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]], 1594323/1562500 = {{monzo| -2 13 -8 }}. The canonical extension to the 7-limit is by interpreting the generator as a slightly flattened [[~]][[28/27]] so that a flat [[~]][[6/5]] is found at 5 generators, corresponding to tempering [[126/125]], the [[octaphore]] and the [[hemimage comma]].
: This revision was by author [[User:Gedankenwelt|Gedankenwelt]] and made on <tt>2013-10-30 12:51:55 UTC</tt>.<br>
: The original revision id was <tt>464521456</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 ==
Comma: 1594323/1562500
By noticing that the generator is very close to [[28/27]] we find the extension to the 7-limit by tempering the [[octaphore]] (which finds [[~]][[9/7]] at 7 gens and [[~]][[4/3]] at 8 gens, hence its name) and [[126/125]] (finding [[~]][[6/5]] at 5 gens). From this we can observe that the most natural extension is by equating adjacent [[superparticular interval]]s, by tempering the [[square-particular]]s between them, leading to its [[S-expression]]-based comma list of {[[676/675|S26]], [[729/728|S27]], [[784/783|S28]], [[841/840|S29]]}, to which experimentation shows we can find a reasonable mapping for prime 43 at -11 gens while all other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both.


POTE generator: ~250/243 = 62.458
[[Subgroup]]: 2.3.5


Map: [&lt;1 2 3|, &lt;0 -8 -13|]
[[Comma list]]: 1594323/1562500
EDOs: 19, 58, 77, 96, 173, 269
Badness: 0.1505


=Alicorn=
{{Mapping|legend=1| 1 2 3 | 0 -8 -13 }}
Commas: 126/125, 10976/10935


POTE generator: ~28/27 = 62.279
[[Optimal tuning]]s:
* [[CTE]]: 2 = 1\1, ~250/243 =  62.441
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458


Map: [&lt;1 2 3 4|, &lt;0 -8 -13 -23|]
{{Optimal ET sequence|legend=1| 19, 58, 77, 96, 173, 269 }}
EDOs: 19, 58, 77, 96d
Badness: 0.0409


==11-limit==
[[Badness]]:
Commas: 126/125, 540/539, 896/891
* Smith: 0.150487
* Dirichlet: 3.530


POTE generator: ~28/27 = 62.101
== Septimal unicorn ==
{{See also| Octaphore }}


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
[[Subgroup]]: 2.3.5.7
EDOs: 19, 58
Badness: 0.0392


==13-limit==
[[Comma list]]: [[126/125]], [[10976/10935]]
Commas: 126/125, 144/143, 196/195, 676/675


POTE generator: ~28/27 = 62.119
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}


Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
[[Optimal tuning]]s:
EDOs: 19, 58
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
Badness: 0.0237
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278


==Camahueto==
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
Commas: 126/125, 10976/10935, 385/384


POTE generator: ~28/27 = 62.431
Badness:  
* Smith: 0.040913
* Dirichlet: 1.035


Map: [&lt;1 2 3 4 2|, &lt;0 -8 -13 -23 28|]
=== 2.3.5.7.13 subgroup ===
EDOs: 19, 58, 77, 96d
[[Subgroup]]: 2.3.5.7.13
Badness: 0.0659


==13-limit==
[[Comma list]]: [[126/125]], [[351/350]], [[676/675]]
Commas: 126/125, 196/195, 385/384, 676/675


POTE generator: ~28/27 = 62.434
{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}


Map: [&lt;1 2 3 4 2 5|, &lt;0 -8 -13 -23 28 -25|]
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
EDOs: 19, 58, 77, 96d
Badness: 0.0362


=Monocerus=
{{Optimal ET sequence|legend=1| 19, 39df, 58, 77, 212cf }}
Commas: 126/125, 243/242, 5488/5445


POTE generator: ~28/27 = 62.292
Badness (Sintel): 0.590


Map: [&lt;2 4 6 8 9|, &lt;0 -8 -13 -23 -20|]
==== 2.3.5.7.13.29 subgroup ====
EDOs: 58, 96d, 154, 212ce, 366de
[[Subgroup]]: 2.3.5.7.13.29
Badness: 0.0528


==13-limit==
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]]
Commas: 126/125, 196/195, 364/363, 676/675


