Unicorn family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''unicorn family''' tempers out the [[unicorn comma]] ({{monzo|legend=1| -2 13 -8 }}, [[ratio]]: 1594323/1562500).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-04-26 19:49:21 UTC</tt>.<br>
: The original revision id was <tt>325787840</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]]


The unicorn family tempers out the unicorn comma, 1594323/1562500 = |-2 13 -8&gt;.
== Unicorn ==
Unicorn divides the [[4/3|perfect fourth]] into eight equal parts, three for [[10/9]] and five for [[6/5]]. Its [[ploidacot]] is omega-octacot.  


=Unicorn=
[[Subgroup]]: 2.3.5
Comma: 1594323/1562500


POTE generator: ~250/243 = 62.458
[[Comma list]]: 1594323/1562500


Map: [&lt;1 2 3|, &lt;0 -8 -13|]
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
EDOs: 19, 58, 77, 96, 173, 269
: mapping generators: ~2, ~250/243
Badness: 0.1505


=Alicorn=
[[Optimal tuning]]s:
Commas: 126/125, 10976/10935
* [[WE]]: ~2 = 1200.0889{{c}}, ~250/243 = 62.4623{{c}}
: [[error map]]: {{val| +0.089 -1.476 +1.943 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 62.4494{{c}}
: error map: {{val| 0.000 -1.550 +1.844 }}


POTE generator: ~28/27 = 62.279
{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
[[Badness]] (Sintel): 3.53
EDOs: 19, 58, 77, 96d
Badness: 0.0409


==11-limit==
== Septimal unicorn ==
Commas: 126/125, 540/539, 896/891
The canonical extension to the 7-limit is by interpreting the generator as a very slightly flattened [[~]][[28/27]], corresponding to tempering out 126/125, the [[starling comma]], and 10976/10935, the [[hemimage comma]]. As it equates the perfect fourth with a stack of eight 28/27's, it also tempers out the [[octaphore comma]] in the [[2.3.7 subgroup]].


POTE generator: ~28/27 = 62.101
The hemifourth, reached by four generator steps, can be interpreted as ~[[15/13]]. This gives rise to a natural [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] extension where [[196/195]] and [[676/675]] vanish.  


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
From 10976/10935 = ([[784/783]])<sup>2</sup>⋅([[841/840]]) we obtain an add-29 extension for free, so the generator triples as 28/27~[[29/28]]~[[30/29]], and as 196/195 vanishes already, so does [[729/728]], leading to its [[S-expression]]-based comma list of {[[676/675|S26]], [[729/728|S27]], [[784/783|S28]], [[841/840|S29]]}.  
EDOs: 19, 58
Badness: 0.0392


==13-limit==
Experimentation shows we can find a reasonable mapping for prime 43 at -11 generator steps, tempering out [[216/215]]. All other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both.
Commas: 126/125, 144/143, 196/195, 676/675


POTE generator: ~28/27 = 62.119
In all the subgroups above, the optimum is between [[58edo]] and [[77edo]], though a notable tuning not appearing in the [[optimal ET sequence]]s here is [[96edo]] using the 96d val (with a 963{{c}} [[~]][[7/4]] similar to that of [[meanpop]]), which sacrifices the accuracy of prime 7 in favour of a more accurate 5.  


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
=== 7-limit ===
EDOs: 19, 58
[[Subgroup]]: 2.3.5.7
Badness: 0.0237


==Camahueto==
[[Comma list]]: 126/125, 10976/10935
Commas: 126/125, 10976/10935, 385/384


POTE generator: ~28/27 = 62.431
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
[[Optimal tuning]]s:  
EDOs: 19, 58, 77, 96d
* [[WE]]: ~2 = 1199.6949{{c}}, ~28/27 = 62.2621{{c}}
Badness: 0.0659
: [[error map]]: {{val| -0.305 -0.662 +3.364 -2.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/27 = 62.2996{{c}}
: error map: {{val| 0.000 -0.352 +3.792 -1.717 }}


