Porcupine family: Difference between revisions

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


Line 10: Line 11:
{{Main| Porcupine }}
{{Main| Porcupine }}


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 21: Line 22:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.166
* [[WE]]: ~2 = 1199.5444{{c}}, ~10/9 = 163.8881{{c}}
: [[error map]]: {{val| 0.000 +5.547 -7.143 }}
: [[error map]]: {{val| -0.456 +5.469 -7.121 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 163.950
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
: error map: {{val| 0.000 +6.194 -6.065 }}
: error map: {{val| 0.000 +5.859 -6.624 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 32: Line 33:
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}


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


=== Overview to extensions ===
=== Overview to extensions ===
==== 7-limit extensions ====
==== 7-limit extensions ====
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. That means
The second comma defines which [[7-limit]] family member we are looking at.  
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
* [[#Septimal porcupine|Septimal porcupine]] adds [[64/63]], the archytas comma, for a tuning between 15edo and 22edo;
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].
* [[#Porky|Porky]] adds [[225/224]], the marvel comma, for a tuning between 22edo and 29edo;
* [[#Coendou|Coendou]] adds [[525/512]], the avicennma, for a tuning sharp of 29edo.  


Temperaments discussed elsewhere include [[7th-octave temperaments #Jamesbond|jamesbond]].
Those all share the same generator with porcupine.
 
[[#Nautilus|nautilus]] tempers out [[49/48]] and splits the generator in two. [[#Hedgehog|hedgehog]] tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.
 
Temperaments discussed elsewhere include:
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].


==== Subgroup extensions ====
==== Subgroup extensions ====
Noting that {{nowrap| 250/243 {{=}} [[Square superparticular#Top-heavy lopsided commas|S10<sup>2</sup>⋅S11]] {{=}} ([[55/54]])⋅([[100/99]]) }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below.
Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|S11]] }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below.


=== 2.3.5.11 subgroup (porkypine) ===
=== 2.3.5.11 subgroup (porkypine) ===
Line 52: Line 60:
Comma list: 55/54, 100/99
Comma list: 55/54, 100/99


Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
Subgroup-val mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


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]


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.887
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 164.078
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}


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


Badness (Smith): 0.0097
Badness (Sintel): 0.303


==== Undecimation ====
==== Undecimation ====
Line 71: Line 77:
Comma list: 55/54, 100/99, 512/507
Comma list: 55/54, 100/99, 512/507


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


: sval mapping generators: ~2, ~65/44
: mapping generators: ~2, ~88/65


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~88/65 = 518.086
* WE: ~2 = 1199.4791{{c}}, ~88/65 = 517.9845{{c}}
* POTE: ~2 = 1200.000, ~88/65 = 518.209
* CWE: ~2 = 1200.0000{{c}}, ~88/65 = 518.1740{{c}}


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


Badness (Smith): 0.0305
Badness (Sintel): 1.21


== 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 93: Line 99:


{{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 = 1200.000, ~10/9 = 163.203
* [[WE]]: ~2 = 1197.8178{{c}}, ~10/9 = 162.5839{{c}}
: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
: [[error map]]: {{val| -2.182 +5.929 -5.780 +2.313 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 162.880
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
: error map: {{val| 0.000 +9.405 -0.714 +8.455 }}
: 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 115: Line 119:
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}


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


=== 11-limit ===
=== 11-limit ===
Line 125: Line 129:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.105
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 162.747
* 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 (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
Line 138: Line 142:
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


Badness (Smith): 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


Line 148: Line 154:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.442
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 162.708
* 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 (unchanged-interval) basis: 2.11
: unchanged-interval (eigenmonzo) basis: 2.11


Tuning ranges:  
Tuning ranges:  
Line 160: Line 166:
* 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|legend=0| 7, 15, 22f, 37f }}
{{Optimal ET sequence|legend=0| 7, 15, 22f }}


Badness (Smith): 0.021276
Badness (Sintel): 0.879


==== Porcupinefish ====
==== Porcupinefish ====
Line 174: Line 180:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 162.636
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 162.277
* 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 (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]


