Xenharmonic Wiki

Porcupine family: Difference between revisions

← Older edit
VisualWikitext
m Cleanup (2/3)
m Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "
 
(86 intermediate revisions by 14 users not shown)
Line 1: Line 1:
{{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 ==
{{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]]:  
* valid 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)
* nice range: [157.821, 166.015]
* 5-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* strict range: [157.821, 166.015]
 
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
 
[[Badness]] (Sintel): 0.722


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


[[Badness]]: 0.0308
Those all share the same generator with porcupine.  


== Extensions ==
[[#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.  
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
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].


= Septimal porcupine =
Temperaments discussed elsewhere include:
{{main| Porcupine }}
* ''[[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]]


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.
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]]:  
* valid 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)
* nice range: [157.821, 166.015]
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* strict range: [160.000, 163.636]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]
 
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}


{{Val list|legend=1| 7, 15, 22, 59, 81bd, 140bbd }}
[[Badness]] (Sintel): 1.04


[[Badness]]: 0.0411
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
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:  
* valid 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)
* nice range: [150.637, 182.404]
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
* strict range: [160.000, 163.636]


Vals: {{val list| 7, 15, 22, 37, 59 }}
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


Badness: 0.0217
Badness (Sintel): 0.713
 
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.
 
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
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: ~10/9 = 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:  
* valid range: [160.000, 163.636] (15 to 22f)
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* nice range: [138.573, 182.404]
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
* strict 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


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: ~10/9 = 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:  
* valid range: [160.000, 162.162] (15 to 37)
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
* nice range: [150.637, 182.404]
* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
* strict 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 ====
Subgroup: 2.3.5.7.11.13


=== Pourcup ===
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: ~10/9 = 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 ====
Subgroup: 2.3.5.7.11.13


=== Porkpie ===
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: ~10/9 = 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.


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


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -1 }}]
Comma list: 55/54, 64/63, 100/99, 169/168


{{Multival|legend=1| 3 5 1 1 -7 -12 }}
{{Mapping|legend=0| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}
: mapping generators: ~2, ~35/26


[[POTE generator]]: ~8/7 = 158.868
Optimal tunings:
* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}


[[Minimax tuning]]:
{{Optimal ET sequence|legend=0| 7, 30, 37, 81bd }}
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 5/4


{{Val list|legend=1| 7, 8d, 15d }}
Badness (Sintel): 1.68


[[Badness]]: 0.0449
== Opossum ==
{{Main| Opossum }}


== 11-limit ==
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.
Comma list: 22/21, 36/35, 80/77


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -3 -5 -1 -4 }}]
[[Subgroup]]: 2.3.5.7


POTE generator: ~8/7 = 158.750
[[Comma list]]: 28/27, 126/125


Vals: {{val list| 7, 8d, 15d }}
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


Badness: 0.0268
[[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 }}


= Porky =
[[Minimax tuning]]:
[[Comma list]]: 225/224, 250/243
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


[[Mapping]]: [{{val| 1 2 3 5 }}, {{val| 0 -3 -5 -16 }}]
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


{{Multival|legend=1| 3 5 16 1 17 23 }}
[[Badness]] (Sintel): 1.03


[[POTE generator]]: ~10/9 = 164.412
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Minimax tuning]]:  
Comma list: 28/27, 55/54, 77/75
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 7/5


{{Val list|legend=1| 7d, 15d, 22, 29, 51, 73c }}
{{Mapping|legend=0| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


[[Badness]]: 0.0544
Optimal tunings:  
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}


== 11-limit ==
Minimax tuning:
Comma list: 55/54, 100/99, 225/224
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Mapping: [{{val| 1 2 3 5 4 }}, {{val| 0 -3 -5 -16 -4 }}]
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


POTE generator: ~10/9 = 164.552
Badness (Sintel): 0.738


Minimax tuning:
=== 13-limit ===
* 11-odd-limit eigenmonzo: 7/5
Subgroup: 2.3.5.7.11.13


Vals: {{val list| 7d, 15d, 22, 29, 51, 73ce }}
Comma list: 28/27, 40/39, 55/54, 66/65


Badness: 0.0273
{{Mapping|legend=0| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


== 13-limit ==
Optimal tunings:
Comma list: 55/54, 65/64, 91/90, 100/99
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


Mapping: [{{val| 1 2 3 5 4 3 }}, {{val| 0 -3 -5 -16 -4 5 }}]
Minimax tuning:  
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


POTE generator: ~10/9 = 164.953
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


Vals: {{val list| 7d, 22, 29, 51f, 80cdeff }}
Badness (Sintel): 0.801


Badness: 0.0265
== Porky ==
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.  


