Porcupine family: Difference between revisions

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The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.
{{Technical data page}}
 
The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.  
Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 3 5 1 }} for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.
 
Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.
 
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. That means
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].
 
Temperaments discussed elsewhere include [[Dicot family #Jamesbond|jamesbond]].


== Porcupine ==
== Porcupine ==
{{Main| Porcupine }}
{{Main| Porcupine }}
The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, {{nowrap| (10/9)<sup>3</sup> {{=}} (4/3)⋅(250/243) }}, and {{nowrap| (10/9)<sup>5</sup> {{=}} (8/5)⋅(250/243)<sup>2</sup> }}. Its [[ploidacot]] is omega-tricot. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 30: Line 21:
: mapping generators: ~2, ~10/9
: mapping generators: ~2, ~10/9


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 164.1659
[[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 }}


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


{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}


[[Badness]]: 0.030778
[[Badness]] (Smith): 0.030778
 
=== 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.
 
Those all share the same generator with porcupine.
 
[[#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.
 
Temperaments discussed elsewhere include:
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
 
==== Subgroup extensions ====
Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|S11]] }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below.


=== 2.3.5.11 subgroup (porkypine) ===
=== 2.3.5.11 subgroup (porkypine) ===
Line 50: Line 64:
Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}


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


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.8867
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1200.000, ~11/10 = 164.078


{{Optimal ET sequence|legend=1| 7, 15, 22, 73ce, 95ce }}
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}


Badness: 0.0097
Badness (Smith): 0.0097


==== Undecimation ====
==== Undecimation ====
Line 67: Line 83:
: sval mapping generators: ~2, ~65/44
: sval mapping generators: ~2, ~65/44


Optimal tuning (CTE): ~2 = 1\1, ~88/65 = 518.2094
Optimal tunings:
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1200.000, ~88/65 = 518.209


{{Optimal ET sequence|legend=1| 7, 23bc, 30, 37, 44 }}
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}


Badness: 0.0305
Badness (Smith): 0.0305


== Septimal porcupine ==
== Septimal porcupine ==
{{Main| Porcupine }}
{{Main| Porcupine }}


Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.
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
[[Subgroup]]: 2.3.5.7
Line 84: Line 102:
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


{{Multival|legend=1| 3 5 -6 1 -18 -28 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~10/9 = 163.203
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 163.2032
: [[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]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 98: Line 118:
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]


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


[[Badness]]: 0.041057
[[Badness]] (Smith): 0.041057


=== 11-limit ===
=== 11-limit ===
Line 111: Line 130:
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.1055
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1200.000, ~11/10 = 162.747


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


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
* 11-odd-limit diamond monotone and tradeoff: ~11/10 = [160.000, 163.636]


{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59 }}
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


Badness: 0.021562
Badness (Smith): 0.021562
 
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.


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


Line 133: Line 155:
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.4425
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.442
* POTE: ~2 = 1200.000, ~11/10 = 162.708


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


Tuning ranges:  
Tuning ranges:  
Line 143: Line 167:
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (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]
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]
* 13-odd-limit diamond monotone and tradeoff: ~11/10 = [160.000, 163.636]
* 15-odd-limit diamond monotone and tradeoff: ~11/10 = 163.636


{{Optimal ET sequence|legend=1| 7, 15, 22f, 37f }}
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}


Badness: 0.021276
Badness (Smith): 0.021276


==== Porcupinefish ====
==== Porcupinefish ====
Line 159: Line 181:
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 162.6361
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 162.636
* POTE: ~2 = 1200.000, ~11/10 = 162.277


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


Tuning ranges:  
Tuning ranges:  
Line 169: Line 193:
* 15-odd-limit diamond monotone: ~10/9 = 162.162 (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]
* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
* 13-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 162.162]
* 15-odd-limit diamond monotone and tradeoff: ~10/9 = 162.162


{{Optimal ET sequence|legend=1| 15, 22, 37 }}
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Badness: 0.025314
Badness (Smith): 0.025314


==== Pourcup ====
==== Pourcup ====
Line 183: Line 205:
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.3781
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.378
* POTE: ~2 = 1200.000, ~11/10 = 162.482


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


{{Optimal ET sequence|legend=1| 15f, 22f, 37, 59f }}
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}


Badness: 0.035130
Badness (Smith): 0.035130


==== Porkpie ====
==== Porkpie ====
Line 200: Line 224:
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.6778
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 163.678
* POTE: ~2 = 1200.000, ~11/10 = 163.688


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


{{Optimal ET sequence|legend=1| 7, 15f, 22 }}
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}


