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m →Hedgehog: the difference in accuracy between echidna and hedgehog is astounding. pls at least let me emphasize it with "much more" instead of "more". the link to echidna is already obscured from the "see also".
→Porcupine: base the sharpness on 4/3 rather than 3/2 (see talk). Hystrix isn't actually flat of 8d
 
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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 [[7th-octave temperaments #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 31: 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 ===
==== 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 51: 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 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 ====
Line 71: Line 84:


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
Line 88: Line 101:


{{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]]:  
Line 108: Line 121:
{{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


Line 141: Line 156:


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 153: 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 ====
Line 167: Line 182:


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 179: 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 ====
Line 191: Line 206:


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 ====
Line 210: Line 225:


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 ===
Line 248: 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: {{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 ===
Line 264: 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: {{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
Line 281: Line 302:


{{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 ===
Line 304: Line 325:


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 ===
Line 323: Line 344:


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
* [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 338: 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 361: 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 380: 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 ==
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 399: 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 422: 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 }}
 
[[Optimal tuning]]s:
* [[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 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.  


== Hedgehog ==
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.  
{{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 461: 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 480: Line 486:


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 495: 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 510: 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 525: Line 531:


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 543: 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 562: Line 570:


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 577: 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 592: 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 603: 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 610: 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 629: Line 639:


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 644: 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 659: 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 678: 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 693: 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"