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Porcupine family: Difference between revisions

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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.  
The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.  


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{{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, (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.
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 21: Line 22:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/9 = 164.1659
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.166
* [[POTE]]: ~2 = 1\1, ~10/9 = 163.950
: [[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]


{{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 ===
=== Overview to extensions ===
==== 7-limit extensions ====
==== 7-limit extensions ====
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
The second comma defines which [[7-limit]] family member we are looking at.  
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
* [[#Septimal porcupine|Septimal porcupine]] adds [[64/63]], the archytas comma, for a tuning between 15edo and 22edo;
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].
* [[#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.  


Temperaments discussed elsewhere include [[7th-octave temperaments #Jamesbond|jamesbond]].
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 ====
==== Subgroup extensions ====
Notice {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below.  
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) ===
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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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 163.8867
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1\1, ~11/10 = 164.0777
* POTE: ~2 = 1200.000, ~11/10 = 164.078


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


Badness: 0.0097
Badness (Smith): 0.0097


==== Undecimation ====
==== Undecimation ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~88/65 = 518.0865
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1\1, ~88/65 = 518.2094
* POTE: ~2 = 1200.000, ~88/65 = 518.209


Optimal ET sequence: {{Optimal ET sequence| 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
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{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/9 = 163.2032
* [[CTE]]: ~2 = 1200.000, ~10/9 = 163.203
* [[POTE]]: ~2 = 1\1, ~10/9 = 162.880
: [[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]]:  
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{{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 ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 163.1055
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1\1, ~11/10 = 162.747
* 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 (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
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* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]


Optimal ET sequence: {{Optimal ET sequence| 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


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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 163.4425
* CTE: ~2 = 1200.000, ~11/10 = 163.442
* POTE: ~2 = 1\1, ~11/10 = 162.708
* 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 (unchanged-interval) basis: 2.11
: unchanged-interval (eigenmonzo) basis: 2.11


Tuning ranges:  
Tuning ranges:  
Line 156: Line 168:
* 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]


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


Badness: 0.021276
Badness (Smith): 0.021276


==== Porcupinefish ====
==== Porcupinefish ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 162.6361
* CTE: ~2 = 1200.000, ~11/10 = 162.636
* POTE: ~2 = 1\1, ~11/10 = 162.277
* 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 (unchanged-interval) basis: 2.13/11
: unchanged-interval (eigenmonzo) basis: 2.13/11


Tuning ranges:  
Tuning ranges:  
Line 182: Line 194:
* 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]


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


Badness: 0.025314
Badness (Smith): 0.025314


==== Pourcup ====
==== Pourcup ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 163.3781
* CTE: ~2 = 1200.000, ~11/10 = 163.378
* POTE: ~2 = 1\1, ~11/10 = 162.482
* 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 (unchanged-interval) basis: 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/7


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


Badness: 0.035130
Badness (Smith): 0.035130


==== Porkpie ====
==== Porkpie ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 163.6778
* CTE: ~2 = 1200.000, ~11/10 = 163.678
* POTE: ~2 = 1\1, ~11/10 = 163.688
* 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 (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Optimal ET sequence: {{Optimal ET sequence| 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
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{{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 ===
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Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


{{Multival|legend=1| 3 5 9 4 1 6 -4 7 -8 -20 }}
Optimal tunings:
 
* CTE: ~2 = 1200.000, ~11/10 = 161.365
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646
* 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: {{Optimal ET sequence| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Badness: 0.022325
Badness (Smith): 0.022325


=== 13-limit ===
=== 13-limit ===
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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: {{Optimal ET sequence| 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
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{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/9 = 164.3913
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
* [[POTE]]: ~2 = 1\1, ~10/9 = 164.412
: [[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 ===
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Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
{{Multival|legend=1| 3 5 16 4 1 17 -4 23 -8 -44 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 164.3207
* CTE: ~2 = 1200.000, ~11/10 = 164.321
* POTE: ~2 = 1\1, ~11/10 = 164.552
* 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 (unchanged-interval) basis: 2.7/5
: unchanged-interval (eigenmonzo) basis: 2.7/5


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


Badness: 0.027268
Badness (Smith): 0.027268


=== 13-limit ===
=== 13-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 164.4782
* CTE: ~2 = 1200.000, ~11/10 = 164.478
* POTE: ~2 = 1\1, ~11/10 = 164.953
* POTE: ~2 = 1200.000, ~11/10 = 164.953