POTE generator: ~28/27 = 62.301
{{Mapping|legend=1| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}


Map: [&lt;2 4 6 8 9 10|, &lt;0 -8 -13 -23 -20 -25|]
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.334
EDOs: 58, 96d, 154, 366cef
Badness: 0.0288


=Qilin=
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cfn }}
Commas: 126/125, 176/175, 10976/10935


POTE generator: ~28/27 = 62.196
Badness (Sintel): 0.487


Map: [&lt;1 2 3 4 6|, &lt;0 -8 -13 -23 -49|]
==== 2.3.5.7.13.29.43 subgroup ====
EDOs: 58, 77, 135c, 193c, 328c
A notable tuning of unicorn not appearing in the [[optimal ET sequence]] here is [[96edo]] using the 96d val (with a 963[[cent|¢]] [[~]][[7/4]] similar to that of [[meanpop]]), an alternative to [[77edo]] that sacrifices the accuracy of prime 7 in favour of a more accurate [[5/4]] and [[43/32]].
Badness: 0.0414


==13-limit==
[[Subgroup]]: 2.3.5.7.13.29.43
Commas: 126/125, 176/175, 196/195, 2200/2197


POTE generator: ~28/27 = 62.197
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]], 216/215


Map: [&lt;1 2 3 4 6 5|, &lt;0 -8 -13 -23 -49 -25|]
{{Mapping|legend=1| 1 2 3 4 5 6 6 | 0 -8 -13 -23 -25 -22 -11 }}
EDOs: 58, 77, 135c, 193cf, 328cf
Badness: 0.0228


=Rhinoceros=
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
Commas: 49/48, 4375/4374


POTE generator: ~21/20 = 62.920
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 135c, 212cfn }}


Map: [&lt;1 2 3 3]|, &lt;0 -8 -13 -4|]
Badness (Sintel): 0.514
Wedgie: &lt;&lt;8 13 4 2 -16 -27||
EDOs: 19
Badness: 0.0819


==11-limit==  
=== Alicorn ===
Commas: 49/48, 100/99, 126/121
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 62.874
Comma list: 126/125, 540/539, 896/891


Map: [&lt;1 2 3 3 4]|, &lt;0 -8 -13 -4 -10|]
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
EDOs: 19
Badness: 0.0593


==13-limit==
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101
Commas: 49/48, 78/77, 100/99, 126/121


POTE generator: ~21/20 = 63.043
{{Optimal ET sequence|legend=1| 19, 39d, 58 }}


Map: [&lt;1 2 3 3 4 4]|, &lt;0 -8 -13 -4 -10 -6|]
Badness: 0.039156
EDOs: 19
 
Badness: 0.0393</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;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;The unicorn family tempers out the unicorn comma, 1594323/1562500 = |-2 13 -8&amp;gt;.&lt;br /&gt;
Comma list: 126/125, 144/143, 196/195, 676/675
&lt;br /&gt;
 
&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;
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}
Comma: 1594323/1562500&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.119
POTE generator: ~250/243 = 62.458&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 39df, 58 }}
Map: [&amp;lt;1 2 3|, &amp;lt;0 -8 -13|]&lt;br /&gt;
 
EDOs: 19, 58, 77, 96, 173, 269&lt;br /&gt;
Badness: 0.023667
Badness: 0.1505&lt;br /&gt;
 
&lt;br /&gt;
=== Camahueto ===
&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;
Subgroup: 2.3.5.7.11
Commas: 126/125, 10976/10935&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 385/384, 10976/10935
POTE generator: ~28/27 = 62.279&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}
Map: [&amp;lt;1 2 3 4|, &amp;lt;0 -8 -13 -23|]&lt;br /&gt;
 
EDOs: 19, 58, 77, 96d&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431
Badness: 0.0409&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}
&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;
Badness: 0.065940
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.101&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;1 2 3 4 3|, &amp;lt;0 -8 -13 -23 9|]&lt;br /&gt;
 
EDOs: 19, 58&lt;br /&gt;
Comma list: 126/125, 196/195, 385/384, 676/675
Badness: 0.0392&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
&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;
 
Commas: 126/125, 144/143, 196/195, 676/675&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.434
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.119&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 58e, 77, 96d, 173d }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 3 5|, &amp;lt;0 -8 -13 -23 9 -25|]&lt;br /&gt;
Badness: 0.036155
EDOs: 19, 58&lt;br /&gt;
 
