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Porwell temperaments: 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 is a collection of [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[porwell comma]] ({{monzo|legend=1| 11 1 -3 -2 }}, [[ratio]]: [[6144/6125]]).  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-01-10 16:20:10 UTC</tt>.<br>
: The original revision id was <tt>571568031</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]]
----
This family of temperaments tempers out the //porwell comma//, |11 1 -3 -2&gt; = 6144/6125, and includes hemiwuerschmidt, orwell, amity, valentine, porcupine, hendecatonic, shrutar, hexadecimal, grendel, hemikleismic, mohajira, twothirdtonic and nessafof.


=Hendecatonic=
Temperaments discussed elsewhere are:
Commas: 6144/6125, 10976/10935
* ''[[Armodue (temperament)|Armodue]]'' (+36/35) → [[Mavila family #Armodue|Mavila family]]
* [[Mohajira]] (+81/80) → [[Meantone family #Mohajira|Meantone family]]
* ''[[Hemischis]]'' (+19683/19600) → [[Schismatic family #Hemischis|Schismatic family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Porcupine|Porcupine family]]
* ''[[Alphatrident]]'' (+14348907/14336000) → [[Alphatricot family #Alphatrident|Alphatricot family]]
* ''[[Shrutar]]'' (+245/243) → [[Diaschismic family #Shrutar|Diaschismic family]]
* [[Amity]] (+4375/4374 or 5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Orwell]] (+225/224) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Twilight]]'' (+{{monzo| 19 -22 2 4 }}) → [[Undim family #Twilight|Undim family]]
* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]]
* ''[[Decimaleap]]'' (+{{monzo| 15 -18 1 4 }}) → [[Quintaleap family #Decimaleap|Quintaleap family]]
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Bison]]'' (+78732/78125) → [[Sensipent family #Bison|Sensipent family]]
* ''[[Quinkee]]'' (+1029/1000) → [[Cloudy clan #Quinkee|Cloudy clan]]
* ''[[Hemiwürschmidt]]'' (+2401/2400 or 3136/3125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Septisuperfourth]]'' (+118098/117649) → [[Escapade family #Septisuperfourth|Escapade family]]
* ''[[Hemimabila]]'' (+117649/116640) → [[Mabila family #Hemimabila|Mabila family]]
* ''[[Countermiracle]]'' (+823543/819200) → [[Quince clan #Countermiracle|Quince clan]]
* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]]


[[POTE tuning|POTE generator]]: ~3/2 = 703.054
Considered below are hendecatonic, nessafof, grendel, twothirdtonic, aufo, absurdity, polypyth, whoops, dodifo, and icositritonic, in the order of increasing [[badness]].  


Map: [&lt;11 0 43 -4], &lt;0 1 -1 2|]
== Hendecatonic ==
Wedgie: &lt;&lt;11 -11 22 -43 4 82||
: ''For the 5-limit version, see [[11th-octave temperaments #Hendecapent]].''
EDOs: 22, 55, 77, [[99edo|99]]
Badness: 0.0411


==11-limit==
The hendecatonic temperament has a period of 1/11 octave, which represents [[16/15]] and four times of which represent [[9/7]]. It tempers out 10976/10935, the [[hemimage comma]], and may be described as the {{nowrap| 22 & 99 }} temperament, with [[99edo]] giving an almost perfect tuning.
Commas: 121/120, 176/175, 10976/10935


[[POTE tuning|POTE generator]]: ~3/2 = 702.636
[[Subgroup]]: 2.3.5.7


Map: [&lt;11 0 43 -4 38], &lt;0 1 -1 2 0|]
[[Comma list]]: 6144/6125, 10976/10935
EDOs: 22, 55, 77, 99, 176e, 275e
Badness: 0.0461


==Icosidillic==
{{Mapping|legend=1| 11 0 43 -4 | 0 1 -1 2 }}
Commas: 3388/3375, 6144/6125, 9801/9800
: mapping generators: ~16/15, ~3


[[POTE tuning|POTE generator]]: 702.914
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 109.0526{{c}}, ~3/2 = 702.8069{{c}}
: [[error map]]: {{val| -0.421 +0.431 +0.563 -0.265 }}
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 702.9705{{c}}
: error map: {{val| 0.000 +1.015 +1.625 +0.751 }}


Map: [&lt;22 0 86 -8 111|, &lt;0 1 -1 2 -1|]
{{Optimal ET sequence|legend=1| 22, 55, 77, 99 }}
EDOs: 22, 154, 176, 198, 968, 1166
Badness: 0.0577


=Hemischis=
[[Badness]] (Sintel): 1.04
Commas: 6144/6125, 19683/19600


POTE generator: ~81/70 = 249.203
=== Hendecaton ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 15 -17|, &lt;0 2 -16 25|]
Comma list: 121/120, 176/175, 10976/10935
Wedgie: &lt;&lt;2 -16 25 -30 34 103||
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313
Badness: 0.0458


==11-limit==
Mapping: {{mapping| 11 0 43 -4 38 | 0 1 -1 2 0 }}
Commas: 540/539, 8019/8000, 5632/5625


POTE generator: ~81/70 = 249.199
Optimal tunings:  
* WE: ~16/15 = 109.0977{{c}}, ~3/2 = 702.6801{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.6484{{c}}


Map: [&lt;1 0 15 -17 51|, &lt;0 2 -16 25 -60|]
{{Optimal ET sequence|legend=0| 22, 55, 77, 99 }}
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313
Badness: 0.0363


