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Porcupine family: Difference between revisions

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**Imported revision 312933686 - Original comment: **
→Porcupine: base the sharpness on 4/3 rather than 3/2 (see talk). Hystrix isn't actually flat of 8d
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
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| de = Porcupine
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| en = Porcupine family
: The original revision id was <tt>312933686</tt>.<br>
| es =
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<h4>Original Wikitext content:</h4>
{{Technical data page}}
<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 '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.  
----
The 5-limit parent comma for the porcupine family is 250/243, the maximal [[diesis]] or porcupine comma. Its [[monzo]] is |1 -5 3&gt;, and flipping that yields &lt;&lt;3 5 1|| for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10_9|10/9]] interval, and that three of these add up to a fourth, with two more giving the minor sixth. In fact, (10/9)^3 = 4/3 * 250/243, and (10/9)^5 = 8/5 * (250/243)^2. 3/22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.


[[POTE tuning|POTE generator]]: 163.950
== Porcupine ==
{{Main| Porcupine }}


Map: [&lt;1 2 3|, &lt;0 -3 -5|]
The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, {{nowrap| (10/9)<sup>3</sup> {{=}} (4/3)⋅(250/243) }}, and {{nowrap| (10/9)<sup>5</sup> {{=}} (8/5)⋅(250/243)<sup>2</sup> }}. Its [[ploidacot]] is omega-tricot. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.


EDOs: [[15edo|15]], [[22edo|22]], [[95edo|95c]], [[117edo|117bc]], [[139edo|139bc]], [[161edo|161bc]], [[183edo|183bc]]
[[Subgroup]]: 2.3.5


==Seven limit children==
[[Comma list]]: 250/243
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. That means [[64_63|64/63]], the [[Archyta's comma]], for [[Porcupine family#Porcupine|porcupine]], [[36_35|36/35]], the [[septimal quarter tone]], for [[Porcupine family#Hystrix|hystrix]], [[50_49|50/49]], the [[jubilisma]], for [[Porcupine family#Hedgehog|hedgehog]], and [[49_48|49/48]], the [[slendro diesis]], for [[Porcupine family#Nautilus|nautilus]].


=Porcupine=
{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}
Porcupine, with wedgie &lt;&lt;3 5 -6 1 -18 -28||, uses six of its minor tone generator steps to get to [[7_4|7/4]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.


Commas: 250/243, 64/63
: mapping generators: ~2, ~10/9


[[POTE tuning|POTE generator]]: ~10/9 = 162.880
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.166
: [[error map]]: {{val| 0.000 +5.547 -7.143 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 163.950
: error map: {{val| 0.000 +6.194 -6.065 }}


Map: [&lt;1 2 3 2|, &lt;0 -3 -5 6|]
[[Tuning ranges]]:  
EDOs: 22, [[59edo|59]], [[81edo|81bd]], [[140edo|140bd]]
* [[5-odd-limit]] [[diamond monotone]]: ~10/9 = [150.000, 171.429] (1\8 to 1\7)
* 5-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]


==11-limit==
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
Commas: 55/54, 64/63, 100/99


POTE generator: ~10/9 = 162.747
[[Badness]] (Smith): 0.030778


Map: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]
=== Overview to extensions ===
EDOs: [[7edo|7]], 15, 22, [[37edo|37]], [[59edo|59]]
==== 7-limit extensions ====
Badness: 0.0217
The second comma defines which [[7-limit]] family member we are looking at.
* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;  
* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;
* [[#Septimal porcupine|Septimal porcupine]] adds [[64/63]], the archytas comma, for a tuning between 15edo and 22edo;
* [[#Porky|Porky]] adds [[225/224]], the marvel comma, for a tuning between 22edo and 29edo;
* [[#Coendou|Coendou]] adds [[525/512]], the avicennma, for a tuning sharp of 29edo.  


=Hystrix=
Those all share the same generator with porcupine.  
Hystrix, with wedgie &lt;&lt;3 5 1 1 -7 -12||, provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits.


Commas: 36/35, 160/147
[[#Nautilus|nautilus]] tempers out [[49/48]] and splits the generator in two. [[#Hedgehog|hedgehog]] tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.


