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{{interwiki
{{Technical data page}}
| de = Porcupine
The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.  
| en = Porcupine family
| es =
| ja =
}}
The 5-limit parent comma for the '''porcupine family''' is [[250/243]], the maximal diesis or porcupine comma. Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 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)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.


= Porcupine =
== Porcupine ==
Subgroup: 2.3.5
{{Main| Porcupine }}
 
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.
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 250/243
[[Comma list]]: 250/243


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


[[POTE generator]]: ~27/25 = 163.950
[[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 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [150.000, 171.429] (1\8 to 1\7)
* [[5-odd-limit]] [[diamond monotone]]: ~10/9 = [150.000, 171.429] (1\8 to 1\7)
* [[diamond purer]] range: [157.821, 166.015]
* 5-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* [[diamond nice]] range: [157.821, 166.015]


{{Val list|legend=1| 7, 15, 22, 95c, 117bc, 139bc, 161bc, 183bc }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}


[[Badness]]: 0.0308
[[Badness]] (Sintel): 0.722


== Extensions ==
=== Overview to extensions ===
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. That means
==== 7-limit extensions ====
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
The second comma defines which [[7-limit]] family member we are looking at.  
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].
* [[#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.  


= Septimal porcupine =
Those all share the same generator with porcupine.
{{main| Porcupine }}


Porcupine 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.
[[#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.  


Subgroup: 2.3.5.7
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]]
 
The rest are considered in this page.
 
==== 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.
 
=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11
 
Comma list: 55/54, 100/99
 
{{Mapping|legend=2| 1 2 3 4 | 0 -3 -5 -4 }}
 
{{Mapping|legend=4| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
 
Optimal tunings:
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}
 
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}
 
Badness (Sintel): 0.303
 
== Septimal porcupine ==
{{Main| Porcupine }}
 
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.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 250/243
[[Comma list]]: 64/63, 250/243


[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -3 -5 6 }}]
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
 
{{Multival|legend=1| 3 5 -6 1 -18 -28 }}


[[POTE generator]]: ~10/9 = 162.880
[[Optimal tuning]]s:
* [[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 9/7
* [[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


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


{{Val list|legend=1| 7, 15, 22, 59, 81bd, 140bbd }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}


[[Badness]]: 0.0411
[[Badness]] (Sintel): 1.04


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99


Mapping: [{{val| 1 2 3 2 4 }}, {{val| 0 -3 -5 6 -4 }}]
{{Mapping|legend=0| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


POTE generator: ~11/10 = 162.747
Optimal tunings:  
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit eigenmonzo: 9/7
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [160.000, 163.636] (2\15 to 3\22)
* 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* [[diamond purer]] range: [150.637, 182.404]
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
* [[diamond nice]] range: [160.000, 163.636]
 
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


Vals: {{val list| 7, 15, 22, 37, 59 }}
Badness (Sintel): 0.713


Badness: 0.0217
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.  


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 55/54, 64/63, 66/65
Comma list: 40/39, 55/54, 64/63, 66/65


Mapping: [{{val| 1 2 3 2 4 4 }}, {{val| 0 -3 -5 6 -4 -2 }}]
{{Mapping|legend=0| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


POTE generator: ~11/10 = 162.708
Optimal tunings:  
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit eigenmonzo: 11/8
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [160.000, 163.636] (15 to 22f)
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* [[diamond purer]] range: [138.573, 182.404]
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
* [[diamond nice]] range: [160.000, 163.636]
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]


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


Badness: 0.0213
Badness (Sintel): 0.879


=== Porcupinefish ===
==== Porcupinefish ====
{{see also| The Biosphere }}
{{See also| The Biosphere }}


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 108: Line 156:
Comma list: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99


Mapping: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]
{{Mapping|legend=0| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


POTE generator: ~11/10 = 162.277
Optimal tunings:  
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}


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


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [160.000, 162.162] (15 to 37)
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
* [[diamond purer]] range: [150.637, 182.404]
* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
* [[diamond nice]] range: [160.000, 162.162]
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]


