Xenharmonic Wiki

Porcupine family: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
+cat
→Porcupine: base the sharpness on 4/3 rather than 3/2 (see talk). Hystrix isn't actually flat of 8d
 
(70 intermediate revisions by 13 users not shown)
Line 1: Line 1:
<span style="display: block; text-align: right;">[[de:Porcupine]]</span>
{{interwiki
-----
| de = Porcupine
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 &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)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. 3\22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.
| en = Porcupine family
| es =
| 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]]: ~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 }}


EDOs: {{EDOs| 7, 15, 22, 95c, 117bc, 139bc, 161bc, 183bc }}
: mapping generators: ~2, ~10/9


Badness: 0.0308
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.166
: [[error map]]: {{val| 0.000 +5.547 -7.143 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 163.950
: error map: {{val| 0.000 +6.194 -6.065 }}


== Seven limit children ==
[[Tuning ranges]]:
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. That means [[64/63]], the Archytas comma, for [[#Porcupine|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]].
* [[5-odd-limit]] [[diamond monotone]]: ~10/9 = [150.000, 171.429] (1\8 to 1\7)
* 5-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]


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


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|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]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].


[[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: {{EDOs| 7, 15, 22, 59, 81bd, 140bbd }}
Sval 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 ==
: gencom: [2 10/9; 55/54, 100/99]
Commas: 55/54, 64/63, 100/99


valid range: [160.000, 163.636] (15 to 22)
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1200.000, ~11/10 = 164.078


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


strict range: [160.000, 163.636]
Badness (Smith): 0.0097


POTE generator: ~10/9 = 162.747
==== Undecimation ====
Subgroup: 2.3.5.11.13


11-limit minimax eigenmonzo: 9/7
Comma list: 55/54, 100/99, 512/507


Map: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]
Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}


EDOs: {{EDOs| 7, 15, 22, 37, 59 }}
: sval mapping generators: ~2, ~65/44


Badness: 0.0217
Optimal tunings:
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1200.000, ~88/65 = 518.209
 
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
 
Badness (Smith): 0.0305
 
== 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
 
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 163.203
: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 162.880
: error map: {{val| 0.000 +9.405 -0.714 +8.455 }}
 
[[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
 
[[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]
 
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
 
[[Badness]] (Smith): 0.041057
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 64/63, 100/99
 
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1200.000, ~11/10 = 162.747
 
Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7
 
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]
 
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}
 
Badness (Smith): 0.021562
 
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 55/54, 64/63, 66/65
 
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.442
* POTE: ~2 = 1200.000, ~11/10 = 162.708
 
Minimax tuning:
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11
 
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]
 
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}
 
Badness (Smith): 0.021276
 
==== Porcupinefish ====
{{See also| The Biosphere }}
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 64/63, 91/90, 100/99
 
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 162.636
* POTE: ~2 = 1200.000, ~11/10 = 162.277
 
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
 
Tuning ranges:
* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
 
{{Optimal ET sequence|legend=0| 15, 22, 37 }}
 
Badness (Smith): 0.025314
 
==== Pourcup ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 64/63, 100/99, 196/195
 
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.378
* POTE: ~2 = 1200.000, ~11/10 = 162.482
 
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
 
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
 
Badness (Smith): 0.035130
 
==== Porkpie ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 64/63, 65/63, 100/99
 
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.678
* POTE: ~2 = 1200.000, ~11/10 = 163.688
 
Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7
 
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
 
Badness (Smith): 0.026043
 
== Opossum ==
Opossum can be described as {{nowrap| 8d & 15 }}. Tempering out [[28/27]], the perfect fifth of three generator steps is conflated with not [[32/21]] as in porcupine but [[14/9]]. Three such fifths or nine generator steps octave reduced give a flat 7/4. 2\15 is a good generator.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 28/27, 126/125
 
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
 
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
 
[[Badness]] (Smith): 0.040650
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 28/27, 55/54, 77/75
 
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.365
* POTE: ~2 = 1200.000, ~11/10 = 159.807
 
Minimax tuning:
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
 
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
 
Badness (Smith): 0.022325


=== 13-limit ===
=== 13-limit ===
Commas: 40/39, 55/54, 64/63, 66/65
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 40/39, 55/54, 66/65
 
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.631
* POTE: ~2 = 1200.000, ~11/10 = 158.805
 
Minimax tuning:
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


valid range: [160.000, 163.636] (15 to 22f)
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


nice range: [138.573, 182.404]
Badness (Smith): 0.019389


strict range: [160.000, 163.636]
== Porky ==
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.  