[[POTE tuning|POTE generator]]: 158.868
Temperaments discussed elsewhere include:
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
==== Subgroup extensions ====
Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|S11]] }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below.


EDOs: 10d, 12, 13d, 15
=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11


=Hedgehog=
Comma list: 55/54, 100/99
Hedgehog, with wedgie &lt;&lt;6 10 10 2 -1 -5||, has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out 245/243, the sensamagic comma. 22et provides the obvious tuning, but if you are looking for an alternative, you could try the &lt;146 232 338 411| val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14 note MOS gives scope for harmony while stopping well short of 22.


Commas: 50/49, 245/243
Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


[[POTE tuning|POTE generator]]: ~9/7 = 435.648
Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
: gencom: [2 10/9; 55/54, 100/99]
Wedgie: &lt;&lt;6 10 10 2 -1 -5||
EDOs: 22, [[146edo|146]]
Badness: 0.0440


==11-limit==
Optimal tunings:
Commas: 50/49, 55/54, 99/98
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1200.000, ~11/10 = 164.078


POTE generator: ~9/7 = 435.386
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Badness (Smith): 0.0097
EDOs: 14c, 22, 58ce, 80ce, 102cde
Badness: 0.0231


==Hedgepig==
==== Undecimation ====
Commas: 50/49, 245/243, 385/384
Subgroup: 2.3.5.11.13


POTE generator: ~9/7 = 435.425
Comma list: 55/54, 100/99, 512/507


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}
EDOs: 22, 80c, 102cd, 124cd
Badness: 0.0684


===Music===
: sval mapping generators: ~2, ~65/44
[[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3|Phobos Light]] by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.


=Nautilus=  
Optimal tunings:
Commas: 49/48, 250/243
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1200.000, ~88/65 = 518.209


Pote generator: ~21/20 = 82.505
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
Badness (Smith): 0.0305
Wedgie: &lt;&lt;6 10 3 2 -12 -21||
EDOs: 10, 15, 19, [[29edo|29]], [[102edo|102cd]]


==11-limit==  
== Septimal porcupine ==
Commas: 49/48, 55/54, 245/242
{{Main| Porcupine }}


POTE generator: ~21/20 = 82.504
Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. Here, we share the same mapping of 7/4 in terms of fifths as [[archy]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.  


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
[[Subgroup]]: 2.3.5.7
EDOs: 10e, 14c, 15, 19, 22d, 29, 102cde


==13-limit==
[[Comma list]]: 64/63, 250/243
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
[[Optimal tuning]]s:  
EDOs: 10e, 15f, 17d, 19, 22d, 29, 102cde
* [[CTE]]: ~2 = 1200.000, ~10/9 = 163.203
: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 162.880
: error map: {{val| 0.000 +9.405 -0.714 +8.455 }}


[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3|Nautilus Reverie]] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]]
[[Minimax tuning]]:
=Ammonite=
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
Commas: 250/243, 686/675
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


POTE generator: ~9/7 = 454.448
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~10/9 = [160.000, 163.636] (2\15 to 3\22)
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
Wedgie: &lt;&lt;9 15 19 3 5 2||
EDOs: 29, 37, 66
Badness: 0.1077


==11-limit==
[[Badness]] (Smith): 0.041057
Commas: 55/54, 100/99, 686/675


POTE generator: ~9/7 = 454.512
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
Comma list: 55/54, 64/63, 100/99
EDOs: 29, 37, 66
Badness: 0.0457


==13-limit==
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
Commas: 55/54, 91/90, 100/99, 169/168


POTE generator: ~13/10 = 454.429
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1200.000, ~11/10 = 162.747


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
Minimax tuning:  
EDOs: 29, 37, 66
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
Badness: 0.0272
: unchanged-interval (eigenmonzo) basis: 2.9/7


=Porky=
Tuning ranges:
Commas: 225/224, 250/243
* 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]


POTE generator: ~10/9 = 164.412
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
Badness (Smith): 0.021562
Wedgie: &lt;&lt;3 5 16 1 17 23||
EDOS: 7, 8, 15, 22, 29, 51, 73
Badness: 0.0544


==11-limit==  
==== Porcupinefowl ====
Commas: 55/54, 100/99, 225/224
This extension used to be ''tridecimal porcupine''.