==13-limit==
Badness (Sintel): 1.52
Commas: 351/350, 540/539, 676/675, 4096/4095


POTE generator: ~15/13 = 249.199
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 15 -17 51 14|, &lt;0 2 -16 25 -60 -13|]
Comma list: 121/120, 176/175, 351/350, 4459/4455
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313
Badness: 0.0208


==17-limit==
Mapping: {{mapping| 11 0 43 -4 38 93 | 0 1 -1 2 0 -3 }}
Commas: 351/350, 442/441, 561/560, 676/675, 4096/4095


POTE generator: ~15/13 = 249.190
Optimal tunings:
* WE: ~16/15 = 109.1092{{c}}, ~3/2 = 702.4093{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.2930{{c}}


Map: [&lt;1 0 15 -17 51 14|, &lt;0 2 -16 25 -60 -13|]
{{Optimal ET sequence|legend=0| 22, 55, 77, 99 }}
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 679
Badness: 0.0211


=Twothirdtonic=
Badness (Sintel): 1.66
Commas: 686/675, 6144/6125


POTE generator: ~15/14 = 130.401
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 3 2 4|, &lt;0 -13 3 -11|]
Comma list: 121/120, 154/153, 176/175, 273/272, 2025/2023
Wedgie: &lt;&lt;13 -3 11 -35 -19 34||
EDOs: 9, 10, 19, 28, 37, 46
Badness: 0.0996


==11-limit==
Mapping: {{mapping| 11 0 43 -4 38 93 45 | 0 1 -1 2 0 -3 0 }}
Commas: 121/120, 176/175, 686/675


POTE generator: ~15/14 = 130.430
Optimal tunings:
* WE: ~16/15 = 109.0933{{c}}, ~3/2 = 702.3170{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 702.3017{{c}}


Map: [&lt;1 3 2 4 4|, &lt;0 -13 3 -11 -5|]
{{Optimal ET sequence|legend=0| 22, 55, 77, 99, 176eg }}
EDOs: 9, 10, 19, 28, 37, 46
Badness: 0.0408


==13-limit==
Badness (Sintel): 1.48
Commas: 91/90, 121/120, 169/168, 176/175


POTE generator: ~15/14 = 130.409
=== Cohendecatonic ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 3 2 4 4 5|, &lt;0 -13 3 -11 -5 -12|]
Comma list: 540/539, 896/891, 4375/4356
EDOs: 9, 10, 19, 28, 37, 46
Badness: 0.0259


=Nessafof=
Mapping: {{mapping| 11 0 43 -4 73 | 0 1 -1 2 -2 }}
Commas: 6144/6125, 250047/250000


POTE generator: ~35/32 = 157.420
Optimal tunings:  
* WE: ~16/15 = 109.0237{{c}}, ~3/2 = 703.2522{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.6563{{c}}


Map: [&lt;3 2 5 10|, &lt;0 7 5 -4|]
{{Optimal ET sequence|legend=0| 22, 77e, 99e, 121, 220e }}
Wedgie: &lt;&lt;21 15 -12 -25 -78 -70||
EDOs: 15, 69, 84, 99, 282, 381
Badness: 0.0450


=Septisuperfourth=
Badness (Sintel): 1.26
Commas: 6144/6125, 118098/117649


POTE generator: ~48/35 = 544.680
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 4 4 7|, &lt;0 -9 7 -15|]
Comma list: 352/351, 364/363, 540/539, 625/624
Wedgie: &lt;&lt;18 -14 30 -64 -3 109||
EDOs: 22, 86, 108, 130, 152, 282
Badness: 0.0592


==11-limit==
Mapping: {{mapping| 11 0 43 -4 73 128 | 0 1 -1 2 -2 -5 }}
Commas: 540/539, 4000/3993, 5632/5625


POTE generator: ~48/35 = 544.696
Optimal tunings:  
* WE: ~16/15 = 109.0189{{c}}, ~3/2 = 703.4228{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9248{{c}}


Map: [&lt;2 4 4 7 6|, &lt;0 -9 7 -15 10|]
{{Optimal ET sequence|legend=0| 22, 99ef, 121, 341bdeeff }}
EDOs: 22, 86, 108, 130, 152, 282, 434de, 716de
Badness: 0.0246


==13-limit==
Badness (Sintel): 1.49
Commas: 540/539 729/728 4000/3993 21168/21125


POTE generator: ~48/35 = 544.675
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 4 4 7 6 11|, &lt;0 -9 7 -15 10 -39|]
Comma list: 256/255, 352/351, 364/363, 375/374, 540/539
EDOs: 130, 282
Badness: 0.0229


==Septisuperquad==
Mapping: {{mapping| 11 0 43 -4 73 128 45 | 0 1 -1 2 -2 -5 0 }}
Commas: 351/350, 364/363, 540/539, 5632/5625


POTE generator: ~48/35 = 544.641
Optimal tunings:  
* WE: ~16/15 = 109.0159{{c}}, ~3/2 = 703.3932{{c}}
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.9110{{c}}


Map: [&lt;2 4 4 7 6 5|, &lt;0 -9 7 -15 10 26|]
{{Optimal ET sequence|legend=0| 22, 99ef, 121, 220efg, 341bdeeffgg }}
EDOs: 22, 108, 130
Badness: 0.0330


=Whoops=
Badness (Sintel): 1.15
Commas: 6144/6125, 244140625/243045684


POTE generator: ~441/320 = 560.519
=== Icosidillic ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 17 14 -7|, &lt;0 -33 -25 21|]
Comma list: 3388/3375, 6144/6125, 9801/9800
Wedgie: &lt;&lt;33 25 -21 -37 -126 -119||
EDOs: 15, 122d, 137, 152, 608d, 623bd, 775bcd
Badness: 0.1758


