Porcupine family: Difference between revisions

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<span style="display: block; text-align: right;">[[de:Porcupine]]</span>
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| 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
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{{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]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 10:23, 29 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The porcupine family of temperaments tempers out the porcupine comma, 250/243, also called the maximal diesis.

Porcupine

Main article: 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, (10/9)3 = (4/3)⋅(250/243), and (10/9)5 = (8/5)⋅(250/243)2. Its ploidacot is omega-tricot. 3\22 is a very recommendable generator, and mos scales of 7, 8 and 15 notes make for some nice scale possibilities.

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

mapping generators: ~2, ~10/9

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 164.166
error map: ⟨0.000 +5.547 -7.143]
  • POTE: ~2 = 1200.000, ~10/9 = 163.950
error map: ⟨0.000 +6.194 -6.065]

Tuning ranges:

Optimal ET sequence: 7, 15, 22, 95c

Badness (Smith): 0.030778

Overview to extensions

7-limit extensions

The second comma defines which 7-limit family member we are looking at.

  • Hystrix adds 36/35, the mint comma, for an exotemperament tuning around 8d-edo;
  • Opossum adds 28/27, the trienstonic comma, for a tuning between 8d-edo and 15edo;
  • Septimal porcupine adds 64/63, the archytas comma, for a tuning between 15edo and 22edo;
  • Porky adds 225/224, the marvel comma, for a tuning between 22edo and 29edo;
  • Coendou adds 525/512, the avicennma, for a tuning sharp of 29edo.

Those all share the same generator with porcupine.

nautilus tempers out 49/48 and splits the generator in two. hedgehog tempers out 50/49 with a semi-octave period. Finally, ammonite tempers out 686/675 and ceratitid tempers out 1728/1715. Those split the generator in three.

Temperaments discussed elsewhere include:

Subgroup extensions

Noting that 250/243 = (55/54)⋅(100/99) = S102⋅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

Sval mapping: [⟨1 2 3 4], ⟨0 -3 -5 -4]]

Gencom mapping: [⟨1 2 3 0 4], ⟨0 -3 -5 0 -4]]

gencom: [2 10/9; 55/54, 100/99]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 163.887
  • POTE: ~2 = 1200.000, ~11/10 = 164.078

Optimal ET sequence: 7, 15, 22, 73ce, 95ce

Badness (Smith): 0.0097

Undecimation

Subgroup: 2.3.5.11.13

Comma list: 55/54, 100/99, 512/507

Sval mapping: [⟨1 5 8 8 2], ⟨0 -6 -10 -8 3]]

sval mapping generators: ~2, ~65/44

Optimal tunings:

  • CTE: ~2 = 1200.000, ~88/65 = 518.086
  • POTE: ~2 = 1200.000, ~88/65 = 518.209

Optimal ET sequence: 7, 23bc, 30, 37, 44

Badness (Smith): 0.0305

Septimal porcupine

Main article: 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: [⟨1 2 3 2], ⟨0 -3 -5 6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 163.203
error map: ⟨0.000 +8.435 -2.330 +10.394]
  • POTE: ~2 = 1200.000, ~10/9 = 162.880
error map: ⟨0.000 +9.405 -0.714 +8.455]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5
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: 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: [⟨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 = [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: 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: [⟨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 = [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: 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: [⟨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 = [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: 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: [⟨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 = [1/14 0 0 -1/14 0 1/14⟩
unchanged-interval (eigenmonzo) basis: 2.13/7

Optimal ET sequence: 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: [⟨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 = [1/6 -1/6 0 1/12⟩
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 7, 15f, 22

Badness (Smith): 0.026043

Opossum

Opossum can be described as 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: [⟨1 2 3 4], ⟨0 -3 -5 -9]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 161.306
error map: ⟨0.000 +14.126 +7.155 -20.583]
  • POTE: ~2 = 1200.000, ~10/9 = 159.691
error map: ⟨0.000 +18.971 +15.229 -6.048]

Minimax tuning:

Optimal ET sequence: 7d, 8d, 15

Badness (Smith): 0.040650

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 55/54, 77/75

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: 7d, 8d, 15

Badness (Smith): 0.022325

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 55/54, 66/65

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

Optimal ET sequence: 7d, 8d, 15, 38bceff

Badness (Smith): 0.019389

Porky

Porky can be described as 22 & 29, suggesting a less sharp perfect fifth. 7\51 is a good generator.

