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Hemimage 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 [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-18 15:20:44 UTC</tt>.<br>
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
: The original revision id was <tt>490298826</tt>.<br>
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
: The revision comment was: <tt></tt><br>
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
<h4>Original Wikitext content:</h4>
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
<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]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


== Chromat ==
The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[Amity family|amity extension]] with third-octave period.


Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&gt; = 10976/10935.
[[Subgroup]]: 2.3.5.7


=Commatic=
[[Comma list]]: 10976/10935, 235298/234375


==7-limit==
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
Commas: 10976/10935, 50421/50000


POTE generator: ~81/80 = 20.377
: mapping generators: ~63/50, ~28/27


Map: [&lt;2 3 4 5|, &lt;0 5 19 18|]
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528
Wedgie: &lt;&lt;10 38 36 37 29 -23||
EDOs: 58, 118, 294, 412d, 530d
Badness: 0.0843


==11-limit==
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
Commas: 441/440, 3388/3375, 8019/8000


POTE generator: ~81/80 = 20.390
[[Badness]]: 0.057499


Map: [&lt;2 3 4 5 6|, &lt;0 5 19 18 27|]
=== 11-limit ===
EDOs: 58, 118, 294, 412d
Subgroup: 2.3.5.7.11
Badness: 0.0305


=Chromat=
Comma list: 441/440, 4375/4356, 10976/10935
Commas: 10976/10935, 235298/234375


POTE generator: ~28/27 = 60.528
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


Map: [&lt;3 4 5 6|, &lt;0 5 13 16|]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
Wedgie: &lt;&lt;15 39 48 27 34 2||
EDOs: 60, 99, 258, 357, 456
Badness: 0.0575


=Degrees=
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
Commas: 10976/10935, 390625/388962


POTE generator: ~3/2 = 703.015
Badness: 0.050379


Map: [&lt;20 0 -17 -39|, &lt;0 1 2 3|]
==== 13-limit ====
Wedgie: &lt;&lt;20 40 60 17 39 27||
Subgroup: 2.3.5.7.11.13
EDOs: 60, 80, 140, 640b, 780b, 920b
Badness: 0.1065


==11-limit==
Comma list: 364/363, 441/440, 625/624, 10976/10935
Commas: 1331/1323, 1375/1372, 2200/2187


POTE generator: ~3/2 = 703.231
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


Map: [&lt;20 0 -17 -39 -26|, &lt;0 1 2 3 3|]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428
EDOs: 80, 140, 360, 500be, 860bde
Badness: 0.0468


==13-limit==
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
Commas: 325/324, 352/351, 1001/1000, 1331/1323


POTE generator: ~3/2 = 703.080
Badness: 0.046006


Map: [&lt;20 0 -17 -39 -26 74|, &lt;0 1 2 3 3 0|]
===== 17-limit =====
EDOs: 60e, 80, 140, 500be, 640be, 780be
Subgroup: 2.3.5.7.11.13.17
Badness: 0.0327


=Subfourth=
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
Commas: 10976/10935, 65536/64827


POTE generator: ~21/16 = 475.991
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


Map: [&lt;1 0 17 4|, &lt;0 4 -37 -3|]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438
EDOs: 58, 121, 179, 300bd, 479bcd
Badness: 0.1407


==11-limit==
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
Commas: 540/539, 896/891, 12005/11979


POTE generator: ~21/16 = 475.995
Badness: 0.031678


Map: [&lt;1 0 17 4 11|, &lt;0 4 -37 -3 -19|]
==== Catachrome ====
EDOs: 58, 121, 179e, 300bde
Subgroup: 2.3.5.7.11.13
Badness: 0.0453


==13-limit==
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
Commas: 352/351, 364/363, 540/539, 676/675


POTE generator: ~21/16 = 475.996
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


Map: [&lt;1 0 17 4 11 16|, &lt;0 4 -37 -3 -19 -31|]
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378
EDOs: 58, 121, 179ef, 300bdef
Badness: 0.0238


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Badness: 0.043844


===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


</pre></div>
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
<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;Hemimage temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commatic"&gt;Commatic&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Commatic-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Commatic-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377
&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Chromat"&gt;Chromat&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Degrees"&gt;Degrees&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Degrees-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Degrees-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
Badness: 0.030218
&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Subfourth"&gt;Subfourth&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Subfourth-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
==== Chromic ====
&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Subfourth-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;br /&gt;
Comma list: 196/195, 352/351, 729/728, 1875/1859
&lt;br /&gt;
 
Hemimage temperaments temper out the hemimage comma, |5 -7 -1 3&amp;gt; = 10976/10935.&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Commatic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Commatic&lt;/h1&gt;
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Commatic-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
Commas: 10976/10935, 50421/50000&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.049857
POTE generator: ~81/80 = 20.377&lt;br /&gt;
 
&lt;br /&gt;
===== 17-limit =====
Map: [&amp;lt;2 3 4 5|, &amp;lt;0 5 19 18|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
Wedgie: &amp;lt;&amp;lt;10 38 36 37 29 -23||&lt;br /&gt;
 
EDOs: 58, 118, 294, 412d, 530d&lt;br /&gt;
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
Badness: 0.0843&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Commatic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
 
Commas: 441/440, 3388/3375, 8019/8000&lt;br /&gt;
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459
&lt;br /&gt;
 
POTE generator: ~81/80 = 20.390&lt;br /&gt;
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
&lt;br /&gt;
 
Map: [&amp;lt;2 3 4 5 6|, &amp;lt;0 5 19 18 27|]&lt;br /&gt;
Badness: 0.031043
EDOs: 58, 118, 294, 412d&lt;br /&gt;
 