POTE generator: ~10/9 = 162.708
[[Subgroup]]: 2.3.5.7


13- and 15-limit minimax eigenmonzo: 11/8
[[Comma list]]: 225/224, 250/243


Map: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


EDOs: {{EDOs| 7, 15, 22f, 37f }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}


Badness: 0.0213
[[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


=== Porcupinefish ===
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
{{see also| The Biosphere }}


Commas: 55/54, 64/63, 91/90, 100/99
[[Badness]] (Smith): 0.054389


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, 100/99, 225/224


strict range: [160.000, 162.162]
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


POTE generator: ~10/9 = 162.277
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 164.321
* POTE: ~2 = 1200.000, ~11/10 = 164.552


13- and 15-limit minimax eigenmonzo: 13/11
Minimax tuning:
* 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 (Smith): 0.027268
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
Comma list: 55/54, 65/64, 91/90, 100/99


EDOs: {{EDOs| 15, 22, 37, 59, 96b }}
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


Badness: 0.0253
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 164.478
* POTE: ~2 = 1200.000, ~11/10 = 164.953


=== Pourcup ===
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
Commas: 55/54, 64/63, 100/99, 196/195


POTE generator: ~10/9 = 162.482
Badness (Smith): 0.026543


13- and 15-limit minimax eigenmonzo: 13/7
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]
== 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.


EDOs: {{EDOs| 15f, 22f, 37 }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0351
[[Comma list]]: 250/243, 525/512


=== Porkpie ===
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
Commas: 55/54, 64/63, 65/63, 100/99


POTE generator: ~10/9 = 163.688
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}


13- and 15-limit minimax eigenmonzo: 9/7
[[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


Map: [&lt;1 2 3 2 4 3|, &lt;0 -3 -5 6 -4 5|]
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}


EDOs: {{EDOs| 7, 15f, 22 }}
[[Badness]] (Smith): 0.118344


Badness: 0.0260
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Hystrix =
Comma list: 55/54, 100/99, 525/512
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.


Commas: 36/35, 160/147
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


[[POTE generator]]: ~8/7 = 158.868
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 165.925
* POTE: ~2 = 1200.000, ~11/10 = 165.981


7- and 9-limit minimax eigenmonzo: 5/4
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 -3 -5 -1|]
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}


EDOs: {{EDOs| 7, 8d, 15d }}
Badness (Smith): 0.049669


Badness: 0.0449
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 11-limit ==
Comma list: 55/54, 65/64, 100/99, 105/104
Commas: 22/21, 36/35, 80/77


POTE generator: ~8/7 = 158.750
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


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


EDOs: {{EDOs| 7, 8d, 15d }}
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


Badness: 0.0268
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}


= Porky =
Badness (Smith): 0.030233
Commas: 225/224, 250/243


POTE generator: ~10/9 = 164.412
== 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.


7- and 9-limit minimax eigenmonzo: 7/5
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
[[Comma list]]: 36/35, 160/147


Wedgie: &lt;&lt;3 5 16 1 17 23||
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


EDOs: {{EDOs| 7d, 15d, 22, 29, 51, 73c }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}


Badness: 0.0544
[[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


== 11-limit ==
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Commas: 55/54, 100/99, 225/224


POTE generator: ~10/9 = 164.552
[[Badness]] (Smith): 0.044944


11-limit minimax eigenmonzo: 7/5
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]
Comma list: 22/21, 36/35, 80/77


EDOs: {{EDOs| 7d, 15d, 22, 29, 51, 73ce }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Badness: 0.0273
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 164.768
* POTE: ~2 = 1200.000, ~11/10 = 158.750


== 13-limit ==
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}
Commas: 55/54, 65/64, 91/90, 100/99


POTE generator: ~10/9 = 164.953
Badness (Smith): 0.026790


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
== Hedgehog ==
{{See also| Sensamagic clan | Stearnsmic clan }}


EDOs: {{EDOs| 7d, 22, 29, 51f, 80cdeff }}
Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid omega-tricot.