Badness: 0.0237&lt;br /&gt;
=== Qilin ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&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;
 
Commas: 126/125, 10976/10935, 385/384&lt;br /&gt;
Comma list: 126/125, 176/175, 10976/10935
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.431&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 2|, &amp;lt;0 -8 -13 -23 28|]&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196
EDOs: 19, 58, 77, 96d&lt;br /&gt;
 
Badness: 0.0659&lt;br /&gt;
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193c, 328cc }}
&lt;br /&gt;
 
&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;
Badness: 0.041426
Commas: 126/125, 196/195, 385/384, 676/675&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
POTE generator: ~28/27 = 62.434&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 2 5|, &amp;lt;0 -8 -13 -23 28 -25|]&lt;br /&gt;
Comma list: 126/125, 176/175, 196/195, 2200/2197
EDOs: 19, 58, 77, 96d&lt;br /&gt;
 
Badness: 0.0362&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
&lt;br /&gt;
 
&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;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.197
Commas: 126/125, 243/242, 5488/5445&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 58, 77, 135c, 193cf, 328ccff }}
POTE generator: ~28/27 = 62.292&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.022842
Map: [&amp;lt;2 4 6 8 9|, &amp;lt;0 -8 -13 -23 -20|]&lt;br /&gt;
 
EDOs: 58, 96d, 154, 212ce, 366de&lt;br /&gt;
=== Monocerus ===
Badness: 0.0528&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&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;
Comma list: 126/125, 243/242, 5488/5445
Commas: 126/125, 196/195, 364/363, 676/675&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}
POTE generator: ~28/27 = 62.301&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292
Map: [&amp;lt;2 4 6 8 9 10|, &amp;lt;0 -8 -13 -23 -20 -25|]&lt;br /&gt;
 
EDOs: 58, 96d, 154, 366cef&lt;br /&gt;
{{Optimal ET sequence|legend=1| 58, 96d, 154, 212ce, 366cce }}
Badness: 0.0288&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.052757
&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;
 
Commas: 126/125, 176/175, 10976/10935&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~28/27 = 62.196&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 196/195, 364/363, 676/675
Map: [&amp;lt;1 2 3 4 6|, &amp;lt;0 -8 -13 -23 -49|]&lt;br /&gt;
 
EDOs: 58, 77, 135c, 193c, 328c&lt;br /&gt;
Mapping: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}
Badness: 0.0414&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301
&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;
 
Commas: 126/125, 176/175, 196/195, 2200/2197&lt;br /&gt;
{{Optimal ET sequence|legend=1| 58, 96d, 154, 366ccef }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.197&lt;br /&gt;
Badness: 0.028795
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 6 5|, &amp;lt;0 -8 -13 -23 -49 -25|]&lt;br /&gt;
== Rhinoceros ==
EDOs: 58, 77, 135c, 193cf, 328cf&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Badness: 0.0228&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 49/48, 4375/4374
&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;
 
Commas: 49/48, 4375/4374&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
&lt;br /&gt;
 
POTE generator: ~21/20 = 62.920&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3]|, &amp;lt;0 -8 -13 -4|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 1c, 19 }}
Wedgie: &amp;lt;&amp;lt;8 13 4 2 -16 -27||&lt;br /&gt;
 
EDOs: 19&lt;br /&gt;
[[Badness]]: 0.081864
Badness: 0.0819&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
&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;
Subgroup: 2.3.5.7.11
Commas: 49/48, 100/99, 126/121&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 49/48, 100/99, 126/121
POTE generator: ~21/20 = 62.874&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping|  1 2 3 3 4 | 0 -8 -13 -4 -10 }}
Map: [&amp;lt;1 2 3 3 4]|, &amp;lt;0 -8 -13 -4 -10|]&lt;br /&gt;
 
EDOs: 19&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.874
Badness: 0.0593&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 1ce, 19 }}
&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;
 
Commas: 49/48, 78/77, 100/99, 126/121&lt;br /&gt;
Badness: 0.059319
&lt;br /&gt;
 
