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Porcupine family: Difference between revisions

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**Imported revision 468004376 - 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>
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| en = Porcupine family
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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
Badness: 0.0308


==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 [[64_63|Archytas 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:
7- and 9-limit minimax eigenmonzo: 9/7
* [[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)
Badness: 0.0411
* 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
11-limit minimax eigenmonzo: 9/7


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.  


==13-limit==
Those all share the same generator with porcupine.
Commas: 40/39, 55/54, 64/63, 66/65


POTE generator: ~10/9 = 162.708
[[#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.
13- and 15-limit minimax eigenmonzo: 11/8


Map: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]
Temperaments discussed elsewhere include:  
EDOs: 7, 15, 22f, 37f
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
Badness: 0.0213
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].


==Porcupinefish==  
==== Subgroup extensions ====
See also: [[The Biosphere]]
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.
Commas: 55/54, 64/63, 91/90, 100/99


POTE generator: ~10/9 = 162.277
=== 2.3.5.11 subgroup (porkypine) ===
13- and 15-limit minimax eigenmonzo: 13/11
Subgroup: 2.3.5.11


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
Comma list: 55/54, 100/99
EDOs: 15, 22, 37, 59, 96b
&lt;span style="background-color: #ffffff;"&gt;Badness: 0.0253&lt;/span&gt;
Badness: 0.0253


==Pourcup==
Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
Commas: 55/54, 64/63, 100/99, 196/195


POTE generator: ~10/9 = 162.482
Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
13- and 15-limit minimax eigenmonzo: 13/7


Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]
: gencom: [2 10/9; 55/54, 100/99]
EDOs: 15f, 22f, 37
Badness: 0.0351


==Porkpie==
Optimal tunings:
Commas: 55/54, 64/63, 65/63, 100/99
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1200.000, ~11/10 = 164.078


POTE generator: ~10/9 = 163.688
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}
13- and 15-limit minimax eigenmonzo: 9/7


Map: [&lt;1 2 3 2 4 3|, &lt;0 -3 -5 6 -4 5|]
Badness (Smith): 0.0097
EDOs: 7, 15f, 22
Badness: 0.0260


=Hystrix=  
==== Undecimation ====
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.
Subgroup: 2.3.5.11.13


Commas: 36/35, 160/147
Comma list: 55/54, 100/99, 512/507


[[POTE tuning|POTE generator]]: 158.868
Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}
7- and 9-limit minimax eigenmonzo: 5/4


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
: sval mapping generators: ~2, ~65/44


EDOs: 10d, 12, 13d, 15
Optimal tunings:  
Badness: 0.0449
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1200.000, ~88/65 = 518.209


=Porky=
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
Commas: 225/224, 250/243


POTE generator: ~10/9 = 164.412
Badness (Smith): 0.0305
7- and 9-limit minimax eigenmonzo: 7/5


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
== Septimal porcupine ==
Wedgie: &lt;&lt;3 5 16 1 17 23||
{{Main| Porcupine }}
EDOS: 7, 8, 15, 22, 29, 51, 73
Badness: 0.0544


==11-limit==
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.
Commas: 55/54, 100/99, 225/224


POTE generator: ~10/9 = 164.552
[[Subgroup]]: 2.3.5.7
11-limit minimax eigenmonzo: 7/5


Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]
[[Comma list]]: 64/63, 250/243
EDOs: 7, 8, 15, 22, 29, 51, 73
Badness: 0.0273


==13-limit==
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
Commas: 55/54, 65/64, 91/90, 100/99


POTE generator: ~10/9 = 164.953
[[Optimal tuning]]s:
* [[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 }}


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
[[Minimax tuning]]:  
EDOs: 22, 29, 51f, 80cdef
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
Badness: 0.0265
: [[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


=Coendou=  
[[Tuning ranges]]:
Commas: 250/243, 525/512
* 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]


POTE generator: ~10/9 = 166.041
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
7- and 9-limit minimax eigenmonzo: 3/2


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
[[Badness]] (Smith): 0.041057
Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
EDOs: 7, 29, 65c, 94cd
Badness: 0.1183


==11-limit==  
=== 11-limit ===
Commas: 55/54, 100/99, 525/512
Subgroup: 2.3.5.7.11


