Porcupine family: Difference between revisions

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


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 27: Line 12:


{{Mapping|legend=1| 1 2 3 | 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:
* [[WE]]: ~2 = 1199.5444{{c}}, ~10/9 = 163.8881{{c}}
: [[error map]]: {{val| -0.456 +5.469 -7.121 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
: error map: {{val| 0.000 +5.859 -6.624 }}


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


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


[[Badness]]: 0.030778
[[Badness]] (Sintel): 0.722


=== 2.3.5.11 subgroup (porkypine) ===
=== Overview to extensions ===
Subgroup: 2.3.5.11
==== 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.  


Comma list: 55/54, 100/99
Those all share the same generator with porcupine.


Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
[[#Nautilus|Nautilus]] tempers out [[49/48]] and splits the generator in two. 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.


Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
Temperaments discussed elsewhere include:  
* ''[[Oxygen]]'' (+21/20) → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Stearnsmic clan #Hedgehog|Stearnsmic clan]]
* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]


: gencom: [2 10/9; 55/54, 100/99]
The rest are considered in this page.


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.8867
==== 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.


{{Optimal ET sequence|legend=1| 7, 15, 22, 73ce, 95ce }}
=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11


Badness: 0.0097
Comma list: 55/54, 100/99
 
==== Undecimation ====
Subgroup: 2.3.5.11.13
 
Comma list: 55/54, 100/99, 512/507


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


: sval mapping generators: ~2, ~65/44
{{Mapping|legend=4| 1 2 3 0 4 | 0 -3 -5 0 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~88/65 = 518.2094
Optimal tunings:
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}


{{Optimal ET sequence|legend=1| 7, 23bc, 30, 37, 44 }}
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}


Badness: 0.0305
Badness (Sintel): 0.303


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


Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.
Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. Here, we share the same mapping of 7/4 in terms of fifths as [[archy]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 84: Line 79:
{{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:
 
* [[WE]]: ~2 = 1197.8178{{c}}, ~10/9 = 162.5839{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 163.2032
: [[error map]]: {{val| -2.182 +5.929 -5.780 +2.313 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 98: Line 95:
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]


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


[[Badness]]: 0.041057
[[Badness]] (Sintel): 1.04


=== 11-limit ===
=== 11-limit ===
Line 109: Line 105:
Comma list: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99


Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
{{Mapping|legend=0| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.1055
Optimal tunings:
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}


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


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


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


Badness: 0.021562
Badness (Sintel): 0.713
 
==== 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: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
{{Mapping|legend=0| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.4425
Optimal tunings:
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}


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


Tuning ranges:  
Tuning ranges:  
Line 143: Line 144:
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]
* 13-odd-limit diamond monotone and tradeoff: ~11/10 = [160.000, 163.636]
* 15-odd-limit diamond monotone and tradeoff: ~11/10 = 163.636


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


Badness: 0.021276
Badness (Sintel): 0.879


==== Porcupinefish ====
==== Porcupinefish ====
Line 157: Line 156:
Comma list: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99


Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
{{Mapping|legend=0| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 162.6361
Optimal tunings:
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}


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


Tuning ranges:  
Tuning ranges:  
* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
* 13-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 162.162]
* 15-odd-limit diamond monotone and tradeoff: ~10/9 = 162.162


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


Badness: 0.025314
Badness (Sintel): 1.05


==== Pourcup ====
==== Pourcup ====
Line 181: Line 180:
Comma list: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195


Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
{{Mapping|legend=0| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.3781
Optimal tunings:
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}


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


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


Badness: 0.035130
Badness (Sintel): 1.45


==== Porkpie ====
==== Porkpie ====
Line 198: Line 199:
Comma list: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99


Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
{{Mapping|legend=0| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.6778
Optimal tunings:
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}


Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: Eigenmonzo basis (unchanged-interval basis): 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7
 
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
 
Badness (Sintel): 1.08
 
==== Undecimation ====
This weak extension is used by most patent vals that support porcupine and tune 13 well.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 64/63, 100/99, 169/168
 
{{Mapping|legend=0| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}
: mapping generators: ~2, ~35/26
 
