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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 14:43:47 UTC</tt>.<br>
: The original revision id was <tt>212007488</tt>.<br>
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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=
[[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
==== Full 7-limit extensions ====
EDOs: 19, 27, 46, 249, 295
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.  
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]]'' → [[Canopic clan #Browser|Canopic 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===
==== Subgroup extensions ====
Commas: 91/90, 126/125, 169/168, 385/384
The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts, which is important as it is the difference between 5/4 and 4/3 (whose product is 5/3 at 2 gens). This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions_2|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's fine-tempered 5-limit structure.


[[POTE tuning|POTE generator]]: ~9/7 = 443.321
This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 40/31~31/24 such that 24:31:40 is tuned as a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.


Map: [&lt;1 6 8 11 -6 10|, &lt;0 -7 -9 -13 15 -10|]
[[Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap| 65 + 111 = 176g }} is almost patent val.
EDOs: 8, 19, 27, 46, 157
Badness: 0.0256


==Sensis==
== Sensi ==
Commas: 56/55, 100/99, 245/243
{{Main| Sensi }}


[[POTE tuning|POTE generator]]:  443.962
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.  


Map: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
=== Septimal sensi ===
EDOs: 19, 27, 73, 100
[[Subgroup]]: 2.3.5.7
Badness: 0.0287


===13-limit===
[[Comma list]]: 126/125, 245/243
Commas: 56/55, 78/77, 91/90, 100/99


[[POTE tuning|POTE generator]]: 443.945
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}


Map: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
[[Optimal tuning]]s:  
EDOs: 19, 27, 73, 100
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
Badness: 0.0200
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}


==Sensus==
[[Minimax tuning]]:
Commas: 126/125, 176/175, 245/243
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


POTE generator: ~9/7 = 443.626
[[Tuning ranges of regular temperaments|Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
* 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
* 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
* 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]


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


===13-limit===
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
Commas: 91/90, 126/125, 169/168, 352/351


POTE generator: ~9/7 = 443.559
[[Badness]] (Sintel): 0.648


Map: [&lt;1 6 8 11 23 10|, &lt;0 -7 -9 -13 -31 -10|]
==== 2.3.5.7.13 subgroup (sensation) ====
EDOs: 8, 19, 27, 46, 303
Subgroup: 2.3.5.7.13
Badness: 0.0208


=Heinz=
Comma list: 91/90, 126/125, 169/168
Commas: 78732/78125, 1029/1024


POTE generator: ~48/35 = 546.815
{{Mapping|legend=0| 1 -1 -1 -2 0| 0 7 9 13 10 }}


Map: [&lt;1 13 17 -1|, &lt;0 -21 -27 7|]
Optimal tunings:  
EDOs: 46, 103, 149
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
Badness: 0.1154
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
Commas: 385/384, 441/440, 88208/87500


POTE generator: ~11/8 = 547.631
Badness (Sintel): 0.484


Map: [&lt;1 13 17 -1 4|, &lt;0 -21 -27 7 -1|]
=== Sensor ===
EDOs: 46, 103, 149
Subgroup: 2.3.5.7.11
Badness: 0.0424


==13-limit==
Comma list: 126/125, 245/243, 385/384
Commas: 351/350, 385/384, 441/440, 847/845


POTE generator: ~11/8 = 547.629
{{Mapping|legend=0| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}


Map: [&lt;1 13 17 -1 4 -5|, &lt;0 -21 -27 7 -1 16|]
Optimal tunings:  
EDOs: 46, 103, 149
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
Badness: 0.0258
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}


==17-limit==</pre></div>
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}
<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;Sensipent family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:24:&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:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;a href="#Sensipent"&gt;Sensipent&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#Sensi"&gt;Sensi&lt;/a&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;!-- 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;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Heinz"&gt;Heinz&lt;/a&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;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;
Badness (Sintel): 1.25
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;br /&gt;
 
&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;
==== 13-limit ====
Comma: 78732/78125&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 162/125 =&lt;br /&gt;
Comma list: 91/90, 126/125, 169/168, 385/384
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8|, &amp;lt;0 -7 -9|]&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
EDOs: 8, 19, 46, 65, 539&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Sensi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Sensi&lt;/h1&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}
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;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 126/125, 245/243&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.06
7-limit minimax&lt;br /&gt;
 
[|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;
==== 17-limit ====
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
9-limit minimax&lt;br /&gt;
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
[|1 0 0 0&amp;gt;, |2/5 14/5 -7/5 0&amp;gt;, &lt;br /&gt;
 
|4/5 18/5 -9/5 0&amp;gt;, |3/5 26/5 -13/5 0&amp;gt;]&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 9/5&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 443.383&lt;br /&gt;
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
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;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11|, &amp;lt;0 -7 -9 -13|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 27, 46 }}
&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;
Badness (Sintel): 1.17
Badness: 0.0256&lt;br /&gt;
 
&lt;br /&gt;
=== Sensus ===
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Sensi-Sensor"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Sensor&lt;/h2&gt;
Subgroup: 2.3.5.7.11
Commas: 126/125, 245/243, 385/384&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 176/175, 245/243
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 443.294&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
Map: [&amp;lt;1 6 8 11 -6|, &amp;lt;0 -7 -9 -13 15|]&lt;br /&gt;
 
EDOs: 8, 19, 27, 46, 111, 157&lt;br /&gt;
Optimal tunings:  
Badness: 0.0379&lt;br /&gt;
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Sensi-Sensor-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h3&gt;
 