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


Badness (Smith): 0.025314
Badness (Sintel): 1.05


==== Pourcup ====
==== Pourcup ====
Line 198: Line 204:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.378
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 162.482
* 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 (unchanged-interval) basis: 2.13/7
: unchanged-interval (eigenmonzo) basis: 2.13/7


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


Badness (Smith): 0.035130
Badness (Sintel): 1.45


==== Porkpie ====
==== Porkpie ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 163.678
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 163.688
* 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 (unchanged-interval) basis: 2.9/7
: unchanged-interval (eigenmonzo) basis: 2.9/7


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


Badness (Smith): 0.026043
Badness (Sintel): 1.08


== 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 236: Line 244:


{{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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
* [[WE]]: ~2 = 1195.7927{{c}}, ~10/9 = 159.1315{{c}}
: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
: [[error map]]: {{val| -4.207 +12.236 +5.407 -17.838 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}
: 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]] (Smith): 0.040650
[[Badness]] (Sintel): 1.03


=== 11-limit ===
=== 11-limit ===
Line 258: Line 264:


Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
{{Multival|legend=1| 3 5 9 4 1 6 -4 7 -8 -20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 161.365
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 159.807
* 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=0| 7d, 8d, 15 }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Badness (Smith): 0.022325
Badness (Sintel): 0.738


=== 13-limit ===
=== 13-limit ===
Line 280: Line 284:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 161.631
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 158.805
* 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=0| 7d, 8d, 15, 38bceff }}
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


Badness (Smith): 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 298: 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 = 1200.000, ~10/9 = 164.391
* [[WE]]: ~2 = 1200.0685{{c}}, ~10/9 = 164.4215{{c}}
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
: [[error map]]: {{val| +0.068 +4.917 -8.216 +0.772 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}
: 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]] (Smith): 0.054389
[[Badness]] (Sintel): 1.38


=== 11-limit ===
=== 11-limit ===
Line 321: Line 323:


Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
Wedgie: {{multival| 3 5 16 4 1 17 -4 23 -8 -44 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 164.321
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 164.552
* 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 (unchanged-interval) basis: 2.7/5
: unchanged-interval (eigenmonzo) basis: 2.7/5


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


Badness (Smith): 0.027268
Badness (Sintel): 0.901


=== 13-limit ===
=== 13-limit ===
Line 344: Line 344:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 164.478
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 164.953
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


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


Badness (Smith): 0.026543
Badness (Sintel): 1.10


; Music
; Music
Line 355: Line 355:


== 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 362: 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 = 1200.000, ~10/9 = 166.094
* [[WE]]: ~2 = 1202.6772{{c}}, ~10/9 = 166.4110{{c}}
: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
: [[error map]]: {{val| +2.678 +4.166 -10.337 -2.806 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}
: 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]] (Smith): 0.118344
[[Badness]] (Sintel): 2.99


=== 11-limit ===
=== 11-limit ===
Line 387: Line 385:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 165.925
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 165.981
* 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 (unchanged-interval) basis: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


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


Badness (Smith): 0.049669
Badness (Sintel): 1.64


=== 13-limit ===
=== 13-limit ===
Line 406: Line 404:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 166.046
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 165.974
* 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 (unchanged-interval) basis: 2.3
: unchanged-interval (eigenmonzo) basis: 2.3


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


Badness (Smith): 0.030233
Badness (Sintel): 1.25


== Hystrix ==
== Hystrix ==
Line 425: 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 = 1200.000, ~10/9 = 165.185
* [[WE]]: ~2 = 1187.8599{{c}}, ~10/9 = 157.2605{{c}}
: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
: [[error map]]: {{val| -12.140 +1.983 -9.037 +37.493 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}
: 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]] (Smith): 0.044944
[[Badness]] (Sintel): 1.14


=== 11-limit ===
=== 11-limit ===
Line 450: Line 446:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~11/10 = 164.768
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* POTE: ~2 = 1200.000, ~11/10 = 158.750
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}


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


Badness (Smith): 0.026790
Badness (Sintel): 0.886
 
== Oxygen ==
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
 
[[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 = 1200.000, ~10/9 = 161.341
: [[error map]]: {{val| 0.000 +14.023 +6.982 -91.507 }}
* [[POTE]]: ~2 = 1200.000, ~10/9 = 169.112
: error map: {{val| 0.000 -9.291 -31.873 -107.050 }}
 
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}
 
[[Badness]] (Smith): 0.059866


== Hedgehog ==
== Hedgehog ==
{{See also| Sensamagic clan | Stearnsmic clan }}
{{See also| Sensamagic clan | 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.  
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 alpha-tricot.
 