==11-limit==
Mapping: {{mapping| 22 0 86 -8 111 | 0 1 -1 2 -1 }}
Commas: 6144/6125, 3025/3024, 4000/3993
: mapping generators: ~33/32, ~3


POTE generator: ~242/175 = 560.519
Optimal tunings:  
* WE: ~33/32 = 54.5305{{c}}, ~3/2 = 702.7206{{c}}
* CWE: ~33/32 = 54.5455{{c}}, ~3/2 = 702.8829{{c}}


Map: [&lt;1 17 14 -7 10|, &lt;0 -33 -25 21 -14|]
{{Optimal ET sequence|legend=0| 22, 154, 176, 198 }}
EDOs: 15, 122d, 137, 152, 608de, 623bde, 775bcde
Badness: 0.0437


=Polypyth=
Badness (Sintel): 1.84
Commas: 6144/6125, 179200/177147


POTE generator: ~3/2 = 704.174
== Nessafof ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nessafof]].''


Map: [&lt;1 0 -31 52|, &lt;0 1 21 -31|]
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's short post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101089.html Yahoo! Tuning Group | ''Some more unclassified temperaments'']</ref>, nessafof adds the [[landscape comma]] and has a third-octave period. The name actually refers to the fact that it has a neutral-second generator, and that a semi-augmented fourth, stacked five times, makes 5/1<ref name="petr's long post"/>.  
EDOs: 46, 121, 167, 288b, 455bcd, 743bcd
Badness: 0.13800


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 896/891, 2200/2187, 6144/6125


POTE generator: ~3/2 = 704.177
[[Comma list]]: 6144/6125, 250047/250000


Map: [&lt;1 0 -31 52 59|, &lt;0 1 21 -31 -35|]
{{Mapping|legend=1| 3 2 5 10 | 0 7 5 -4 }}
EDOs: 46, 121, 167, 288be, 455bcde
: mapping generators: ~63/50, ~35/32
Badness: 0.0511


==13-limit==
[[Optimal tuning]]s:
Commas: 325/324, 352/351, 364/363, 1716/1715
* [[WE]]: ~63/50 = 399.9023{{c}}, ~35/32 = 157.4418{{c}}
: [[error map]]: {{val| -0.293 -0.057 +0.407 +0.430 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~35/32 = 157.4658{{c}}
: error map: {{val| 0.000 +0.306 1.016 +1.311 }}


POTE generator: ~3/2 = 704.168
{{Optimal ET sequence|legend=1| 15, 54b, 69, 84, 99, 282, 381 }}


Map: [&lt;1 0 -31 52 59 64|, &lt;0 1 21 -31 -35 -38|]
[[Badness]] (Sintel): 1.14
EDOs: 46, 121, 167, 288be
Badness: 0.0303


==17-limit==
=== Nessa ===
Commas: 256/255, 325/324, 352/351, 364/363, 1716/1715
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 704.168
Comma list: 441/440, 1344/1331, 4375/4356


Map: [&lt;1 0 -31 52 59 64 39|, &lt;0 1 21 -31 -35 -38 -22|]
Mapping: {{mapping| 3 2 5 10 10 | 0 7 5 -4 1 }}
EDOs: 46, 121, 167, 288beg
Badness: 0.0191


Optimal tunings:
* WE: ~44/35 = 399.7815{{c}}, ~35/32 = 157.4527{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.5109{{c}}


{{Optimal ET sequence|legend=0| 15, 69, 84, 99e }}


Badness (Sintel): 1.61


</pre></div>
==== 13-limit ====
<h4>Original HTML content:</h4>
Subgroup: 2.3.5.7.11.13
<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;Porwell temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:42:&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:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;a href="#Hendecatonic"&gt;Hendecatonic&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; | &lt;a href="#Hemischis"&gt;Hemischis&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt; | &lt;a href="#Twothirdtonic"&gt;Twothirdtonic&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt; | &lt;a href="#Nessafof"&gt;Nessafof&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt; | &lt;a href="#Septisuperfourth"&gt;Septisuperfourth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt; | &lt;a href="#Whoops"&gt;Whoops&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt; | &lt;a href="#Polypyth"&gt;Polypyth&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;hr /&gt;
Comma list: 144/143, 364/363, 441/440, 625/624
This family of temperaments tempers out the &lt;em&gt;porwell comma&lt;/em&gt;, |11 1 -3 -2&amp;gt; = 6144/6125, and includes hemiwuerschmidt, orwell, amity, valentine, porcupine, hendecatonic, shrutar, hexadecimal, grendel, hemikleismic, mohajira, twothirdtonic and nessafof.&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 2 5 10 10 6 | 0 7 5 -4 1 13 }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Hendecatonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Hendecatonic&lt;/h1&gt;
 
Commas: 6144/6125, 10976/10935&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~44/35 = 399.7595{{c}}, ~35/32 = 157.3348{{c}}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 703.054&lt;br /&gt;
* CWE: ~44/35 = 400.0000{{c}}, ~35/32 = 157.3955{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;11 0 43 -4], &amp;lt;0 1 -1 2|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15, 69, 84, 99ef, 183ef, 282eeff }}
Wedgie: &amp;lt;&amp;lt;11 -11 22 -43 4 82||&lt;br /&gt;
 
EDOs: 22, 55, 77, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;&lt;br /&gt;
Badness (Sintel): 1.55
Badness: 0.0411&lt;br /&gt;
 