POTE generator: ~10/9 = 165.981
Comma list: 55/54, 64/63, 100/99
11-limit minimax eigenmonzo: 3/2


Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
EDOs: 7, 29, 65ce, 94cde
Badness: 0.0497


==13-limit==
Optimal tunings:
Commas: 55/54, 65/64, 100/99, 105/104
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1200.000, ~11/10 = 162.747


POTE generator: ~10/9 = 165.974
Minimax tuning:
13- and 15-limit minimax eigenmonzo: 3/2
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
Tuning ranges:  
EDOs: 7, 29, 65cef, 94cdef
* 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
Badness: 0.0302
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]


=Hedgehog=
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}
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
Badness (Smith): 0.021562


[[POTE tuning|POTE generator]]: ~9/7 = 435.648
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.  


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
Subgroup: 2.3.5.7.11.13
Wedgie: &lt;&lt;6 10 10 2 -1 -5||
EDOs: 22, [[146edo|146]]
Badness: 0.0440


==11-limit==
Comma list: 40/39, 55/54, 64/63, 66/65
Commas: 50/49, 55/54, 99/98


POTE generator: ~9/7 = 435.386
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Optimal tunings:  
EDOs: 14c, 22, 58ce, 80ce, 102cde
* CTE: ~2 = 1200.000, ~11/10 = 163.442
Badness: 0.0231
* POTE: ~2 = 1200.000, ~11/10 = 162.708


==13-limit==
Minimax tuning:
Commas: 50/49, 55/54, 65/63, 99/98
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11


POTE generator: ~9/7 = 435.861
Tuning ranges:
* 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]


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}
EDOs: 14cf, 22
Badness: 0.0215


==Urchin==
Badness (Smith): 0.021276
Commas: 40/39, 50/49, 55/54, 66/65


POTE generator: ~9/7 = 437.078
==== Porcupinefish ====
{{See also| The Biosphere }}


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
Subgroup: 2.3.5.7.11.13
EDOs: 14c, 22f
Badness: 0.0252


==Hedgepig==
Comma list: 55/54, 64/63, 91/90, 100/99
Commas: 50/49, 245/243, 385/384


POTE generator: ~9/7 = 435.425
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
Optimal tunings:  
EDOs: 22, 80c, 102cd, 124cd
* CTE: ~2 = 1200.000, ~11/10 = 162.636
Badness: 0.0684
* POTE: ~2 = 1200.000, ~11/10 = 162.277


===Music===
Minimax tuning:
[[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3|Phobos Light]] by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


=Nautilus=  
Tuning ranges:
Commas: 49/48, 250/243
* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]


POTE generator: ~21/20 = 82.505
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
Badness (Smith): 0.025314
Wedgie: &lt;&lt;6 10 3 2 -12 -21||
EDOs: 10, 15, 19, [[29edo|29]], [[102edo|102cd]]


==11-limit==  
==== Pourcup ====
Commas: 49/48, 55/54, 245/242
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 82.504
Comma list: 55/54, 64/63, 100/99, 196/195


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
EDOs: 10e, 14c, 15, 19, 22d, 29, 102cde


==13-limit==
Optimal tunings:
Commas: 49/48, 55/54, 91/90, 100/99
* CTE: ~2 = 1200.000, ~11/10 = 163.378
* POTE: ~2 = 1200.000, ~11/10 = 162.482


POTE generator: ~21/20 = 62.530
Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
EDOs: 10e, 15f, 17d, 19, 22d, 29, 102cde
Badness: 0.0223


==Belauensis==
Badness (Smith): 0.035130
Commas: 40/39, 49/48, 55/54, 66/65


POTE generator: ~21/20 = ~14/13 = 81.759
==== Porkpie ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]
Comma list: 55/54, 64/63, 65/63, 100/99
EDOs: 14c, 15, 29f, 44df
Badness: 0.0298


[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3|Nautilus Reverie]] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]]
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
=Ammonite=
Commas: 250/243, 686/675


POTE generator: ~9/7 = 454.448
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.678
* POTE: ~2 = 1200.000, ~11/10 = 163.688


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
Minimax tuning:  
Wedgie: &lt;&lt;9 15 19 3 5 2||
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
EDOs: 29, 37, 66
: unchanged-interval (eigenmonzo) basis: 2.9/7
Badness: 0.1077