Optimal tunings:
* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}


{{Optimal ET sequence|legend=1| 7, 15f, 22 }}
{{Optimal ET sequence|legend=0| 7, 30, 37, 81bd }}


Badness: 0.026043
Badness (Sintel): 1.68


== 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.  
{{Main| Opossum }}
 
Opossum can be described as {{nowrap| 8d & 15 }}. Tempering out [[28/27]], the perfect fifth of three generator steps is conflated with not [[32/21]] as in porcupine but [[14/9]]. Three such fifths or nine generator steps octave reduced give a flat 7/4. 2\15 is a good generator.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 219: Line 242:
{{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:
 
* [[WE]]: ~2 = 1195.7927{{c}}, ~10/9 = 159.1315{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
: [[error map]]: {{val| -4.207 +12.236 +5.407 -17.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}


[[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]] (Sintel): 1.03


=== 11-limit ===
=== 11-limit ===
Line 235: Line 260:
Comma list: 28/27, 55/54, 77/75
Comma list: 28/27, 55/54, 77/75


Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
{{Mapping|legend=0| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646
Optimal tunings:
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}


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


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


Badness: 0.022325
Badness (Sintel): 0.738


=== 13-limit ===
=== 13-limit ===
Line 251: Line 278:
Comma list: 28/27, 40/39, 55/54, 66/65
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|legend=0| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312
Optimal tunings:
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


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


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


Badness: 0.019389
Badness (Sintel): 0.801


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 271: Line 300:
{{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:
 
* [[WE]]: ~2 = 1200.0685{{c}}, ~10/9 = 164.4215{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 164.3913
: [[error map]]: {{val| +0.068 +4.917 -8.216 +0.772 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}


[[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, 51, 73c }}


[[Badness]]: 0.054389
[[Badness]] (Sintel): 1.38


=== 11-limit ===
=== 11-limit ===
Line 288: Line 319:
Comma list: 55/54, 100/99, 225/224
Comma list: 55/54, 100/99, 225/224


Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
{{Mapping|legend=0| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.3207
Optimal tunings:
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}


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


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


Badness: 0.027268
Badness (Sintel): 0.901


=== 13-limit ===
=== 13-limit ===
Line 305: Line 338:
Comma list: 55/54, 65/64, 91/90, 100/99
Comma list: 55/54, 65/64, 91/90, 100/99


Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.4782
Optimal tunings:
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


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


Badness: 0.026543
Badness (Sintel): 1.10
 
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 322: Line 360:
{{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:
 
* [[WE]]: ~2 = 1202.6772{{c}}, ~10/9 = 166.4110{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 166.0938
: [[error map]]: {{val| +2.678 +4.166 -10.337 -2.806 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}


[[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 }}


[[Badness]]: 0.118344
[[Badness]] (Sintel): 2.99


=== 11-limit ===
=== 11-limit ===
Line 339: Line 379:
Comma list: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 525/512


Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
{{Mapping|legend=0| 1 2 3 1 4 | 0 -3 -5 13 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 165.9246
Optimal tunings:
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


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


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


Badness: 0.049669
Badness (Sintel): 1.64


=== 13-limit ===
=== 13-limit ===
Line 356: Line 398:
Comma list: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 100/99, 105/104


Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 166.0459
Optimal tunings:
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


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


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


Badness: 0.030233
Badness (Sintel): 1.25


== Hystrix ==
== Hystrix ==
Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits.
Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix is very high in [[error]] due to the large disparity between typical porcupine generators and a justly-tuned 8/7, and is usually considered an [[exotemperament]]. A generator of 2\15 or 9\68 can be used for hystrix.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 377: Line 421:
{{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:
 
* [[WE]]: ~2 = 1187.8599{{c}}, ~10/9 = 157.2605{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 165.1845
: [[error map]]: {{val| -12.140 +1.983 -9.037 +37.493 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}


[[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]] (Sintel): 1.14


=== 11-limit ===
=== 11-limit ===
Line 394: Line 440:
Comma list: 22/21, 36/35, 80/77
Comma list: 22/21, 36/35, 80/77


Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
 
Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.7684


{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
Optimal tunings:
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}


Badness: 0.026790
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


== Oxygen ==
Badness (Sintel): 0.886
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.  