&lt;br /&gt;
=== Fof ===
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Hendecatonic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11
Commas: 121/120, 176/175, 10976/10935&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 121/120, 176/175, 250047/250000
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~3/2 = 702.636&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 2 5 10 8 | 0 7 5 -4 6 }}
Map: [&amp;lt;11 0 43 -4 38], &amp;lt;0 1 -1 2 0|]&lt;br /&gt;
 
EDOs: 22, 55, 77, 99, 176e, 275e&lt;br /&gt;
Optimal tunings:  
Badness: 0.0461&lt;br /&gt;
* WE: ~63/50 = 400.0266{{c}}, ~12/11 = 157.5301{{c}}
&lt;br /&gt;
* CWE: ~63/50 = 400.0000{{c}}, ~12/11 = 157.5240{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Hendecatonic-Icosidillic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Icosidillic&lt;/h2&gt;
 
Commas: 3388/3375, 6144/6125, 9801/9800&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15, 69e, 84e, 99 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 702.914&lt;br /&gt;
Badness (Sintel): 2.26
&lt;br /&gt;
 
Map: [&amp;lt;22 0 86 -8 111|, &amp;lt;0 1 -1 2 -1|]&lt;br /&gt;
== Grendel ==
EDOs: 22, 154, 176, 198, 968, 1166&lt;br /&gt;
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Counterwürschmidt]].''
Badness: 0.0577&lt;br /&gt;
 
&lt;br /&gt;
Grendel tempers out 16875/16807, the [[mirkwai comma]], and may be described as the {{nowrap| 31 & 152 }} temperament. [[152edo]], [[183edo]] and especially [[335edo]] serve as good tunings.
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Hemischis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Hemischis&lt;/h1&gt;
 
Commas: 6144/6125, 19683/19600&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
POTE generator: ~81/70 = 249.203&lt;br /&gt;
[[Comma list]]: 6144/6125, 16875/16807
&lt;br /&gt;
 
Map: [&amp;lt;1 0 15 -17|, &amp;lt;0 2 -16 25|]&lt;br /&gt;
{{Mapping|legend=1| 1 -14 3 -6 | 0 23 -1 13 }}
Wedgie: &amp;lt;&amp;lt;2 -16 25 -30 34 103||&lt;br /&gt;
: mapping generators: ~2, ~8/5
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313&lt;br /&gt;
 
Badness: 0.0458&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1199.7348{{c}}, ~8/5 = 812.9574{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Hemischis-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
: [[error map]]: {{val| -0.265 -0.220 -0.067 +1.212 }}
Commas: 540/539, 8019/8000, 5632/5625&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 813.1311{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +0.059 +0.555 +1.878 }}
POTE generator: ~81/70 = 249.199&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 90, 121, 152, 335d, 822dd }}
Map: [&amp;lt;1 0 15 -17 51|, &amp;lt;0 2 -16 25 -60|]&lt;br /&gt;
 
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313&lt;br /&gt;
[[Badness]] (Sintel): 1.31
Badness: 0.0363&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Hemischis-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11
Commas: 351/350, 540/539, 676/675, 4096/4095&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 540/539, 1375/1372, 5632/5625
POTE generator: ~15/13 = 249.199&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -14 3 -6 -25 | 0 23 -1 13 42 }}
Map: [&amp;lt;1 0 15 -17 51 14|, &amp;lt;0 2 -16 25 -60 -13|]&lt;br /&gt;
 
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 313&lt;br /&gt;
Optimal tunings:  
Badness: 0.0208&lt;br /&gt;
* WE: ~2 = 1199.7355{{c}}, ~8/5 = 812.9622{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1353{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Hemischis-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;17-limit&lt;/h2&gt;
 
Commas: 351/350, 442/441, 561/560, 676/675, 4096/4095&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152, 335d, 487d }}
&lt;br /&gt;
 
POTE generator: ~15/13 = 249.190&lt;br /&gt;
Badness (Sintel): 0.656
&lt;br /&gt;
 
Map: [&amp;lt;1 0 15 -17 51 14|, &amp;lt;0 2 -16 25 -60 -13|]&lt;br /&gt;
=== 13-limit ===
EDOs: 5, 9, 14, 19, 24, 29, 53, 130, 183, 679&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0211&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 352/351, 540/539, 625/624, 1375/1372
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Twothirdtonic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Twothirdtonic&lt;/h1&gt;
 
Commas: 686/675, 6144/6125&lt;br /&gt;
Mapping: {{mapping| 1 -14 3 -6 -25 22 | 0 23 -1 13 42 -27 }}
&lt;br /&gt;
 
POTE generator: ~15/14 = 130.401&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.4412{{c}}, ~8/5 = 812.7956{{c}}
Map: [&amp;lt;1 3 2 4|, &amp;lt;0 -13 3 -11|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1209{{c}}
Wedgie: &amp;lt;&amp;lt;13 -3 11 -35 -19 34||&lt;br /&gt;
 
EDOs: 9, 10, 19, 28, 37, 46&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152f, 273def, 425deff }}
Badness: 0.0996&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.03
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Twothirdtonic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
 
Commas: 121/120, 176/175, 686/675&lt;br /&gt;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
POTE generator: ~15/14 = 130.430&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 256/255, 352/351, 625/624, 715/714, 1275/1274
Map: [&amp;lt;1 3 2 4 4|, &amp;lt;0 -13 3 -11 -5|]&lt;br /&gt;
 