POTE generator: ~10/9 = 164.552
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]
Comma list: 40/39, 55/54, 64/63, 66/65
EDOs: 7, 8, 15, 22, 29, 51, 73
Badness: 0.0273


=Coendou=
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
Commas: 250/243, 525/512


POTE generator: ~10/9 = 166.041
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.442
* POTE: ~2 = 1200.000, ~11/10 = 162.708


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
Minimax tuning:  
Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
EDOs: 7, 29, 65c, 94cd
: unchanged-interval (eigenmonzo) basis: 2.11
Badness: 0.1183


==11-limit==
Tuning ranges:
Commas: 55/54, 100/99, 525/512
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]


POTE generator: ~10/9 = 165.981
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}


Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]
Badness (Smith): 0.021276
EDOs: 7, 29, 65ce, 94cde
Badness: 0.0497


==13-limit==
==== Porcupinefish ====
Commas: 55/54, 65/64, 100/99, 105/104
{{See also| The Biosphere }}


POTE generator: ~10/9 = 165.974
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
Comma list: 55/54, 64/63, 91/90, 100/99
EDOs: 7, 29, 65cef, 94cdef
 
Badness: 0.0302
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
</pre></div>
 
<h4>Original HTML content:</h4>
Optimal tunings:  
<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;Porcupine family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:38:&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:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt; | &lt;a href="#Porcupine"&gt;Porcupine&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#Hystrix"&gt;Hystrix&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt; | &lt;a href="#Hedgehog"&gt;Hedgehog&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt; | &lt;a href="#Nautilus"&gt;Nautilus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt; | &lt;a href="#Ammonite"&gt;Ammonite&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt; | &lt;a href="#Porky"&gt;Porky&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt; | &lt;a href="#Coendou"&gt;Coendou&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;
* CTE: ~2 = 1200.000, ~11/10 = 162.636
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;hr /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 162.277
The 5-limit parent comma for the porcupine family is 250/243, the maximal &lt;a class="wiki_link" href="/diesis"&gt;diesis&lt;/a&gt; or porcupine comma. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |1 -5 3&amp;gt;, and flipping that yields &amp;lt;&amp;lt;3 5 1|| for the &lt;a class="wiki_link" href="/wedgie"&gt;wedgie&lt;/a&gt;. This tells us the &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt; is a minor whole tone, the &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; interval, and that three of these add up to a fourth, with two more giving the minor sixth. In fact, (10/9)^3 = 4/3 * 250/243, and (10/9)^5 = 8/5 * (250/243)^2. 3/22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:  
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 163.950&lt;br /&gt;
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.13/11
Map: [&amp;lt;1 2 3|, &amp;lt;0 -3 -5|]&lt;br /&gt;
 
&lt;br /&gt;
Tuning ranges:  
EDOs: &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/95edo"&gt;95c&lt;/a&gt;, &lt;a class="wiki_link" href="/117edo"&gt;117bc&lt;/a&gt;, &lt;a class="wiki_link" href="/139edo"&gt;139bc&lt;/a&gt;, &lt;a class="wiki_link" href="/161edo"&gt;161bc&lt;/a&gt;, &lt;a class="wiki_link" href="/183edo"&gt;183bc&lt;/a&gt;&lt;br /&gt;
* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
&lt;br /&gt;
* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; family member we are looking at. That means &lt;a class="wiki_link" href="/64_63"&gt;64/63&lt;/a&gt;, the &lt;a class="wiki_link" href="/Archyta%27s%20comma"&gt;Archyta's comma&lt;/a&gt;, for &lt;a class="wiki_link" href="/Porcupine%20family#Porcupine"&gt;porcupine&lt;/a&gt;, &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;, the &lt;a class="wiki_link" href="/septimal%20quarter%20tone"&gt;septimal quarter tone&lt;/a&gt;, for &lt;a class="wiki_link" href="/Porcupine%20family#Hystrix"&gt;hystrix&lt;/a&gt;, &lt;a class="wiki_link" href="/50_49"&gt;50/49&lt;/a&gt;, the &lt;a class="wiki_link" href="/jubilisma"&gt;jubilisma&lt;/a&gt;, for &lt;a class="wiki_link" href="/Porcupine%20family#Hedgehog"&gt;hedgehog&lt;/a&gt;, and &lt;a class="wiki_link" href="/49_48"&gt;49/48&lt;/a&gt;, the &lt;a class="wiki_link" href="/slendro%20diesis"&gt;slendro diesis&lt;/a&gt;, for &lt;a class="wiki_link" href="/Porcupine%20family#Nautilus"&gt;nautilus&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15, 22, 37 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Porcupine"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Porcupine&lt;/h1&gt;
 