Commas: 91/90, 126/125, 169/168, 385/384&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 443.321&lt;br /&gt;
Badness (Sintel): 0.975
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11 -6 10|, &amp;lt;0 -7 -9 -13 15 -10|]&lt;br /&gt;
==== 13-limit ====
EDOs: 8, 19, 27, 46, 157&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0256&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 126/125, 169/168, 352/351
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Sensi-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;
{{Mapping|legend=0| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;:  443.962&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
Map: [&amp;lt;1 6 8 11 6|, &amp;lt;0 -7 -9 -13 -4|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}
EDOs: 19, 27, 73, 100&lt;br /&gt;
 
Badness: 0.0287&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Sensi-Sensis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h3&gt;
Badness (Sintel): 0.859
Commas: 56/55, 78/77, 91/90, 100/99&lt;br /&gt;
 
&lt;br /&gt;
==== 17-limit ====
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 443.945&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;br /&gt;
 
Map: [&amp;lt;1 6 8 11 6 10|, &amp;lt;0 -7 -9 -13 -4 -10|]&lt;br /&gt;
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
EDOs: 19, 27, 73, 100&lt;br /&gt;
 
Badness: 0.0200&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Sensi-Sensus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Sensus&lt;/h2&gt;
Optimal tunings:  
Commas: 126/125, 176/175, 245/243&lt;br /&gt;
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}
POTE generator: ~9/7 = 443.626&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}
Map: [&amp;lt;1 6 8 11 23|, &amp;lt;0 -7 -9 -13 -31|]&lt;br /&gt;
 
EDOs: 8, 19, 27, 46, 165&lt;br /&gt;
Badness (Sintel): 0.827
Badness: 0.0295&lt;br /&gt;
 
&lt;br /&gt;
=== Sensis ===
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Sensi-Sensus-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h3&gt;
Subgroup: 2.3.5.7.11
Commas: 91/90, 126/125, 169/168, 352/351&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 100/99, 245/243
POTE generator: ~9/7 = 443.559&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -1 -2 2| 0 7 9 13 4 }}
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;
Optimal tunings:  
Badness: 0.0208&lt;br /&gt;
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}
&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;
 
Commas: 78732/78125, 1029/1024&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
&lt;br /&gt;
 
POTE generator: ~48/35 = 546.815&lt;br /&gt;
Badness (Sintel): 0.948
&lt;br /&gt;
 
Map: [&amp;lt;1 13 17 -1|, &amp;lt;0 -21 -27 7|]&lt;br /&gt;
==== 13-limit ====
EDOs: 46, 103, 149&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.1154&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 56/55, 78/77, 91/90, 100/99
&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;
{{Mapping|legend=0| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
&lt;br /&gt;
 
POTE generator: ~11/8 = 547.631&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
Map: [&amp;lt;1 13 17 -1 4|, &amp;lt;0 -21 -27 7 -1|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}
EDOs: 46, 103, 149&lt;br /&gt;
 
Badness: 0.0424&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Heinz-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
Badness (Sintel): 0.827
Commas: 351/350, 385/384, 441/440, 847/845&lt;br /&gt;
 
&lt;br /&gt;
=== Sensa ===
POTE generator: ~11/8 = 547.629&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 13 17 -1 4 -5|, &amp;lt;0 -21 -27 7 -1 16|]&lt;br /&gt;
Comma list: 55/54, 77/75, 99/98
EDOs: 46, 103, 149&lt;br /&gt;
 
Badness: 0.0258&lt;br /&gt;
{{Mapping|legend=0| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Heinz-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;17-limit&lt;/h2&gt;
Optimal tunings:
&lt;/body&gt;&lt;/html&gt;</pre></div>
* 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|legend=0| 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)). (As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)
 
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 126/125, 245/243
 
{{Mapping|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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|legend=0| 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
 
== Subgroup extensions ==
=== Sensipent (2.3.5.31) ===
Subgroup: 2.3.5.31
 
Comma list: 961/960, 2511/2500
 
{{Mapping|legend=2| 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
 
=== Sensible (2.3.5.11) ===
{{See also| Sensipent #Sensible interval table }}
 
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 for 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.
 
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) }}.
 
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]]. A notable non-patent val is [[84edo|84g]].
 
Finally, despite not appearing in the final optimal ET sequence, [[241edo]] is a good tuning for it via the val {{nowrap| 65 + 176g {{=}} 241g. }} (Specifically, in the sequence of edos 111, 176, 241, 65 we have increasingly good 5-limit, an essentially-just 11/8 in 111 that gets slightly sharper, a 23/16 that gets slightly flatter and becomes essentially-just in 65, and a 31 that is essentially-just in 241, leaving only the sharp 17 as encouraging a tuning closer to 111 than 65, and with more unusual complex subgroup harmony extensions (or simpler but less accurate ones) available in most of these tunings.)
 
Subgroup: 2.3.5.11
 
Comma list: 6912/6875, 8019/8000
 
{{Mapping|legend=2| 1 -1 -1 9 | 0 7 9 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
 
Badness (Sintel): 0.728
 
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17
 
Comma list: 256/255, 1089/1088, 1377/1375
 
{{Mapping|legend=2| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.639
 
==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23
 
Comma list: 256/255, 576/575, 1089/1088, 1377/1375
 
{{Mapping|legend=2| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.555
 
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31
 
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
 
{{Mapping|legend=2| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
 
Optimal tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.490
 
=== Sendai (2.3.5.23) ===
{{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). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence. It is ideal in tuning for ambiguating the generator's 2.3.5.31 interpretation as 40/31~31/24, being between these (and hence also a good tuning of 2.3.5.31 sensipent at the cost of some 5-limit accuracy).
 
{{Todo|inline=1|complete section|comment=Add the data for the intermediate subgroups. }}
 
==== 2.3.5.23.29.31 subgroup ====
Subgroup: 2.3.5.23.29.31
 
Comma list: 465/464, 576/575, 621/620, 900/899
 
{{Mapping|legend=2| 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:Sensipent family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
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