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Octagar 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]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-27 14:54:23 UTC</tt>.<br>
: The original revision id was <tt>504840724</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]]


Octagar temperaments temper out the octagar comma, |5 -4 3 -2&gt; = 4000/3969.
Temperaments discussed elsewhere are:
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
* [[Superkleismic]] (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]
* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
* ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]


=Slendi=
Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
Commas: 4000/3969, 84035/82944


POTE generator: ~49/48 = 29.181
== Pluto ==
{{Distinguish| Plutus }}


Map: [&lt;1 2 3 3|, &lt;0 -17 -28 -8|]
Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[~]][[10/7]] [[tritone]], seven of which give the [[interval class]] of [[3/1|3]], and the [[ploidacot]] for this temperament is gamma-heptacot. [[121edo|59\121]] is about perfect as a tuning.  
Wedgie: &lt;&lt;17 28 8 5 -35 -60||
EDOs: 41, 206, 247bc, 288bcd, 329bcd
Badness: 0.1429


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 100/99, 245/242, 46656/46585


POTE generator: ~49/48 = 29.180
[[Comma list]]: 4000/3969, 10976/10935


Map: [&lt;1 2 3 3 4|, &lt;0 -17 -28 -8 -22|]
{{Mapping|legend=1| 1 -2 -11 -10 | 0 7 26 25 }}
EDOs: 41, 206e
: mapping generators: ~2, ~10/7
Badness: 0.0632


==13-limit==
[[Optimal tuning]]s:
Commas: 100/99, 144/143, 245/242, 343/338
* [[WE]]: ~2 = 1199.4720{{c}}, ~10/7 = 614.5826{{c}}
: [[error map]]: {{val| -0.528 +1.179 -1.358 +1.020 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.8282{{c}}
: error map: {{val| 0.000 +1.843 -0.780 +1.880 }}


POTE generator: ~49/48 = 29.125
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 283d, 404bd }}


Map: [&lt;1 2 3 3 4 4|, &lt;0 -17 -28 -8 -22 -12|]
[[Badness]] (Sintel): 1.46
EDOs: 41, 124ef, 165ef
Badness: 0.0436


=Slithy=
=== 11-limit ===
Commas: 4000/3969, 420175/419904
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 89.282
Comma list: 540/539, 896/891, 1375/1372


Map: [&lt;1 3 5 4|, &lt;0 -19 -36 -16|]
Mapping: {{mapping| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}
EDOs: 27, 94, 121, 215, 336d
Badness: 0.1075


==11-limit==
Optimal tunings:
Commas: 540/539, 2200/2187, 4375/4356
* WE: ~2 = 1200.1835{{c}}, ~10/7 = 614.4677{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8616{{c}}


POTE generator: ~21/20 = 89.291
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}


Map: [&lt;1 3 5 4 8|, &lt;0 -19 -36 -16 -61|]
Badness (Sintel): 0.987
EDOs: 94, 121, 215, 336de, 551bde
Badness: 0.0459


==13-limit==
==== 13-limit ====
Commas: 325/324, 352/351, 540/539, 1575/1573
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 89.286
Comma list: 325/324, 352/351, 364/363, 540/539


Map: [&lt;1 3 5 4 8 4|, &lt;0 -19 -36 -16 -61 -4|]
Mapping: {{mapping| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}
EDOs: 94, 121, 215, 336def
 
Badness: 0.0285
Optimal tunings:
</pre></div>
* WE: ~2 = 1199.2469{{c}}, ~10/7 = 614.4908{{c}}
<h4>Original HTML content:</h4>
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8684{{c}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Octagar temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&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:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Slendi"&gt;Slendi&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Slithy"&gt;Slithy&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;br /&gt;
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
Octagar temperaments temper out the octagar comma, |5 -4 3 -2&amp;gt; = 4000/3969.&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.06
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Slendi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Slendi&lt;/h1&gt;
 
Commas: 4000/3969, 84035/82944&lt;br /&gt;
===== 17-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
POTE generator: ~49/48 = 29.181&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -17 -28 -8|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;17 28 8 5 -35 -60||&lt;br /&gt;
Mapping: {{mapping| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}
EDOs: 41, 206, 247bc, 288bcd, 329bcd&lt;br /&gt;
 
Badness: 0.1429&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* WE: ~2 = 1199.2009{{c}}, ~10/7 = 614.4746{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Slendi-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;11-limit&lt;/h2&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8813{{c}}
Commas: 100/99, 245/242, 46656/46585&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
POTE generator: ~49/48 = 29.180&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.09
Map: [&amp;lt;1 2 3 3 4|, &amp;lt;0 -17 -28 -8 -22|]&lt;br /&gt;
 