Subgroup: 2.3.5.7

Comma list: 225/224, 250/243

Mapping: [⟨1 2 3 5], ⟨0 -3 -5 -16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 164.391
error map: ⟨0.000 +4.871 -8.270 +0.913]
  • POTE: ~2 = 1200.000, ~10/9 = 164.412
error map: ⟨0.000 +4.809 -8.375 +0.580]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 29, 51, 73c

Badness (Smith): 0.054389

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 5 4], ⟨0 -3 -5 -16 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 164.321
  • POTE: ~2 = 1200.000, ~11/10 = 164.552

Minimax tuning:

  • 11-odd-limit: ~11/10 = [2/11 0 1/11 -1/11⟩
unchanged-interval (eigenmonzo) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 51

Badness (Smith): 0.027268

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 5 4 3], ⟨0 -3 -5 -16 -4 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 164.478
  • POTE: ~2 = 1200.000, ~11/10 = 164.953

Optimal ET sequence: 7d, 22, 29, 51f, 80cdeff

Badness (Smith): 0.026543

Music

Coendou

Coendou can be described as 29 & 36c, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.

Subgroup: 2.3.5.7

Comma list: 250/243, 525/512

Mapping: [⟨1 2 3 1], ⟨0 -3 -5 13]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 166.094
error map: ⟨0.000 -0.236 -16.783 -9.607]
  • POTE: ~2 = 1200.000, ~10/9 = 166.041
error map: ⟨0.000 -0.077 -16.516 -10.299]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65c, 94cd

Badness (Smith): 0.118344

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 1 4], ⟨0 -3 -5 13 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 165.925
  • POTE: ~2 = 1200.000, ~11/10 = 165.981

Minimax tuning:

  • 11-odd-limit: ~11/10 = [2/3 -1/3⟩
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65ce

Badness (Smith): 0.049669

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 1 4 3], ⟨0 -3 -5 13 -4 5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 166.046
  • POTE: ~2 = 1200.000, ~11/10 = 165.974

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [2/3 -1/3⟩
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65cef

Badness (Smith): 0.030233

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.

Subgroup: 2.3.5.7

Comma list: 36/35, 160/147

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 165.185
error map: ⟨0.000 +2.491 -12.236 +65.990]
  • POTE: ~2 = 1200.000, ~10/9 = 158.868
error map: ⟨0.000 +21.442 +19.348 +72.306]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 7, 8d, 15d

Badness (Smith): 0.044944

11-limit

Subgroup: 2.3.5.7.11

Comma list: 22/21, 36/35, 80/77

Mapping: [⟨1 2 3 3 4], ⟨0 -3 -5 -1 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 164.768
  • POTE: ~2 = 1200.000, ~11/10 = 158.750

Optimal ET sequence: 7, 8d, 15d

Badness (Smith): 0.026790

Hedgehog

See also: Sensamagic clan and Stearnsmic clan

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.

22edo provides an obvious tuning, which happens to be the only patent-val tuning, but if you are looking for an alternative you could try the ⟨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.

Subgroup: 2.3.5.7

Comma list: 50/49, 245/243

Mapping: [⟨2 1 1 2], ⟨0 3 5 5]]

mapping generators: ~7/5, ~9/7

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 435.258
error map: ⟨0.000 +3.819 -10.024 +7.464]
  • POTE: ~7/5 = 600.000, ~9/7 = 435.648
error map: ⟨0.000 +4.989 -8.074 +9.414]

Optimal ET sequence: 8d, 14c, 22

Badness (Smith): 0.043983

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 99/98

Mapping: [⟨2 1 1 2 4], ⟨0 3 5 5 4]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 435.528
  • POTE: ~7/5 = 600.000, ~9/7 = 435.386

Optimal ET sequence: 8d, 14c, 22, 58ce

Badness (Smith): 0.023095

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 1 1 2 4 3], ⟨0 3 5 5 4 6]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 436.309
  • POTE: ~7/5 = 600.000, ~9/7 = 435.861

