Xenharmonic Wiki

Porcupine family: Difference between revisions

← Older edit
VisualWikitext
Mike Battaglia (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, emailconfirmed, Interface administrators, Sysops
1,399 edits
m Text replacement - "(\[\[\:(dev|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"
m →Porcupine: update a link
 
(86 intermediate revisions by 15 users not shown)
Line 1: Line 1:
<span style="display: block; text-align: right;">[[:de:Porcupine|Deutsch]]</span>
{{interwiki
|  
| de = Porcupine
__FORCETOC__
| en = Porcupine family
-----
| es =
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.
| ja =
}}
{{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 }}
: mapping generators: ~2, ~10/9


EDOs: [[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|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
=== 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|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]]'' → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
* ''[[Jamesbond]]'' → [[Whitewood family #Jamesbond|Whitewood family]]


[[POTE_tuning|POTE generator]]: ~10/9 = 162.880
==== 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.


7- and 9-limit minimax eigenmonzo: 9/7
=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11


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


EDOs: 22, [[59edo|59]], [[81edo|81bd]], [[140edo|140bd]]
Subgroup-val mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


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


==11-limit==
Optimal tunings:
Commas: 55/54, 64/63, 100/99
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}


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


nice range: [150.637, 182.404]
Badness (Sintel): 0.303


strict range: [160.000, 163.636]
==== Undecimation ====
Subgroup: 2.3.5.11.13


POTE generator: ~10/9 = 162.747
Comma list: 55/54, 100/99, 512/507


11-limit minimax eigenmonzo: 9/7
Subgroup-val mapping: {{mapping| 1 -1 -2 0 5 | 0 6 10 8 -3 }}


Map: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]
: mapping generators: ~2, ~88/65


EDOs: [[7edo|7]], 15, 22, [[37edo|37]], [[59edo|59]]
Optimal tunings:  
* WE: ~2 = 1199.4791{{c}}, ~88/65 = 517.9845{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/65 = 518.1740{{c}}


Badness: 0.0217
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}


==13-limit==
Badness (Sintel): 1.21
Commas: 40/39, 55/54, 64/63, 66/65


valid range: [160.000, 163.636] (15 to 22f)
== Septimal porcupine ==
{{Main| Porcupine }}


nice range: [138.573, 182.404]
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.


strict range: [160.000, 163.636]
[[Subgroup]]: 2.3.5.7


POTE generator: ~10/9 = 162.708
[[Comma list]]: 64/63, 250/243


13- and 15-limit minimax eigenmonzo: 11/8
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


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


EDOs: 7, 15, 22f, 37f
[[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


Badness: 0.0213
[[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]


==Porcupinefish==
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
See also: [[The_Biosphere|The Biosphere]]


Commas: 55/54, 64/63, 91/90, 100/99
[[Badness]] (Sintel): 1.04


valid range: [160.000, 162.162] (15 to 37)
=== 11-limit ===
Subgroup: 2.3.5.7.11


nice range: [150.637, 182.404]
Comma list: 55/54, 64/63, 100/99


strict range: [160.000, 162.162]
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


POTE generator: ~10/9 = 162.277
Optimal tunings:  
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}


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


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
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]


EDOs: 15, 22, 37, 59, 96b
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


<span style="background-color: #ffffff;">Badness: 0.0253</span>
Badness (Sintel): 0.713


Badness: 0.0253
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.  


==Pourcup==
Subgroup: 2.3.5.7.11.13
Commas: 55/54, 64/63, 100/99, 196/195


POTE generator: ~10/9 = 162.482
Comma list: 40/39, 55/54, 64/63, 66/65


13- and 15-limit minimax eigenmonzo: 13/7
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]
Optimal tunings:  
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}


EDOs: 15f, 22f, 37
Minimax tuning:  
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11


Badness: 0.0351
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]


==Porkpie==
{{Optimal ET sequence|legend=0| 7, 15, 22f }}
Commas: 55/54, 64/63, 65/63, 100/99


POTE generator: ~10/9 = 163.688
Badness (Sintel): 0.879


13- and 15-limit minimax eigenmonzo: 9/7
==== Porcupinefish ====
{{See also| The Biosphere }}