EDOs: 9, 10, 19, 28, 37, 46&lt;br /&gt;
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 | 0 23 -1 13 42 -27 -22 }}
Badness: 0.0408&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Twothirdtonic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;13-limit&lt;/h2&gt;
* WE: ~2 = 1199.3029{{c}}, ~8/5 = 812.7156{{c}}
Commas: 91/90, 121/120, 169/168, 176/175&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1843{{c}}
&lt;br /&gt;
 
POTE generator: ~15/14 = 130.409&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}
&lt;br /&gt;
 
Map: [&amp;lt;1 3 2 4 4 5|, &amp;lt;0 -13 3 -11 -5 -12|]&lt;br /&gt;
Badness (Sintel): 1.09
EDOs: 9, 10, 19, 28, 37, 46&lt;br /&gt;
 
Badness: 0.0259&lt;br /&gt;
=== 19-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Nessafof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Nessafof&lt;/h1&gt;
 
Commas: 6144/6125, 250047/250000&lt;br /&gt;
Comma list: 256/255, 352/351, 375/374, 400/399, 456/455, 715/714
&lt;br /&gt;
 
POTE generator: ~35/32 = 157.420&lt;br /&gt;
Mapping: {{mapping| 1 -14 3 -6 -25 22 19 30 | 0 23 -1 13 42 -27 -22 -38 }}
&lt;br /&gt;
 
Map: [&amp;lt;3 2 5 10|, &amp;lt;0 7 5 -4|]&lt;br /&gt;
Optimal tunings:  
Wedgie: &amp;lt;&amp;lt;21 15 -12 -25 -78 -70||&lt;br /&gt;
* WE: ~2 = 1199.3587{{c}}, ~8/5 = 812.7462{{c}}
EDOs: 15, 69, 84, 99, 282, 381&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~8/5 = 813.1796{{c}}
Badness: 0.0450&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 90e, 121, 152fg, 273defgg }}
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Septisuperfourth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Septisuperfourth&lt;/h1&gt;
 
Commas: 6144/6125, 118098/117649&lt;br /&gt;
Badness (Sintel): 1.12
&lt;br /&gt;
 
POTE generator: ~48/35 = 544.680&lt;br /&gt;
== Twothirdtonic ==
&lt;br /&gt;
Twothirdtonic tempers out 686/675, the [[senga]], in addition to the porwell comma, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.
Map: [&amp;lt;2 4 4 7|, &amp;lt;0 -9 7 -15|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;18 -14 30 -64 -3 109||&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 22, 86, 108, 130, 152, 282&lt;br /&gt;
 
Badness: 0.0592&lt;br /&gt;
[[Comma list]]: 686/675, 6144/6125
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Septisuperfourth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}
Commas: 540/539, 4000/3993, 5632/5625&lt;br /&gt;
: mapping generators: ~2, ~28/15
&lt;br /&gt;
 
POTE generator: ~48/35 = 544.696&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
Map: [&amp;lt;2 4 4 7 6|, &amp;lt;0 -9 7 -15 10|]&lt;br /&gt;
: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
EDOs: 22, 86, 108, 130, 152, 282, 434de, 716de&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
Badness: 0.0246&lt;br /&gt;
: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Septisuperfourth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}
Commas: 540/539 729/728 4000/3993 21168/21125&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.52
POTE generator: ~48/35 = 544.675&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
Map: [&amp;lt;2 4 4 7 6 11|, &amp;lt;0 -9 7 -15 10 -39|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 130, 282&lt;br /&gt;
 
Badness: 0.0229&lt;br /&gt;
Comma list: 121/120, 176/175, 686/675
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Septisuperfourth-Septisuperquad"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Septisuperquad&lt;/h2&gt;
Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}
Commas: 351/350, 364/363, 540/539, 5632/5625&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~48/35 = 544.641&lt;br /&gt;
* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}
Map: [&amp;lt;2 4 4 7 6 5|, &amp;lt;0 -9 7 -15 10 26|]&lt;br /&gt;
 
EDOs: 22, 108, 130&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
Badness: 0.0330&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.35
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Whoops"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Whoops&lt;/h1&gt;
 
Commas: 6144/6125, 244140625/243045684&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~441/320 = 560.519&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 91/90, 121/120, 169/168, 176/175
Map: [&amp;lt;1 17 14 -7|, &amp;lt;0 -33 -25 21|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;33 25 -21 -37 -126 -119||&lt;br /&gt;
Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}
EDOs: 15, 122d, 137, 152, 608d, 623bd, 775bcd&lt;br /&gt;
 
Badness: 0.1758&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Whoops-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;11-limit&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}
Commas: 6144/6125, 3025/3024, 4000/3993&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}
POTE generator: ~242/175 = 560.519&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.07
Map: [&amp;lt;1 17 14 -7 10|, &amp;lt;0 -33 -25 21 -14|]&lt;br /&gt;
 
EDOs: 15, 122d, 137, 152, 608de, 623bde, 775bcde&lt;br /&gt;
== Semaja ==
Badness: 0.0437&lt;br /&gt;
{{See also| Llywelynsmic clan }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="Polypyth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Polypyth&lt;/h1&gt;
Cryptically named by [[Petr Pařízek]] in 2011, semaja adds the [[gariboh comma]] to the comma list, and may be described as the {{nowrap| 37 & 53 }} temperament. Its [[ploidacot]] is gamma-19-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). The name actually refers to the fact that two of its ~[[8/7]] generator steps reach a ~[[13/10]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
Commas: 6144/6125, 179200/177147&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
POTE generator: ~3/2 = 704.174&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 3125/3087, 6144/6125
Map: [&amp;lt;1 0 -31 52|, &amp;lt;0 1 21 -31|]&lt;br /&gt;
 