Vals: {{val list| 15, 22, 37, 59, 96b }}
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Badness: 0.0253
Badness (Sintel): 1.05


=== Pourcup ===
==== Pourcup ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195


Mapping: [{{val| 1 2 3 2 4 1 }}, {{val| 0 -3 -5 6 -4 20 }}]
{{Mapping|legend=0| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


POTE generator: ~11/10 = 162.482
Optimal tunings:  
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}


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


Vals: {{val list| 15f, 22f, 37 }}
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}


Badness: 0.0351
Badness (Sintel): 1.45


=== Porkpie ===
==== Porkpie ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99


Mapping: [{{val| 1 2 3 2 4 3 }}, {{val| 0 -3 -5 6 -4 5 }}]
{{Mapping|legend=0| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


POTE generator: ~11/10 = 163.688
Optimal tunings:  
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}


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


Vals: {{val list| 7, 15f, 22 }}
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}


Badness: 0.0260
Badness (Sintel): 1.08


= Hystrix =
==== Undecimation ====
Hystrix 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.
This weak extension is used by most patent vals that support porcupine and tune 13 well.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 36/35, 160/147
 
[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -1 }}]
 
{{Multival|legend=1| 3 5 1 1 -7 -12 }}
 
[[POTE generator]]: ~10/9 = 158.868
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 5/4
 
{{Val list|legend=1| 7, 8d, 15d }}
 
[[Badness]]: 0.0449


== 11-limit ==
Comma list: 55/54, 64/63, 100/99, 169/168
Subgroup: 2.3.5.7.11


Comma list: 22/21, 36/35, 80/77
{{Mapping|legend=0| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}
: mapping generators: ~2, ~35/26


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -3 -5 -1 -4 }}]
Optimal tunings:
* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}


POTE generator: ~11/10 = 158.750
{{Optimal ET sequence|legend=0| 7, 30, 37, 81bd }}


Vals: {{val list| 7, 8d, 15d }}
Badness (Sintel): 1.68


Badness: 0.0268
== Opossum ==
{{Main| Opossum }}


= Porky =
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.  
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 250/243
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 3 5 }}, {{val| 0 -3 -5 -16 }}]
[[Comma list]]: 28/27, 126/125


{{Multival|legend=1| 3 5 16 1 17 23 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


[[POTE generator]]: ~10/9 = 164.412
[[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 7/5
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


{{Val list|legend=1| 7d, 15d, 22, 29, 51, 73c }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


[[Badness]]: 0.0544
[[Badness]] (Sintel): 1.03


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 100/99, 225/224
Comma list: 28/27, 55/54, 77/75


Mapping: [{{val| 1 2 3 5 4 }}, {{val| 0 -3 -5 -16 -4 }}]
{{Mapping|legend=0| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


POTE generator: ~11/10 = 164.552
Optimal tunings:  
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit eigenmonzo: 7/5
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Vals: {{val list| 7d, 15d, 22, 29, 51, 73ce }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Badness: 0.0273
Badness (Sintel): 0.738


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 65/64, 91/90, 100/99
Comma list: 28/27, 40/39, 55/54, 66/65
 
{{Mapping|legend=0| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Mapping: [{{val| 1 2 3 5 4 3 }}, {{val| 0 -3 -5 -16 -4 5 }}]
Optimal tunings:  
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


POTE generator: ~11/10 = 164.953
Minimax tuning:  
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Vals: {{val list| 7d, 22, 29, 51f, 80cdeff }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


Badness: 0.0265
Badness (Sintel): 0.801


= Coendou =
== Porky ==
Subgroup: 2.3.5.7
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.  