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


[[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 ===
Comma list: 55/54, 100/99, 525/512
Subgroup: 2.3.5.7.11
 
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: ~10/9 = 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
 
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}
 
Badness (Sintel): 0.901
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{val list| 7, 29, 65ce, 94cde }}
Comma list: 55/54, 65/64, 91/90, 100/99


Badness: 0.0497
{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


== 13-limit ==
Optimal tunings:
Comma list: 55/54, 65/64, 100/99, 105/104
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


Mapping: [{{val| 1 2 3 1 4 3 }}, {{val| 0 -3 -5 13 -4 5 }}]
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}


POTE generator: ~10/9 = 165.974
Badness (Sintel): 1.10


Minimax tuning:
; Music
* 13- and 15-odd-limit eigenmonzo: 3/2
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


Vals: {{val list| 7, 29, 65cef, 94cdef }}
== 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.


Badness: 0.0302
[[Subgroup]]: 2.3.5.7


= Hedgehog =
[[Comma list]]: 250/243, 525/512
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.


[[Comma list]]: 50/49, 245/243
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


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


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


[[POTE generator]]: ~9/7 = 435.648
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c }}


{{Val list|legend=1| 8d, 14c, 22, 146bccdd }}
[[Badness]] (Sintel): 2.99


[[Badness]]: 0.0440
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 55/54, 100/99, 525/512
Comma list: 50/49, 55/54, 99/98


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


POTE generator: ~9/7 = 435.386
Optimal tunings:  
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


Vals: {{val list| 14c, 22, 58ce, 80ce, 102cde }}
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


Badness: 0.0231
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}
 
Badness (Sintel): 1.64


=== 13-limit ===
=== 13-limit ===
Comma list: 50/49, 55/54, 65/63, 99/98
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 65/64, 100/99, 105/104
 
{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
 
Optimal tunings:
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


Mapping: [{{val| 2 1 1 2 4 3 }}, {{val| 0 3 5 5 4 6 }}]
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


POTE generator: ~9/7 = 435.861
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}


Vals: {{val list| 14cf, 22 }}
Badness (Sintel): 1.25


Badness: 0.0215
== 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.


=== Urchin ===
[[Subgroup]]: 2.3.5.7
Comma list: 40/39, 50/49, 55/54, 66/65


Mapping: [{{val| 2 1 1 2 4 6 }}, {{val| 0 3 5 5 4 2 }}]
[[Comma list]]: 36/35, 160/147


POTE generator: ~9/7 = 437.078
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


Vals: {{val list| 14c, 22f }}
[[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 }}


Badness: 0.0252
[[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


== Hedgepig ==
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Comma list: 50/49, 245/243, 385/384


Mapping: [{{val| 2 1 1 2 12 }}, {{val| 0 3 5 5 -7 }}]
[[Badness]] (Sintel): 1.14


POTE generator: ~9/7 = 435.425
=== 11-limit ===
Subgroup: 2.3.5.7.11


Vals: {{val list| 22, 80c, 102cd, 124cd }}
Comma list: 22/21, 36/35, 80/77


Badness: 0.0684
{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


; Music
Optimal tunings:
[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by [[Chris Vaisvil]] in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}
 
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}
 
Badness (Sintel): 0.886
 
== 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


= Nautilus =
[[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
 
[[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 }}


{{Multival|legend=1| 6 10 3 2 -12 -21 }}
{{Optimal ET sequence|legend=1| 14c, 15, 29 }}


[[POTE generator]]: ~21/20 = 82.505
[[Badness]] (Sintel): 1.45


{{Val list|legend=1| 15, 29, 43cd, 44d, 59d, 73cd, 102cd }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
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 }}


POTE generator: ~21/20 = 82.504
Optimal tunings:  
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}


Vals: {{val list| 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde }}
{{Optimal ET sequence|legend=0| 14c, 15, 29 }}
 
Badness (Sintel): 0.860
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
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 }}
 
Optimal tunings:
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}


POTE generator: ~21/20 = 62.530
{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}


Vals: {{val list| 15f, 29, 43cde, 44d, 59df, 73cde, 102cde }}
Badness (Sintel): 0.921


Badness: 0.0223
==== Belauensis ====
Subgroup: 2.3.5.7.11.13


=== Belauensis ===
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 = ~14/13 = 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
Comma list: 250/243, 686/675


Mapping: [{{val| 1 5 8 10 }}, {{val| 0 -9 -15 -19 }}]
[[Comma list]]: 250/243, 686/675


{{Multival|legend=1| 9 15 19 3 5 2 }}
{{Mapping|legend=1| 1 -4 -7 -9 | 0 9 15 19 }}


POTE generator: ~9/7 = 454.448
: mapping generators: ~2, ~14/9


{{Val list|legend=1| 29, 37, 66 }}
[[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 }}


Badness: 0.1077
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
 
[[Badness]] (Sintel): 2.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
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 ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
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.
 
[[Subgroup]]: 2.3.5.7


= Ceratitid =
[[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 ===
Comma list: 55/54, 100/99, 5324/5145
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -9 -15 -4 -12 }}]
Comma list: 55/54, 100/99, 352/343


POTE generator: ~36/35 = 54.376
{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


Vals: {{val list| 22 }}
Optimal tunings:  
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}


Badness: 0.0513
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
 
Badness (Sintel): 1.70
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
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"