==11-limit==
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
Commas: 55/54, 100/99, 686/675


POTE generator: ~9/7 = 454.512
Badness (Smith): 0.026043


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
== Opossum ==
EDOs: 29, 37, 66
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.  
Badness: 0.0457


==13-limit==
[[Subgroup]]: 2.3.5.7
Commas: 55/54, 91/90, 100/99, 169/168


POTE generator: ~13/10 = 454.429
[[Comma list]]: 28/27, 126/125


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
EDOs: 29, 37, 66
Badness: 0.0272


=Ceratitid=  
[[Optimal tuning]]s:
Commas: 250/243, 1728/1715
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}


POTE generator: ~36/35 = 54.384
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
Wedgie: &lt;&lt;9 15 4 3 -19 -33||
EDOs: 22
Badness: 0.115


==11-limit==
[[Badness]] (Smith): 0.040650
Commas: 55/54, 100/99, 5324/5145


POTE generator: ~36/35 = 54.376
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
Comma list: 28/27, 55/54, 77/75
EDOs: 22
Badness: 0.0513


==13-limit==
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
Commas: 55/54, 65/63, 100/99, 352/343


POTE generator: ~36/35 = 54.665
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.365
* POTE: ~2 = 1200.000, ~11/10 = 159.807


Map: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]
Minimax tuning:  
EDOs: 22
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
Badness: 0.0447</pre></div>
 
<h4>Original HTML content:</h4>
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
<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:60:&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:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt; | &lt;a href="#Porcupine"&gt;Porcupine&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt; | &lt;a href="#Hystrix"&gt;Hystrix&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt; | &lt;a href="#Porky"&gt;Porky&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt;&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&gt; | &lt;a href="#Coendou"&gt;Coendou&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&gt;&lt;!-- ws:end:WikiTextTocRule:73 --&gt;&lt;!-- ws:start:WikiTextTocRule:74: --&gt;&lt;!-- ws:end:WikiTextTocRule:74 --&gt;&lt;!-- ws:start:WikiTextTocRule:75: --&gt; | &lt;a href="#Hedgehog"&gt;Hedgehog&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&gt;&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&gt;&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt;&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt; | &lt;a href="#Nautilus"&gt;Nautilus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&gt;&lt;!-- ws:end:WikiTextTocRule:83 --&gt;&lt;!-- ws:start:WikiTextTocRule:84: --&gt;&lt;!-- ws:end:WikiTextTocRule:84 --&gt;&lt;!-- ws:start:WikiTextTocRule:85: --&gt; | &lt;a href="#Ammonite"&gt;Ammonite&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:85 --&gt;&lt;!-- ws:start:WikiTextTocRule:86: --&gt;&lt;!-- ws:end:WikiTextTocRule:86 --&gt;&lt;!-- ws:start:WikiTextTocRule:87: --&gt;&lt;!-- ws:end:WikiTextTocRule:87 --&gt;&lt;!-- ws:start:WikiTextTocRule:88: --&gt; | &lt;a href="#Ceratitid"&gt;Ceratitid&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:88 --&gt;&lt;!-- ws:start:WikiTextTocRule:89: --&gt;&lt;!-- ws:end:WikiTextTocRule:89 --&gt;&lt;!-- ws:start:WikiTextTocRule:90: --&gt;&lt;!-- ws:end:WikiTextTocRule:90 --&gt;&lt;!-- ws:start:WikiTextTocRule:91: --&gt;
 
&lt;!-- ws:end:WikiTextTocRule:91 --&gt;&lt;hr /&gt;
Badness (Smith): 0.022325
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;
=== 13-limit ===
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 163.950&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3|, &amp;lt;0 -3 -5|]&lt;br /&gt;
Comma list: 28/27, 40/39, 55/54, 66/65
&lt;br /&gt;
 
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;
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
Badness: 0.0308&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&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;
* CTE: ~2 = 1200.000, ~11/10 = 161.631
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="/64_63"&gt;Archytas 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;
* POTE: ~2 = 1200.000, ~11/10 = 158.805
&lt;br /&gt;
 
&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;
Minimax tuning:
&lt;a class="wiki_link" href="/Porcupine"&gt;Porcupine&lt;/a&gt;, 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;
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
&lt;br /&gt;
 
