 351/350


POTE generator: ~7/5 = 549.958
Badness: 0.162592


Map: [&lt;1 7 11 12 17 14|, &lt;0 -10 -16 -17 -25 -19|]
==== 13-limit ====
EDOs: 24
Subgroup: 2.3.5.7.11.13
Badness: 0.0499


Comma list: 245/243, 275/273, 847/845, 1331/1323


</pre></div>
Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
<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:30:&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:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;a href="#Bohpier"&gt;Bohpier&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;!-- 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="#Sensa"&gt;Sensa&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&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;a href="#Salsa"&gt;Salsa&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#Pycnic"&gt;Pycnic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt; | &lt;a href="#Cohemiripple"&gt;Cohemiripple&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822
&lt;!-- ws:end:WikiTextTocRule:46 --&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;
{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}
&lt;br /&gt;
 
&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;
Badness: 0.082158
Comma: 1220703125/1162261467&lt;br /&gt;
 
&lt;br /&gt;
== Salsa ==
POTE generator: ~27/25 = 146.476&lt;br /&gt;
{{See also| Schismatic family }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0|, &amp;lt;0 13 19|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 8, 41, 131, 172, 213c&lt;br /&gt;
 
Badness: 0.8605&lt;br /&gt;
[[Comma list]]: 245/243, 32805/32768
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Bohpier-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 245/243, 3125/3087&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~27/25 = 146.474&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
Map: [&amp;lt;1 0 0 0|, &amp;lt;0 13 19 23|]&lt;br /&gt;
 
&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;
[[Badness]]: 0.080152
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;
 
EDTs: &lt;a class="wiki_link" href="/13edt"&gt;13&lt;/a&gt;&lt;br /&gt;
=== 11-limit ===
&lt;a class="wiki_link" href="/Badness"&gt;Badness&lt;/a&gt;: 0.0682&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bohpier-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Comma list: 243/242, 245/242, 385/384
Commas: 100/99, 245/243, 1344/1331&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
POTE generator: ~12/11 = 146.545&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.014
Map: [&amp;lt;1 0 0 0 0|, &amp;lt;0 13 19 23 12|]&lt;br /&gt;
 
EDOs: 41, 90e, 131e&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}
Badness: 0.0339&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.039444
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bohpier-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
 
Commas: 100/99, 144/143, 245/243, 275/273&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~12/11 = 146.603&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 105/104, 144/143, 243/242, 245/242
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;
Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
Badness: 0.0249&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Bohpier-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Music&lt;/h2&gt;
 
By &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}
&lt;!-- ws:start:WikiTextUrlRule:197: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:197 --&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextUrlRule:198: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:198 --&gt;&lt;br /&gt;
Badness: 0.030793
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Sensa"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Sensa&lt;/h1&gt;
== Pycnic ==
Commas: 245/243, 65625/65536&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
&lt;br /&gt;
 
POTE generator: ~28/27 = 55.122&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.
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 4|, &amp;lt;0 -9 7 -26|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Wedgie: &amp;lt;&amp;lt;9 -7 26 -32 16 80||&lt;br /&gt;
 
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;
[[Comma list]]: 245/243, 525/512
Badness: 0.0887&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 3 -1 8 | 0 -3 7 -11 }}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Sensa-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11-limit&lt;/h2&gt;
 
Commas: 245/243, 385/384, 4000/3993&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~45/32 = 567.720
&lt;br /&gt;
 
POTE generator: ~28/27 = 55.126&lt;br /&gt;
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 4 3|, &amp;lt;0 -9 7 -26 10|]&lt;br /&gt;
[[Badness]]: 0.073735
EDOs: 22, 43, 65, 87, 109, 196, 283&lt;br /&gt;
 
Badness: 0.0358&lt;br /&gt;
== Superthird ==
&lt;br /&gt;
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Sensa-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
 
