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<span style="display: block; text-align: right;">[[de:Porcupine]]</span>
{{Technical data page}}
-----
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]] 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.


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]]: ~27/25 = 163.950
[[Comma list]]: 250/243


Map: [&lt;1 2 3|, &lt;0 -3 -5|]
{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}
: mapping generators: ~2, ~10/9


EDOs: [[7edo|7]], [[15edo|15]], [[22edo|22]], [[95edo|95c]], [[117edo|117bc]], [[139edo|139bc]], [[161edo|161bc]], [[183edo|183bc]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5444{{c}}, ~10/9 = 163.8881{{c}}
: [[error map]]: {{val| -0.456 +5.469 -7.121 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
: error map: {{val| 0.000 +5.859 -6.624 }}


Badness: 0.0308
[[Tuning ranges]]:  
* [[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]


==Seven limit children==
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
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]].


=Porcupine=
[[Badness]] (Sintel): 0.722
[[Porcupine]], with wedgie &lt;&lt;3 5 -6 1 -18 -28||, uses six of its minor tone generator steps to get to [[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: 64/63, 250/243
=== 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.


valid range: [160.000, 163.636] (15 to 22)
Those all share the same generator with porcupine.  


nice range: [157.821, 166.015]
[[#Nautilus|Nautilus]] tempers out [[49/48]] and splits the generator in two. 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.  


strict range: [160.000, 163.636]
Temperaments discussed elsewhere include:  
* ''[[Oxygen]]'' (+21/20) → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Stearnsmic clan #Hedgehog|Stearnsmic clan]]
* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]


[[POTE_tuning|POTE generator]]: ~10/9 = 162.880
The rest are considered in this page.  


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: 7, 15, 22, [[59edo|59]], [[81edo|81bd]], [[140edo|140bbd]]
Comma list: 55/54, 100/99


Badness: 0.0411
{{Mapping|legend=2| 1 2 3 4 | 0 -3 -5 -4 }}


==11-limit==
{{Mapping|legend=4| 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)
Optimal tunings:  
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}


nice range: [150.637, 182.404]
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}


strict range: [160.000, 163.636]
Badness (Sintel): 0.303


POTE generator: ~10/9 = 162.747
== Septimal porcupine ==
{{Main| Porcupine }}


11-limit minimax eigenmonzo: 9/7
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 2 4|, &lt;0 -3 -5 6 -4|]
[[Subgroup]]: 2.3.5.7


EDOs: [[7edo|7]], 15, 22, [[37edo|37]], [[59edo|59]]
[[Comma list]]: 64/63, 250/243


Badness: 0.0217
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


==13-limit==
[[Optimal tuning]]s:
Commas: 40/39, 55/54, 64/63, 66/65
* [[WE]]: ~2 = 1197.8178{{c}}, ~10/9 = 162.5839{{c}}
: [[error map]]: {{val| -2.182 +5.929 -5.780 +2.313 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}


valid range: [160.000, 163.636] (15 to 22f)
[[Minimax tuning]]:
* [[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


nice range: [138.573, 182.404]
[[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]


strict range: [160.000, 163.636]
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}


POTE generator: ~10/9 = 162.708
[[Badness]] (Sintel): 1.04


13- and 15-limit minimax eigenmonzo: 11/8
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]
Comma list: 55/54, 64/63, 100/99


EDOs: 7, 15, 22f, 37f
{{Mapping|legend=0| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Badness: 0.0213
Optimal tunings:  
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}


==Porcupinefish==
Minimax tuning:
See also: [[The_Biosphere|The Biosphere]]
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


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


valid range: [160.000, 162.162] (15 to 37)
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


nice range: [150.637, 182.404]
Badness (Sintel): 0.713


strict range: [160.000, 162.162]
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.  