Porcupine, with wedgie &amp;lt;&amp;lt;3 5 -6 1 -18 -28||, uses six of its minor tone generator steps to get to &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;. For this to work you need a small minor tone such as &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.&lt;br /&gt;
Badness (Smith): 0.025314
&lt;br /&gt;
 
Commas: 250/243, 64/63&lt;br /&gt;
==== Pourcup ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 162.880&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 64/63, 100/99, 196/195
Map: [&amp;lt;1 2 3 2|, &amp;lt;0 -3 -5 6|]&lt;br /&gt;
 
EDOs: 22, &lt;a class="wiki_link" href="/59edo"&gt;59&lt;/a&gt;, &lt;a class="wiki_link" href="/81edo"&gt;81bd&lt;/a&gt;, &lt;a class="wiki_link" href="/140edo"&gt;140bd&lt;/a&gt;&lt;br /&gt;
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Porcupine-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 55/54, 64/63, 100/99&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 163.378
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 162.482
POTE generator: ~10/9 = 162.747&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:  
Map: [&amp;lt;1 2 3 2 4|, &amp;lt;0 -3 -5 6 -4|]&lt;br /&gt;
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
EDOs: &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, 15, 22, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/59edo"&gt;59&lt;/a&gt;&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.13/7
Badness: 0.0217&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Hystrix"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Hystrix&lt;/h1&gt;
 
Hystrix, with wedgie &amp;lt;&amp;lt;3 5 1 1 -7 -12||, provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;. They can try the even sharper fifth of hystrix in &lt;a class="wiki_link" href="/68edo"&gt;68edo&lt;/a&gt; and see how that suits.&lt;br /&gt;
Badness (Smith): 0.035130
&lt;br /&gt;
 
Commas: 36/35, 160/147&lt;br /&gt;
==== Porkpie ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 158.868&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 64/63, 65/63, 100/99
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -3 -5 -1|]&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
EDOs: 10d, 12, 13d, 15&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Hedgehog"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Hedgehog&lt;/h1&gt;
* CTE: ~2 = 1200.000, ~11/10 = 163.678
Hedgehog, with wedgie &amp;lt;&amp;lt;6 10 10 2 -1 -5||, has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out 245/243, the sensamagic comma. 22et provides the obvious tuning, but if you are looking for an alternative, you could try the &amp;lt;146 232 338 411| val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14 note MOS gives scope for harmony while stopping well short of 22.&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 163.688
&lt;br /&gt;
 
Commas: 50/49, 245/243&lt;br /&gt;
Minimax tuning:
&lt;br /&gt;
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 435.648&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.9/7
&lt;br /&gt;
 
Map: [&amp;lt;2 1 1 2|, &amp;lt;0 3 5 5|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
Wedgie: &amp;lt;&amp;lt;6 10 10 2 -1 -5||&lt;br /&gt;
 
EDOs: 22, &lt;a class="wiki_link" href="/146edo"&gt;146&lt;/a&gt;&lt;br /&gt;
Badness (Smith): 0.026043
Badness: 0.0440&lt;br /&gt;
 
&lt;br /&gt;
== Opossum ==
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Hedgehog-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;11-limit&lt;/h2&gt;
Opossum can be described as {{nowrap| 8d & 15 }}. Tempering out [[28/27]], the perfect fifth of three generator steps is conflated with not [[32/21]] as in porcupine but [[14/9]]. Three such fifths or nine generator steps octave reduced give a flat 7/4. 2\15 is a good generator.
Commas: 50/49, 55/54, 99/98&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
POTE generator: ~9/7 = 435.386&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 28/27, 126/125
Map: [&amp;lt;2 1 1 2 4|, &amp;lt;0 3 5 5 4|]&lt;br /&gt;
 