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 490: 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 = 600.000, ~9/7 = 435.258
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[POTE]]: ~7/5 = 600.000, ~9/7 = 435.648
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.989 -8.074 +9.414 }}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}


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


[[Badness]] (Smith): 0.043983
[[Badness]] (Sintel): 1.11


=== 11-limit ===
=== 11-limit ===
Line 509: Line 484:


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 435.528
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* POTE: ~7/5 = 600.000, ~9/7 = 435.386
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}


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


Badness (Smith): 0.023095
Badness (Sintel): 0.764


==== 13-limit ====
==== 13-limit ====
Line 528: Line 501:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 436.309
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* POTE: ~7/5 = 600.000, ~9/7 = 435.861
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}


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


Badness (Smith): 0.021516
Badness (Sintel): 0.889


==== Urchin ====
==== Urchin ====
Line 543: Line 516:


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 435.186
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* POTE: ~7/5 = 600.000, ~9/7 = 437.078
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}


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


Badness (Smith): 0.025233
Badness (Sintel): 1.04


=== Hedgepig ===
=== Hedgepig ===
Line 556: Line 529:


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~7/5 = 600.000, ~9/7 = 435.329
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* POTE: ~7/5 = 600.000, ~9/7 = 435.425
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}


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


Badness (Smith): 0.068406
Badness (Sintel): 2.26


; Music
; Music
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|hedgehog[14]]], 22edo tuning.
* [https://web.archive.org/web/20240624173512/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 578: 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 = 1200.000, ~21/20 = 81.914
* [[WE]]: ~2 = 1202.1642{{c}}, ~21/20 = 82.6542{{c}}
: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
: [[error map]]: {{val| +2.164 +6.448 -6.364 -10.296 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 82.505
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
: error map: {{val| 0.000 +3.012 -11.368 -16.342 }}
: 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]] (Smith): 0.057420
[[Badness]] (Sintel): 1.45


=== 11-limit ===
=== 11-limit ===
Line 597: Line 568:


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 81.802
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* POTE: ~2 = 1200.000, ~21/20 = 82.504
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}


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


Badness (Smith): 0.026023
Badness (Sintel): 0.860


==== 13-limit ====
==== 13-limit ====
Line 616: Line 585:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 81.912
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* POTE: ~2 = 1200.000, ~21/20 = 82.530
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}


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


Badness (Smith): 0.022285
Badness (Sintel): 0.921


==== Belauensis ====
==== Belauensis ====
Line 631: Line 600:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 82.034
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* POTE: ~2 = 1200.000, ~21/20 = 81.759
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}


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


Badness (Smith): 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


{{Multival|legend=1| 9 15 19 3 5 2 }}
: mapping generators: ~2, ~14/9


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
* [[WE]]: ~2 = 1199.3342{{c}}, ~14/9 = 745.1379{{c}}
: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
: [[error map]]: {{val| -0.666 +6.949 -4.584 -5.213 }}
* [[POTE]]: ~2 = 1200.000, ~9/7 = 454.448
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
: error map: {{val| 0.000 +8.009 -3.040 -3.346 }}
: 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]] (Smith): 0.107686
[[Badness]] (Sintel): 2.73


=== 11-limit ===
=== 11-limit ===
Line 667: Line 637:
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: {{mapping| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}
 
Wedgie: {{multival| 9 15 19 12 3 5 -12 2 -24 -32 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 454.505
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* POTE: ~2 = 1200.000, ~9/7 = 454.512
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}


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


Badness (Smith): 0.045694
Badness (Sintel): 1.51


=== 13-limit ===
=== 13-limit ===
Line 684: Line 652:
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: {{mapping| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/10 = 454.480
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* POTE: ~2 = 1200.000, ~13/10 = 454.529
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}