==13-limit==
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
Commas: 126/125, 196/195, 385/384, 676/675


POTE generator: ~28/27 = 62.434
[[Badness]] (Sintel): 1.04


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
==== 2.3.5.7.13 subgroup ====
EDOs: 19, 58, 77, 96d
Subgroup: 2.3.5.7.13
Badness: 0.0362


=Monocerus=
Comma list: 126/125, 196/195, 676/675
Commas: 126/125, 243/242, 5488/5455


POTE generator: ~28/27 = 62.292
Subgroup-val mapping: {{mapping| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}


Map: [&lt;2 4 6 8 9|, &lt;0 -8 -13 -23 -20|]
Optimal tunings:  
EDOs: 58, 96d, 154, 212ce, 366de
* WE: ~2 = 1199.7013{{c}}, ~28/27 = 62.2785{{c}}
Badness: 0.0528
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3151{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19, 39df, 58, 77, 212cf }}
Commas: 126/125, 196/195, 364/363, 676/675


POTE generator: ~28/27 = 62.301
Badness (Sintel): 0.590


Map: [&lt;2 4 6 8 9 10|, &lt;0 -8 -13 -23 -20 -25|]
==== 2.3.5.7.13.29 subgroup ====
EDOs: 58, 96d, 154, 366cef
Subgroup: 2.3.5.7.13.29
Badness: 0.0288


=Qilin=
Comma list: 126/125, 196/195, 261/260, 377/375
Commas: 126/125, 176/175, 10976/10935


POTE generator: ~28/27 = 62.196
Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}


Map: [&lt;1 2 3 4 6|, &lt;0 -8 -13 -23 -49|]
Optimal tunings:  
EDOs: 58, 77, 135c, 193c, 328c
* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{c}}
Badness: 0.0414
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3141{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}
Commas: 126/125, 176/175, 196/195, 2200/2197


POTE generator: ~28/27 = 62.197
Badness (Sintel): 0.487


Map: [&lt;1 2 3 4 6 5|, &lt;0 -8 -13 -23 -49 -25|]
=== Alicorn ===
EDOs: 58, 77, 135c, 193cf, 328cf
Subgroup: 2.3.5.7.11
Badness: 0.0228


=Rhinoceros=
Comma list: 126/125, 540/539, 896/891
Commas: 49/48, 4375/4374


POTE generator: ~21/20 = 62.920
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}


Map: [&lt;1 2 3 3]|, &lt;0 -8 -13 -4|]
Optimal tunings:  
Wedgie: &lt;&lt;8 13 4 2 -16 -27||
* WE: ~2 = 1198.6510{{c}}, ~28/27 = 62.0316{{c}}
EDOs: 19
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1435{{c}}
Badness: 0.0819


==11-limit==
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}
Commas: 49/48, 100/99, 126/121


POTE generator: ~21/20 = 62.874
Badness (Sintel): 1.29


Map: [&lt;1 2 3 3 4]|, &lt;0 -8 -13 -4 -10|]
==== 13-limit ====
EDOs: 19
Subgroup: 2.3.5.7.11.13
Badness: 0.0593


==13-limit==
Comma list: 126/125, 144/143, 196/195, 676/675
Commas: 49/48, 78/77, 100/99, 126/121


POTE generator: ~21/20 = 63.043
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}


Map: [&lt;1 2 3 3 4 4]|, &lt;0 -8 -13 -4 -10 -6|]
Optimal tunings:  
EDOs: 19
* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}
Badness: 0.0393
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}
</pre></div>
 
<h4>Original HTML content:</h4>
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
<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;Unicorn family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:26:&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:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;a href="#Unicorn"&gt;Unicorn&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Alicorn"&gt;Alicorn&lt;/a&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="#Monocerus"&gt;Monocerus&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;a href="#Qilin"&gt;Qilin&lt;/a&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; | &lt;a href="#Rhinoceros"&gt;Rhinoceros&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;br /&gt;
Badness (Sintel): 0.978
The unicorn family tempers out the unicorn comma, 1594323/1562500 = |-2 13 -8&amp;gt;.&lt;br /&gt;
 
