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Sensipent 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>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-19 14:37:48 UTC</tt>.<br>
: The original revision id was <tt>212006708</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]]


=Sensipent=
[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.
Comma: 78732/78125


[[POTE tuning|POTE generator]]: 162/125 =
== Sensipent ==
{{Main| Sensipent }}


Map: [&lt;1 6 8|, &lt;0 -7 -9|]
The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its [[ploidacot]] is beta-heptacot ([[pergen]] (P8, ccP5/7)) and its color name is Sepguti.
EDOs: 8, 19, 46, 65, 539


[[Subgroup]]: 2.3.5


=Sensi temperament=
[[Comma list]]: 78732/78125
Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.


[[Comma|Commas]]: 126/125, 245/243
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
: mapping generators: ~2, ~162/125


7-limit minimax
[[Optimal tuning]]s:
[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&gt;]
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
[[Eigenmonzo|Eigenmonzos]]: 2, 7
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}


9-limit minimax
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}
[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,  
|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 9/5


[[POTE tuning|POTE generator]]: ~9/7 = 443.383
[[Badness]] (Sintel): 0.826
Algebraic generator: Calista, the [[Algebraic number|real root]] of x^7-2x^2-1, at 340.6467 cents.  


Map: [&lt;1 6 8 11|, &lt;0 -7 -9 -13|]
=== Overview to extensions ===
[[Generator|Generators]]: 2, 14/9
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.  
EDOs: 19, 27, 46, 249, 295
Badness: 0.0256


==Sensor==
Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.
Commas: 126/125, 245/243, 385/384


[[POTE tuning|POTE generator]]: ~9/7 = 443.294
Temperaments discussed elsewhere include:
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)


Map: [&lt;1 6 8 11 -6|, &lt;0 -7 -9 -13 15|]
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
EDOs: 8, 19, 27, 46, 111, 157
Badness: 0.0379


===13-limit===
=== Sensible ===
Commas: 91/90, 126/125, 169/168, 385/384
{{See also| Sensipent #Sensible interval table }}


[[POTE tuning|POTE generator]]: ~9/7 = 443.321
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.


Map: [&lt;1 6 8 11 -6 10|, &lt;0 -7 -9 -13 15 -10|]
The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.
EDOs: 8, 19, 27, 46, 157
Badness: 0.0256


==Sensis==
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
Commas: 56/55, 100/99, 245/243


[[POTE tuning|POTE generator]]: 443.962
Subgroup: 2.3.5.11


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
Comma list: 8019/8000, 16384/16335
EDOs: 19, 27, 73, 100
Badness: 0.0287


===13-limit===
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
Commas: 56/55, 78/77, 91/90, 100/99


[[POTE tuning|POTE generator]]: 443.945
: mapping generators: ~2, ~128/99


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
Optimal tunings:  
EDOs: 19, 27, 73, 100
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
Badness: 0.0200
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}


==Sensus==
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
Commas: 126/125, 176/175, 245/243


POTE generator: ~9/7 = 443.626
Badness (Sintel): 0.728


Map: [&lt;1 6 8 11 23|, &lt;0 -7 -9 -13 -31|]
==== 2.3.5.11.17 subgroup ====
EDOs: 8, 19, 27, 46, 165
Subgroup: 2.3.5.11.17
Badness: 0.0295


===13-limit===
Comma list: 256/255, 1089/1088, 1377/1375
Commas: 91/90, 126/125, 169/168, 352/351


POTE generator: ~9/7 = 443.559
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}


Map: [&lt;1 6 8 11 23 10|, &lt;0 -7 -9 -13 -31 -10|]
: mapping generators: ~2, ~22/17
EDOs: 8, 19, 27, 46, 303
Badness: 0.0208


=Heinz=
Optimal tunings:
Commas: 78732/78125, 1029/1024
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}


POTE generator: ~48/35 = 546.815
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


Map: [&lt;1 13 17 -1|, &lt;0 -21 -27 7|]
Badness (Sintel): 0.639
EDOs: 46, 103, 149
Badness: 0.1154


==11-limit==
==== 2.3.5.11.17.23 subgroup ====
Commas: 385/384, 441/440, 88208/87500
Subgroup: 2.3.5.11.17.23