POTE generator: ~10/9 = 162.277
Subgroup: 2.3.5.7.11.13


13- and 15-limit minimax eigenmonzo: 13/11
Comma list: 40/39, 55/54, 64/63, 66/65


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
{{Mapping|legend=0| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


EDOs: 15, 22, 37, 59, 96b
Optimal tunings:  
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}


Badness: 0.0253
Minimax tuning:  
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11


==Pourcup==
Tuning ranges:
Commas: 55/54, 64/63, 100/99, 196/195
* 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 = 162.482
{{Optimal ET sequence|legend=0| 7, 15, 22f }}


13- and 15-limit minimax eigenmonzo: 13/7
Badness (Sintel): 0.879


Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]
==== Porcupinefish ====
{{See also| The Biosphere }}


EDOs: 15f, 22f, 37
Subgroup: 2.3.5.7.11.13


Badness: 0.0351
Comma list: 55/54, 64/63, 91/90, 100/99


==Porkpie==
{{Mapping|legend=0| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
Commas: 55/54, 64/63, 65/63, 100/99


POTE generator: ~10/9 = 163.688
Optimal tunings:  
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}


13- and 15-limit minimax eigenmonzo: 9/7
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 2 4 3|, &lt;0 -3 -5 6 -4 5|]
Tuning ranges:  
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]


EDOs: 7, 15f, 22
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Badness: 0.0260
Badness (Sintel): 1.05


=Hystrix=
==== Pourcup ====
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: 2.3.5.7.11.13


Commas: 36/35, 160/147
Comma list: 55/54, 64/63, 100/99, 196/195


[[POTE_tuning|POTE generator]]: ~8/7 = 158.868
{{Mapping|legend=0| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


7- and 9-limit minimax eigenmonzo: 5/4
Optimal tunings:  
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
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, 8d, 15d
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}


Badness: 0.0449
Badness (Sintel): 1.45


==11-limit==
==== Porkpie ====
Commas: 22/21, 36/35, 80/77
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 158.750
Comma list: 55/54, 64/63, 65/63, 100/99


Map: [&lt;1 2 3 3 4|, &lt;0 -3 -5 -1 -4|]
{{Mapping|legend=0| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


EDOs: 7, 8d, 15d
Optimal tunings:  
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}


Badness: 0.0268
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


=Porky=
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
Commas: 225/224, 250/243


POTE generator: ~10/9 = 164.412
Badness (Sintel): 1.08


7- and 9-limit minimax eigenmonzo: 7/5
==== Undecimation ====
This weak extension is used by most patent vals that support porcupine and tune 13 well.


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
Subgroup: 2.3.5.7.11.13


Wedgie: &lt;&lt;3 5 16 1 17 23||
Comma list: 55/54, 64/63, 100/99, 169/168


EDOs: 7d, 15d, 22, 29, 51, 73c
{{Mapping|legend=0| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}
: mapping generators: ~2, ~35/26


Badness: 0.0544
Optimal tunings:
* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 7, 30, 37, 81bd }}
Commas: 55/54, 100/99, 225/224


POTE generator: ~10/9 = 164.552
Badness (Sintel): 1.68


11-limit minimax eigenmonzo: 7/5
== Opossum ==
{{Main| Opossum }}


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


EDOs: 7d, 15d, 22, 29, 51, 73ce
[[Subgroup]]: 2.3.5.7


Badness: 0.0273
[[Comma list]]: 28/27, 126/125


==13-limit==
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
Commas: 55/54, 65/64, 91/90, 100/99


POTE generator: ~10/9 = 164.953
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.7927{{c}}, ~10/9 = 159.1315{{c}}
: [[error map]]: {{val| -4.207 +12.236 +5.407 -17.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


EDOs: 7d, 22, 29, 51f, 80cdeff
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


Badness: 0.0265
[[Badness]] (Sintel): 1.03


=Coendou=
=== 11-limit ===
Commas: 250/243, 525/512
Subgroup: 2.3.5.7.11


POTE generator: ~10/9 = 166.041
Comma list: 28/27, 55/54, 77/75


7- and 9-limit minimax eigenmonzo: 3/2
{{Mapping|legend=0| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
Optimal tunings:  
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}


Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
Minimax tuning:  
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


EDOs: 7, 29, 65c, 94cd
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Badness: 0.1183
Badness (Sintel): 0.738


==11-limit==
=== 13-limit ===
Commas: 55/54, 100/99, 525/512
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/9 = 165.981
Comma list: 28/27, 40/39, 55/54, 66/65


11-limit minimax eigenmonzo: 3/2
{{Mapping|legend=0| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]
Optimal tunings:  
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


EDOs: 7, 29, 65ce, 94cde
Minimax tuning:  
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Badness: 0.0497
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


==13-limit==
Badness (Sintel): 0.801
Commas: 55/54, 65/64, 100/99, 105/104


POTE generator: ~10/9 = 165.974
== Porky ==
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.  


13- and 15-limit minimax eigenmonzo: 3/2
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
[[Comma list]]: 225/224, 250/243


EDOs: 7, 29, 65cef, 94cdef
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


Badness: 0.0302
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0685{{c}}, ~10/9 = 164.4215{{c}}
: [[error map]]: {{val| +0.068 +4.917 -8.216 +0.772 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}


=Hedgehog=
[[Minimax tuning]]:
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.
* [[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


Commas: 50/49, 245/243
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51, 73c }}


[[POTE_tuning|POTE generator]]: ~9/7 = 435.648
[[Badness]] (Sintel): 1.38


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;6 10 10 2 -1 -5||
Comma list: 55/54, 100/99, 225/224


EDOs: 22, [[146edo|146]]
{{Mapping|legend=0| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Badness: 0.0440
Optimal tunings:  
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}


==11-limit==
Minimax tuning:
Commas: 50/49, 55/54, 99/98
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: unchanged-interval (eigenmonzo) basis: 2.7/5


POTE generator: ~9/7 = 435.386
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Badness (Sintel): 0.901


EDOs: 14c, 22, 58ce, 80ce, 102cde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0231
Comma list: 55/54, 65/64, 91/90, 100/99


==13-limit==
{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
Commas: 50/49, 55/54, 65/63, 99/98


POTE generator: ~9/7 = 435.861
Optimal tunings:  
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}


EDOs: 14cf, 22
Badness (Sintel): 1.10


Badness: 0.0215
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


==Urchin==
== Coendou ==
Commas: 40/39, 50/49, 55/54, 66/65
Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.


POTE generator: ~9/7 = 437.078
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
[[Comma list]]: 250/243, 525/512


EDOs: 14c, 22f
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


Badness: 0.0252
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.6772{{c}}, ~10/9 = 166.4110{{c}}
: [[error map]]: {{val| +2.678 +4.166 -10.337 -2.806 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}


==Hedgepig==
[[Minimax tuning]]:
Commas: 50/49, 245/243, 385/384
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


POTE generator: ~9/7 = 435.425
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c }}


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
[[Badness]] (Sintel): 2.99


EDOs: 22, 80c, 102cd, 124cd
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0684
Comma list: 55/54, 100/99, 525/512


===Music===
{{Mapping|legend=0| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
[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
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


POTE generator: ~21/20 = 82.505
Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}


Wedgie: &lt;&lt;6 10 3 2 -12 -21||
Badness (Sintel): 1.64


EDOs: 15, [[29edo|29]], 43cd, 44d, 59d, 73cd, [[102edo|102cd]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 55/54, 65/64, 100/99, 105/104
Commas: 49/48, 55/54, 245/242


POTE generator: ~21/20 = 82.504
{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
Optimal tunings:  
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


EDOs: 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


==13-limit==
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
Badness (Sintel): 1.25


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
== 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.