POTE generator: ~21/20 = 63.043&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Map: [&amp;lt;1 2 3 3 4 4]|, &amp;lt;0 -8 -13 -4 -10 -6|]&lt;br /&gt;
 
EDOs: 19&lt;br /&gt;
Comma list: 49/48, 78/77, 100/99, 126/121
Badness: 0.0393&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -8 -13 -4 -10 -6 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 63.043
 
{{Optimal ET sequence|legend=1| 1ce, 19 }}
 
Badness: 0.039343
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Unicorn family| ]] <!-- main article -->
[[Category:Unicorn| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 00:34, 24 June 2025

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, 1594323/1562500 = [-2 13 -8⟩. The canonical extension to the 7-limit is by interpreting the generator as a slightly flattened ~28/27 so that a flat ~6/5 is found at 5 generators, corresponding to tempering 126/125, the octaphore and the hemimage comma.

Unicorn

By noticing that the generator is very close to 28/27 we find the extension to the 7-limit by tempering the octaphore (which finds ~9/7 at 7 gens and ~4/3 at 8 gens, hence its name) and 126/125 (finding ~6/5 at 5 gens). From this we can observe that the most natural extension is by equating adjacent superparticular intervals, by tempering the square-particulars between them, leading to its S-expression-based comma list of {S26, S27, S28, S29}, to which experimentation shows we can find a reasonable mapping for prime 43 at -11 gens while all other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both.

Subgroup: 2.3.5

Comma list: 1594323/1562500

Mapping: [⟨1 2 3], ⟨0 -8 -13]]

Optimal tunings:

  • CTE: 2 = 1\1, ~250/243 = 62.441
  • POTE: 2 = 1\1, ~250/243 = 62.458

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

Badness:

  • Smith: 0.150487
  • Dirichlet: 3.530

Septimal unicorn

See also: Octaphore

Subgroup: 2.3.5.7

Comma list: 126/125, 10976/10935

Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]

Optimal tunings:

  • CTE: 2 = 1\1, ~28/27 = 62.324
  • POTE: 2 = 1\1, ~28/27 = 62.278

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

Badness:

  • Smith: 0.040913
  • Dirichlet: 1.035

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 126/125, 351/350, 676/675

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

Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.339

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, 729/728, 784/783, 841/840

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

Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.334

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

Badness (Sintel): 0.487

2.3.5.7.13.29.43 subgroup

A notable tuning of unicorn not appearing in the optimal ET sequence here is 96edo using the 96d val (with a 963¢ ~7/4 similar to that of meanpop), an alternative to 77edo that sacrifices the accuracy of prime 7 in favour of a more accurate 5/4 and 43/32.

Subgroup: 2.3.5.7.13.29.43

Comma list: 126/125, 729/728, 784/783, 841/840, 216/215

Mapping: [⟨1 2 3 4 5 6 6], ⟨0 -8 -13 -23 -25 -22 -11]]

Optimal tuning (CTE): 2 = 1\1, ~28/27 = 62.339

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

Badness (Sintel): 0.514

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.101

Optimal ET sequence: 19, 39d, 58

Badness: 0.039156

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.119

Optimal ET sequence: 19, 39df, 58

Badness: 0.023667

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.431

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

Badness: 0.065940

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.434

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

Badness: 0.036155

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.196

Optimal ET sequence: 58, 77, 135c, 193c, 328cc

Badness: 0.041426

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.197

Optimal ET sequence: 58, 77, 135c, 193cf, 328ccff

Badness: 0.022842

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 tuning (POTE): ~2 = 1\1, ~28/27 = 62.292

Optimal ET sequence: 58, 96d, 154, 212ce, 366cce

Badness: 0.052757

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 676/675

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

Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.301

Optimal ET sequence: 58, 96d, 154, 366ccef

Badness: 0.028795

Rhinoceros

Subgroup: 2.3.5.7

Comma list: 49/48, 4375/4374

Mapping: [⟨1 2 3 3], ⟨0 -8 -13 -4]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 62.920

Optimal ET sequence: 1c, 19

Badness: 0.081864

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 tuning (POTE): ~2 = 1\1, ~21/20 = 62.874

Optimal ET sequence: 1ce, 19

Badness: 0.059319

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 tuning (POTE): ~2 = 1\1, ~21/20 = 63.043

Optimal ET sequence: 1ce, 19

Badness: 0.039343

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