&lt;br /&gt;
=== Camahueto ===
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Unicorn"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Unicorn&lt;/h1&gt;
Subgroup: 2.3.5.7.11
Comma: 1594323/1562500&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 385/384, 10976/10935
POTE generator: ~250/243 = 62.458&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}
Map: [&amp;lt;1 2 3|, &amp;lt;0 -8 -13|]&lt;br /&gt;
 
EDOs: 19, 58, 77, 96, 173, 269&lt;br /&gt;
Optimal tunings:  
Badness: 0.1505&lt;br /&gt;
* WE: ~2 = 1200.5186{{c}}, ~28/27 = 62.4576{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4252{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Alicorn"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Alicorn&lt;/h1&gt;
 
Commas: 126/125, 10976/10935&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.279&lt;br /&gt;
Badness (Sintel): 2.18
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4|, &amp;lt;0 -8 -13 -23|]&lt;br /&gt;
==== 13-limit ====
EDOs: 19, 58, 77, 96d&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.0409&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 196/195, 385/384, 676/675
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Alicorn-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
 
Commas: 126/125, 540/539, 896/891&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.101&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
Map: [&amp;lt;1 2 3 4 3|, &amp;lt;0 -8 -13 -23 9|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}
EDOs: 19, 58&lt;br /&gt;
 
Badness: 0.0392&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 77, 96d, 173d }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Alicorn-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
Badness (Sintel): 1.49
Commas: 126/125, 144/143, 196/195, 676/675&lt;br /&gt;
 
&lt;br /&gt;
=== Qilin ===
POTE generator: ~28/27 = 62.119&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 3 5|, &amp;lt;0 -8 -13 -23 9 -25|]&lt;br /&gt;
Comma list: 126/125, 176/175, 10976/10935
EDOs: 19, 58&lt;br /&gt;
 
Badness: 0.0237&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Alicorn-Camahueto"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Camahueto&lt;/h2&gt;
Optimal tunings:  
Commas: 126/125, 10976/10935, 385/384&lt;br /&gt;
* WE: ~2 = 1199.3865{{c}}, ~28/27 = 62.1645{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}
POTE generator: ~28/27 = 62.431&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193c }}
Map: [&amp;lt;1 2 3 4 2|, &amp;lt;0 -8 -13 -23 28|]&lt;br /&gt;
 
EDOs: 19, 58, 77, 96d&lt;br /&gt;
Badness (Sintel): 1.37
Badness: 0.0659&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Alicorn-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
Subgroup: 2.3.5.7.11.13
Commas: 126/125, 196/195, 385/384, 676/675&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 176/175, 196/195, 2200/2197
POTE generator: ~28/27 = 62.434&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
Map: [&amp;lt;1 2 3 4 2 5|, &amp;lt;0 -8 -13 -23 28 -25|]&lt;br /&gt;
 
EDOs: 19, 58, 77, 96d&lt;br /&gt;
Optimal tunings:  
Badness: 0.0362&lt;br /&gt;
* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Monocerus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Monocerus&lt;/h1&gt;
 
Commas: 126/125, 243/242, 5488/5455&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.292&lt;br /&gt;
Badness (Sintel): 0.944
&lt;br /&gt;
 
Map: [&amp;lt;2 4 6 8 9|, &amp;lt;0 -8 -13 -23 -20|]&lt;br /&gt;
=== Monocerus ===
EDOs: 58, 96d, 154, 212ce, 366de&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.0528&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 243/242, 5488/5445
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Monocerus-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;13-limit&lt;/h2&gt;
 
Commas: 126/125, 196/195, 364/363, 676/675&lt;br /&gt;
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.301&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~99/70 = 599.8223{{c}}, ~28/27 = 62.2737{{c}}
Map: [&amp;lt;2 4 6 8 9 10|, &amp;lt;0 -8 -13 -23 -20 -25|]&lt;br /&gt;
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}
EDOs: 58, 96d, 154, 366cef&lt;br /&gt;
 
