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→Porcupine: base the sharpness on 4/3 rather than 3/2 (see talk). Hystrix isn't actually flat of 8d
 
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<span style="display: block; text-align: right;">[[de:Porcupine]]</span>
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| en = Porcupine family
The 5-limit parent comma for the porcupine family is 250/243, the maximal [[diesis|diesis]] or porcupine comma. Its [[monzo|monzo]] is |1 -5 3&gt;, and flipping that yields &lt;&lt;3 5 1|| for the [[wedgie|wedgie]]. This tells us the [[generator|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.
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{{Technical data page}}
The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.  


valid range: [150.000, 171.429] (8 to 7)
== Porcupine ==
{{Main| Porcupine }}


nice range: [157.821, 166.015]
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.


strict range: [157.821, 166.015]
[[Subgroup]]: 2.3.5


[[POTE_tuning|POTE generator]]: 163.950
[[Comma list]]: 250/243


Map: [&lt;1 2 3|, &lt;0 -3 -5|]
{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}


EDOs: [[15edo|15]], [[22edo|22]], [[95edo|95c]], [[117edo|117bc]], [[139edo|139bc]], [[161edo|161bc]], [[183edo|183bc]]
: mapping generators: ~2, ~10/9


Badness: 0.0308
[[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 }}


==Seven limit children==
[[Tuning ranges]]:
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit|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|septimal quarter tone]], for [[Porcupine_family#Hystrix|hystrix]], [[50/49|50/49]], the [[jubilisma|jubilisma]], for [[Porcupine_family#Hedgehog|hedgehog]], and [[49/48|49/48]], the [[slendro_diesis|slendro diesis]], for [[Porcupine_family#Nautilus|nautilus]].
* [[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]


=Porcupine=
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
[[Porcupine|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|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
[[Badness]] (Smith): 0.030778


valid range: [160.000, 163.636] (15 to 22)
=== Overview to extensions ===
==== 7-limit extensions ====
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.  


nice range: [157.821, 166.015]
Those all share the same generator with porcupine.  


strict range: [160.000, 163.636]
[[#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]]: ~10/9 = 162.880
Temperaments discussed elsewhere include:
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].


7- and 9-limit minimax eigenmonzo: 9/7
==== 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.


Map: [&lt;1 2 3 2|, &lt;0 -3 -5 6|]
=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11


EDOs: 22, [[59edo|59]], [[81edo|81bd]], [[140edo|140bd]]
Comma list: 55/54, 100/99


Badness: 0.0411
Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


==11-limit==
Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
Commas: 55/54, 64/63, 100/99


valid range: [160.000, 163.636] (15 to 22)
: gencom: [2 10/9; 55/54, 100/99]


nice range: [150.637, 182.404]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1200.000, ~11/10 = 164.078


strict range: [160.000, 163.636]
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}


POTE generator: ~10/9 = 162.747
Badness (Smith): 0.0097


11-limit minimax eigenmonzo: 9/7
==== Undecimation ====
Subgroup: 2.3.5.11.13


Map: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]
Comma list: 55/54, 100/99, 512/507


EDOs: [[7edo|7]], 15, 22, [[37edo|37]], [[59edo|59]]
Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}


Badness: 0.0217
: sval mapping generators: ~2, ~65/44


==13-limit==
Optimal tunings:
Commas: 40/39, 55/54, 64/63, 66/65
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1200.000, ~88/65 = 518.209


valid range: [160.000, 163.636] (15 to 22f)
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}


nice range: [138.573, 182.404]
Badness (Smith): 0.0305


strict range: [160.000, 163.636]
== Septimal porcupine ==
{{Main| Porcupine }}


POTE generator: ~10/9 = 162.708
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.  


13- and 15-limit minimax eigenmonzo: 11/8
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]
[[Comma list]]: 64/63, 250/243


EDOs: 7, 15, 22f, 37f
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


Badness: 0.0213
[[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 }}


==Porcupinefish==
[[Minimax tuning]]:
See also: [[The_Biosphere|The Biosphere]]
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[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


Commas: 55/54, 64/63, 91/90, 100/99
[[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]


valid range: [160.000, 162.162] (15 to 37)
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}


nice range: [150.637, 182.404]
[[Badness]] (Smith): 0.041057


strict range: [160.000, 162.162]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~10/9 = 162.277
Comma list: 55/54, 64/63, 100/99


13- and 15-limit minimax eigenmonzo: 13/11
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1200.000, ~11/10 = 162.747


EDOs: 15, 22, 37, 59, 96b
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


<span style="background-color: #ffffff;">Badness: 0.0253</span>
Tuning ranges:
* 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]


Badness: 0.0253
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


==Pourcup==
Badness (Smith): 0.021562
Commas: 55/54, 64/63, 100/99, 196/195


POTE generator: ~10/9 = 162.482
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.  


