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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>2011-11-03 17:45:29 UTC</tt>.<br>
: The original revision id was <tt>271691490</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</pre></div>
 
<h4>Original HTML content:</h4>
Comma list: 126/125, 196/195, 261/260, 377/375
<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:16:&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:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Unicorn"&gt;Unicorn&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Alicorn"&gt;Alicorn&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Monocerus"&gt;Monocerus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;br /&gt;
Subgroup-val mapping: {{mapping| 1 2 3 4 5 6 | 0 -8 -13 -23 -25 -22 }}
The unicorn family tempers out the unicorn comma, 1594323/1562500 = |-2 13 -8&amp;gt;.&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1199.7715{{c}}, ~28/27 = 62.2860{{c}}
Comma: 1594323/1562500&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.3141{{c}}
&lt;br /&gt;
 
POTE generator: ~250/243 = 62.458&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 39dfj, 58, 77, 212cf }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3|, &amp;lt;0 -8 -13|]&lt;br /&gt;
Badness (Sintel): 0.487
EDOs: 19, 58, 77, 96, 173, 269&lt;br /&gt;
 
Badness: 0.1505&lt;br /&gt;
=== Alicorn ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
&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;
Comma list: 126/125, 540/539, 896/891
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.279&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 3 | 0 -8 -13 -23 9 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4|, &amp;lt;0 -8 -13 -23|]&lt;br /&gt;
Optimal tunings:  
EDOs: 19, 58, 77, 96d&lt;br /&gt;
* WE: ~2 = 1198.6510{{c}}, ~28/27 = 62.0316{{c}}
Badness: 0.0409&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1435{{c}}
&lt;br /&gt;
 
&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;
{{Optimal ET sequence|legend=0| 19, 39d, 58 }}
Commas: 126/125, 540/539, 896/891&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.29
POTE generator: ~28/27 = 62.101&lt;br /&gt;
 
&lt;br /&gt;
==== 13-limit ====
Map: [&amp;lt;1 2 3 4 3|, &amp;lt;0 -8 -13 -23 9|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 19, 58&lt;br /&gt;
 
Badness: 0.0392&lt;br /&gt;
Comma list: 126/125, 144/143, 196/195, 676/675
&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;
Mapping: {{mapping| 1 2 3 4 3 5 | 0 -8 -13 -23 9 -25 }}
Commas: 126/125, 144/143, 196/195, 676/675&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~28/27 = 62.119&lt;br /&gt;
* WE: ~2 = 1198.6298{{c}}, ~28/27 = 62.0480{{c}}
&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.1636{{c}}
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;
{{Optimal ET sequence|legend=0| 19, 39df, 58 }}
Badness: 0.0237&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.978
&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;
=== Camahueto ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~28/27 = 62.431&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 126/125, 385/384, 10976/10935
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;
Mapping: {{mapping| 1 2 3 4 2 | 0 -8 -13 -23 28 }}
Badness: 0.0659&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* WE: ~2 = 1200.5186{{c}}, ~28/27 = 62.4576{{c}}
Commas: 126/125, 196/195, 385/384, 676/675&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4252{{c}}
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.434&lt;br /&gt;
{{Optimal ET sequence|legend=0| 19, 58e, 77, 96d, 173d }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 4 2 5|, &amp;lt;0 -8 -13 -23 28 -25|]&lt;br /&gt;
Badness (Sintel): 2.18
EDOs: 19, 58, 77, 96d&lt;br /&gt;
 
Badness: 0.0362&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
Comma list: 126/125, 196/195, 385/384, 676/675
&lt;br /&gt;
 
POTE generator: ~28/27 = 62.292&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 2 5 | 0 -8 -13 -23 28 -25 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 4 6 8 9|, &amp;lt;0 -8 -13 -23 -20|]&lt;br /&gt;
Optimal tunings:  
EDOs: 58, 96d, 154, 212ce, 366de&lt;br /&gt;
* WE: ~2 = 1200.5004{{c}}, ~28/27 = 62.4603{{c}}
Badness: 0.0528&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.4277{{c}}
&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;
{{Optimal ET sequence|legend=0| 19, 77, 96d, 173d }}
Commas: 126/125, 196/195, 364/363, 676/675&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.49
POTE generator: ~28/27 = 62.301&lt;br /&gt;
 
&lt;br /&gt;
=== Qilin ===
Map: [&amp;lt;2 4 6 8 9 10|, &amp;lt;0 -8 -13 -23 -20 -25|]&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOs: 58, 96d, 154, 366cef&lt;br /&gt;
 
Badness: 0.0288&lt;/body&gt;&lt;/html&gt;</pre></div>
Comma list: 126/125, 176/175, 10976/10935
 
Mapping: {{mapping| 1 2 3 4 6 | 0 -8 -13 -23 -49 }}
 
Optimal tunings:  
* WE: ~2 = 1199.3865{{c}}, ~28/27 = 62.1645{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2199{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 3 4 6 5 | 0 -8 -13 -23 -49 -25 }}
 
Optimal tunings:  
* WE: ~2 = 1199.2874{{c}}, ~28/27 = 62.1601{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/27 = 62.2251{{c}}
 
{{Optimal ET sequence|legend=0| 19e, …, 58, 135c, 193cf }}
 
Badness (Sintel): 0.944
 
=== 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 tunings:  
* WE: ~99/70 = 599.8223{{c}}, ~28/27 = 62.2737{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~28/27 = 62.3180{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 4 6 8 9 10 | 0 -8 -13 -23 -20 -25 }}
 
Optimal tunings:  
* WE: ~55/39 = 599.8267{{c}}, ~28/27 = 62.2833{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~28/27 = 62.3262{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 4 6 8 9 10 9 | 0 -8 -13 -23 -20 -25 -8 }}
 
Optimal tunings:  
* WE: ~17/12 = 600.0715{{c}}, ~28/27 = 62.3767{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/27 = 62.3605{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{nowrap| 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|legend=1| 1 2 3 3 | 0 -8 -13 -4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.1031{{c}}, ~21/20 = 63.0830{{c}}
: [[error map]]: {{val| +3.103 -0.412 +2.917 -11.848 }}
* [[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]]
Retrieved from "https://en.xen.wiki/w/Unicorn_family"