Badness: 0.0288&lt;br /&gt;
{{Optimal ET sequence|legend=0| 38d, 58, 96d, 154, 212ce }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Qilin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Qilin&lt;/h1&gt;
Badness (Sintel): 1.74
Commas: 126/125, 176/175, 10976/10935&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
POTE generator: ~28/27 = 62.196&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 6|, &amp;lt;0 -8 -13 -23 -49|]&lt;br /&gt;
Comma list: 126/125, 196/195, 243/242, 364/363
EDOs: 58, 77, 135c, 193c, 328c&lt;br /&gt;
 
Badness: 0.0414&lt;br /&gt;
Mapping: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Qilin-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;13-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 126/125, 176/175, 196/195, 2200/2197&lt;br /&gt;
* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}
&lt;br /&gt;
* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}
POTE generator: ~28/27 = 62.197&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}
Map: [&amp;lt;1 2 3 4 6 5|, &amp;lt;0 -8 -13 -23 -49 -25|]&lt;br /&gt;
 
EDOs: 58, 77, 135c, 193cf, 328cf&lt;br /&gt;
Badness (Sintel): 1.19
Badness: 0.0228&lt;br /&gt;
 
&lt;br /&gt;
==== 17-limit ====
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Rhinoceros"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Rhinoceros&lt;/h1&gt;
Subgroup: 2.3.5.7.11.13.17
Commas: 49/48, 4375/4374 &lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 196/195, 221/220, 243/242, 289/288
POTE generator: ~21/20 = 62.920&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}
Map: [&amp;lt;1 2 3 3]|, &amp;lt;0 -8 -13 -4|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;8 13 4 2 -16 -27||&lt;br /&gt;
Optimal tunings:
EDOs: 19&lt;br /&gt;
* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}
Badness: 0.0819&lt;br /&gt;
* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Rhinoceros-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 38df, 58, 96d, 154 }}
Commas: 49/48, 100/99, 126/121&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.24
POTE generator: ~21/20 = 62.874&lt;br /&gt;
 
&lt;br /&gt;
== Rhinoceros ==
Map: [&amp;lt;1 2 3 3 4]|, &amp;lt;0 -8 -13 -4 -10|]&lt;br /&gt;
Rhinoceros tempers out the [[semaphoresma]] and the [[ragisma]], identifying the generator as a [[~]][[21/20]], rather than the [[28/27]] of septimal unicorn. It may be described as the {{nowrap| 19 & 20c }} temperament. [[19edo]] is a good tuning, though [[39edo]] is a possible alternative.
EDOs: 19&lt;br /&gt;
 
Badness: 0.0593&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Rhinoceros-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;13-limit&lt;/h2&gt;
[[Comma list]]: 49/48, 4375/4374
Commas: 49/48, 78/77, 100/99, 126/121&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
POTE generator: ~21/20 = 63.043&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
Map: [&amp;lt;1 2 3 3 4 4]|, &amp;lt;0 -8 -13 -4 -10 -6|]&lt;br /&gt;
* [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}
EDOs: 19&lt;br /&gt;
: [[error map]]: {{val| +3.103 -0.412 +2.917 -11.848 }}
Badness: 0.0393&lt;/body&gt;&lt;/html&gt;</pre></div>
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 62.6687{{c}}
: error map: {{val| 0.000 -3.305 -1.007 -19.501 }}
 
{{Optimal ET sequence|legend=1| 1c, …, 18bcd, 19 }}
 
[[Badness]] (Sintel): 2.07
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 100/99, 126/121
 
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}
 
Optimal tunings:  
* WE: ~2 = 1201.1458{{c}}, ~21/20 = 62.9338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.7783{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 18bcd, 19 }}
 
Badness (Sintel): 1.96
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 78/77, 100/99, 126/121
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}
 
Optimal tunings:  
* WE: ~2 = 1202.2047{{c}}, ~21/20 = 63.1591{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 62.8727{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 18bcdf, 19 }}
 
Badness (Sintel): 1.63
 
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:25, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The unicorn family tempers out the unicorn comma (monzo: [-2 13 -8⟩, ratio: 1594323/1562500).