EDOs: 41, 206e&lt;br /&gt;
===== 19-limit =====
Badness: 0.0632&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Slendi-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit&lt;/h2&gt;
Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
Commas: 100/99, 144/143, 245/242, 343/338&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}
POTE generator: ~49/48 = 29.125&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
Map: [&amp;lt;1 2 3 3 4 4|, &amp;lt;0 -17 -28 -8 -22 -12|]&lt;br /&gt;
* WE: ~2 = 1199.2833{{c}}, ~10/7 = 614.5237{{c}}
EDOs: 41, 124ef, 165ef&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8876{{c}}
Badness: 0.0436&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Slithy"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Slithy&lt;/h1&gt;
 
Commas: 4000/3969, 420175/419904&lt;br /&gt;
Badness (Sintel): 1.07
&lt;br /&gt;
 
POTE generator: ~21/20 = 89.282&lt;br /&gt;
==== Orcus ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;1 3 5 4|, &amp;lt;0 -19 -36 -16|]&lt;br /&gt;
 
EDOs: 27, 94, 121, 215, 336d&lt;br /&gt;
Comma list: 144/143, 196/195, 275/273, 896/891
Badness: 0.1075&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Slithy-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
 
Commas: 540/539, 2200/2187, 4375/4356&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1198.7372{{c}}, ~10/7 = 614.2418{{c}}
POTE generator: ~21/20 = 89.291&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8502{{c}}
&lt;br /&gt;
 
Map: [&amp;lt;1 3 5 4 8|, &amp;lt;0 -19 -36 -16 -61|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 39df, 41, 80f, 121ff }}
EDOs: 94, 121, 215, 336de, 551bde&lt;br /&gt;
 
Badness: 0.0459&lt;br /&gt;
Badness (Sintel): 1.38
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Slithy-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
=== Plutino ===
Commas: 325/324, 352/351, 540/539, 1575/1573&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
POTE generator: ~21/20 = 89.286&lt;br /&gt;
Comma list: 100/99, 245/242, 10976/10935
&lt;br /&gt;
 
Map: [&amp;lt;1 3 5 4 8 4|, &amp;lt;0 -19 -36 -16 -61 -4|]&lt;br /&gt;
Mapping: {{mapping| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}
EDOs: 94, 121, 215, 336def&lt;br /&gt;
 
Badness: 0.0285&lt;/body&gt;&lt;/html&gt;</pre></div>
Optimal tunings:  
* WE: ~2 = 1199.7793{{c}}, ~10/7 = 614.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7074{{c}}
 
{{Optimal ET sequence|legend=0| 39dee, 41 }}
 
Badness (Sintel): 1.92
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 196/195, 245/242, 729/728
 
Mapping: {{mapping| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}
 
Optimal tunings:  
* WE: ~2 = 1199.0328{{c}}, ~10/7 = 614.2729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7277{{c}}
 
{{Optimal ET sequence|legend=0| 39deef, 41 }}
 
Badness (Sintel): 1.66
 
== Slithy ==
Slithy tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 27 & 94 }} temperament with a [[ploidacot]] signature of 17-sheared 19-cot. [[94edo]], [[121edo]] and [[215edo]] can all be used as tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 420175/419904
 
{{Mapping|legend=1| 1 -16 -31 -12 | 0 19 36 16 }}
: mapping generators: ~2, ~40/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5464{{c}}, ~40/21 = 1110.2981{{c}}
: [[error map]]: {{val| -0.454 +0.996 -1.521 +1.387 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7026{{c}}
: error map: {{val| 0.000 +1.394 -1.021 +2.415 }}
 
{{Optimal ET sequence|legend=1| 27, 67c, 94, 121, 215, 336d }}
 
[[Badness]] (Sintel): 2.72
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 2200/2187, 4375/4356
 
Mapping: {{mapping| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}
 
Optimal tunings:  
* WE: ~2 = 1199.5884{{c}}, ~40/21 = 1110.3276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.6969{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 67ce, 94, 121, 215, 336de }}
 
Badness (Sintel): 1.52
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 540/539, 1575/1573
 
Mapping: {{mapping| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}
 
Optimal tunings:
* WE: ~2 = 1199.5032{{c}}, ~40/21 = 1110.2545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7037{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 67ce, 94, 121, 215, 336def }}
 
Badness (Sintel): 1.18
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 325/324, 352/351, 375/374, 540/539, 595/594
 
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}
 
Optimal tunings:
* WE: ~2 = 1199.5051{{c}}, ~40/21 = 1110.2555{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7033{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 67ceg, 94, 121, 215, 336defg }}
 
