Xenharmonic Wiki

Porcupine family: Difference between revisions

← Older edit
VisualWikitext
m →Hedgehog: +links
→Porcupine: base the sharpness on 4/3 rather than 3/2 (see talk). Hystrix isn't actually flat of 8d
 
(48 intermediate revisions by 10 users not shown)
Line 5: Line 5:
| ja =  
| ja =  
}}
}}
The 5-limit parent comma for the '''porcupine family''' is [[250/243]], the maximal diesis or porcupine comma. Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{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 fourth, with two more giving the minor sixth. In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.
{{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 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]].
 
All these 7-limit extensions notably share the same 2.3.5.11 subgroup, ''porkypine''.
 
Temperaments discussed elsewhere include [[opossum]], [[Septisemi temperaments #Oxygen|oxygen]], and [[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 24: Line 17:
[[Comma list]]: 250/243
[[Comma list]]: 250/243


[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}]
{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}


: 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]


{{Val list|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 44: Line 60:
Comma list: 55/54, 100/99
Comma list: 55/54, 100/99


Sval mapping: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -4 }}]
Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


Gencom mapping: [{{val| 1 2 3 0 4 }}, {{val| 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 GPV sequence: {{val list| 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 61: Line 79:
Comma list: 55/54, 100/99, 512/507
Comma list: 55/54, 100/99, 512/507


Sval mapping: [{{val| 1 5 8 8 2 }}, {{val| 0 -6 -10 -8 3 }}]
Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}


: 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 GPV sequence: {{val list| 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 80: Line 100:
[[Comma list]]: 64/63, 250/243
[[Comma list]]: 64/63, 250/243


[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 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]]: 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]]: 2.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 96: 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]


{{Val list|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 107: Line 128:
Comma list: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99


Mapping: [{{val| 1 2 3 2 4 }}, {{val| 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: 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 GPV sequence: {{val list| 7, 15, 22, 37, 59 }}
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}
 
Badness (Smith): 0.021562


Badness: 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


Comma list: 40/39, 55/54, 64/63, 66/65
Comma list: 40/39, 55/54, 64/63, 66/65


Mapping: [{{val| 1 2 3 2 4 4 }}, {{val| 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: 2.11
: unchanged-interval (eigenmonzo) basis: 2.11


Tuning ranges:  
Tuning ranges:  
Line 141: 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 GPV sequence: {{val list| 7, 15, 22f, 37f }}
{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}


Badness: 0.021276
Badness (Smith): 0.021276


==== Porcupinefish ====
==== Porcupinefish ====
Line 155: Line 179:
Comma list: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99


Mapping: [{{val| 1 2 3 2 4 6 }}, {{val| 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: 2.13/11
: unchanged-interval (eigenmonzo) basis: 2.13/11


Tuning ranges:  
Tuning ranges:  
Line 167: 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 GPV sequence: {{val list| 15, 22, 37 }}
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Badness: 0.025314
Badness (Smith): 0.025314


==== Pourcup ====
==== Pourcup ====
Line 179: Line 203:
Comma list: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195


Mapping: [{{val| 1 2 3 2 4 1 }}, {{val| 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: 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/7


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


Badness: 0.035130
Badness (Smith): 0.035130


==== Porkpie ====
==== Porkpie ====
Line 196: Line 222:
Comma list: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99


Mapping: [{{val| 1 2 3 2 4 3 }}, {{val| 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: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Optimal GPV sequence: {{val list| 7, 15f, 22 }}
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}


Badness: 0.026043
Badness (Smith): 0.026043


== Hystrix ==
== Opossum ==
Hystrix provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits.
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


[[Comma list]]: 36/35, 160/147
[[Comma list]]: 28/27, 126/125
 
[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -1 }}]


{{Multival|legend=1| 3 5 1 1 -7 -12 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 165.1845
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}


[[Minimax tuning]]:  
[[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]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
: [[Eigenmonzo basis]]: 2.5


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


[[Badness]]: 0.044944
[[Badness]] (Smith): 0.040650


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 22/21, 36/35, 80/77
Comma list: 28/27, 55/54, 77/75
 
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~11/10 = 161.365
* POTE: ~2 = 1200.000, ~11/10 = 159.807
 
Minimax tuning:
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
 
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
 
Badness (Smith): 0.022325
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 28/27, 40/39, 55/54, 66/65
 
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -3 -5 -1 -4 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 161.631
* POTE: ~2 = 1200.000, ~11/10 = 158.805


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.7684
Minimax tuning:
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Optimal GPV sequence: {{val list| 7, 8d, 15d }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


Badness: 0.026790
Badness (Smith): 0.019389


== Porky ==
== Porky ==
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


[[Comma list]]: 225/224, 250/243
[[Comma list]]: 225/224, 250/243


[[Mapping]]: [{{val| 1 2 3 5 }}, {{val| 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]] ([[CTE]]): ~2 = 1\1, ~10/9 = 164.3913
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}


[[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]]: 2.7/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5


{{Val list|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 266: Line 322:
Comma list: 55/54, 100/99, 225/224
Comma list: 55/54, 100/99, 225/224


Mapping: [{{val| 1 2 3 5 4 }}, {{val| 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: 2.7/5
: unchanged-interval (eigenmonzo) basis: 2.7/5


Optimal GPV sequence: {{val list| 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 283: Line 341:
Comma list: 55/54, 65/64, 91/90, 100/99
Comma list: 55/54, 65/64, 91/90, 100/99