Badness: 0.0305&lt;br /&gt;
=== Hemichromat ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Chromat"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Chromat&lt;/h1&gt;
 
Commas: 10976/10935, 235298/234375&lt;br /&gt;
Comma list: 3025/3024, 10976/10935, 102487/102400
&lt;br /&gt;
 
POTE generator: ~28/27 = 60.528&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
&lt;br /&gt;
 
Map: [&amp;lt;3 4 5 6|, &amp;lt;0 5 13 16|]&lt;br /&gt;
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
Wedgie: &amp;lt;&amp;lt;15 39 48 27 34 2||&lt;br /&gt;
 
EDOs: 60, 99, 258, 357, 456&lt;br /&gt;
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
Badness: 0.0575&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.067173
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Degrees"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Degrees&lt;/h1&gt;
 
Commas: 10976/10935, 390625/388962&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~3/2 = 703.015&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
Map: [&amp;lt;20 0 -17 -39|, &amp;lt;0 1 2 3|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;20 40 60 17 39 27||&lt;br /&gt;
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
EDOs: 60, 80, 140, 640b, 780b, 920b&lt;br /&gt;
 
Badness: 0.1065&lt;br /&gt;
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Degrees-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
Commas: 1331/1323, 1375/1372, 2200/2187&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.033420
POTE generator: ~3/2 = 703.231&lt;br /&gt;
 
&lt;br /&gt;
== Bisupermajor ==
Map: [&amp;lt;20 0 -17 -39 -26|, &amp;lt;0 1 2 3 3|]&lt;br /&gt;
{{See also| Very high accuracy temperaments #Kwazy }}
EDOs: 80, 140, 360, 500be, 860bde&lt;br /&gt;
 
Badness: 0.0468&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Degrees-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit&lt;/h2&gt;
[[Comma list]]: 10976/10935, 65625/65536
Commas: 325/324, 352/351, 1001/1000, 1331/1323&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
POTE generator: ~3/2 = 703.080&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~1225/864, ~192/175
Map: [&amp;lt;20 0 -17 -39 -26 74|, &amp;lt;0 1 2 3 3 0|]&lt;br /&gt;
 
EDOs: 60e, 80, 140, 500be, 640be, 780be&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
Badness: 0.0327&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Subfourth"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Subfourth&lt;/h1&gt;
 
Commas: 10976/10935, 65536/64827&lt;br /&gt;
[[Badness]]: 0.065492
&lt;br /&gt;
 
POTE generator: ~21/16 = 475.991&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 0 17 4|, &amp;lt;0 4 -37 -3|]&lt;br /&gt;
 
EDOs: 58, 121, 179, 300bd, 479bcd&lt;br /&gt;
Comma list: 385/384, 3388/3375, 9801/9800
Badness: 0.1407&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Subfourth-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 896/891, 12005/11979&lt;br /&gt;
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
&lt;br /&gt;
 
POTE generator: ~21/16 = 475.995&lt;br /&gt;
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 17 4 11|, &amp;lt;0 4 -37 -3 -19|]&lt;br /&gt;
Badness: 0.032080
EDOs: 58, 121, 179e, 300bde&lt;br /&gt;
 
Badness: 0.0453&lt;br /&gt;
== Bicommatic ==
&lt;br /&gt;
Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Subfourth-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;13-limit&lt;/h2&gt;
 
Commas: 352/351, 364/363, 540/539, 676/675&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
POTE generator: ~21/16 = 475.996&lt;br /&gt;
[[Comma list]]: 10976/10935, 50421/50000
&lt;br /&gt;
 
Map: [&amp;lt;1 0 17 4 11 16|, &amp;lt;0 4 -37 -3 -19 -31|]&lt;br /&gt;
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
EDOs: 58, 121, 179ef, 300bdef&lt;br /&gt;
 
Badness: 0.0238&lt;/body&gt;&lt;/html&gt;</pre></div>
: mapping generators: ~567/400, ~81/80
 
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
 
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
 
[[Badness]]: 0.084317
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 3388/3375, 8019/8000
 
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390
 
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
 
Badness: 0.030461
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 729/728, 1001/1000
 
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
 
Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427
 
{{Optimal ET sequence|legend=1| 58, 118, 176f }}
 
Badness: 0.026336
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
 
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
 
Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
 
{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}
 
Badness: 0.022396
 
== Degrees ==
{{ See also | 20th-octave temperaments }}
Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.  
 
An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
 
By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.
 
By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 390625/388962
 
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
 
: mapping generators: ~28/27, ~3
 
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)
 
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
 
[[Badness]]: 0.106471
 
Badness (Sintel): 2.694
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1331/1323, 1375/1372, 2200/2187
 
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
 
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)
 
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}
 
Badness: 0.046770
 
Badness (Sintel): 1.546
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 1001/1000, 1331/1323
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
 
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness: 0.032718
 
Badness (Sintel): 1.352
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 1.171
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 1.273
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)
 
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
 
Badness (Sintel): 1.209
 
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)
 
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
 
Badness (Sintel): 1.134
 
=== no-31's 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)
 
{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
 
Badness (Sintel): 1.127
 
=== no-31's 41-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
 
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
 
Badness (Sintel): 1.100
 
== Squarschmidt ==
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 61 4 -29 }}
 
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
 
: mapping generators: ~2, ~98304/78125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
 
[[Badness]]: 0.218314
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 29360128/29296875
 
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
 
[[Badness]]: 0.132821
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 5632/5625, 10976/10935
 
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
 
Badness: 0.038186
 
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
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