[[Subgroup]]: 2.3.5.7
== Nautilus ==
 
Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.  
[[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]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3408
 
{{Optimal ET sequence|legend=1| 7d }}
 
[[Badness]]: 0.059866
 
== Hedgehog ==
{{See also| Sensamagic clan }}
{{See also| Stearnsmic clan }}
 
Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. 22edo provides the obvious tuning, but if you are looking for an alternative, you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 50/49, 245/243
 
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
 
: mapping generators: ~7/5, ~9/7
 
{{Multival|legend=1| 6 10 10 2 -1 -5 }}
 
[[Optimal tuning]] ([[CTE]]): ~7/5 = 1\2, ~9/7 = 435.2580
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
 
[[Badness]]: 0.043983
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 55/54, 99/98
 
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
 
Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.5281
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22, 58ce }}
 
Badness: 0.023095
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 55/54, 65/63, 99/98
 
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 ET sequence|legend=1| 8d, 14cf, 22 }}
 
Badness: 0.021516
 
==== Urchin ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 50/49, 55/54, 66/65
 
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 ET sequence|legend=1| 14c, 22f }}
 
Badness: 0.025233
 
=== Hedgepig ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 245/243, 385/384
 
Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
 
Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.3289
 
{{Optimal ET sequence|legend=1| 22 }}
 
Badness: 0.068406
 
; Music
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] in [[hedgehog14|hedgehog[14]]] to 22edo.


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


Line 505: Line 461:
: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 6 10 3 2 -12 -21 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1202.1642{{c}}, ~21/20 = 82.6542{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~21/20 = 81.9143
: [[error map]]: {{val| +2.164 +6.448 -6.364 -10.296 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}


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


[[Badness]]: 0.057420
[[Badness]] (Sintel): 1.45


=== 11-limit ===
=== 11-limit ===
Line 518: Line 476:
Comma list: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242


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


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.8017
Optimal tunings:
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}


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


Badness: 0.026023
Badness (Sintel): 0.860


==== 13-limit ====
==== 13-limit ====
Line 531: Line 491:
Comma list: 49/48, 55/54, 91/90, 100/99
Comma list: 49/48, 55/54, 91/90, 100/99


Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.9123
Optimal tunings:
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}


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


Badness: 0.022285
Badness (Sintel): 0.921


==== Belauensis ====
==== Belauensis ====
Line 544: Line 506:
Comma list: 40/39, 49/48, 55/54, 66/65
Comma list: 40/39, 49/48, 55/54, 66/65


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


Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 82.0342
Optimal tunings:
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}


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


Badness: 0.029816
Badness (Sintel): 1.23


; Music
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]
* [https://web.archive.org/web/20201127013840/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones]]


== Ammonite ==
== Ammonite ==
{{See also|Subgroup temperaments #Ammon}}
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|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}
{{Mapping|legend=1| 1 -4 -7 -9 | 0 9 15 19 }}


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~14/9


{{Multival|legend=1| 9 15 19 3 5 2 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.3342{{c}}, ~14/9 = 745.1379{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~9/7 = 454.5500
: [[error map]]: {{val| -0.666 +6.949 -4.584 -5.213 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}


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


[[Badness]]: 0.107686
[[Badness]] (Sintel): 2.73


=== 11-limit ===
=== 11-limit ===
Line 577: Line 546:
Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
{{Mapping|legend=0| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 454.5050
Optimal tunings:
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}


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


Badness: 0.045694
Badness (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 590: Line 561:
Comma list: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168


Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}
{{Mapping|legend=0| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}


Optimal tuning (CTE): ~2 = 1\1, ~13/10 = 454.4798
Optimal tunings:
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}


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


Badness: 0.027168
Badness (Sintel): 1.12


== Ceratitid ==
== Ceratitid ==
Ceratitid adds 1728/1715 to the comma list and may be described as the {{nowrap| 21c & 22 }} temperament. Its ploidacot is omega-enneacot. [[22edo]] provides an obvious tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 607: Line 582:
: mapping generators: ~2, ~36/35
: mapping generators: ~2, ~36/35


{{Multival|legend=1| 9 15 4 3 -19 -33 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1197.6274{{c}}, ~36/35 = 54.2770{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 54.8040
: [[error map]]: {{val| -2.373 +4.807 -7.586 +6.948 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}