Optimal ET sequence: 8d, 14cf, 22

Badness (Smith): 0.021516

Urchin

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 1 1 2 4 6], ⟨0 3 5 5 4 2]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 435.186
  • POTE: ~7/5 = 600.000, ~9/7 = 437.078

Optimal ET sequence: 14c, 22f

Badness (Smith): 0.025233

Hedgepig

Subgroup: 2.3.5.7.11

Comma list: 50/49, 245/243, 385/384

Mapping: [⟨2 1 1 2 12], ⟨0 3 5 5 -7]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 435.329
  • POTE: ~7/5 = 600.000, ~9/7 = 435.425

Optimal ET sequence: 22

Badness (Smith): 0.068406

Music

Nautilus

Nautilus tempers out 49/48 and may be described as the 14c & 15 temperament. Its ploidacot is omega-hexacot.

Subgroup: 2.3.5.7

Comma list: 49/48, 250/243

Mapping: [⟨1 2 3 3], ⟨0 -6 -10 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 81.914
error map: ⟨0.000 +6.559 -5.457 -14.569]
  • POTE: ~2 = 1200.000, ~21/20 = 82.505
error map: ⟨0.000 +3.012 -11.368 -16.342]

Optimal ET sequence: 14c, 15, 29, 44d

Badness (Smith): 0.057420

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -6 -10 -3 -8]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 81.802
  • POTE: ~2 = 1200.000, ~21/20 = 82.504

Optimal ET sequence: 14c, 15, 29, 44d

Badness (Smith): 0.026023

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 5], ⟨0 -6 -10 -3 -8 -19]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 81.912
  • POTE: ~2 = 1200.000, ~21/20 = 82.530

Optimal ET sequence: 14cf, 15, 29, 44d

Badness (Smith): 0.022285

Belauensis

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 4], ⟨0 -6 -10 -3 -8 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 82.034
  • POTE: ~2 = 1200.000, ~21/20 = 81.759

Optimal ET sequence: 14c, 15, 29f, 44dff

Badness (Smith): 0.029816

Music

Ammonite

Ammonite adds 686/675 to the comma list and may be described as the 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: [⟨1 5 8 10], ⟨0 -9 -15 -19]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 454.550
error map: ⟨0.000 +7.095 -4.564 -5.276]
  • POTE: ~2 = 1200.000, ~9/7 = 454.448
error map: ⟨0.000 +8.009 -3.040 -3.346]

Optimal ET sequence: 8d, 21cd, 29, 37, 66

Badness (Smith): 0.107686

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 5 8 10 8], ⟨0 -9 -15 -19 -12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 454.505
  • POTE: ~2 = 1200.000, ~9/7 = 454.512

Optimal ET sequence: 8d, 21cde, 29, 37, 66

Badness (Smith): 0.045694

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 5 8 10 8 9], ⟨0 -9 -15 -19 -12 -14]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/10 = 454.480
  • POTE: ~2 = 1200.000, ~13/10 = 454.529

Optimal ET sequence: 8d, 21cdef, 29, 37, 66

Badness (Smith): 0.027168

Ceratitid

Ceratitid adds 1728/1715 to the comma list and may be described as the 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: [⟨1 2 3 3], ⟨0 -9 -15 -4]]

mapping generators: ~2, ~36/35

Optimal tunings:

  • CTE: ~2 = 1200.000, ~36/35 = 54.804
error map: ⟨0.000 +4.809 -8.374 +11.958]
  • POTE: ~2 = 1200.000, ~36/35 = 54.384
error map: ⟨0.000 +8.585 -2.081 +13.636]

Optimal ET sequence: 1c, 21c, 22

Badness (Smith): 0.115304

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -9 -15 -4 -12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~36/35 = 54.702
  • POTE: ~2 = 1200.000, ~36/35 = 54.376

Optimal ET sequence: 1ce, 21ce, 22

Badness (Smith): 0.051319

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 4], ⟨0 -9 -15 -4 -12 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~36/35 = 54.575
  • POTE: ~2 = 1200.000, ~36/35 = 54.665

Optimal ET sequence: 1ce, 21cef, 22

Badness (Smith): 0.044739

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