13- and 15-limit minimax eigenmonzo: 13/7
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]
Comma list: 40/39, 55/54, 64/63, 66/65


EDOs: 15f, 22f, 37
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Badness: 0.0351
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.442
* POTE: ~2 = 1200.000, ~11/10 = 162.708


==Porkpie==
Minimax tuning:
Commas: 55/54, 64/63, 65/63, 100/99
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11


POTE generator: ~10/9 = 163.688
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]


13- and 15-limit minimax eigenmonzo: 9/7
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}


Map: [&lt;1 2 3 2 4 3|, &lt;0 -3 -5 6 -4 5|]
Badness (Smith): 0.021276


EDOs: 7, 15f, 22
==== Porcupinefish ====
{{See also| The Biosphere }}


Badness: 0.0260
Subgroup: 2.3.5.7.11.13


=Hystrix=
Comma list: 55/54, 64/63, 91/90, 100/99
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|15edo]]. They can try the even sharper fifth of hystrix in [[68edo|68edo]] and see how that suits.


Commas: 36/35, 160/147
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


[[POTE_tuning|POTE generator]]: 158.868
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 162.636
* POTE: ~2 = 1200.000, ~11/10 = 162.277


7- and 9-limit minimax eigenmonzo: 5/4
Minimax tuning:
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
Tuning ranges:  
* 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]


EDOs: 10d, 12, 13d, 15
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Badness: 0.0449
Badness (Smith): 0.025314


=Porky=
==== Pourcup ====
Commas: 225/224, 250/243
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/9 = 164.412
Comma list: 55/54, 64/63, 100/99, 196/195


7- and 9-limit minimax eigenmonzo: 7/5
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.378
* POTE: ~2 = 1200.000, ~11/10 = 162.482


Wedgie: &lt;&lt;3 5 16 1 17 23||
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


EDOS: 7, 8, 15, 22, 29, 51, 73
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}


Badness: 0.0544
Badness (Smith): 0.035130


==11-limit==
==== Porkpie ====
Commas: 55/54, 100/99, 225/224
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/9 = 164.552
Comma list: 55/54, 64/63, 65/63, 100/99


11-limit minimax eigenmonzo: 7/5
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.678
* POTE: ~2 = 1200.000, ~11/10 = 163.688


EDOs: 7, 8, 15, 22, 29, 51, 73
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


Badness: 0.0273
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}


==13-limit==
Badness (Smith): 0.026043
Commas: 55/54, 65/64, 91/90, 100/99


POTE generator: ~10/9 = 164.953
== Opossum ==
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.  


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
[[Subgroup]]: 2.3.5.7


EDOs: 22, 29, 51f, 80cdef
[[Comma list]]: 28/27, 126/125


Badness: 0.0265
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


=Coendou=
[[Optimal tuning]]s:
Commas: 250/243, 525/512
* [[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: ~10/9 = 166.041
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


7- and 9-limit minimax eigenmonzo: 3/2
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
[[Badness]] (Smith): 0.040650


Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 7, 29, 65c, 94cd
Comma list: 28/27, 55/54, 77/75


Badness: 0.1183
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


==11-limit==
Optimal tunings:
Commas: 55/54, 100/99, 525/512
* CTE: ~2 = 1200.000, ~11/10 = 161.365
* POTE: ~2 = 1200.000, ~11/10 = 159.807


POTE generator: ~10/9 = 165.981
Minimax tuning:  
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


11-limit minimax eigenmonzo: 3/2
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]
Badness (Smith): 0.022325


EDOs: 7, 29, 65ce, 94cde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0497
Comma list: 28/27, 40/39, 55/54, 66/65


==13-limit==
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
Commas: 55/54, 65/64, 100/99, 105/104


POTE generator: ~10/9 = 165.974
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.631
* POTE: ~2 = 1200.000, ~11/10 = 158.805


13- and 15-limit minimax eigenmonzo: 3/2
Minimax tuning:
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


EDOs: 7, 29, 65cef, 94cdef
Badness (Smith): 0.019389


Badness: 0.0302
== Porky ==
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.  


=Hedgehog=
[[Subgroup]]: 2.3.5.7
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;[http://tel.wikispaces.com/146_232_338 146 232 338] 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
[[Comma list]]: 225/224, 250/243


[[POTE_tuning|POTE generator]]: ~9/7 = 435.648
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}


Wedgie: &lt;&lt;6 10 10 2 -1 -5||
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5


EDOs: 22, [[146edo|146]]
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}


Badness: 0.0440
[[Badness]] (Smith): 0.054389


==11-limit==
=== 11-limit ===
Commas: 50/49, 55/54, 99/98
Subgroup: 2.3.5.7.11


POTE generator: ~9/7 = 435.386
Comma list: 55/54, 100/99, 225/224


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


EDOs: 14c, 22, 58ce, 80ce, 102cde
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 164.321
* POTE: ~2 = 1200.000, ~11/10 = 164.552


