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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:genewardsmith|genewardsmith]] and made on <tt>2011-09-07 19:15:56 UTC</tt>.<br>
: The original revision id was <tt>251740674</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 ==
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.


=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
Badness: 0.1505


=Alicorn=
[[Optimal tuning]]s:
Commas: 126/125, 10976/10935
* [[CTE]]: 2 = 1\1, ~250/243 = 62.441
* [[POTE]]: 2 = 1\1, ~250/243 = 62.458


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]]:
EDOs: 19, 58, 77
* Smith: 0.150487
Badness: 0.0409
* Dirichlet: 3.530


==11-limit==
== Septimal unicorn ==
Commas: 126/125, 540/539, 896/891
{{See also| Octaphore }}


POTE generator: ~28/27 = 62.101
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 4 3|, &lt;0 -8 -13 -23 9|]
[[Comma list]]: [[126/125]], [[10976/10935]]
EDOs: 19, 58
Badness: 0.0392


==13-limit==
{{Mapping|legend=1| 1 2 3 4 | 0 -8 -13 -23 }}
Commas: 126/125, 144/143, 196/195, 676/675


POTE generator: ~28/27 = 62.119
[[Optimal tuning]]s:
Map: [&lt;1 2 3 4 3 5|, &lt;0 -8 -13 -23 9 -25|]
* [[CTE]]: 2 = 1\1, ~28/27 = 62.324
EDOs: 19, 58
* [[POTE]]: 2 = 1\1, ~28/27 = 62.278
Badness: 0.0237</pre></div>
 
<h4>Original HTML content:</h4>
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c }}
<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:8:&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:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Unicorn"&gt;Unicorn&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Alicorn"&gt;Alicorn&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;br /&gt;
Badness:
The unicorn family tempers out the unicorn comma, 1594323/1562500 = |-2 13 -8&amp;gt;.&lt;br /&gt;
* Smith: 0.040913
&lt;br /&gt;
* Dirichlet: 1.035
&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;
 
Comma: 1594323/1562500&lt;br /&gt;
=== 2.3.5.7.13 subgroup ===
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.13
POTE generator: ~250/243 = 62.458&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: [[126/125]], [[351/350]], [[676/675]]
Map: [&amp;lt;1 2 3|, &amp;lt;0 -8 -13|]&lt;br /&gt;
 
EDOs: 19, 58, 77, 96, 173, 269&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 4 5 | 0 -8 -13 -23 -25 }}
Badness: 0.1505&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.339
&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=1| 19, 39df, 58, 77, 212cf }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.279&lt;br /&gt;
Badness (Sintel): 0.590
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4|, &amp;lt;0 -8 -13 -23|]&lt;br /&gt;
==== 2.3.5.7.13.29 subgroup ====
EDOs: 19, 58, 77&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.13.29
Badness: 0.0409&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]]
&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|legend=1| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.101&lt;br /&gt;
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~28/27 = 62.334
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 3|, &amp;lt;0 -8 -13 -23 9|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 39dfj, 58, 77, 212cfn }}
EDOs: 19, 58&lt;br /&gt;
 
Badness: 0.0392&lt;br /&gt;
Badness (Sintel): 0.487
&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;
==== 2.3.5.7.13.29.43 subgroup ====
Commas: 126/125, 144/143, 196/195, 676/675&lt;br /&gt;
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]].
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.119&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.13.29.43
Map: [&amp;lt;1 2 3 4 3 5|, &amp;lt;0 -8 -13 -23 9 -25|]&lt;br /&gt;
 
EDOs: 19, 58&lt;br /&gt;
[[Comma list]]: [[126/125]], [[729/728]], [[784/783]], [[841/840]], 216/215
Badness: 0.0237&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Mapping|legend=1| 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|legend=1| 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: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.101
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 19, 39df, 58 }}
 
Badness: 0.023667
 
=== Camahueto ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 385/384, 10976/10935
 
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.431
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 19, 58e, 77, 96d, 173d }}
 
Badness: 0.036155
 
=== Qilin ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 176/175, 10976/10935
 
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.196
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 58, 77, 135c, 193cf, 328ccff }}
 
Badness: 0.022842
 
=== Monocerus ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 5488/5445
 
Mapping: {{mapping| 2 4 6 8 9 | 0 -8 -13 -23 -20 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.292
 
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 58, 96d, 154, 366ccef }}
 
Badness: 0.028795
 
== Rhinoceros ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 4375/4374
 
{{Mapping|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 62.920
 
{{Optimal ET sequence|legend=1| 1c, 19 }}
 
[[Badness]]: 0.081864
 
=== 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 tuning (POTE): ~2 = 1\1, ~21/20 = 62.874
 
{{Optimal ET sequence|legend=1| 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: {{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]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"