Unicorn

Unicorn divides the perfect fourth into eight equal parts, three for 10/9 and five for 6/5. Its ploidacot is omega-octacot.

Subgroup: 2.3.5

Comma list: 1594323/1562500

Mapping: [⟨1 2 3], ⟨0 -8 -13]]

mapping generators: ~2, ~250/243

Optimal tunings:

  • WE: ~2 = 1200.0889 ¢, ~250/243 = 62.4623 ¢
error map: ⟨+0.089 -1.476 +1.943]
  • CWE: ~2 = 1200.0000 ¢, ~250/243 = 62.4494 ¢
error map: ⟨0.000 -1.550 +1.844]

Optimal ET sequence: 19, 58, 77, 96, 173, 269

Badness (Sintel): 3.53

Septimal unicorn

The canonical extension to the 7-limit is by interpreting the generator as a very slightly flattened ~28/27, corresponding to tempering out 126/125, the starling comma, and 10976/10935, the hemimage comma. As it equates the perfect fourth with a stack of eight 28/27's, it also tempers out the octaphore comma in the 2.3.7 subgroup.

The hemifourth, reached by four generator steps, can be interpreted as ~15/13. This gives rise to a natural 2.3.5.7.13-subgroup extension where 196/195 and 676/675 vanish.

From 10976/10935 = (784/783)2⋅(841/840) we obtain an add-29 extension for free, so the generator triples as 28/27~29/28~30/29, and as 196/195 vanishes already, so does 729/728, leading to its S-expression-based comma list of {S26, S27, S28, S29}.

Experimentation shows we can find a reasonable mapping for prime 43 at -11 generator steps, tempering out 216/215. All other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both.

In all the subgroups above, the optimum is between 58edo and 77edo, though a notable tuning not appearing in the optimal ET sequences here is 96edo using the 96d val (with a 963 ¢ ~7/4 similar to that of meanpop), which sacrifices the accuracy of prime 7 in favour of a more accurate 5.

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 10976/10935

Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]

Optimal tunings:

  • WE: ~2 = 1199.6949 ¢, ~28/27 = 62.2621 ¢
error map: ⟨-0.305 -0.662 +3.364 -2.074]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2996 ¢
error map: ⟨0.000 -0.352 +3.792 -1.717]

Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c

Badness (Sintel): 1.04

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 126/125, 196/195, 676/675

Subgroup-val mapping: [⟨1 2 3 4 5], ⟨0 -8 -13 -23 -25]]

Optimal tunings:

  • WE: ~2 = 1199.7013 ¢, ~28/27 = 62.2785 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3151 ¢

Optimal ET sequence: 19, 39df, 58, 77, 212cf

Badness (Sintel): 0.590

2.3.5.7.13.29 subgroup

Subgroup: 2.3.5.7.13.29

Comma list: 126/125, 196/195, 261/260, 377/375

Subgroup-val mapping: [⟨1 2 3 4 5 6], ⟨0 -8 -13 -23 -25 -22]]

Optimal tunings:

  • WE: ~2 = 1199.7715 ¢, ~28/27 = 62.2860 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3141 ¢

Optimal ET sequence: 19, 39dfj, 58, 77, 212cf

Badness (Sintel): 0.487

Alicorn

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 896/891

Mapping: [⟨1 2 3 4 3], ⟨0 -8 -13 -23 9]]

Optimal tunings:

  • WE: ~2 = 1198.6510 ¢, ~28/27 = 62.0316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.1435 ¢

Optimal ET sequence: 19, 39d, 58

Badness (Sintel): 1.29

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 676/675

Mapping: [⟨1 2 3 4 3 5], ⟨0 -8 -13 -23 9 -25]]

Optimal tunings:

  • WE: ~2 = 1198.6298 ¢, ~28/27 = 62.0480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.1636 ¢