Badness: 0.0231
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: unchanged-interval (eigenmonzo) basis: 2.7/5


==13-limit==
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}
Commas: 50/49, 55/54, 65/63, 99/98


POTE generator: ~9/7 = 435.861
Badness (Smith): 0.027268


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 14cf, 22
Comma list: 55/54, 65/64, 91/90, 100/99


Badness: 0.0215
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


==Urchin==
Optimal tunings:
Commas: 40/39, 50/49, 55/54, 66/65
* CTE: ~2 = 1200.000, ~11/10 = 164.478
* POTE: ~2 = 1200.000, ~11/10 = 164.953


POTE generator: ~9/7 = 437.078
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
Badness (Smith): 0.026543


EDOs: 14c, 22f
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


Badness: 0.0252
== Coendou ==
Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.  


==Hedgepig==
[[Subgroup]]: 2.3.5.7
Commas: 50/49, 245/243, 385/384


POTE generator: ~9/7 = 435.425
[[Comma list]]: 250/243, 525/512


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


EDOs: 22, 80c, 102cd, 124cd
[[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 }}


Badness: 0.0684
[[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


===Music===
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.


=Nautilus=
[[Badness]] (Smith): 0.118344
Commas: 49/48, 250/243


POTE generator: ~21/20 = 82.505
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
Comma list: 55/54, 100/99, 525/512


Wedgie: &lt;&lt;6 10 3 2 -12 -21||
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


EDOs: 15, [[29edo|29]], 43cd, 44d, 59d, 73cd, [[102edo|102cd]]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 165.925
* POTE: ~2 = 1200.000, ~11/10 = 165.981


==11-limit==
Minimax tuning:
Commas: 49/48, 55/54, 245/242
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


POTE generator: ~21/20 = 82.504
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
Badness (Smith): 0.049669


EDOs: 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 55/54, 65/64, 100/99, 105/104
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 166.046
* POTE: ~2 = 1200.000, ~11/10 = 165.974


EDOs: 15f, 29, 43cde, 44d, 59df, 73cde, 102cde
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


Badness: 0.0223
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}


==Belauensis==
Badness (Smith): 0.030233
Commas: 40/39, 49/48, 55/54, 66/65


POTE generator: ~21/20 = ~14/13 = 81.759
== 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.


Map: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]
[[Subgroup]]: 2.3.5.7


EDOs: 14c, 15, 29f, 44df
[[Comma list]]: 36/35, 160/147


Badness: 0.0298
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]]
[[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 }}


=Ammonite=
[[Minimax tuning]]:
Commas: 250/243, 686/675
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


POTE generator: ~9/7 = 454.448
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
[[Badness]] (Smith): 0.044944


Wedgie: &lt;&lt;9 15 19 3 5 2||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 29, 37, 66
Comma list: 22/21, 36/35, 80/77


Badness: 0.1077
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


==11-limit==
Optimal tunings:
Commas: 55/54, 100/99, 686/675
* CTE: ~2 = 1200.000, ~11/10 = 164.768
* POTE: ~2 = 1200.000, ~11/10 = 158.750


POTE generator: ~9/7 = 454.512
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
Badness (Smith): 0.026790


EDOs: 29, 37, 66
== Hedgehog ==
{{See also| Sensamagic clan | Stearnsmic clan }}


Badness: 0.0457
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.  


==13-limit==
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.
Commas: 55/54, 91/90, 100/99, 169/168


POTE generator: ~13/10 = 454.429
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
[[Comma list]]: 50/49, 245/243


EDOs: 29, 37, 66
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}


Badness: 0.0272
: mapping generators: ~7/5, ~9/7


=Ceratitid=
[[Optimal tuning]]s:
Commas: 250/243, 1728/1715
* [[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 }}


POTE generator: ~36/35 = 54.384
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
[[Badness]] (Smith): 0.043983


Wedgie: &lt;&lt;9 15 4 3 -19 -33||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 22
Comma list: 50/49, 55/54, 99/98


Badness: 0.115
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}


==11-limit==
Optimal tunings:
Commas: 55/54, 100/99, 5324/5145
* CTE: ~7/5 = 600.000, ~9/7 = 435.528
* POTE: ~7/5 = 600.000, ~9/7 = 435.386


POTE generator: ~36/35 = 54.376
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
Badness (Smith): 0.023095


EDOs: 22
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0513
Comma list: 50/49, 55/54, 65/63, 99/98


==13-limit==
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
Commas: 55/54, 65/63, 100/99, 352/343


POTE generator: ~36/35 = 54.665
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 436.309
* POTE: ~7/5 = 600.000, ~9/7 = 435.861


Map: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}


EDOs: 22
Badness (Smith): 0.021516


Badness: 0.0447
==== Urchin ====
[[Category:family]]
Subgroup: 2.3.5.7.11.13
[[Category:porcupine]]
 
[[Category:theory]]
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"