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


Badness: 0.026543
Badness (Smith): 0.026543


; Music
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – Palace scale in 29edo tuning
* [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 348: Line 362:


{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/9 = 166.0938
* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
* [[POTE]]: ~2 = 1\1, ~10/9 = 166.041
: [[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 371: Line 385:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 165.9246
* CTE: ~2 = 1200.000, ~11/10 = 165.925
* POTE: ~2 = 1\1, ~11/10 = 165.981
* 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 (unchanged-interval) basis: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


Optimal ET sequence: {{Optimal ET sequence| 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 390: Line 404:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 166.0459
* CTE: ~2 = 1200.000, ~11/10 = 166.046
* POTE: ~2 = 1\1, ~11/10 = 165.974
* 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 (unchanged-interval) basis: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


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


Badness: 0.030233
Badness (Smith): 0.030233


== Hystrix ==
== Hystrix ==
Line 409: Line 423:


{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~10/9 = 165.1845
* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
* [[POTE]]: ~2 = 1\1, ~10/9 = 158.868
: [[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 432: Line 446:


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


Optimal ET sequence: {{Optimal ET sequence| 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]]s:
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.  
* [[CTE]]: ~2 = 1\1, ~10/9 = 161.3408
* [[POTE]]: ~2 = 1\1, ~10/9 = 169.112
 
{{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. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). 22edo provides the obvious (i.e the only [[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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 471: Line 467:


: mapping generators: ~7/5, ~9/7
: mapping generators: ~7/5, ~9/7
{{Multival|legend=1| 6 10 10 2 -1 -5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 1\2, ~9/7 = 435.2580
* [[CTE]]: ~7/5 = 600.000, ~9/7 = 435.258
* [[POTE]]: ~7/5 = 1\2, ~9/7 = 435.648
: [[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 488: Line 484:


Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
{{Multival|legend=1| 6 10 10 8 2 -1 -8 -5 -16 -12 }}


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


Optimal ET sequence: {{Optimal ET sequence| 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 507: Line 501:


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


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


Badness: 0.021516
Badness (Smith): 0.021516


==== Urchin ====
==== Urchin ====
Line 522: Line 516:


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


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


Badness: 0.025233
Badness (Smith): 0.025233


=== Hedgepig ===
=== Hedgepig ===
Line 535: Line 529:


Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
{{Multival|legend=1| 6 10 10 -14 2 -1 -43 -5 -67 -74 }}


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


Optimal ET sequence: {{Optimal ET sequence| 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 557: Line 551:


: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20
{{Multival|legend=1| 6 10 3 2 -12 -21 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~21/20 = 81.9143
* [[CTE]]: ~2 = 1200.000, ~21/20 = 81.914
* [[POTE]]: ~2 = 1\1, ~21/20 = 82.505
: [[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 574: Line 568:


Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
{{Multival|legend=1| 6 10 3 8 2 -12 -8 -21 -16 12 }}


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


Optimal ET sequence: {{Optimal ET sequence| 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 593: Line 585:


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


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


Badness: 0.022285
Badness (Smith): 0.022285


==== Belauensis ====
==== Belauensis ====
Line 608: Line 600:


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


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


Badness: 0.029816
Badness (Smith): 0.029816


; Music
; Music
Line 619: 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 626: Line 620:


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 9 15 19 3 5 2 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~9/7 = 454.5500
* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
* [[POTE]]: ~2 = 1\1, ~9/7 = 454.448
: [[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 643: Line 637:


Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
{{Multival|legend=1| 9 15 19 12 3 5 -12 2 -24 -32 }}


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


Optimal ET sequence: {{Optimal ET sequence| 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 662: Line 654:


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


Optimal ET sequence: {{Optimal ET sequence| 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 677: Line 671:


: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35
{{Multival|legend=1| 9 15 4 3 -19 -33 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~36/35 = 54.8040
* [[CTE]]: ~2 = 1200.000, ~36/35 = 54.804
* [[POTE]]: ~2 = 1\1, ~36/35 = 54.384
: [[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 696: Line 690:


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


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


Badness: 0.051319
Badness (Smith): 0.051319


=== 13-limit ===
=== 13-limit ===
Line 711: Line 705:


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


Optimal ET sequence: {{Optimal ET sequence| 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]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"