POTE generator: ~11/8 = 547.631
Comma list: 256/255, 576/575, 1089/1088, 1377/1375


Map: [&lt;1 13 17 -1 4|, &lt;0 -21 -27 7 -1|]
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
EDOs: 46, 103, 149
 
Badness: 0.0424
Optimal tunings:  
Badness:</pre></div>
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
<h4>Original HTML content:</h4>
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
<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;Sensipent family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&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:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Sensipent"&gt;Sensipent&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Sensi temperament"&gt;Sensi temperament&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&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;a href="#Heinz"&gt;Heinz&lt;/a&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;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Sensipent"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Sensipent&lt;/h1&gt;
 
Comma: 78732/78125&lt;br /&gt;
Badness (Sintel): 0.555
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 162/125 =&lt;br /&gt;
==== 2.3.5.11.17.23.31 subgroup ====
&lt;br /&gt;
Subgroup: 2.3.5.11.17.23.31
Map: [&amp;lt;1 6 8|, &amp;lt;0 -7 -9|]&lt;br /&gt;
 
EDOs: 8, 19, 46, 65, 539&lt;br /&gt;
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
&lt;br /&gt;
 
&lt;br /&gt;
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Sensi temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Sensi temperament&lt;/h1&gt;
 
Sensi tempers out 686/675, 245/243 and 4375/4374 in addition to 126/125, and can be described as the 19&amp;amp;27 temperament. It has as a generator half of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 13-limit sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt; is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available.&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 245/243&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
&lt;br /&gt;
 
7-limit minimax&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
[|1 0 0 0&amp;gt;, |1/13 0 0 7/13&amp;gt;, |5/13 0 0 9/13&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7&lt;br /&gt;
Badness (Sintel): 0.490
&lt;br /&gt;
 
9-limit minimax&lt;br /&gt;
== Sensi ==
[|1 0 0 0&amp;gt;, |2/5 14/5 -7/5 0&amp;gt;, &lt;br /&gt;
{{Main| Sensi }}
|4/5 18/5 -9/5 0&amp;gt;, |3/5 26/5 -13/5 0&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 9/5&lt;br /&gt;
Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 443.383&lt;br /&gt;
=== Septimal sensi ===
Algebraic generator: Calista, the &lt;a class="wiki_link" href="/Algebraic%20number"&gt;real root&lt;/a&gt; of x^7-2x^2-1, at 340.6467 cents. &lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11|, &amp;lt;0 -7 -9 -13|]&lt;br /&gt;
[[Comma list]]: 126/125, 245/243
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 14/9&lt;br /&gt;
 
EDOs: 19, 27, 46, 249, 295&lt;br /&gt;
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
Badness: 0.0256&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Sensi temperament-Sensor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Sensor&lt;/h2&gt;
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
Commas: 126/125, 245/243, 385/384&lt;br /&gt;
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 443.294&lt;br /&gt;
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11 -6|, &amp;lt;0 -7 -9 -13 15|]&lt;br /&gt;
[[Minimax tuning]]:
EDOs: 8, 19, 27, 46, 111, 157&lt;br /&gt;
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
Badness: 0.0379&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
&lt;br /&gt;
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Sensi temperament-Sensor-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h3&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
Commas: 91/90, 126/125, 169/168, 385/384&lt;br /&gt;
 
&lt;br /&gt;
[[Tuning ranges of regular temperaments|Tuning ranges]]:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 443.321&lt;br /&gt;
* 7-odd-limit [[diamond monotone]]: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
&lt;br /&gt;
* 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
Map: [&amp;lt;1 6 8 11 -6 10|, &amp;lt;0 -7 -9 -13 15 -10|]&lt;br /&gt;
* 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
EDOs: 8, 19, 27, 46, 157&lt;br /&gt;
* 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]
Badness: 0.0256&lt;br /&gt;
 
&lt;br /&gt;
[[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Sensi temperament-Sensis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Sensis&lt;/h2&gt;
 
Commas: 56/55, 100/99, 245/243&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 443.962&lt;br /&gt;
[[Badness]] (Sintel): 0.648
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11 6|, &amp;lt;0 -7 -9 -13 -4|]&lt;br /&gt;
==== 2.3.5.7.13 subgroup (sensation) ====
EDOs: 19, 27, 73, 100&lt;br /&gt;
Subgroup: 2.3.5.7.13
Badness: 0.0287&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 126/125, 169/168
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Sensi temperament-Sensis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h3&gt;
 