EDOs: 14c, 22, 58ce, 80ce, 102cde&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
Badness: 0.0231&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Hedgehog-Hedgepig"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Hedgepig&lt;/h2&gt;
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
Commas: 50/49, 245/243, 385/384&lt;br /&gt;
: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
POTE generator: ~9/7 = 435.425&lt;br /&gt;
: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 1 1 2 12|, &amp;lt;0 3 5 5 -7|]&lt;br /&gt;
[[Minimax tuning]]:  
EDOs: 22, 80c, 102cd, 124cd&lt;br /&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
Badness: 0.0684&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Hedgehog-Hedgepig-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Music&lt;/h3&gt;
 
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3" rel="nofollow"&gt;Phobos Light&lt;/a&gt; by Chris Vaisvil in Hedgehog[14] &lt;a class="wiki_link" href="/hedgehog14"&gt;tuned&lt;/a&gt; to 22edo.&lt;br /&gt;
[[Badness]] (Smith): 0.040650
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Nautilus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Nautilus&lt;/h1&gt;
=== 11-limit ===
Commas: 49/48, 250/243&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Pote generator: ~21/20 = 82.505&lt;br /&gt;
Comma list: 28/27, 55/54, 77/75
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -6 -10 -3|]&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
Wedgie: &amp;lt;&amp;lt;6 10 3 2 -12 -21||&lt;br /&gt;
 
EDOs: 10, 15, 19, &lt;a class="wiki_link" href="/29edo"&gt;29&lt;/a&gt;, &lt;a class="wiki_link" href="/102edo"&gt;102cd&lt;/a&gt;&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 161.365
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Nautilus-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
* POTE: ~2 = 1200.000, ~11/10 = 159.807
Commas: 49/48, 55/54, 245/242&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:  
POTE generator: ~21/20 = 82.504&lt;br /&gt;
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3 4|, &amp;lt;0 -6 -10 -3 -8|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
EDOs: 10e, 14c, 15, 19, 22d, 29, 102cde&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.022325
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Nautilus-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
 
Commas: 49/48, 55/54, 91/90, 100/99&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~21/20 = 62.530&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 28/27, 40/39, 55/54, 66/65
Map: [&amp;lt;1 2 3 3 4 5|, &amp;lt;0 -6 -10 -3 -8 -19|]&lt;br /&gt;
 
EDOs: 10e, 15f, 17d, 19, 22d, 29, 102cde&lt;br /&gt;
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
&lt;br /&gt;
 
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3" rel="nofollow"&gt;Nautilus Reverie&lt;/a&gt; by &lt;a class="wiki_link" href="/IgliashonJones"&gt;Igliashon Calvin Jones-Coolidge&lt;/a&gt;&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Ammonite"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Ammonite&lt;/h1&gt;
* CTE: ~2 = 1200.000, ~11/10 = 161.631
Commas: 250/243, 686/675&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 158.805
&lt;br /&gt;
 
POTE generator: ~9/7 = 454.448&lt;br /&gt;
Minimax tuning:  
&lt;br /&gt;
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
Map: [&amp;lt;1 5 8 10|, &amp;lt;0 -9 -15 -19|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;9 15 19 3 5 2||&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}
EDOs: 29, 37, 66&lt;br /&gt;
 
Badness: 0.1077&lt;br /&gt;
Badness (Smith): 0.019389
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Ammonite-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
== Porky ==
Commas: 55/54, 100/99, 686/675&lt;br /&gt;
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.
&lt;br /&gt;
 
POTE generator: ~9/7 = 454.512&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 5 8 10 8|, &amp;lt;0 -9 -15 -19 -12|]&lt;br /&gt;
[[Comma list]]: 225/224, 250/243
EDOs: 29, 37, 66&lt;br /&gt;
 
Badness: 0.0457&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Ammonite-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
[[Optimal tuning]]s:  
Commas: 55/54, 91/90, 100/99, 169/168&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
&lt;br /&gt;
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
POTE generator: ~13/10 = 454.429&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
&lt;br /&gt;
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}
Map: [&amp;lt;1 5 8 10 8 9|, &amp;lt;0 -9 -15 -19 -12 -14|]&lt;br /&gt;
 