Optimal ET sequence: 19, 39df, 58

Badness (Sintel): 0.978

Camahueto

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 10976/10935

Mapping: [⟨1 2 3 4 2], ⟨0 -8 -13 -23 28]]

Optimal tunings:

  • WE: ~2 = 1200.5186 ¢, ~28/27 = 62.4576 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.4252 ¢

Optimal ET sequence: 19, 58e, 77, 96d, 173d

Badness (Sintel): 2.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 676/675

Mapping: [⟨1 2 3 4 2 5], ⟨0 -8 -13 -23 28 -25]]

Optimal tunings:

  • WE: ~2 = 1200.5004 ¢, ~28/27 = 62.4603 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.4277 ¢

Optimal ET sequence: 19, 77, 96d, 173d

Badness (Sintel): 1.49

Qilin

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 10976/10935

Mapping: [⟨1 2 3 4 6], ⟨0 -8 -13 -23 -49]]

Optimal tunings:

  • WE: ~2 = 1199.3865 ¢, ~28/27 = 62.1645 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2199 ¢

Optimal ET sequence: 19e, …, 58, 135c, 193c

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 2200/2197

Mapping: [⟨1 2 3 4 6 5], ⟨0 -8 -13 -23 -49 -25]]

Optimal tunings:

  • WE: ~2 = 1199.2874 ¢, ~28/27 = 62.1601 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2251 ¢

Optimal ET sequence: 19e, …, 58, 135c, 193cf

Badness (Sintel): 0.944

Monocerus

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 5488/5445

Mapping: [⟨2 4 6 8 9], ⟨0 -8 -13 -23 -20]]

Optimal tunings:

  • WE: ~99/70 = 599.8223 ¢, ~28/27 = 62.2737 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.3180 ¢

Optimal ET sequence: 38d, 58, 96d, 154, 212ce

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242, 364/363

Mapping: [⟨2 4 6 8 9 10], ⟨0 -8 -13 -23 -20 -25]]

Optimal tunings:

  • WE: ~55/39 = 599.8267 ¢, ~28/27 = 62.2833 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~28/27 = 62.3262 ¢

Optimal ET sequence: 38df, 58, 96d, 154

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 196/195, 221/220, 243/242, 289/288

Mapping: [⟨2 4 6 8 9 10 9], ⟨0 -8 -13 -23 -20 -25 -8]]

Optimal tunings:

  • WE: ~17/12 = 600.0715 ¢, ~28/27 = 62.3767 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~28/27 = 62.3605 ¢

Optimal ET sequence: 38df, 58, 96d, 154

Badness (Sintel): 1.24

Rhinoceros

Rhinoceros tempers out the semaphoresma and the ragisma, identifying the generator as a ~21/20, rather than the 28/27 of septimal unicorn. It may be described as the 19 & 20c temperament. 19edo is a good tuning, though 39edo is a possible alternative.

Subgroup: 2.3.5.7

Comma list: 49/48, 4375/4374

Mapping: [⟨1 2 3 3], ⟨0 -8 -13 -4]]

Optimal tunings:

  • WE: ~2 = 1203.1031 ¢, ~21/20 = 63.0830 ¢
error map: ⟨+3.103 -0.412 +2.917 -11.848]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.6687 ¢
error map: ⟨0.000 -3.305 -1.007 -19.501]

Optimal ET sequence: 1c, …, 18bcd, 19

Badness (Sintel): 2.07

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 126/121

Mapping: [⟨1 2 3 3 4], ⟨0 -8 -13 -4 -10]]

Optimal tunings:

  • WE: ~2 = 1201.1458 ¢, ~21/20 = 62.9338 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.7783 ¢

Optimal ET sequence: 1ce, 18bcd, 19

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 78/77, 100/99, 126/121

Mapping: [⟨1 2 3 3 4 4], ⟨0 -8 -13 -4 -10 -6]]

Optimal tunings:

  • WE: ~2 = 1202.2047 ¢, ~21/20 = 63.1591 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.8727 ¢

Optimal ET sequence: 1ce, 18bcdf, 19

Badness (Sintel): 1.63

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