Badness: 0.0265
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.  


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


POTE generator: ~10/9 = 166.041
[[Comma list]]: 50/49, 245/243


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


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
: mapping generators: ~7/5, ~9/7


Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000, ~9/7 = 435.258
: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
* [[POTE]]: ~7/5 = 600.000, ~9/7 = 435.648
: error map: {{val| 0.000 +4.989 -8.074 +9.414 }}


EDOs: {{EDOs| 7, 29, 65c, 94cd }}
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}


Badness: 0.1183
[[Badness]] (Smith): 0.043983


== 11-limit ==
=== 11-limit ===
Commas: 55/54, 100/99, 525/512
Subgroup: 2.3.5.7.11


POTE generator: ~10/9 = 165.981
Comma list: 50/49, 55/54, 99/98


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


Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 435.528
* POTE: ~7/5 = 600.000, ~9/7 = 435.386


EDOs: {{EDOs| 7, 29, 65ce, 94cde }}
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}


Badness: 0.0497
Badness (Smith): 0.023095


== 13-limit ==
==== 13-limit ====
Commas: 55/54, 65/64, 100/99, 105/104
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/9 = 165.974
Comma list: 50/49, 55/54, 65/63, 99/98


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


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 436.309
* POTE: ~7/5 = 600.000, ~9/7 = 435.861


EDOs: {{EDOs| 7, 29, 65cef, 94cdef }}
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}


Badness: 0.0302
Badness (Smith): 0.021516


= Hedgehog =
==== Urchin ====
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. 22edo provides the obvious tuning, but if you are looking for an alternative, you could try the &lt;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.
Subgroup: 2.3.5.7.11.13


Commas: 50/49, 245/243
Comma list: 40/39, 50/49, 55/54, 66/65


[[POTE_tuning|POTE generator]]: ~9/7 = 435.648
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 435.186
* POTE: ~7/5 = 600.000, ~9/7 = 437.078


Wedgie: &lt;&lt;6 10 10 2 -1 -5||
{{Optimal ET sequence|legend=0| 14c, 22f }}


EDOs: {{EDOs| 8d, 14c, 22, 146bccdd }}
Badness (Smith): 0.025233


Badness: 0.0440
=== Hedgepig ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 50/49, 245/243, 385/384
Commas: 50/49, 55/54, 99/98


POTE generator: ~9/7 = 435.386
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 435.329
* POTE: ~7/5 = 600.000, ~9/7 = 435.425


EDOs: {{EDOs| 14c, 22, 58ce, 80ce, 102cde }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.0231
Badness (Smith): 0.068406


=== 13-limit ===
; Music
Commas: 50/49, 55/54, 65/63, 99/98
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|hedgehog[14]]], 22edo tuning.


POTE generator: ~9/7 = 435.861
== Nautilus ==
Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.  


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 14cf, 22 }}
[[Comma list]]: 49/48, 250/243


Badness: 0.0215
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}


=== Urchin ===
: mapping generators: ~2, ~21/20
Commas: 40/39, 50/49, 55/54, 66/65


POTE generator: ~9/7 = 437.078
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~21/20 = 81.914
: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 82.505
: error map: {{val| 0.000 +3.012 -11.368 -16.342 }}


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}


EDOs: {{EDOs| 14c, 22f }}
[[Badness]] (Smith): 0.057420


Badness: 0.0252
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Hedgepig ==
Comma list: 49/48, 55/54, 245/242
Commas: 50/49, 245/243, 385/384


POTE generator: ~9/7 = 435.425
Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 81.802
* POTE: ~2 = 1200.000, ~21/20 = 82.504


EDOs: {{EDOs| 22, 80c, 102cd, 124cd }}
{{Optimal ET sequence|legend=0| 14c, 15, 29, 44d }}


Badness: 0.0684
Badness (Smith): 0.026023


== Music ==
==== 13-limit ====
[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.
Subgroup: 2.3.5.7.11.13


= Nautilus =
Comma list: 49/48, 55/54, 91/90, 100/99
Commas: 49/48, 250/243


POTE generator: ~21/20 = 82.505
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 81.912
* POTE: ~2 = 1200.000, ~21/20 = 82.530