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


EDOs: 7, 15f, 22
Comma list: 55/54, 64/63, 91/90, 100/99


Badness: 0.0260
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


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


Commas: 36/35, 160/147
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


[[POTE_tuning|POTE generator]]: 158.868
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]


7- and 9-limit minimax eigenmonzo: 5/4
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
Badness (Sintel): 1.05


EDOs: 10d, 12, 13d, 15
==== Pourcup ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0449
Comma list: 55/54, 64/63, 100/99, 196/195


=Porky=
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
Commas: 225/224, 250/243


POTE generator: ~10/9 = 164.412
Optimal tunings:  
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}


7- and 9-limit minimax eigenmonzo: 7/5
Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}


Wedgie: &lt;&lt;3 5 16 1 17 23||
Badness (Sintel): 1.45


EDOS: 7, 8, 15, 22, 29, 51, 73
==== Porkpie ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0544
Comma list: 55/54, 64/63, 65/63, 100/99


==11-limit==
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
Commas: 55/54, 100/99, 225/224


POTE generator: ~10/9 = 164.552
Optimal tunings:  
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}


11-limit minimax eigenmonzo: 7/5
Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}


EDOs: 7, 8, 15, 22, 29, 51, 73
Badness (Sintel): 1.08


Badness: 0.0273
== Opossum ==
{{Main| Opossum }}


==13-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.
Commas: 55/54, 65/64, 91/90, 100/99


POTE generator: ~10/9 = 164.953
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
[[Comma list]]: 28/27, 126/125


EDOs: 22, 29, 51f, 80cdef
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


Badness: 0.0265
[[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 }}


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


POTE generator: ~10/9 = 166.041
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


7- and 9-limit minimax eigenmonzo: 3/2
[[Badness]] (Sintel): 1.03


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
Comma list: 28/27, 55/54, 77/75


EDOs: 7, 29, 65c, 94cd
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Badness: 0.1183
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:
Commas: 55/54, 100/99, 525/512
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


POTE generator: ~10/9 = 165.981
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


11-limit minimax eigenmonzo: 3/2
Badness (Sintel): 0.738


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


EDOs: 7, 29, 65ce, 94cde
Comma list: 28/27, 40/39, 55/54, 66/65


Badness: 0.0497
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


==13-limit==
Optimal tunings:
Commas: 55/54, 65/64, 100/99, 105/104
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


POTE generator: ~10/9 = 165.974
Minimax tuning:  
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


13- and 15-limit minimax eigenmonzo: 3/2
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
Badness (Sintel): 0.801


EDOs: 7, 29, 65cef, 94cdef
== Porky ==
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.


Badness: 0.0302
[[Subgroup]]: 2.3.5.7


=Hedgehog=
[[Comma list]]: 225/224, 250/243
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
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


[[POTE_tuning|POTE generator]]: ~9/7 = 435.648
[[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 }}


Map: [&lt;2 1 1 2|, &lt;0 3 5 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


Wedgie: &lt;&lt;6 10 10 2 -1 -5||
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51, 73c }}


EDOs: 22, [[146edo|146]]
[[Badness]] (Sintel): 1.38


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


==11-limit==
Comma list: 55/54, 100/99, 225/224
Commas: 50/49, 55/54, 99/98


POTE generator: ~9/7 = 435.386
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Optimal tunings:  
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}


EDOs: 14c, 22, 58ce, 80ce, 102cde
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: unchanged-interval (eigenmonzo) basis: 2.7/5


Badness: 0.0231
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}


==13-limit==
Badness (Sintel): 0.901
Commas: 50/49, 55/54, 65/63, 99/98


POTE generator: ~9/7 = 435.861
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
Comma list: 55/54, 65/64, 91/90, 100/99


EDOs: 14cf, 22
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


Badness: 0.0215
Optimal tunings:  
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


==Urchin==
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
Commas: 40/39, 50/49, 55/54, 66/65


POTE generator: ~9/7 = 437.078
Badness (Sintel): 1.10


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


EDOs: 14c, 22f
== 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: 0.0252
[[Subgroup]]: 2.3.5.7


==Hedgepig==
[[Comma list]]: 250/243, 525/512
Commas: 50/49, 245/243, 385/384


POTE generator: ~9/7 = 435.425
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
[[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 }}


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


Badness: 0.0684
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c }}


===Music===
[[Badness]] (Sintel): 2.99
[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.