Mapping: [{{val| 1 2 3 5 4 3 }}, {{val| 0 -3 -5 -16 -4 5 }}]
Mapping: {{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 tuning (CTE): ~2 = 1\1, ~11/10 = 164.4782
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}


Optimal GPV sequence: {{val list| 7d, 22, 29, 51f, 80cdeff }}
Badness (Smith): 0.026543


Badness: 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 {{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


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


[[Mapping]]: [{{val| 1 2 3 1 }}, {{val| 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]] ([[CTE]]): ~2 = 1\1, ~10/9 = 166.0938
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}


[[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]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


{{Val list|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 315: Line 382:
Comma list: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 525/512


Mapping: [{{val| 1 2 3 1 4 }}, {{val| 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: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


Optimal GPV sequence: {{val list| 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 332: Line 401:
Comma list: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 100/99, 105/104


Mapping: [{{val| 1 2 3 1 4 3 }}, {{val| 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: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3
 
{{Optimal ET sequence|legend=0| 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|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
 
[[Badness]] (Smith): 0.044944
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 22/21, 36/35, 80/77
 
Mapping: {{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 GPV sequence: {{val list| 7, 22d, 29, 65cef }}
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


Badness: 0.030233
Badness (Smith): 0.026790


== Hedgehog ==
== Hedgehog ==
{{See also| Sensamagic clan }}
{{See also| Sensamagic clan | Stearnsmic 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)). Its ploidacot is diploid omega-tricot.


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.
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 354: Line 464:
[[Comma list]]: 50/49, 245/243
[[Comma list]]: 50/49, 245/243


[[Mapping]]: [{{val| 2 1 1 2 }}, {{val| 0 3 5 5 }}]
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}


: 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 }}


{{Val list|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 371: Line 483:
Comma list: 50/49, 55/54, 99/98
Comma list: 50/49, 55/54, 99/98


Mapping: [{{val| 2 1 1 2 4 }}, {{val| 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 GPV sequence: {{val list| 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 384: Line 498:
Comma list: 50/49, 55/54, 65/63, 99/98
Comma list: 50/49, 55/54, 65/63, 99/98


Mapping: [{{val| 2 1 1 2 4 3 }}, {{val| 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 GPV sequence: {{val list| 8d, 14cf, 22 }}
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}


Badness: 0.021516
Badness (Smith): 0.021516


==== Urchin ====
==== Urchin ====
Line 397: Line 513:
Comma list: 40/39, 50/49, 55/54, 66/65
Comma list: 40/39, 50/49, 55/54, 66/65


Mapping: [{{val| 2 1 1 2 4 6 }}, {{val| 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 GPV sequence: {{val list| 14c, 22f }}
{{Optimal ET sequence|legend=0| 14c, 22f }}


Badness: 0.025233
Badness (Smith): 0.025233


=== Hedgepig ===
=== Hedgepig ===
Line 410: Line 528:
Comma list: 50/49, 245/243, 385/384
Comma list: 50/49, 245/243, 385/384


Mapping: [{{val| 2 1 1 2 12 }}, {{val| 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 GPV sequence: {{val list| 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


[[Comma list]]: 49/48, 250/243
[[Comma list]]: 49/48, 250/243


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -6 -10 -3 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}


: 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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 443: Line 567:
Comma list: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 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 GPV sequence: {{val list| 14c, 15, 29 }}
{{Optimal ET sequence|legend=0| 14c, 15, 29, 44d }}


Badness: 0.026023
Badness (Smith): 0.026023


==== 13-limit ====
==== 13-limit ====
Line 456: Line 582:
Comma list: 49/48, 55/54, 91/90, 100/99
Comma list: 49/48, 55/54, 91/90, 100/99


Mapping: [{{val| 1 2 3 3 4 5 }}, {{val| 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 GPV sequence: {{val list| 14cf, 15, 29 }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29, 44d }}


Badness: 0.022285
Badness (Smith): 0.022285


==== Belauensis ====
==== Belauensis ====
Line 469: Line 597:
Comma list: 40/39, 49/48, 55/54, 66/65
Comma list: 40/39, 49/48, 55/54, 66/65


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 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 GPV sequence: {{val list| 14c, 15 }}
{{Optimal ET sequence|legend=0| 14c, 15, 29f, 44dff }}


Badness: 0.029816
Badness (Smith): 0.029816


; Music
; Music
Line 481: 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


[[Comma list]]: 250/243, 686/675
[[Comma list]]: 250/243, 686/675


[[Mapping]]: [{{val| 1 5 8 10 }}, {{val| 0 -9 -15 -19 }}]
{{Mapping|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}


: 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 }}


{{Val list|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 502: Line 636:
Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


Mapping: [{{val| 1 5 8 10 8 }}, {{val| 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 GPV sequence: {{val list| 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 515: Line 651:
Comma list: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168


Mapping: [{{val| 1 5 8 10 8 9 }}, {{val| 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 GPV sequence: {{val list| 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


[[Comma list]]: 250/243, 1728/1715
[[Comma list]]: 250/243, 1728/1715


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -9 -15 -4 }}]
{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}


: 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 }}


{{Val list|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 545: Line 687:
Comma list: 55/54, 100/99, 352/343
Comma list: 55/54, 100/99, 352/343


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 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 GPV sequence: {{val list| 1ce, 21ce, 22 }}
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}


Badness: 0.051319
Badness (Smith): 0.051319


=== 13-limit ===
=== 13-limit ===
Line 558: Line 702:
Comma list: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 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 GPV sequence: {{val list| 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]]
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"