EDOs: 29, 37, 66&lt;br /&gt;
[[Minimax tuning]]:  
Badness: 0.0272&lt;br /&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Porky"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Porky&lt;/h1&gt;
 
Commas: 225/224, 250/243&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
&lt;br /&gt;
 
POTE generator: ~10/9 = 164.412&lt;br /&gt;
[[Badness]] (Smith): 0.054389
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 5|, &amp;lt;0 -3 -5 -16|]&lt;br /&gt;
=== 11-limit ===
Wedgie: &amp;lt;&amp;lt;3 5 16 1 17 23||&lt;br /&gt;
Subgroup: 2.3.5.7.11
EDOS: 7, 8, 15, 22, 29, 51, 73&lt;br /&gt;
 
Badness: 0.0544&lt;br /&gt;
Comma list: 55/54, 100/99, 225/224
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Porky-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
Commas: 55/54, 100/99, 225/224&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~10/9 = 164.552&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 164.321
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 164.552
Map: [&amp;lt;1 2 3 5 4|, &amp;lt;0 -3 -5 -16 -4|]&lt;br /&gt;
 
EDOs: 7, 8, 15, 22, 29, 51, 73&lt;br /&gt;
Minimax tuning:  
Badness: 0.0273&lt;br /&gt;
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.7/5
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="Coendou"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Coendou&lt;/h1&gt;
 
Commas: 250/243, 525/512&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}
&lt;br /&gt;
 
POTE generator: ~10/9 = 166.041&lt;br /&gt;
Badness (Smith): 0.027268
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 1|, &amp;lt;0 -3 -5 13|]&lt;br /&gt;
=== 13-limit ===
Wedgie: &amp;lt;&amp;lt;3 5 -13 1 -29 -44||&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 7, 29, 65c, 94cd&lt;br /&gt;
 
Badness: 0.1183&lt;br /&gt;
Comma list: 55/54, 65/64, 91/90, 100/99
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Coendou-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;11-limit&lt;/h2&gt;
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
Commas: 55/54, 100/99, 525/512&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~10/9 = 165.981&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 164.478
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 164.953
Map: [&amp;lt;1 2 3 1 4|, &amp;lt;0 -3 -5 13 -4|]&lt;br /&gt;
 
EDOs: 7, 29, 65ce, 94cde&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
Badness: 0.0497&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.026543
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Coendou-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;13-limit&lt;/h2&gt;
 
Commas: 55/54, 65/64, 100/99, 105/104&lt;br /&gt;
; Music
&lt;br /&gt;
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning
POTE generator: ~10/9 = 165.974&lt;br /&gt;
 
&lt;br /&gt;
== Coendou ==
Map: [&amp;lt;1 2 3 1 4 3|, &amp;lt;0 -3 -5 13 -4 5|]&lt;br /&gt;
Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.
EDOs: 7, 29, 65cef, 94cdef&lt;br /&gt;
 
Badness: 0.0302&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250/243, 525/512
 
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
 
[[Badness]] (Smith): 0.118344
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 100/99, 525/512
 
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 165.925
* POTE: ~2 = 1200.000, ~11/10 = 165.981
 
Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3
 
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}
 
Badness (Smith): 0.049669
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 65/64, 100/99, 105/104
 
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 166.046
* POTE: ~2 = 1200.000, ~11/10 = 165.974
 
Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3
 
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
 
Badness (Smith): 0.030233
 
== Hystrix ==
Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix is very high in [[error]] due to the large disparity between typical porcupine generators and a justly-tuned 8/7, and is usually considered an [[exotemperament]]. A generator of 2\15 or 9\68 can be used for hystrix.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 160/147
 
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}
 
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
 
[[Badness]] (Smith): 0.044944
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 22/21, 36/35, 80/77
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 164.768
* POTE: ~2 = 1200.000, ~11/10 = 158.750
 
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}
 
Badness (Smith): 0.026790
 
== Hedgehog ==
{{See also| Sensamagic clan | Stearnsmic clan }}
 
Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid omega-tricot.
 