Commas: 250/243, 64/63&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~10/9 = 162.880&lt;br /&gt;
Badness (Smith): 0.019389
7- and 9-limit minimax eigenmonzo: 9/7&lt;br /&gt;
 
&lt;br /&gt;
== Porky ==
Map: [&amp;lt;1 2 3 2|, &amp;lt;0 -3 -5 6|]&lt;br /&gt;
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.
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;
 
Badness: 0.0411&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&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;
[[Comma list]]: 225/224, 250/243
Commas: 55/54, 64/63, 100/99&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
POTE generator: ~10/9 = 162.747&lt;br /&gt;
 
11-limit minimax eigenmonzo: 9/7&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
Map: [&amp;lt;1 2 3 2 4|, &amp;lt;0 -3 -5 6 -4|]&lt;br /&gt;
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
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;
* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
Badness: 0.0217&lt;br /&gt;
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Porcupine-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
[[Minimax tuning]]:
Commas: 40/39, 55/54, 64/63, 66/65&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
POTE generator: ~10/9 = 162.708&lt;br /&gt;
 
13- and 15-limit minimax eigenmonzo: 11/8&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 2 4 4|, &amp;lt;0 -3 -5 6 -4 -2|]&lt;br /&gt;
[[Badness]] (Smith): 0.054389
EDOs: 7, 15, 22f, 37f&lt;br /&gt;
 
Badness: 0.0213&lt;br /&gt;
=== 11-limit ===
&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="Porcupine-Porcupinefish"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Porcupinefish&lt;/h2&gt;
 
See also: &lt;a class="wiki_link" href="/The%20Biosphere"&gt;The Biosphere&lt;/a&gt;&lt;br /&gt;
Comma list: 55/54, 100/99, 225/224
Commas: 55/54, 64/63, 91/90, 100/99&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
POTE generator: ~10/9 = 162.277&lt;br /&gt;
 
13- and 15-limit minimax eigenmonzo: 13/11&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 164.321
Map: [&amp;lt;1 2 3 2 4 6|, &amp;lt;0 -3 -5 6 -4 -17|]&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 164.552
EDOs: 15, 22, 37, 59, 96b&lt;br /&gt;
 
&lt;span style="background-color: #ffffff;"&gt;Badness: 0.0253&lt;/span&gt;&lt;br /&gt;
Minimax tuning:  
Badness: 0.0253&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:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Porcupine-Pourcup"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Pourcup&lt;/h2&gt;
 
Commas: 55/54, 64/63, 100/99, 196/195&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}
&lt;br /&gt;
 
POTE generator: ~10/9 = 162.482&lt;br /&gt;
Badness (Smith): 0.027268
13- and 15-limit minimax eigenmonzo: 13/7&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
Map: [&amp;lt;1 2 3 2 4 1|, &amp;lt;0 -3 -5 6 -4 20|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 15f, 22f, 37&lt;br /&gt;
 
Badness: 0.0351&lt;br /&gt;
Comma list: 55/54, 65/64, 91/90, 100/99
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Porcupine-Porkpie"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Porkpie&lt;/h2&gt;
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
Commas: 55/54, 64/63, 65/63, 100/99&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
POTE generator: ~10/9 = 163.688&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 164.478
13- and 15-limit minimax eigenmonzo: 9/7&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 164.953
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 2 4 3|, &amp;lt;0 -3 -5 6 -4 5|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
EDOs: 7, 15f, 22&lt;br /&gt;
 
Badness: 0.0260&lt;br /&gt;
Badness (Smith): 0.026543
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Hystrix"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Hystrix&lt;/h1&gt;
; Music
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;
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning
&lt;br /&gt;
 
Commas: 36/35, 160/147&lt;br /&gt;
== Coendou ==
&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.
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 158.868&lt;br /&gt;
 
7- and 9-limit minimax eigenmonzo: 5/4&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -3 -5 -1|]&lt;br /&gt;
[[Comma list]]: 250/243, 525/512
&lt;br /&gt;
 