EDOs: 46, 121, 167, 288b, 455bcd, 743bcd&lt;br /&gt;
{{Mapping|legend=1| 1 -2 1 3 | 0 19 7 -1 }}
Badness: 0.13800&lt;br /&gt;
: mapping generators: ~2, ~8/7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Polypyth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;11-limit&lt;/h2&gt;
[[Optimal tuning]]s:  
Commas: 896/891, 2200/2187, 6144/6125&lt;br /&gt;
* [[WE]]: ~2 = 1199.4860{{c}}, ~8/7 = 226.3864{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.514 +0.415 -2.123 +3.246 }}
POTE generator: ~3/2 = 704.177&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.4697{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +0.970 -1.026 +4.704 }}
Map: [&amp;lt;1 0 -31 52 59|, &amp;lt;0 1 21 -31 -35|]&lt;br /&gt;
 
EDOs: 46, 121, 167, 288be, 455bcde&lt;br /&gt;
{{Optimal ET sequence|legend=1| 16, 37, 53, 196d }}
Badness: 0.0511&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.71
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Polypyth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;13-limit&lt;/h2&gt;
 
Commas: 325/324, 352/351, 364/363, 1716/1715&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~3/2 = 704.168&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 121/120, 176/175, 3125/3087
Map: [&amp;lt;1 0 -31 52 59 64|, &amp;lt;0 1 21 -31 -35 -38|]&lt;br /&gt;
 
EDOs: 46, 121, 167, 288be&lt;br /&gt;
Mapping: {{mapping| 1 -2 1 3 1 | 0 19 7 -1 13 }}
Badness: 0.0303&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Polypyth-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;17-limit&lt;/h2&gt;
* WE: ~2 = 1199.9818{{c}}, ~8/7 = 226.4821{{c}}
Commas: 256/255, 325/324, 352/351, 364/363, 1716/1715&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4851{{c}}
&lt;br /&gt;
 
POTE generator: ~3/2 = 704.168&lt;br /&gt;
{{Optimal ET sequence|legend=0| 16, 37, 53 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -31 52 59 64 39|, &amp;lt;0 1 21 -31 -35 -38 -22|]&lt;br /&gt;
Badness (Sintel): 1.98
EDOs: 46, 121, 167, 288beg&lt;br /&gt;
 
Badness: 0.0191&lt;/body&gt;&lt;/html&gt;</pre></div>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 169/168, 176/175, 275/273
 
Mapping: {{mapping| 1 -2 1 3 1 2 | 0 19 7 -1 13 9 }}
 
Optimal tunings:  
* WE: ~2 = 1200.1020{{c}}, ~8/7 = 226.4987{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.4822{{c}}
 
{{Optimal ET sequence|legend=0| 16, 37, 53 }}
 
Badness (Sintel): 1.35
 
== Aufo ==
:''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
 
Also named by [[Petr Pařízek]] in 2011, ''aufo'' refers to the augmented fourth, which is a generator of this temperament<ref name="petr's long post"/>. The functional generator however is the [[64/45]] diminished fifth, and like its [[untriton]] variant, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.  
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175616
 
{{Mapping|legend=1| 1 -3 12 -14 | 0 9 -19 33 }}
: mapping generators: ~2, ~64/45
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9758{{c}}, ~64/45 = 611.2055{{c}}
: [[error map]]: {{val| -0.024 -1.303 +0.491 +1.295 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 611.2177{{c}}
: error map: {{val| 0.000 -0.996 +0.551 +1.357 }}
 
{{Optimal ET sequence|legend=1| 53, 161, 214 }}
 
[[Badness]] (Sintel): 3.07
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 176/175, 177147/175616
 
Mapping: {{mapping| 1 -3 12 -14 6 | 0 9 -19 33 -5 }}
 
Optimal tunings:  
* WE: ~2 = 1200.4500{{c}}, ~64/45 = 611.4185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.1918{{c}}
 
{{Optimal ET sequence|legend=0| 53, 108e, 161e }}
 
Badness (Sintel): 2.93
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 176/175, 351/350, 58806/57967
 
Mapping: {{mapping| 1 -3 12 -14 6 20 | 0 9 -19 33 -5 -32 }}
 
Optimal tunings:  
* WE: ~2 = 1200.3134{{c}}, ~64/45 = 611.3715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2118{{c}}
 
{{Optimal ET sequence|legend=0| 53, 108e }}
 
Badness (Sintel): 2.42
 
=== Aufic ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 5632/5625, 72171/71680
 
Mapping: {{mapping| 1 -3 12 -14 33 | 0 9 -19 33 -58 }}
 
Optimal tunings:
* WE: ~2 = 1200.0668{{c}}, ~64/45 = 611.2342{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2000{{c}}
 
{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375 }}
 
Badness (Sintel): 2.48
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845, 4096/4095
 
Mapping: {{mapping| 1 -3 12 -14 33 20 | 0 9 -19 33 -58 -32 }}
 
Optimal tunings:
* WE: ~2 = 1200.0177{{c}}, ~64/45 = 611.2130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/45 = 611.2039{{c}}
 
{{Optimal ET sequence|legend=0| 53, 108, 161, 214, 375 }}
 
Badness (Sintel): 1.61
 
== Absurdity ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Absurdity (5-limit)]].''
{{See also| Fifth-chroma temperaments }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 177147/175000
 