[[Comma list]]: 250/243, 525/512
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 3 1 }}, {{val| 0 -3 -5 13 }}]
[[Comma list]]: 225/224, 250/243


{{Multival|legend=1| 3 5 -13 1 -29 -44 }}
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


[[POTE generator]]: ~10/9 = 166.041
[[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 3/2
* [[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


{{Val list|legend=1| 7, 29, 65c, 94cd }}
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51, 73c }}


[[Badness]]: 0.1183
[[Badness]] (Sintel): 1.38


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 225/224


Mapping: [{{val| 1 2 3 1 4 }}, {{val| 0 -3 -5 13 -4 }}]
{{Mapping|legend=0| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


POTE generator: ~11/10 = 165.981
Optimal tunings:  
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit eigenmonzo: 3/2
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: unchanged-interval (eigenmonzo) basis: 2.7/5


Vals: {{val list| 7, 29, 65ce, 94cde }}
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}


Badness: 0.0497
Badness (Sintel): 0.901


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 91/90, 100/99


Mapping: [{{val| 1 2 3 1 4 3 }}, {{val| 0 -3 -5 13 -4 5 }}]
{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


POTE generator: ~11/10 = 165.974
Optimal tunings:  
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
* 13- and 15-odd-limit eigenmonzo: 3/2


Vals: {{val list| 7, 29, 65cef, 94cdef }}
Badness (Sintel): 1.10


Badness: 0.0302
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


= Hedgehog =
== Coendou ==
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. 22edo provides the obvious tuning, but if you are looking for an alternative, you could try the {{val| 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.
Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 245/243
[[Comma list]]: 250/243, 525/512


[[Mapping]]: [{{val| 2 1 1 2 }}, {{val| 0 3 5 5 }}]
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


{{Multival|legend=1| 6 10 10 2 -1 -5 }}
[[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 }}


[[POTE generator]]: ~9/7 = 435.648
[[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


{{Val list|legend=1| 8d, 14c, 22, 146bccdd }}
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c }}


[[Badness]]: 0.0440
[[Badness]] (Sintel): 2.99


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 50/49, 55/54, 99/98
Comma list: 55/54, 100/99, 525/512
 
{{Mapping|legend=0| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Mapping: [{{val| 2 1 1 2 4 }}, {{val| 0 3 5 5 4 }}]
Optimal tunings:  
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


POTE generator: ~9/7 = 435.386
Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


Vals: {{val list| 14c, 22, 58ce, 80ce, 102cde }}
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}


Badness: 0.0231
Badness (Sintel): 1.64


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 50/49, 55/54, 65/63, 99/98
Comma list: 55/54, 65/64, 100/99, 105/104


Mapping: [{{val| 2 1 1 2 4 3 }}, {{val| 0 3 5 5 4 6 }}]
{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


POTE generator: ~9/7 = 435.861
Optimal tunings:  
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


Vals: {{val list| 14cf, 22 }}
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3
 
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
 
Badness (Sintel): 1.25
 
== 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.


Badness: 0.0215
[[Subgroup]]: 2.3.5.7


=== Urchin ===
[[Comma list]]: 36/35, 160/147
Subgroup: 2.3.5.7.11.13


Comma list: 40/39, 50/49, 55/54, 66/65
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


Mapping: [{{val| 2 1 1 2 4 6 }}, {{val| 0 3 5 5 4 2 }}]
[[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 }}


POTE generator: ~9/7 = 437.078
[[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


Vals: {{val list| 14c, 22f }}
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}


Badness: 0.0252
[[Badness]] (Sintel): 1.14


== Hedgepig ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 50/49, 245/243, 385/384
Comma list: 22/21, 36/35, 80/77


Mapping: [{{val| 2 1 1 2 12 }}, {{val| 0 3 5 5 -7 }}]
{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


POTE generator: ~9/7 = 435.425
Optimal tunings:  
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}


Vals: {{val list| 22, 80c, 102cd, 124cd }}
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


Badness: 0.0684
Badness (Sintel): 0.886


; Music
== Nautilus ==
[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by [[Chris Vaisvil]] in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.
Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.  