Wedgie: &lt;&lt;6 10 3 2 -12 -21||
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 44d }}


EDOs: {{EDOs| 15, 29, 43cd, 44d, 59d, 73cd, 102cd }}
Badness (Smith): 0.022285


== 11-limit ==
==== Belauensis ====
Commas: 49/48, 55/54, 245/242
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 82.504
Comma list: 40/39, 49/48, 55/54, 66/65


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


EDOs: {{EDOs| 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde }}
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 82.034
* POTE: ~2 = 1200.000, ~21/20 = 81.759


=== 13-limit ===
{{Optimal ET sequence|legend=0| 14c, 15, 29f, 44dff }}
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
Badness (Smith): 0.029816


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]


EDOs: {{EDOs| 15f, 29, 43cde, 44d, 59df, 73cde, 102cde }}
== Ammonite ==
Ammonite adds 686/675 to the comma list and may be described as the {{nowrap| 8d & 29 }} temperament. Its ploidacot is epsilon-enneacot. [[37edo]] provides an obvious tuning.


Badness: 0.0223
[[Subgroup]]: 2.3.5.7


=== Belauensis ===
[[Comma list]]: 250/243, 686/675
Commas: 40/39, 49/48, 55/54, 66/65


POTE generator: ~21/20 = ~14/13 = 81.759
{{Mapping|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}


Map: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]
: mapping generators: ~2, ~9/7


EDOs: {{EDOs| 14c, 15, 29f, 44df }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
* [[POTE]]: ~2 = 1200.000, ~9/7 = 454.448
: error map: {{val| 0.000 +8.009 -3.040 -3.346 }}


Badness: 0.0298
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}


[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]]
[[Badness]] (Smith): 0.107686


= Ammonite =
=== 11-limit ===
Commas: 250/243, 686/675
Subgroup: 2.3.5.7.11


POTE generator: ~9/7 = 454.448
Comma list: 55/54, 100/99, 686/675


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}


Wedgie: &lt;&lt;9 15 19 3 5 2||
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 454.505
* POTE: ~2 = 1200.000, ~9/7 = 454.512


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


Badness: 0.1077
Badness (Smith): 0.045694


== 11-limit ==
=== 13-limit ===
Commas: 55/54, 100/99, 686/675
Subgroup: 2.3.5.7.11.13


POTE generator: ~9/7 = 454.512
Comma list: 55/54, 91/90, 100/99, 169/168


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


EDOs: {{EDOs| 29, 37, 66 }}
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/10 = 454.480
* POTE: ~2 = 1200.000, ~13/10 = 454.529


Badness: 0.0457
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}


== 13-limit ==
Badness (Smith): 0.027168
Commas: 55/54, 91/90, 100/99, 169/168


POTE generator: ~13/10 = 454.429
== 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.  


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 29, 37, 66 }}
[[Comma list]]: 250/243, 1728/1715


Badness: 0.0272
{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}


= Ceratitid =
: mapping generators: ~2, ~36/35
Commas: 250/243, 1728/1715


POTE generator: ~36/35 = 54.384
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~36/35 = 54.804
: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
* [[POTE]]: ~2 = 1200.000, ~36/35 = 54.384
: error map: {{val| 0.000 +8.585 -2.081 +13.636 }}


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}


Wedgie: &lt;&lt;9 15 4 3 -19 -33||
[[Badness]] (Smith): 0.115304


EDOs: {{EDOs| 22 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.115
Comma list: 55/54, 100/99, 352/343


== 11-limit ==
Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
Commas: 55/54, 100/99, 5324/5145


POTE generator: ~36/35 = 54.376
Optimal tunings:
* CTE: ~2 = 1200.000, ~36/35 = 54.702
* POTE: ~2 = 1200.000, ~36/35 = 54.376


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}


EDOs: {{EDOs| 22 }}
Badness (Smith): 0.051319


Badness: 0.0513
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== 13-limit ==
Comma list: 55/54, 65/63, 100/99, 352/343
Commas: 55/54, 65/63, 100/99, 352/343


POTE generator: ~36/35 = 54.665
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


Map: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~36/35 = 54.575
* POTE: ~2 = 1200.000, ~36/35 = 54.665


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


Badness: 0.0447
Badness (Smith): 0.044739


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine]]
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"