Badness (Sintel): 1.01
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594
 
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}
 
Optimal tunings:
* WE: ~2 = 1199.6184{{c}}, ~19/10 = 1110.3583{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.7036{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 67cegh, 94, 121, 215 }}
 
Badness (Sintel): 0.998
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483
 
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}
 
Optimal tunings:
* WE: ~2 = 1199.6651{{c}}, ~19/10 = 1110.3928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.6965{{c}}
 
{{Optimal ET sequence|legend=0| 27egi, 94, 121i, 215 }}
 
Badness (Sintel): 1.07
 
== Tridecatonic ==
{{See also| 13th-octave temperaments #Tridecatonic }}
 
Tridecatonic has a period of 1/13 octave, and may be described as {{nowrap| 26 & 39 }}. It tempers out [[devil's tridecalimma]], {{monzo| -11 26 -13 }} in the 5-limit. Its [[ploidacot]] is 13-ploid monocot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 8748/8575
 
{{Mapping|legend=1| 13 0 -11 16 | 0 1 2 1 }}
: mapping generators: ~21/20, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 92.2197{{c}}, ~3/2 = 701.2915{{c}}
: [[error map]]: {{val| -1.144 -1.807 -0.435 +6.837 }}
* [[CWE]]: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2352{{c}}
: error map: {{val| 0.000 -0.720 +0.772 +9.332 }}
 
{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
 
[[Badness]] (Sintel): 3.52
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 99/98, 1944/1925, 2560/2541
 
Mapping: {{mapping| 13 0 -11 16 45 | 0 1 2 1 0 }}
 
Optimal tunings:
* WE: ~21/20 = 92.2333{{c}}, ~3/2 = 701.1371{{c}}
* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2751{{c}}
 
{{Optimal ET sequence|legend=0| 26, 39d, 65d }}
 
Badness (Sintel): 2.09
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 99/98, 144/143, 275/273, 1040/1029
 
Mapping: {{mapping| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}
 
Optimal tunings:
* WE: ~21/20 = 92.2248{{c}}, ~3/2 = 700.7272{{c}}
* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 700.8621{{c}}
 
{{Optimal ET sequence|legend=0| 26, 39df, 65d }}
 
Badness (Sintel): 1.56
 
== Slendi ==
Slendi may be described as the {{nowrap| 41 & 42 }} temperament. It has a generator of [[49/48]], seventeen of which give [[4/3]]. Its [[ploidacot]] is omega-17-cot.
 
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 84035/82944
 
{{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}
: mapping generators: ~2, ~49/48
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2515{{c}}, ~49/48 = 29.1870{{c}}
: [[error map]]: {{val| +0.252 +2.368 -2.796 -1.568 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 29.1715{{c}}
: error map: {{val| 0.000 +2.129 -3.116 -2.198 }}
 
{{Optimal ET sequence|legend=1| 1c, 40c, 41, 206 }}
 
[[Badness]] (Sintel): 3.62
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 46656/46585
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.8027{{c}}, ~49/48 = 29.1751{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1875{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 40c, 41 }}
 
Badness (Sintel): 2.09
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 144/143, 245/242, 343/338
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
 
Optimal tunings:
* WE: ~2 = 1199.0915{{c}}, ~49/48 = 29.1033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1576{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 41 }}
 
Badness (Sintel): 1.80
 
== Dhaivatic ==
Named by [[Xenllium]] in 2023, dhaivatic tempers out the [[rainy comma]] and may be described as the {{nowrap| 15 & 94 }} temperament. Its [[ploidacot]] is epsilon-21-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 2100875/2097152
 
{{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9902{{c}}, ~6/5 = 319.1955{{c}}
: [[error map]]: {{val| -0.010 +1.190 -2.374 +0.769 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 319.1980{{c}}
: error map: {{val| 0.000 +1.202 -2.354 +0.798 }}
 
{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297c }}
 
[[Badness]] (Sintel): 4.26
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 4000/3969
 
Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}
 
Optimal tunings:
* WE: ~2 = 1200.0983{{c}}, ~6/5 = 319.2229{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1985{{c}}
 
{{Optimal ET sequence|legend=0| 15, 64c, 79, 94, 203, 297c }}
 
Badness (Sintel): 1.69
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 275/273, 325/324, 385/384, 1331/1323
 
Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}
 
Optimal tunings:
* WE: ~2 = 1200.0732{{c}}, ~6/5 = 319.1840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1658{{c}}
 
{{Optimal ET sequence|legend=0| 15, 79, 94 }}
 
Badness (Sintel): 1.55
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Octagar_temperaments"