22edo provides an obvious tuning, which happens to be the only [[patent val|patent-val]] tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is [[echidna]], which offers much more accuracy. They merge on 22edo.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 245/243
 
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
 
: mapping generators: ~7/5, ~9/7
 
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~9/7 = 435.258
: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
* [[POTE]]: ~7/5 = 600.000, ~9/7 = 435.648
: error map: {{val| 0.000 +4.989 -8.074 +9.414 }}
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
 
[[Badness]] (Smith): 0.043983
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 55/54, 99/98
 
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
 
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 435.528
* POTE: ~7/5 = 600.000, ~9/7 = 435.386
 
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
 
Badness (Smith): 0.023095
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 55/54, 65/63, 99/98
 
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
 
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 436.309
* POTE: ~7/5 = 600.000, ~9/7 = 435.861
 
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
 
Badness (Smith): 0.021516
 
==== Urchin ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 55/54, 66/65
 
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
 
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 435.186
* POTE: ~7/5 = 600.000, ~9/7 = 437.078
 
{{Optimal ET sequence|legend=0| 14c, 22f }}
 
Badness (Smith): 0.025233
 
=== Hedgepig ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 245/243, 385/384
 
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
 
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 435.329
* POTE: ~7/5 = 600.000, ~9/7 = 435.425
 
{{Optimal ET sequence|legend=0| 22 }}
 
Badness (Smith): 0.068406
 
; Music
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|hedgehog[14]]], 22edo tuning.
 
== Nautilus ==
Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 49/48, 250/243
 
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}
 
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~21/20 = 81.914
: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 82.505
: error map: {{val| 0.000 +3.012 -11.368 -16.342 }}
 
{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}
 
[[Badness]] (Smith): 0.057420
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 49/48, 55/54, 245/242
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 81.802
* POTE: ~2 = 1200.000, ~21/20 = 82.504
 
{{Optimal ET sequence|legend=0| 14c, 15, 29, 44d }}
 
Badness (Smith): 0.026023
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 55/54, 91/90, 100/99
 
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
 
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 81.912
* POTE: ~2 = 1200.000, ~21/20 = 82.530
 
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 44d }}
 
Badness (Smith): 0.022285
 
==== Belauensis ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 49/48, 55/54, 66/65
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
 
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 82.034
* POTE: ~2 = 1200.000, ~21/20 = 81.759
 
{{Optimal ET sequence|legend=0| 14c, 15, 29f, 44dff }}
 
Badness (Smith): 0.029816
 
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]
 
== Ammonite ==
Ammonite adds 686/675 to the comma list and may be described as the {{nowrap| 8d & 29 }} temperament. Its ploidacot is epsilon-enneacot. [[37edo]] provides an obvious tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250/243, 686/675
 
{{Mapping|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}
 
: mapping generators: ~2, ~9/7
 
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
* [[POTE]]: ~2 = 1200.000, ~9/7 = 454.448
: error map: {{val| 0.000 +8.009 -3.040 -3.346 }}
 
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
 
[[Badness]] (Smith): 0.107686
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 100/99, 686/675
 
Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 454.505
* POTE: ~2 = 1200.000, ~9/7 = 454.512
 
{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
 
Badness (Smith): 0.045694
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 91/90, 100/99, 169/168
 
Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}
 
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/10 = 454.480
* POTE: ~2 = 1200.000, ~13/10 = 454.529
 
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
 
Badness (Smith): 0.027168
 
== Ceratitid ==
Ceratitid adds 1728/1715 to the comma list and may be described as the {{nowrap| 21c & 22 }} temperament. Its ploidacot is omega-enneacot. [[22edo]] provides an obvious tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250/243, 1728/1715
 
{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}
 
: mapping generators: ~2, ~36/35
 
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~36/35 = 54.804
: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
* [[POTE]]: ~2 = 1200.000, ~36/35 = 54.384
: error map: {{val| 0.000 +8.585 -2.081 +13.636 }}
 
{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
 
[[Badness]] (Smith): 0.115304
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 100/99, 352/343
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
 
Optimal tunings:  
* CTE: ~2 = 1200.000, ~36/35 = 54.702
* POTE: ~2 = 1200.000, ~36/35 = 54.376
 
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
 
Badness (Smith): 0.051319
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 65/63, 100/99, 352/343
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~36/35 = 54.575
* POTE: ~2 = 1200.000, ~36/35 = 54.665
 
{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
 
Badness (Smith): 0.044739
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"