Commas: 245/243, 352/351, 385/384, 625/624&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
POTE generator: ~28/27 = 55.138&lt;br /&gt;
[[Comma list]]: 245/243, 78125/76832
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 4 3 2|, &amp;lt;0 -9 7 -26 10 37|]&lt;br /&gt;
{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
EDOs: 22, 43, 65, 87, 109, 196, 283&lt;br /&gt;
 
Badness: 0.0317&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Salsa"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Salsa&lt;/h1&gt;
{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
Commas: 245/243, 32805/32768&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]]: 0.139379
POTE generator: ~128/105 = 351.049&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
Map: [&amp;lt;1 1 7 -1|, &amp;lt;0 2 -16 13|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
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;
Comma list: 100/99, 245/243, 78125/76832
Badness: 0.08015&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Salsa-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
 
Commas: 243/242, 245/242, 385/384&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152
&lt;br /&gt;
 
POTE generator: ~11/9 = 351.014&lt;br /&gt;
{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 7 -1 2|, &amp;lt;0 2 -16 13 5|]&lt;br /&gt;
Badness: 0.070917
EDOs: 17, 24, 41, 106d, 147d&lt;br /&gt;
 
Badness: 0.0394&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Salsa-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
 
Commas: 105/104, 144/143, 243/242, 245/242&lt;br /&gt;
Comma list: 100/99, 144/143, 196/195, 1375/1352
&lt;br /&gt;
 
POTE generator: ~11/9 = 351.025&lt;br /&gt;
Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 1 7 -1 2 4|, &amp;lt;0 2 -16 13 5 -1|]&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119
EDOs: 17, 24, 41, 106df, 147df&lt;br /&gt;
 
Badness: 0.0310&lt;br /&gt;
{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Pycnic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Pycnic&lt;/h1&gt;
Badness: 0.052835
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;
 
&lt;br /&gt;
== Superenneadecal ==
Commas: 245/243, 525/512&lt;br /&gt;
Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
&lt;br /&gt;
 
POTE generator: ~45/32 = 567.720&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 3 -1 8|, &amp;lt;0 -3 7 -11|]&lt;br /&gt;
[[Comma list]]: 245/243, 395136/390625
Wedgie: &amp;lt;&amp;lt;3 -7 11 -18 9 45||&lt;br /&gt;
 
EDOs: 17, 19, 55c, 74cd, 93cd&lt;br /&gt;
{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
Badness: 0.0737&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Cohemiripple"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Cohemiripple&lt;/h1&gt;
 
Commas: 245/243, 1323/1250&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
&lt;br /&gt;
 
POTE generator: ~7/5 = 549.944&lt;br /&gt;
[[Badness]]: 0.132311
&lt;br /&gt;
 
Map: [&amp;lt;1 7 11 12|, &amp;lt;0 -10 -16 -17|]&lt;br /&gt;
=== 11-limit ===
Wedgie: &amp;lt;&amp;lt;10 16 17 2 -1 -5||&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 24&lt;br /&gt;
 
Badness: 0.1902&lt;br /&gt;
Comma list: 245/243, 2560/2541, 3773/3750
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Cohemiripple-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
Commas: 77/75, 243/242, 245/242&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667
POTE generator: ~7/5 = 549.945&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
Map: [&amp;lt;1 7 11 12 17|, &amp;lt;0 -10 -16 -17 -25|]&lt;br /&gt;
 
EDOs: 24&lt;br /&gt;
Badness: 0.101496
Badness: 0.0827&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Cohemiripple-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;13-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13
Commas: 66/65, 77/75, 147/143, 351/350&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 196/195, 245/243, 832/825, 1001/1000
POTE generator: ~7/5 = 549.958&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
Map: [&amp;lt;1 7 11 12 17 14|, &amp;lt;0 -10 -16 -17 -25 -19|]&lt;br /&gt;
 
EDOs: 24&lt;br /&gt;
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801
Badness: 0.0499&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{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"