Commas: 56/55, 78/77, 91/90, 100/99&lt;br /&gt;
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 443.945&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
Map: [&amp;lt;1 6 8 11 6 10|, &amp;lt;0 -7 -9 -13 -4 -10|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
EDOs: 19, 27, 73, 100&lt;br /&gt;
 
Badness: 0.0200&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Sensi temperament-Sensus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Sensus&lt;/h2&gt;
Badness (Sintel): 0.484
Commas: 126/125, 176/175, 245/243&lt;br /&gt;
 
&lt;br /&gt;
=== Sensor ===
POTE generator: ~9/7 = 443.626&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11 23|, &amp;lt;0 -7 -9 -13 -31|]&lt;br /&gt;
Comma list: 126/125, 245/243, 385/384
EDOs: 8, 19, 27, 46, 165&lt;br /&gt;
 
Badness: 0.0295&lt;br /&gt;
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Sensi temperament-Sensus-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h3&gt;
Optimal tunings:
Commas: 91/90, 126/125, 169/168, 352/351&lt;br /&gt;
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}
POTE generator: ~9/7 = 443.559&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}
Map: [&amp;lt;1 6 8 11 23 10|, &amp;lt;0 -7 -9 -13 -31 -10|]&lt;br /&gt;
 
EDOs: 8, 19, 27, 46, 303&lt;br /&gt;
Badness (Sintel): 1.25
Badness: 0.0208&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Heinz"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Heinz&lt;/h1&gt;
Subgroup: 2.3.5.7.11.13
Commas: 78732/78125, 1029/1024&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 126/125, 169/168, 385/384
POTE generator: ~48/35 = 546.815&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
Map: [&amp;lt;1 13 17 -1|, &amp;lt;0 -21 -27 7|]&lt;br /&gt;
 
EDOs: 46, 103, 149&lt;br /&gt;
Optimal tunings:
Badness: 0.1154&lt;br /&gt;
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Heinz-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
 
Commas: 385/384, 441/440, 88208/87500&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
&lt;br /&gt;
 
POTE generator: ~11/8 = 547.631&lt;br /&gt;
Badness (Sintel): 1.06
&lt;br /&gt;
 
Map: [&amp;lt;1 13 17 -1 4|, &amp;lt;0 -21 -27 7 -1|]&lt;br /&gt;
==== 17-limit ====
EDOs: 46, 103, 149&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0424&lt;br /&gt;
 
Badness:&lt;/body&gt;&lt;/html&gt;</pre></div>
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
 
Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}
 
Optimal tunings:
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}
 
{{Optimal ET sequence|legend=0| 19, 27, 46 }}
 
Badness (Sintel): 1.17
 
=== Sensus ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 245/243
 
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
 
Optimal tunings:  
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}
 
Badness (Sintel): 0.975
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168, 352/351
 
Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
 
Optimal tunings:  
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}
 
Badness (Sintel): 0.859
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
 
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}
 
Optimal tunings:  
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}
 
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}
 
Badness (Sintel): 0.827
 
=== Sensis ===
Subgroup: 2.3.5.7.11
 
Comma list: 56/55, 100/99, 245/243
 
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
 
Optimal tunings:
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
 
Badness (Sintel): 0.948
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 91/90, 100/99
 
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
 
Optimal tunings:
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
 
Badness (Sintel): 0.827
 
=== Sensa ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 77/75, 99/98
 
Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
 
Optimal tunings:  
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
 
Badness (Sintel): 1.22
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 66/65, 77/75, 143/140
 
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}
 
Optimal tunings:  
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
 
Badness (Sintel): 0.961
 
=== Bisensi ===
Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).
 