=Nautilus=
=== 11-limit ===
Commas: 49/48, 250/243
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 82.505
Comma list: 55/54, 100/99, 525/512


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Wedgie: &lt;&lt;6 10 3 2 -12 -21||
Optimal tunings:  
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


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


==11-limit==
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}
Commas: 49/48, 55/54, 245/242


POTE generator: ~21/20 = 82.504
Badness (Sintel): 1.64


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde
Comma list: 55/54, 65/64, 100/99, 105/104


==13-limit==
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
Optimal tunings:  
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


EDOs: 15f, 29, 43cde, 44d, 59df, 73cde, 102cde
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}


Badness: 0.0223
Badness (Sintel): 1.25


==Belauensis==
== Hystrix ==
Commas: 40/39, 49/48, 55/54, 66/65
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.


POTE generator: ~21/20 = ~14/13 = 81.759
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]
[[Comma list]]: 36/35, 160/147


EDOs: 14c, 15, 29f, 44df
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


Badness: 0.0298
[[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 }}


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


=Ammonite=
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Commas: 250/243, 686/675


POTE generator: ~9/7 = 454.448
[[Badness]] (Sintel): 1.14


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


Wedgie: &lt;&lt;9 15 19 3 5 2||
Comma list: 22/21, 36/35, 80/77


EDOs: 29, 37, 66
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Badness: 0.1077
Optimal tunings:  
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}
Commas: 55/54, 100/99, 686/675


POTE generator: ~9/7 = 454.512
Badness (Sintel): 0.886


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
== Hedgehog ==
{{See also| Sensamagic clan | Stearnsmic clan }}


EDOs: 29, 37, 66
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 alpha-tricot.


Badness: 0.0457
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.  


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


POTE generator: ~13/10 = 454.429
[[Comma list]]: 50/49, 245/243


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}


EDOs: 29, 37, 66
: mapping generators: ~7/5, ~9/7


Badness: 0.0272
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}


=Ceratitid=
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
Commas: 250/243, 1728/1715


POTE generator: ~36/35 = 54.384
[[Badness]] (Sintel): 1.11


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;9 15 4 3 -19 -33||
Comma list: 50/49, 55/54, 99/98


EDOs: 22
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}


Badness: 0.115
Optimal tunings:  
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}


==11-limit==
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
Commas: 55/54, 100/99, 5324/5145


POTE generator: ~36/35 = 54.376
Badness (Sintel): 0.764


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 22
Comma list: 50/49, 55/54, 65/63, 99/98


Badness: 0.0513
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}


==13-limit==
Optimal tunings:
Commas: 55/54, 65/63, 100/99, 352/343
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}


POTE generator: ~36/35 = 54.665
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}


Map: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]
Badness (Sintel): 0.889


EDOs: 22
==== Urchin ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0447
Comma list: 40/39, 50/49, 55/54, 66/65
[[Category:family]]
 
[[Category:porcupine]]
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
[[Category:theory]]
 
Optimal tunings:
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 22f }}
 
Badness (Sintel): 1.04
 
=== 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:
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
 
{{Optimal ET sequence|legend=0| 22 }}
 
Badness (Sintel): 2.26
 
; Music
* [https://web.archive.org/web/20240624173512/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:
* [[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 }}
 
{{Optimal ET sequence|legend=1| 14c, 15, 29 }}
 
[[Badness]] (Sintel): 1.45
 
=== 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:
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 15, 29 }}
 
Badness (Sintel): 0.860
 
==== 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:
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}
 
Badness (Sintel): 0.921
 
==== 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:
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 15 }}
 
Badness (Sintel): 1.23
 
; Music
* [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.
 
[[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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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:Temperament families]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"