EDOs: 10d, 12, 13d, 15&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
Badness: 0.0449&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Porky"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Porky&lt;/h1&gt;
* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
Commas: 225/224, 250/243&lt;br /&gt;
: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
POTE generator: ~10/9 = 164.412&lt;br /&gt;
: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}
7- and 9-limit minimax eigenmonzo: 7/5&lt;br /&gt;
 
&lt;br /&gt;
[[Minimax tuning]]:  
Map: [&amp;lt;1 2 3 5|, &amp;lt;0 -3 -5 -16|]&lt;br /&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
Wedgie: &amp;lt;&amp;lt;3 5 16 1 17 23||&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
EDOS: 7, 8, 15, 22, 29, 51, 73&lt;br /&gt;
 
Badness: 0.0544&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Porky-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;11-limit&lt;/h2&gt;
[[Badness]] (Smith): 0.118344
Commas: 55/54, 100/99, 225/224&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
POTE generator: ~10/9 = 164.552&lt;br /&gt;
Subgroup: 2.3.5.7.11
11-limit minimax eigenmonzo: 7/5&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 100/99, 525/512
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;
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
Badness: 0.0273&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Porky-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;13-limit&lt;/h2&gt;
* CTE: ~2 = 1200.000, ~11/10 = 165.925
Commas: 55/54, 65/64, 91/90, 100/99 &lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 165.981
&lt;br /&gt;
 
POTE generator: ~10/9 = 164.953&lt;br /&gt;
Minimax tuning:  
&lt;br /&gt;
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
Map: [&amp;lt;1 2 3 5 4 3|, &amp;lt;0 -3 -5 -16 -4 5|]&lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.3
EDOs: 22, 29, 51f, 80cdef&lt;br /&gt;
 
Badness: 0.0265&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Coendou"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Coendou&lt;/h1&gt;
Badness (Smith): 0.049669
Commas: 250/243, 525/512&lt;br /&gt;
 
&lt;br /&gt;
=== 13-limit ===
POTE generator: ~10/9 = 166.041&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
7- and 9-limit minimax eigenmonzo: 3/2&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 55/54, 65/64, 100/99, 105/104
Map: [&amp;lt;1 2 3 1|, &amp;lt;0 -3 -5 13|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;3 5 -13 1 -29 -44||&lt;br /&gt;
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
EDOs: 7, 29, 65c, 94cd&lt;br /&gt;
 
Badness: 0.1183&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* CTE: ~2 = 1200.000, ~11/10 = 166.046
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Coendou-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;11-limit&lt;/h2&gt;
* POTE: ~2 = 1200.000, ~11/10 = 165.974
Commas: 55/54, 100/99, 525/512&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:
POTE generator: ~10/9 = 165.981&lt;br /&gt;
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
11-limit minimax eigenmonzo: 3/2 &lt;br /&gt;
: unchanged-interval (eigenmonzo) basis: 2.3
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 1 4|, &amp;lt;0 -3 -5 13 -4|]&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
EDOs: 7, 29, 65ce, 94cde&lt;br /&gt;
 
Badness: 0.0497&lt;br /&gt;
Badness (Smith): 0.030233
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Coendou-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;13-limit&lt;/h2&gt;
== Hystrix ==
Commas: 55/54, 65/64, 100/99, 105/104&lt;br /&gt;
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.
&lt;br /&gt;
 
POTE generator: ~10/9 = 165.974&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
13- and 15-limit minimax eigenmonzo: 3/2&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 36/35, 160/147
Map: [&amp;lt;1 2 3 1 4 3|, &amp;lt;0 -3 -5 13 -4 5|]&lt;br /&gt;
 
EDOs: 7, 29, 65cef, 94cdef&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
Badness: 0.0302&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Hedgehog"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Hedgehog&lt;/h1&gt;
* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
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;
: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
Commas: 50/49, 245/243&lt;br /&gt;
: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~9/7 = 435.648&lt;br /&gt;
[[Minimax tuning]]:  
&lt;br /&gt;
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
Map: [&amp;lt;2 1 1 2|, &amp;lt;0 3 5 5|]&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
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;
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Badness: 0.0440&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Smith): 0.044944
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Hedgehog-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;11-limit&lt;/h2&gt;
 