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


Badness (Smith): 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 702: Line 672:


: 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 = 1200.000, ~36/35 = 54.804
* [[WE]]: ~2 = 1197.6274{{c}}, ~36/35 = 54.2770{{c}}
: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
: [[error map]]: {{val| -2.373 +4.807 -7.586 +6.948 }}
* [[POTE]]: ~2 = 1200.000, ~36/35 = 54.384
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
: error map: {{val| 0.000 +8.585 -2.081 +13.636 }}
: 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]] (Smith): 0.115304
[[Badness]] (Sintel): 2.92


=== 11-limit ===
=== 11-limit ===
Line 723: Line 691:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~36/35 = 54.702
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* POTE: ~2 = 1200.000, ~36/35 = 54.376
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}


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


Badness (Smith): 0.051319
Badness (Sintel): 1.70


=== 13-limit ===
=== 13-limit ===
Line 738: Line 706:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~36/35 = 54.575
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* POTE: ~2 = 1200.000, ~36/35 = 54.665
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}


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


Badness (Smith): 0.044739
Badness (Sintel): 1.85


[[Category:Temperament families]]
[[Category:Temperament families]]

Latest revision as of 04:15, 5 February 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:

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

Undecimation

Subgroup: 2.3.5.11.13

Comma list: 55/54, 100/99, 512/507

Subgroup-val mapping: [⟨1 -1 -2 0 5], ⟨0 6 10 8 -3]]

mapping generators: ~2, ~88/65

Optimal tunings:

  • WE: ~2 = 1199.4791 ¢, ~88/65 = 517.9845 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/65 = 518.1740 ¢

Optimal ET sequence: 7, 23bc, 30, 37, 44

Badness (Sintel): 1.21

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

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

Hedgehog

See also: Sensamagic clan and Stearnsmic clan

Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out 245/243, the sensamagic comma. It is a strong extension of BPS (as BPS has no 2 or sqrt(2)). Its ploidacot is diploid alpha-tricot.

22edo provides an obvious tuning, which happens to be the only patent-val tuning, but if you are looking for an alternative you could try the ⟨146 232 338 411] (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is echidna, which offers much more accuracy. They merge on 22edo.

Subgroup: 2.3.5.7

Comma list: 50/49, 245/243

Mapping: [⟨2 1 1 2], ⟨0 3 5 5]]

mapping generators: ~7/5, ~9/7

Optimal tunings:

  • WE: ~7/5 = 599.6061 ¢, ~9/7 = 435.3620 ¢
error map: ⟨-0.788 +3.737 -9.897 +7.197]
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4483 ¢
error map: ⟨0.000 +4.390 -9.072 +8.416]

Optimal ET sequence: 8d, 14c, 22

Badness (Sintel): 1.11

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 99/98

Mapping: [⟨2 1 1 2 4], ⟨0 3 5 5 4]]

Optimal tunings:

  • WE: ~7/5 = 600.1133 ¢, ~9/7 = 435.4680 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4431 ¢

Optimal ET sequence: 8d, 14c, 22, 58ce

Badness (Sintel): 0.764

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 65/63, 99/98

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

Optimal tunings:

  • WE: ~7/5 = 600.3651 ¢, ~9/7 = 436.1258 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.0483 ¢

Optimal ET sequence: 8d, 14cf, 22

Badness (Sintel): 0.889

Urchin

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 66/65

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

Optimal tunings:

  • WE: ~7/5 = 598.3303 ¢, ~9/7 = 435.8617 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.3485 ¢

Optimal ET sequence: 14c, 22f

Badness (Sintel): 1.04

Hedgepig

Subgroup: 2.3.5.7.11

Comma list: 50/49, 245/243, 385/384

Mapping: [⟨2 1 1 2 12], ⟨0 3 5 5 -7]]

Optimal tunings:

  • WE: ~7/5 = 599.7917 ¢, ~9/7 = 435.2737 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4047 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.26

Music

Nautilus

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

Subgroup: 2.3.5.7

Comma list: 49/48, 250/243

Mapping: [⟨1 2 3 3], ⟨0 -6 -10 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • 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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