Badness: 0.026043
Badness (Smith): 0.026043


== Opossum ==
== Opossum ==
Opossum can be described as 7d & 8d. 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.  
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
[[Subgroup]]: 2.3.5.7
Line 219: Line 245:
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


{{Multival|legend=1| 3 5 9 1 6 7 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
: [[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]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


[[Badness]]: 0.040650
[[Badness]] (Smith): 0.040650


=== 11-limit ===
=== 11-limit ===
Line 237: Line 265:
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.365
* POTE: ~2 = 1200.000, ~11/10 = 159.807


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


{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Badness: 0.022325
Badness (Smith): 0.022325


=== 13-limit ===
=== 13-limit ===
Line 253: Line 283:
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.631
* POTE: ~2 = 1200.000, ~11/10 = 158.805


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 38bceff }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


Badness: 0.019389
Badness (Smith): 0.019389


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 271: Line 303:
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


{{Multival|legend=1| 3 5 16 1 17 23 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 164.3913
: [[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5


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


[[Badness]]: 0.054389
[[Badness]] (Smith): 0.054389


=== 11-limit ===
=== 11-limit ===
Line 290: Line 324:
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.3207
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 164.321
* POTE: ~2 = 1200.000, ~11/10 = 164.552


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


{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51 }}
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}


Badness: 0.027268
Badness (Smith): 0.027268


=== 13-limit ===
=== 13-limit ===
Line 307: Line 343:
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.4782
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 164.478
* POTE: ~2 = 1200.000, ~11/10 = 164.953


{{Optimal ET sequence|legend=1| 7d, 22, 29, 51f, 80cdeff }}
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}


Badness: 0.026543
Badness (Smith): 0.026543
 
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


== Coendou ==
== Coendou ==
Coendou can be described as 7 & 29, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.  
Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 322: Line 363:
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


{{Multival|legend=1| 3 5 -13 1 -29 -44 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 166.0938
: [[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


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


[[Badness]]: 0.118344
[[Badness]] (Smith): 0.118344


=== 11-limit ===
=== 11-limit ===
Line 341: Line 384:
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 165.9246
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 165.925
* POTE: ~2 = 1200.000, ~11/10 = 165.981


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


{{Optimal ET sequence|legend=1| 7, 22d, 29, 65ce }}
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}


Badness: 0.049669
Badness (Smith): 0.049669


=== 13-limit ===
=== 13-limit ===
Line 358: Line 403:
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 166.0459
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 166.046
* POTE: ~2 = 1200.000, ~11/10 = 165.974


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


{{Optimal ET sequence|legend=1| 7, 22d, 29, 65cef }}
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}


Badness: 0.030233
Badness (Smith): 0.030233


== Hystrix ==
== 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, 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.
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
[[Subgroup]]: 2.3.5.7
Line 377: Line 424:
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}


{{Multival|legend=1| 3 5 1 1 -7 -12 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 165.1845
: [[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 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}


[[Badness]]: 0.044944
[[Badness]] (Smith): 0.044944


=== 11-limit ===
=== 11-limit ===
Line 396: Line 445:
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.7684
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 164.768
* POTE: ~2 = 1200.000, ~11/10 = 158.750


{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


Badness: 0.026790
Badness (Smith): 0.026790


== Oxygen ==
== Hedgehog ==
Oxygen is perhaps not meant to be used as a serious temperament of harmony. Its comma basis suggests potential utility to construct [[Fokker block]]s.
{{See also| Sensamagic clan | Stearnsmic clan }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 21/20, 175/162
 
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}


{{Multival|legend=1| 3 5 2 1 -5 -9 }}
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.


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3408
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.  
 
{{Optimal ET sequence|legend=1| 7d }}
 
[[Badness]]: 0.059866
 
== Hedgehog ==
{{See also| Sensamagic clan }}
{{See also| 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. 22edo provides the obvious 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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 433: Line 468:
: mapping generators: ~7/5, ~9/7
: mapping generators: ~7/5, ~9/7


{{Multival|legend=1| 6 10 10 2 -1 -5 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~7/5 = 600.000, ~9/7 = 435.258
[[Optimal tuning]] ([[CTE]]): ~7/5 = 1\2, ~9/7 = 435.2580
: [[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 }}


{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}


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


=== 11-limit ===
=== 11-limit ===
Line 448: Line 485:
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.5281
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 435.528
* POTE: ~7/5 = 600.000, ~9/7 = 435.386


{{Optimal ET sequence|legend=1| 8d, 14c, 22, 58ce }}
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}