Commas: 50/49, 55/54, 99/98&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~9/7 = 435.386&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 22/21, 36/35, 80/77
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: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
Badness: 0.0231&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Hedgehog-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;13-limit&lt;/h2&gt;
* CTE: ~2 = 1200.000, ~11/10 = 164.768
Commas: 50/49, 55/54, 65/63, 99/98&lt;br /&gt;
* POTE: ~2 = 1200.000, ~11/10 = 158.750
&lt;br /&gt;
 
POTE generator: ~9/7 = 435.861&lt;br /&gt;
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}
&lt;br /&gt;
 
Map: [&amp;lt;2 1 1 2 4 3|, &amp;lt;0 3 5 5 4 6|]&lt;br /&gt;
Badness (Smith): 0.026790
EDOs: 14cf, 22&lt;br /&gt;
 
Badness: 0.0215&lt;br /&gt;
== Hedgehog ==
&lt;br /&gt;
{{See also| Sensamagic clan | Stearnsmic clan }}
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Hedgehog-Urchin"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Urchin&lt;/h2&gt;
 
Commas: 40/39, 50/49, 55/54, 66/65&lt;br /&gt;
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.
&lt;br /&gt;
 
POTE generator: ~9/7 = 437.078&lt;br /&gt;
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.
&lt;br /&gt;
 
Map: [&amp;lt;2 1 1 2 4 6|, &amp;lt;0 3 5 5 4 2|]&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
EDOs: 14c, 22f&lt;br /&gt;
 
Badness: 0.0252&lt;br /&gt;
[[Comma list]]: 50/49, 245/243
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Hedgehog-Hedgepig"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Hedgepig&lt;/h2&gt;
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
Commas: 50/49, 245/243, 385/384&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~7/5, ~9/7
POTE generator: ~9/7 = 435.425&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
Map: [&amp;lt;2 1 1 2 12|, &amp;lt;0 3 5 5 -7|]&lt;br /&gt;
* [[CTE]]: ~7/5 = 600.000, ~9/7 = 435.258
EDOs: 22, 80c, 102cd, 124cd&lt;br /&gt;
: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
Badness: 0.0684&lt;br /&gt;
* [[POTE]]: ~7/5 = 600.000, ~9/7 = 435.648
&lt;br /&gt;
: error map: {{val| 0.000 +4.989 -8.074 +9.414 }}
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Hedgehog-Hedgepig-Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&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;
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc20"&gt;&lt;a name="Nautilus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Nautilus&lt;/h1&gt;
[[Badness]] (Smith): 0.043983
Commas: 49/48, 250/243&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
POTE generator: ~21/20 = 82.505&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -6 -10 -3|]&lt;br /&gt;
Comma list: 50/49, 55/54, 99/98
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;
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Nautilus-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;11-limit&lt;/h2&gt;
Optimal tunings:  
Commas: 49/48, 55/54, 245/242&lt;br /&gt;
* CTE: ~7/5 = 600.000, ~9/7 = 435.528
&lt;br /&gt;
* POTE: ~7/5 = 600.000, ~9/7 = 435.386
POTE generator: ~21/20 = 82.504&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
Map: [&amp;lt;1 2 3 3 4|, &amp;lt;0 -6 -10 -3 -8|]&lt;br /&gt;
 
EDOs: 10e, 14c, 15, 19, 22d, 29, 102cde&lt;br /&gt;
Badness (Smith): 0.023095
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Nautilus-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;13-limit&lt;/h2&gt;
==== 13-limit ====
Commas: 49/48, 55/54, 91/90, 100/99&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
POTE generator: ~21/20 = 62.530&lt;br /&gt;
Comma list: 50/49, 55/54, 65/63, 99/98
&lt;br /&gt;
 
Map: [&amp;lt;1 2 3 3 4 5|, &amp;lt;0 -6 -10 -3 -8 -19|]&lt;br /&gt;
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
EDOs: 10e, 15f, 17d, 19, 22d, 29, 102cde&lt;br /&gt;
 
Badness: 0.0223&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* CTE: ~7/5 = 600.000, ~9/7 = 436.309
&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Nautilus-Belauensis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Belauensis&lt;/h2&gt;
* POTE: ~7/5 = 600.000, ~9/7 = 435.861
Commas: 40/39, 49/48, 55/54, 66/65&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
POTE generator: ~21/20 = ~14/13 = 81.759&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.021516
Map: [&amp;lt;1 2 3 3 4 4|, &amp;lt;0 -6 -10 -3 -8 -4|]&lt;br /&gt;
 