{{Mapping|legend=1| 7 0 -17 64 | 0 1 3 -4 }}
: mapping generators: ~972/875, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~972/875 = 171.4382{{c}}, ~3/2 = 700.6247{{c}}
: [[error map]]: {{val| +0.067 -1.263 +1.313 +0.450 }}
* [[CWE]]: ~972/875 = 171.4286{{c}}, ~3/2 = 700.5871{{c}}
: error map: {{val| 0.000 -1.368 +1.162 +0.254 }}
 
{{Optimal ET sequence|legend=1| 77, 84, 161 }}
 
[[Badness]] (Sintel): 3.38
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 72171/71680
 
Mapping: {{mapping| 7 0 -17 64 124 | 0 1 3 -4 -9 }}
 
Optimal tunings:
* WE: ~495/448 = 171.4346{{c}}, ~3/2 = 700.6602{{c}}
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 700.6339{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
Badness (Sintel): 2.70
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 1188/1183, 3584/3575
 
Mapping: {{mapping| 7 0 -17 64 124 37 | 0 1 3 -4 -9 -1 }}
 
Optimal tunings:
* WE: ~72/65 = 171.4223{{c}}, ~3/2 = 700.6036{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6306{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
Badness (Sintel): 1.72
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 1188/1183, 1632/1625
 
Mapping: {{mapping| 7 0 -17 64 124 37 -49 | 0 1 3 -4 -9 -1 7 }}
 
Optimal tunings:  
* WE: ~72/65 = 171.4263{{c}}, ~3/2 = 700.6429{{c}}
* CWE: ~72/65 = 171.4286{{c}}, ~3/2 = 700.6525{{c}}
 
{{Optimal ET sequence|legend=0| 77, 161 }}
 
Badness (Sintel): 1.62
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 | 0 1 3 -4 -9 -1 7 -3 }}
 
Optimal tunings:  
* WE: ~21/19 = 171.4244{{c}}, ~3/2 = 700.6395{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6568{{c}}
 
{{Optimal ET sequence|legend=0| 77, 161 }}
 
Badness (Sintel): 1.36
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 | 0 1 3 -4 -9 -1 7 -3 -4 }}
 
Optimal tunings:  
* WE: ~21/19 = 171.4321{{c}}, ~3/2 = 700.6475{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6325{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
Badness (Sintel): 1.34
 
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 261/260, 276/275, 324/323, 351/350, 441/440, 456/455, 476/475, 495/494
 
Mapping: {{mapping| 7 0 -17 64 124 37 -49 63 76 34 | 0 1 3 -4 -9 -1 7 -3 -4 0 }}
 
Optimal tunings:  
* WE: ~21/19 = 171.4348{{c}}, ~3/2 = 700.6612{{c}}
* CWE: ~21/19 = 171.4286{{c}}, ~3/2 = 700.6351{{c}}
 
{{Optimal ET sequence|legend=0| 77, 84, 161 }}
 
Badness (Sintel): 1.25
 
== Polypyth ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''
 
Polypyth tempers out the same 5-limit comma as [[leapday]], with which it shares the similarly sharp [[3/2|perfect-fifth]] generator, but the porwell comma (6144/6125) rather than the hemifamity comma (5120/5103) is tempered out here. It may be described as the {{nowrap| 46 & 121 }} temperament, and [[121edo]] and [[167edo]] make for good tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 179200/177147
 
{{Mapping|legend=1| 1 0 -31 52 | 0 1 21 -31 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.3465{{c}}, ~3/2 = 703.7905{{c}}
: [[error map]]: {{val| -0.654 +1.182 -0.177 -0.056 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1749{{c}}
: error map: {{val| 0.000 +2.220 +1.359 +1.752 }}
 
{{Optimal ET sequence|legend=1| 46, 121, 167, 288b, 455bcd }}
 
[[Badness]] (Sintel): 3.49
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 896/891, 2200/2187, 6144/6125
 
Mapping: {{mapping| 1 0 -31 52 59 | 0 1 21 -31 -35 }}
 
Optimal tunings:  
* WE: ~2 = 1199.3335{{c}}, ~3/2 = 703.7856{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1812{{c}}
 
{{Optimal ET sequence|legend=0| 46, 121, 167, 288be, 455bcde }}
 
Badness (Sintel): 1.69
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 364/363, 1716/1715
 
Mapping: {{mapping| 1 0 -31 52 59 64 | 0 1 21 -31 -35 -38 }}
 
Optimal tunings:
* WE: ~2 = 1199.3768{{c}}, ~3/2 = 703.8018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1731{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288be }}
 
Badness (Sintel): 1.25
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 325/324, 352/351, 364/363, 1716/1715
 
Mapping: {{mapping| 1 0 -31 52 59 64 39 | 0 1 21 -31 -35 -38 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.3518{{c}}, ~3/2 = 703.7880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1747{{c}}
 
{{Optimal ET sequence|legend=0| 46, 75e, 121, 167, 288beg }}
 
Badness (Sintel): 0.971
 
== Whoops ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Whoosh]].''
 