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 126/125, 245/243
 
Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
 
: mapping generators: ~99/70, ~9/7
 
Optimal tunings:
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}
 
Badness (Sintel): 1.38
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 126/125, 169/168
 
Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
 
Badness (Sintel): 1.09
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
 
Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
 
Badness (Sintel): 0.960
 
=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-14-cot (pergen (P8, ccP5/14)).
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 245/242
 
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}
 
: mapping generators: ~2, ~25/22
 
Optimal tunings:
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}
 
Badness (Sintel): 1.61
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168, 243/242
 
Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}
 
Badness (Sintel): 1.36
 
== Sensei ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6422{{c}}, ~162/125 = 442.9920{{c}}
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}
 
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}
 
[[Badness]] (Sintel): 1.50
 
== Warrior ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2419{{c}}, ~162/125 = 443.0087{{c}}
: [[error map]]: {{val| -0.758 -0.136 +1.523 +0.516 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}
 
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}
 
[[Badness]] (Sintel): 2.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1331/1323, 5120/5103
 
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}
 
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}
 
Badness (Sintel): 1.53
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 847/845, 1331/1323
 
Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}
 
Optimal tunings:
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}
 
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714
 
Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}
 
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}
 
Badness (Sintel): 0.922
 
== Bison ==
Bison has a 1/2-octave period and the generator can be taken as [[~]][[162/125]] or its semi-octave complement, ~[[35/32]]. Its [[ploidacot]] is diploid delta-heptacot ([[pergen]] (P8/2, ccP5/7)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 78732/78125
 
{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
: mapping generators: ~567/400, ~162/125
 
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 599.9413{{c}}, ~162/125 = 443.0320{{c}} (~35/32 = 156.9093{{c}})
: [[error map]]: {{val| -0.117 -0.613 +1.092 +0.091 }}
* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}
 
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
 
[[Badness]] (Sintel): 1.78
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 8019/8000
 
Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}
 
Badness (Sintel): 1.23
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440, 10985/10976
 
Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
 
Optimal tunings:
* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
 
Badness (Sintel): 0.971
 
== Subpental ==
Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c<sup>4</sup>P4/14)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 19683/19600
 
{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
: [[error map]]: {{val| -0.074 -0.611 +1.107 -0.052 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
 
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}
 
[[Badness]] (Sintel): 1.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 3136/3125, 8019/8000
 
Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}
 
Optimal tunings:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
 
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}
 
Badness (Sintel): 1.50
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 676/675, 3136/3125
 
Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}
 
Optimal tunings:
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}
 
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}
 
Badness (Sintel): 0.989
 
== Heinz ==
Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c<sup>9</sup>P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1029/1024, 78732/78125
 
{{Mapping|legend=1| 1 -8 -10 6 | 0 21 27 -7 }}
: mapping generators: ~2, ~48/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}
 
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
 
[[Badness]] (Sintel): 2.92
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 78732/78125
 
Mapping: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}
 
: mapping generators: ~2, ~11/8
 
Optimal tunings:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
 
Badness (Sintel): 1.40
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 385/384, 441/440, 847/845
 
Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}
 
Optimal tunings:
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}
 
Badness (Sintel): 1.07
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845
 
Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}
 
Optimal tunings:
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}
 
Badness (Sintel): 0.941
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
 
Mapping: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}
 
Optimal tunings:
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}
 
Badness (Sintel): 1.16
 
== Trisensory ==
Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1728/1715, 78732/78125
 
{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
: mapping generators: ~63/50, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.8117{{c}}, ~36/35 = 43.1270{{c}}
: [[error map]]: {{val| -0.565 -0.819 +0.700 +2.176 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}
 
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
 
[[Badness]] (Sintel): 2.27
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 78732/78125
 
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
 
Optimal tunings:
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}
 
Badness (Sintel): 1.93
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 540/539, 9295/9261
 
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
 
: mapping generators: ~49/39, ~36/35
 
Optimal tunings:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
 
Badness (Sintel): 1.44
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714
 
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
 
Optimal tunings:
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 
Badness (Sintel): 1.23
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475
 
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
 
Optimal tunings:
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 
Badness (Sintel): 1.12
 
== Other subgroup extensions ==
=== Sensipent (2.3.5.31 subgroup) ===
The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.
 
For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
 
Subgroup: 2.3.5.31
 
Comma list: 961/960, 2511/2500
 
Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
 
{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
 
Badness (Sintel): 0.243
 
=== Sendai ===
{{See also| Sensipent #Sendai interval table }}
 
Sendai is an accurate extension of sensipent with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).
 
Subgroup: 2.3.5.23.29.31
 
Comma list: 465/464, 576/575, 621/620, 900/899
 
Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
 
Badness (Sintel): 0.283
 
[[Category:Temperament families]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Rank 2]]
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