EDOs: 15f, 29, 43cde, 44d, 59df, 73cde, 102cde
[[Subgroup]]: 2.3.5.7


Badness: 0.0223
[[Comma list]]: 36/35, 160/147


==Belauensis==
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
Commas: 40/39, 49/48, 55/54, 66/65


POTE generator: ~21/20 = ~14/13 = 81.759
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1187.8599{{c}}, ~10/9 = 157.2605{{c}}
: [[error map]]: {{val| -12.140 +1.983 -9.037 +37.493 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}


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


EDOs: 14c, 15, 29f, 44df
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}


Badness: 0.0298
[[Badness]] (Sintel): 1.14


[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Ammonite=
Comma list: 22/21, 36/35, 80/77
Commas: 250/243, 686/675


POTE generator: ~9/7 = 454.448
{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
Optimal tunings:  
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}


Wedgie: &lt;&lt;9 15 19 3 5 2||
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


EDOs: 29, 37, 66
Badness (Sintel): 0.886


Badness: 0.1077
== Nautilus ==
Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.  


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 55/54, 100/99, 686/675


POTE generator: ~9/7 = 454.512
[[Comma list]]: 49/48, 250/243


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}


EDOs: 29, 37, 66
: mapping generators: ~2, ~21/20


Badness: 0.0457
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.1642{{c}}, ~21/20 = 82.6542{{c}}
: [[error map]]: {{val| +2.164 +6.448 -6.364 -10.296 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}


==13-limit==
{{Optimal ET sequence|legend=1| 14c, 15, 29 }}
Commas: 55/54, 91/90, 100/99, 169/168


POTE generator: ~13/10 = 454.429
[[Badness]] (Sintel): 1.45


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 29, 37, 66
Comma list: 49/48, 55/54, 245/242


Badness: 0.0272
{{Mapping|legend=0| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}


=Ceratitid=
Optimal tunings:
Commas: 250/243, 1728/1715
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}


POTE generator: ~36/35 = 54.384
{{Optimal ET sequence|legend=0| 14c, 15, 29 }}


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
Badness (Sintel): 0.860


Wedgie: &lt;&lt;9 15 4 3 -19 -33||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 22
Comma list: 49/48, 55/54, 91/90, 100/99


Badness: 0.115
{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


==11-limit==
Optimal tunings:
Commas: 55/54, 100/99, 5324/5145
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}


POTE generator: ~36/35 = 54.376
{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
Badness (Sintel): 0.921


EDOs: 22
==== Belauensis ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0513
Comma list: 40/39, 49/48, 55/54, 66/65


==13-limit==
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
Commas: 55/54, 65/63, 100/99, 352/343


POTE generator: ~36/35 = 54.665
Optimal tunings:  
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}


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


EDOs: 22
Badness (Sintel): 1.23


Badness: 0.0447
; Music
[[Category:family]]
* [https://web.archive.org/web/20201127013840/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones]]
[[Category:porcupine]]
 
[[Category:theory]]
== Ammonite ==
{{See also|Subgroup temperaments #Ammon}}
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 -4 -7 -9 | 0 9 15 19 }}
 
: mapping generators: ~2, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3342{{c}}, ~14/9 = 745.1379{{c}}
: [[error map]]: {{val| -0.666 +6.949 -4.584 -5.213 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}
 
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
 
[[Badness]] (Sintel): 2.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 100/99, 686/675
 
{{Mapping|legend=0| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}
 
Optimal tunings:
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
 
Badness (Sintel): 1.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 91/90, 100/99, 169/168
 
{{Mapping|legend=0| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
 
Badness (Sintel): 1.12
 
== 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:
* [[WE]]: ~2 = 1197.6274{{c}}, ~36/35 = 54.2770{{c}}
: [[error map]]: {{val| -2.373 +4.807 -7.586 +6.948 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}
 
{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
 
[[Badness]] (Sintel): 2.92
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 100/99, 352/343
 
{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
 
Optimal tunings:
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
 
Badness (Sintel): 1.70
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 65/63, 100/99, 352/343
 
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
 
Optimal tunings:
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
 
Badness (Sintel): 1.85
 
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"