EDOs: 14c, 15, 29f, 44df&lt;br /&gt;
==== Urchin ====
Badness: 0.0298&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&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;
Comma list: 40/39, 50/49, 55/54, 66/65
&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc24"&gt;&lt;a name="Ammonite"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Ammonite&lt;/h1&gt;
 
Commas: 250/243, 686/675&lt;br /&gt;
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
&lt;br /&gt;
 
POTE generator: ~9/7 = 454.448&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* CTE: ~7/5 = 600.000, ~9/7 = 435.186
Map: [&amp;lt;1 5 8 10|, &amp;lt;0 -9 -15 -19|]&lt;br /&gt;
* POTE: ~7/5 = 600.000, ~9/7 = 437.078
Wedgie: &amp;lt;&amp;lt;9 15 19 3 5 2||&lt;br /&gt;
 
EDOs: 29, 37, 66&lt;br /&gt;
{{Optimal ET sequence|legend=0| 14c, 22f }}
Badness: 0.1077&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.025233
&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="Ammonite-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;11-limit&lt;/h2&gt;
 
Commas: 55/54, 100/99, 686/675&lt;br /&gt;
=== Hedgepig ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~9/7 = 454.512&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 50/49, 245/243, 385/384
Map: [&amp;lt;1 5 8 10 8|, &amp;lt;0 -9 -15 -19 -12|]&lt;br /&gt;
 
EDOs: 29, 37, 66&lt;br /&gt;
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
Badness: 0.0457&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Ammonite-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;13-limit&lt;/h2&gt;
* CTE: ~7/5 = 600.000, ~9/7 = 435.329
Commas: 55/54, 91/90, 100/99, 169/168&lt;br /&gt;
* POTE: ~7/5 = 600.000, ~9/7 = 435.425
&lt;br /&gt;
 
POTE generator: ~13/10 = 454.429&lt;br /&gt;
{{Optimal ET sequence|legend=0| 22 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 5 8 10 8 9|, &amp;lt;0 -9 -15 -19 -12 -14|]&lt;br /&gt;
Badness (Smith): 0.068406
EDOs: 29, 37, 66&lt;br /&gt;
 
Badness: 0.0272&lt;br /&gt;
; Music
&lt;br /&gt;
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|hedgehog[14]]], 22edo tuning.
&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc27"&gt;&lt;a name="Ceratitid"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;Ceratitid&lt;/h1&gt;
 
Commas: 250/243, 1728/1715&lt;br /&gt;
== Nautilus ==
&lt;br /&gt;
Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.
POTE generator: ~36/35 = 54.384&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Map: [&amp;lt;1 2 3 3|, &amp;lt;0 -9 -15 -4|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;9 15 4 3 -19 -33||&lt;br /&gt;
[[Comma list]]: 49/48, 250/243
EDOs: 22&lt;br /&gt;
 
Badness: 0.115&lt;br /&gt;
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc28"&gt;&lt;a name="Ceratitid-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;11-limit&lt;/h2&gt;
: mapping generators: ~2, ~21/20
Commas: 55/54, 100/99, 5324/5145&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
POTE generator: ~36/35 = 54.376&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~21/20 = 81.914
&lt;br /&gt;
: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
Map: [&amp;lt;1 2 3 3 4|, &amp;lt;0 -9 -15 -4 -12|]&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~21/20 = 82.505
EDOs: 22&lt;br /&gt;
: error map: {{val| 0.000 +3.012 -11.368 -16.342 }}
Badness: 0.0513&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}
&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc29"&gt;&lt;a name="Ceratitid-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;13-limit&lt;/h2&gt;
 
Commas: 55/54, 65/63, 100/99, 352/343&lt;br /&gt;
[[Badness]] (Smith): 0.057420
&lt;br /&gt;
 
POTE generator: ~36/35 = 54.665&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
Map: [&amp;lt;1 2 3 3 4 4|, &amp;lt;0 -9 -15 -4 -12 -7|]&lt;br /&gt;
 
EDOs: 22&lt;br /&gt;
Comma list: 49/48, 55/54, 245/242
Badness: 0.0447&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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]]
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