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


[[Badness]]: 0.115304
[[Badness]] (Sintel): 2.92


=== 11-limit ===
=== 11-limit ===
Line 620: Line 597:
Comma list: 55/54, 100/99, 352/343
Comma list: 55/54, 100/99, 352/343


Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.7019
Optimal tunings:
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}


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


Badness: 0.051319
Badness (Sintel): 1.70


=== 13-limit ===
=== 13-limit ===
Line 633: Line 612:
Comma list: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}


Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.5751
Optimal tunings:
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}


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


Badness: 0.044739
Badness (Sintel): 1.85


[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 17:44, 30 September 2026

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

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

Porcupine

Main article: Porcupine

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

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

mapping generators: ~2, ~10/9

Optimal tunings:

  • WE: ~2 = 1199.5444 ¢, ~10/9 = 163.8881 ¢
error map: ⟨-0.456 +5.469 -7.121]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 164.0621 ¢
error map: ⟨0.000 +5.859 -6.624]

Tuning ranges:

Optimal ET sequence: 7, 15, 22, 95c

Badness (Sintel): 0.722

Overview to extensions

7-limit extensions

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

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

Those all share the same generator with porcupine.

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

Temperaments discussed elsewhere include:

The rest are considered in this page.

Subgroup extensions

Noting that 250/243 = (55/54)⋅(100/99) = S102⋅S11, the temperament thus extends naturally to the 2.3.5.11 subgroup, sometimes known as porkypine, given right below.

2.3.5.11 subgroup (porkypine)

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

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

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

Optimal tunings:

  • WE: ~2 = 1200.3290 ¢, ~11/10 = 164.1227 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 163.9951 ¢

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

Badness (Sintel): 0.303

Septimal porcupine

Main article: Porcupine

Septimal porcupine uses six of its minor tone generator steps to get to 7/4. Here, we share the same mapping of 7/4 in terms of fifths as archy. For this to work you need a small minor tone such as 22edo provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.

Subgroup: 2.3.5.7

Comma list: 64/63, 250/243

Mapping: [⟨1 2 3 2], ⟨0 -3 -5 6]]

Optimal tunings:

  • WE: ~2 = 1197.8178 ¢, ~10/9 = 162.5839 ¢
error map: ⟨-2.182 +5.929 -5.780 +2.313]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 162.9493 ¢
error map: ⟨0.000 +9.197 -1.060 +8.870]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5
unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~10/9 = [160.000, 163.636] (2\15 to 3\22)
  • 7-odd-limit diamond tradeoff: ~10/9 = [157.821, 166.015]
  • 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]

Optimal ET sequence: 7, 15, 22, 37, 59, 81bd

Badness (Sintel): 1.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 100/99

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

Optimal tunings:

  • WE: ~2 = 1198.3250 ¢, ~11/10 = 162.5202 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.8156 ¢

Minimax tuning:

  • 11-odd-limit: ~11/10 = [1/6 -1/6 0 1/12⟩
unchanged-interval (eigenmonzo) basis: 2.9/7

Tuning ranges:

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

Optimal ET sequence: 7, 15, 22, 37, 59

Badness (Sintel): 0.713

Porcupinefowl

This extension used to be tridecimal porcupine.

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1197.0054 ¢, ~11/10 = 162.3022 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.8314 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~10/9 = [1 0 0 0 -1/4⟩
unchanged-interval (eigenmonzo) basis: 2.11

Tuning ranges:

  • 13-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
  • 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
  • 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]

Optimal ET sequence: 7, 15, 22f

Badness (Sintel): 0.879

Porcupinefish

See also: The Biosphere

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 100/99

Mapping: [⟨1 2 3 2 4 6], ⟨0 -3 -5 6 -4 -17]]

Optimal tunings:

  • WE: ~2 = 1198.3206 ¢, ~11/10 = 162.0502 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.3458 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~10/9 = [2/13 0 0 0 1/13 -1/13⟩
unchanged-interval (eigenmonzo) basis: 2.13/11

Tuning ranges:

  • 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
  • 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
  • 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]