Also named by [[Petr Pařízek]] in 2011, whoops is a relatively simple extension to the otherwise very accurate microtemperament known as ''whoosh''<ref name="petr's long post"/>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 244140625/243045684
 
{{Mapping|legend=1| 1 -16 -11 14 | 0 33 25 -21 }}
: mapping generators: ~2, ~640/441
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5944{{c}}, ~640/441 = 639.2648{{c}}
: [[error map]]: {{val| -0.406 +0.272 -0.233 +0.936 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~640/441 = 639.4769{{c}}
: error map: {{val| 0.000 +0.783 +0.609 +2.159 }}
 
{{Optimal ET sequence|legend=1| 15, 122d, 137, 152, 623bdd, 775bcdd, 927bcddd, 1079bcddd }}
 
[[Badness]] (Sintel): 4.45
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4000/3993, 6144/6125
 
Mapping: {{mapping| 1 -16 -11 14 -4 | 0 33 25 -21 14 }}
 
Optimal tunings:
* WE: ~2 = 1199.5936{{c}}, ~175/121 = 639.264{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~175/121 = 639.4770{{c}}
 
{{Optimal ET sequence|legend=0| 15, 122d, 137, 152, 623bdde, 775bcdde, 927bcdddee, 1079bcdddee }}
 
Badness (Sintel): 1.45
 
== Dodifo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Dodifo]].''
 
Also named by [[Petr Pařízek]] in 2011, ''dodifo'' refers to the (tetraptolemaic) double-diminished fourth, which is a generator of this temperament<ref name="petr's long post"/>. The extension here is a less accurate 7-limit interpretation.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 2500000/2470629
 
{{Mapping|legend=1| 1 -23 -4 0 | 0 35 9 4 }}
: mapping generators: ~2, ~80/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6429{{c}}, ~80/49 = 842.6790{{c}}
: [[error map]]: {{val| -0.357 +0.228 -0.774 +1.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9243{{c}}
: error map: {{val| 0.000 +0.396 +0.005 +2.871 }}
 
{{Optimal ET sequence|legend=1| 37, 84, 121, 205 }}
 
[[Badness]] (Sintel): 4.55
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 2560/2541, 4375/4356
 
Mapping: {{mapping| 1 -23 -4 0 14 | 0 35 9 4 -15 }}
 
Optimal tunings:  
* WE: ~2 = 1199.3401{{c}}, ~80/49 = 842.4880{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 842.9457{{c}}
 
{{Optimal ET sequence|legend=0| 37, 84, 121, 326dee }}
 
Badness (Sintel): 2.71
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 625/624, 640/637, 1375/1372
 
Mapping: {{mapping| 1 12 5 4 -1 4 | 0 -35 -9 -4 15 -1 }}
 
Optimal tunings:  
* WE: ~2 = 1199.3410{{c}}, ~13/8 = 842.4885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9466{{c}}
 
{{Optimal ET sequence|legend=0| 37, 84, 121, 326deef }}
 
Badness (Sintel): 1.63
 
== Icositritonic ==
{{See also| 23rd-octave temperaments }}
 
Icositritonic has a period of 1/23 octave, so six period represents [[6/5]] and nine period represents [[21/16]]. It may be described as {{nowrap| 46 & 161 }}. It was named by [[Xenllium]] in 2019 for its number of periods per octave.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 9920232/9765625
 
{{Mapping|legend=1| 23 0 17 101 | 0 1 1 -1 }}
: mapping generators: ~1323/1280, ~3
 
[[Optimal tuning]]s:  
* [[WE]]: ~1323/1280 = 52.1732{{c}}, ~3/2 = 701.0660{{c}}
: [[error map]]: {{val| -0.017 -0.906 +1.679 -0.386 }}
* [[CWE]]: ~1323/1280 = 52.1739{{c}}, ~3/2 = 701.0722{{c}}
: error map: {{val| 0.000 -0.883 +1.715 -0.333 }}
 
{{Optimal ET sequence|legend=1| 46, 115, 161, 207, 368c }}
 
[[Badness]] (Sintel): 4.98
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 35937/35840
 
Mapping: {{mapping| 23 0 17 101 116 | 0 1 1 -1 -1 }}
 
Optimal tunings:  
* WE: ~33/32 = 52.1740{{c}}, ~3/2 = 701.0379{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0370{{c}}
 
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
 
Badness (Sintel): 2.14
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 441/440, 847/845, 3584/3575
 
Mapping: {{mapping| 23 0 17 101 116 158 | 0 1 1 -1 -1 -2 }}
 
Optimal tunings:
* WE: ~33/32 = 52.1724{{c}}, ~3/2 = 701.1310{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1524{{c}}
 
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
 
Badness (Sintel): 1.67
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 441/440, 561/560, 847/845, 1089/1088
 
Mapping: {{mapping| 23 0 17 101 116 158 94 | 0 1 1 -1 -1 -2 0 }}
 
Optimal tunings:
* WE: ~33/32 = 52.1735{{c}}, ~3/2 = 701.1493{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.1549{{c}}
 
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
 
Badness (Sintel): 1.26
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 351/350, 441/440, 456/455, 476/475, 513/512, 847/845
 
Mapping: {{mapping| 23 0 17 101 116 158 94 207 | 0 1 1 -1 -1 -2 0 -3 }}
 
Optimal tunings:
* WE: ~33/32 = 52.1744{{c}}, ~3/2 = 701.0649{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0582{{c}}
 
{{Optimal ET sequence|legend=0| 46, 115, 161, 207, 368c }}
 
Badness (Sintel): 1.31
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 276/275, 351/350, 391/390, 441/440, 456/455, 476/475, 847/845
 
Mapping: {{mapping| 23 0 17 101 116 158 94 207 104 | 0 1 1 -1 -1 -2 0 -3 0 }}
 
Optimal tunings:
* WE: ~33/32 = 52.1768{{c}}, ~3/2 = 701.1259{{c}}
* CWE: ~33/32 = 52.1739{{c}}, ~3/2 = 701.0841{{c}}
 
{{Optimal ET sequence|legend=0| 46, 115, 161, 207 }}
 
Badness (Sintel): 1.27
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Porwell temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porwell_temperaments"