= Nautilus =
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 49/48, 250/243
[[Comma list]]: 49/48, 250/243


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -6 -10 -3 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}
 
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 6 10 3 2 -12 -21 }}
[[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 }}


[[POTE generator]]: ~21/20 = 82.505
{{Optimal ET sequence|legend=1| 14c, 15, 29 }}


{{Val list|legend=1| 15, 29, 43cd, 44d, 59d, 73cd, 102cd }}
[[Badness]] (Sintel): 1.45


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -6 -10 -3 -8 }}]
{{Mapping|legend=0| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
 
Optimal tunings:
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}


POTE generator: ~21/20 = 82.504
{{Optimal ET sequence|legend=0| 14c, 15, 29 }}


Vals: {{val list| 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde }}
Badness (Sintel): 0.860


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 2 3 3 4 5 }}, {{val| 0 -6 -10 -3 -8 -19 }}]
{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


POTE generator: ~21/20 = 62.530
Optimal tunings:  
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}


Vals: {{val list| 15f, 29, 43cde, 44d, 59df, 73cde, 102cde }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}


Badness: 0.0223
Badness (Sintel): 0.921


=== Belauensis ===
==== Belauensis ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -6 -10 -3 -8 -4 }}]
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}


POTE generator: ~21/20 = 81.759
Optimal tunings:  
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}


Vals: {{val list| 14c, 15, 29f, 44df }}
{{Optimal ET sequence|legend=0| 14c, 15 }}


Badness: 0.0298
Badness (Sintel): 1.23


; Music
; Music
[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]
* [https://web.archive.org/web/20201127013840/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones]]
 
== 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.


= Ammonite =
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 250/243, 686/675
[[Comma list]]: 250/243, 686/675


[[Mapping]]: [{{val| 1 5 8 10 }}, {{val| 0 -9 -15 -19 }}]
{{Mapping|legend=1| 1 -4 -7 -9 | 0 9 15 19 }}


{{Multival|legend=1| 9 15 19 3 5 2 }}
: mapping generators: ~2, ~14/9


[[POTE generator]]: ~9/7 = 454.448
[[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 }}


{{Val list|legend=1| 29, 37, 66 }}
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}


[[Badness]]: 0.1077
[[Badness]] (Sintel): 2.73


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


Mapping: [{{val| 1 5 8 10 8 }}, {{val| 0 -9 -15 -19 -12 }}]
{{Mapping|legend=0| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}


POTE generator: ~9/7 = 454.512
Optimal tunings:  
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}


Vals: {{val list| 29, 37, 66 }}
{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}


Badness: 0.0457
Badness (Sintel): 1.51


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168


Mapping: [{{val| 1 5 8 10 8 9 }}, {{val| 0 -9 -15 -19 -12 -14 }}]
{{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}}


POTE generator: ~13/10 = 454.429
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}


Vals: {{val list| 29, 37, 66 }}
Badness (Sintel): 1.12


Badness: 0.0272
== 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.  


= Ceratitid =
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 250/243, 1728/1715
[[Comma list]]: 250/243, 1728/1715


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -9 -15 -4 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}


{{Multival|legend=1| 9 15 4 3 -19 -33 }}
: mapping generators: ~2, ~36/35


[[POTE generator]]: ~36/35 = 54.384
[[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 }}


{{Val list|legend=1| 22 }}
{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}


[[Badness]]: 0.115
[[Badness]] (Sintel): 2.92


== 11-limit ==
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 100/99, 5324/5145
Comma list: 55/54, 100/99, 352/343


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -9 -15 -4 -12 }}]
{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


POTE generator: ~36/35 = 54.376
Optimal tunings:  
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}


Vals: {{val list| 22 }}
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}


Badness: 0.0513
Badness (Sintel): 1.70


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -9 -15 -4 -12 -7 }}]
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


POTE generator: ~36/35 = 54.665
Optimal tunings:  
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}


Vals: {{val list| 22 }}
{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}


Badness: 0.0447
Badness (Sintel): 1.85


[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine]]
[[Category:Temperament families]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"