Optimal ET sequence: 15, 22, 37

Badness (Sintel): 1.05

Pourcup

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 100/99, 196/195

Mapping: [⟨1 2 3 2 4 1], ⟨0 -3 -5 6 -4 20]]

Optimal tunings:

  • WE: ~2 = 1198.0537 ¢, ~11/10 = 162.2183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.4665 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [1/14 0 0 -1/14 0 1/14⟩
unchanged-interval (eigenmonzo) basis: 2.13/7

Optimal ET sequence: 15f, 22f, 37, 59f

Badness (Sintel): 1.45

Porkpie

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 100/99

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

Optimal tunings:

  • WE: ~2 = 1200.0223 ¢, ~11/10 = 163.6908 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 163.6874 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [1/6 -1/6 0 1/12⟩
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 7, 15f, 22

Badness (Sintel): 1.08

Undecimation

This weak extension is used by most patent vals that support porcupine and tune 13 well.

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 100/99, 169/168

Mapping: [⟨1 -1 -2 8 0 5], ⟨0 6 10 -12 8 -3]]

mapping generators: ~2, ~35/26

Optimal tunings:

  • WE: ~2 = 1198.4569 ¢, ~35/26 = 517.9375 ¢
  • CWE: ~2 = 1200.0000 ¢, ~35/26 = 518.5822 ¢

Optimal ET sequence: 7, 30, 37, 81bd

Badness (Sintel): 1.68

Opossum

Main article: Opossum

Opossum can be described as 8d & 15. Tempering out 28/27, the perfect fifth of three generator steps is conflated with not 32/21 as in porcupine but 14/9. Three such fifths or nine generator steps octave reduced give a flat 7/4. 2\15 is a good generator.

Subgroup: 2.3.5.7

Comma list: 28/27, 126/125

Mapping: [⟨1 2 3 4], ⟨0 -3 -5 -9]]

Optimal tunings:

  • WE: ~2 = 1195.7927 ¢, ~10/9 = 159.1315 ¢
error map: ⟨-4.207 +12.236 +5.407 -17.838]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 160.4589 ¢
error map: ⟨0.000 +16.668 +11.392 -12.956]

Minimax tuning:

Optimal ET sequence: 7d, 8d, 15

Badness (Sintel): 1.03

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 4 4], ⟨0 -3 -5 -9 -4]]

Optimal tunings:

  • WE: ~2 = 1196.2331 ¢, ~11/10 = 159.3050 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 160.4644 ¢

Minimax tuning:

  • 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7

Optimal ET sequence: 7d, 8d, 15

Badness (Sintel): 0.738

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 4 4 4], ⟨0 -3 -5 -9 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1193.5447 ¢, ~11/10 = 157.9505 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 159.7600 ¢

Minimax tuning:

  • 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7

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

Badness (Sintel): 0.801

Porky

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

Subgroup: 2.3.5.7

Comma list: 225/224, 250/243

Mapping: [⟨1 2 3 5], ⟨0 -3 -5 -16]]

Optimal tunings:

  • WE: ~2 = 1200.0685 ¢, ~10/9 = 164.4215 ¢
error map: ⟨+0.068 +4.917 -8.216 +0.772]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 164.4060 ¢
error map: ⟨0.000 +4.827 -8.344 +0.678]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.7/5

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

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1200.8706 ¢, ~11/10 = 164.6715 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 164.4810 ¢

Minimax tuning:

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

Optimal ET sequence: 7d, 15d, 22, 51

Badness (Sintel): 0.901

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1202.1557 ¢, ~11/10 = 165.2494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 164.8579 ¢

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

Badness (Sintel): 1.10

Music

Coendou

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

Subgroup: 2.3.5.7

Comma list: 250/243, 525/512

Mapping: [⟨1 2 3 1], ⟨0 -3 -5 13]]

Optimal tunings:

  • WE: ~2 = 1202.6772 ¢, ~10/9 = 166.4110 ¢
error map: ⟨+2.678 +4.166 -10.337 -2.806]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 166.0511 ¢
error map: ⟨0.000 -0.108 -16.569 -10.161]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65c

Badness (Sintel): 2.99

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1203.0245 ¢, ~11/10 = 166.3991 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9714 ¢

Minimax tuning:

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

Optimal ET sequence: 7, 22d, 29, 65ce

Badness (Sintel): 1.64

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1202.9957 ¢, ~11/10 = 166.3885 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9843 ¢

Minimax tuning:

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

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

Badness (Sintel): 1.25

Hystrix

Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix is very high in error due to the large disparity between typical porcupine generators and a justly-tuned 8/7, and is usually considered an exotemperament. A generator of 2\15 or 9\68 can be used for hystrix.