Badness: 0.023095
Badness (Smith): 0.023095


==== 13-limit ====
==== 13-limit ====
Line 461: Line 500:
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 436.3087
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 436.309
* POTE: ~7/5 = 600.000, ~9/7 = 435.861


{{Optimal ET sequence|legend=1| 8d, 14cf, 22 }}
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}


Badness: 0.021516
Badness (Smith): 0.021516


==== Urchin ====
==== Urchin ====
Line 474: Line 515:
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.1856
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 435.186
* POTE: ~7/5 = 600.000, ~9/7 = 437.078


{{Optimal ET sequence|legend=1| 14c, 22f }}
{{Optimal ET sequence|legend=0| 14c, 22f }}


Badness: 0.025233
Badness (Smith): 0.025233


=== Hedgepig ===
=== Hedgepig ===
Line 487: Line 530:
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}


Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.3289
Optimal tunings:
* CTE: ~7/5 = 600.000, ~9/7 = 435.329
* POTE: ~7/5 = 600.000, ~9/7 = 435.425


{{Optimal ET sequence|legend=1| 22 }}
{{Optimal ET sequence|legend=0| 22 }}


Badness: 0.068406
Badness (Smith): 0.068406


; Music
; Music
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] in [[hedgehog14|hedgehog[14]]] to 22edo.
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|hedgehog[14]]], 22edo tuning.


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


Line 505: Line 552:
: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 6 10 3 2 -12 -21 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~21/20 = 81.914
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~21/20 = 81.9143
: [[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 }}


{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}


[[Badness]]: 0.057420
[[Badness]] (Smith): 0.057420


=== 11-limit ===
=== 11-limit ===
Line 520: Line 569:
Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.8017
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 81.802
* POTE: ~2 = 1200.000, ~21/20 = 82.504


{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}
{{Optimal ET sequence|legend=0| 14c, 15, 29, 44d }}


Badness: 0.026023
Badness (Smith): 0.026023


==== 13-limit ====
==== 13-limit ====
Line 533: Line 584:
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.9123
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 81.912
* POTE: ~2 = 1200.000, ~21/20 = 82.530


{{Optimal ET sequence|legend=1| 14cf, 15, 29, 44d }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 44d }}


Badness: 0.022285
Badness (Smith): 0.022285


==== Belauensis ====
==== Belauensis ====
Line 546: Line 599:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 82.0342
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 82.034
* POTE: ~2 = 1200.000, ~21/20 = 81.759


{{Optimal ET sequence|legend=1| 14c, 15, 29f, 44dff }}
{{Optimal ET sequence|legend=0| 14c, 15, 29f, 44dff }}


Badness: 0.029816
Badness (Smith): 0.029816


; Music
; Music
Line 556: Line 611:


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


Line 564: Line 621:
: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 9 15 19 3 5 2 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~9/7 = 454.5500
: [[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 }}


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


[[Badness]]: 0.107686
[[Badness]] (Smith): 0.107686


=== 11-limit ===
=== 11-limit ===
Line 579: Line 638:
Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 454.5050
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 454.505
* POTE: ~2 = 1200.000, ~9/7 = 454.512


{{Optimal ET sequence|legend=1| 8d, 21cde, 29, 37, 66 }}
{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}


Badness: 0.045694
Badness (Smith): 0.045694


=== 13-limit ===
=== 13-limit ===
Line 592: Line 653:
Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}
Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}


Optimal tuning (CTE): ~2 = 1\1, ~13/10 = 454.4798
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/10 = 454.480
* POTE: ~2 = 1200.000, ~13/10 = 454.529


{{Optimal ET sequence|legend=1| 8d, 21cdef, 29, 37, 66 }}
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}


Badness: 0.027168
Badness (Smith): 0.027168


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


Line 607: Line 672:
: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35


{{Multival|legend=1| 9 15 4 3 -19 -33 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~36/35 = 54.804
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 54.8040
: [[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 }}


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


[[Badness]]: 0.115304
[[Badness]] (Smith): 0.115304


=== 11-limit ===
=== 11-limit ===
Line 622: Line 689:
Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.7019
Optimal tunings:
* CTE: ~2 = 1200.000, ~36/35 = 54.702
* POTE: ~2 = 1200.000, ~36/35 = 54.376


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


Badness: 0.051319
Badness (Smith): 0.051319


=== 13-limit ===
=== 13-limit ===
Line 635: Line 704:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.5751
Optimal tunings:
* CTE: ~2 = 1200.000, ~36/35 = 54.575
* POTE: ~2 = 1200.000, ~36/35 = 54.665


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


Badness: 0.044739
Badness (Smith): 0.044739


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine| ]] <!-- key 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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