Subgroup: 2.3.5.7

Comma list: 36/35, 160/147

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1187.8599 ¢, ~10/9 = 157.2605 ¢
error map: ⟨-12.140 +1.983 -9.037 +37.493]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 161.2833 ¢
error map: ⟨0.000 +14.195 +7.270 +69.891]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 7, 8d, 15d

Badness (Sintel): 1.14

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1189.2810 ¢, ~11/10 = 157.3322 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 160.9603 ¢

Optimal ET sequence: 7, 8d, 15d

Badness (Sintel): 0.886

Nautilus

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

Subgroup: 2.3.5.7

Comma list: 49/48, 250/243

Mapping: [⟨1 2 3 3], ⟨0 -6 -10 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1202.1642 ¢, ~21/20 = 82.6542 ¢
error map: ⟨+2.164 +6.448 -6.364 -10.296]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 82.2758 ¢
error map: ⟨0.000 +4.390 -9.072 -15.653]

Optimal ET sequence: 14c, 15, 29

Badness (Sintel): 1.45

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1202.3781 ¢, ~21/20 = 82.6673 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 82.2434 ¢

Optimal ET sequence: 14c, 15, 29

Badness (Sintel): 0.860

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1202.4145 ¢, ~21/20 = 82.6963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 82.3130 ¢

Optimal ET sequence: 14cf, 15, 29

Badness (Sintel): 0.921

Belauensis

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1199.0072 ¢, ~21/20 = 81.6911 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 81.8576 ¢

Optimal ET sequence: 14c, 15

Badness (Sintel): 1.23

Music

Ammonite

See also: Subgroup temperaments § Ammon

Ammonite adds 686/675 to the comma list and may be described as the 8d & 29 temperament. Its ploidacot is epsilon-enneacot. 37edo provides an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 250/243, 686/675

Mapping: [⟨1 -4 -7 -9], ⟨0 9 15 19]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.3342 ¢, ~14/9 = 745.1379 ¢
error map: ⟨-0.666 +6.949 -4.584 -5.213]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 745.4994 ¢
error map: ⟨0.000 +7.540 -3.823 -4.337]

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

Badness (Sintel): 2.73

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -4 -7 -9 -4], ⟨0 9 15 19 12]]

Optimal tunings:

  • WE: ~2 = 1200.0141 ¢, ~14/9 = 745.4971 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 745.4894 ¢

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

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -4 -7 -9 -4 -5], ⟨0 9 15 19 12 14]]

Optimal tunings:

  • WE: ~2 = 1200.2478 ¢, ~14/9 = 745.6252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 745.4904 ¢

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

Badness (Sintel): 1.12

Ceratitid

Ceratitid adds 1728/1715 to the comma list and may be described as the 21c & 22 temperament. Its ploidacot is omega-enneacot. 22edo provides an obvious tuning.

Subgroup: 2.3.5.7

Comma list: 250/243, 1728/1715

Mapping: [⟨1 2 3 3], ⟨0 -9 -15 -4]]

mapping generators: ~2, ~36/35

Optimal tunings:

  • WE: ~2 = 1197.6274 ¢, ~36/35 = 54.2770 ¢
error map: ⟨-2.373 +4.807 -7.586 +6.948]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 54.5489 ¢
error map: ⟨0.000 +7.105 -4.548 +12.978]

Optimal ET sequence: 1c, 21c, 22

Badness (Sintel): 2.92

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1198.2851 ¢, ~36/35 = 54.2986 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 54.4992 ¢

Optimal ET sequence: 1ce, 21ce, 22

Badness (Sintel): 1.70

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1200.3864 ¢, ~36/35 = 54.6830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 54.6396 ¢

Optimal ET sequence: 1ce, 21cef, 22

Badness (Sintel): 1.85

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