Porcupine family: Difference between revisions

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{{interwiki
| de = Porcupine
| en = Porcupine family
| es =
| ja =
}}
{{Technical data page}}
{{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 18: 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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.166{{c}}
* [[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{{c}}, ~10/9 = 163.950{{c}}
* [[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 33: Line 26:
{{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 ===
Line 46: Line 39:
Those all share the same generator with porcupine.  
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.  
[[#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.  


Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* ''[[Oxygen]]'' (+21/20) → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Stearnsmic clan #Hedgehog|Stearnsmic clan]]
* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]
 
The rest are considered in this page.  


==== Subgroup extensions ====
==== Subgroup extensions ====
Line 60: Line 56:
Comma list: 55/54, 100/99
Comma list: 55/54, 100/99


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


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.887{{c}}
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.078{{c}}
* 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 ====
Subgroup: 2.3.5.11.13
 
Comma list: 55/54, 100/99, 512/507
 
Subgroup-val mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}
 
: mapping generators: ~2, ~65/44
 
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~88/65 = 518.086{{c}}
* POTE: ~2 = 1200.000{{c}}, ~88/65 = 518.209{{c}}
 
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
 
Badness (Smith): 0.0305


== Septimal porcupine ==
== Septimal porcupine ==
Line 101: Line 80:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.203{{c}}
* [[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{{c}}, ~10/9 = 162.880{{c}}
* [[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]]:  
Line 119: Line 98:
{{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 126: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.105{{c}}
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.747{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}


Minimax tuning:  
Minimax tuning:  
Line 142: Line 121:
{{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 ====
==== Porcupinefowl ====
Line 151: Line 130:
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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.442{{c}}
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.708{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}


Minimax tuning:  
Minimax tuning:  
Line 166: Line 145:
* 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 177: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 162.636{{c}}
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.277{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}


Minimax tuning:  
Minimax tuning:  
Line 188: Line 167:


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 201: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.378{{c}}
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}


Minimax tuning:  
Minimax tuning:  
Line 213: Line 192:
{{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 ====
Line 220: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.678{{c}}
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 163.688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}


Minimax tuning:  
Minimax tuning:  
Line 232: Line 211:
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}


Badness (Smith): 0.026043
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=0| 7, 30, 37, 81bd }}
 
Badness (Sintel): 1.68


== Opossum ==
== Opossum ==
Line 246: Line 243:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 161.306{{c}}
* [[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{{c}}, ~10/9 = 159.691{{c}}
* [[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]]:  
Line 256: Line 253:
{{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 263: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.365{{c}}
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 159.807{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}


Minimax tuning:  
Minimax tuning:  
Line 274: Line 271:
{{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 281: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.631{{c}}
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


Minimax tuning:  
Minimax tuning:  
Line 292: Line 289:
{{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 ==
Line 304: Line 301:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.391{{c}}
* [[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{{c}}, ~10/9 = 164.412{{c}}
* [[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]]:  
Line 313: Line 310:
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) 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 322: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.321{{c}}
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}


Minimax tuning:  
Minimax tuning:  
Line 334: Line 331:
{{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 341: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.478{{c}}
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.953{{c}}
* 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 364: Line 361:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.094{{c}}
* [[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{{c}}, ~10/9 = 166.041{{c}}
* [[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]]:  
Line 373: Line 370:
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) 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 382: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.925{{c}}
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.981{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


Minimax tuning:  
Minimax tuning:  
Line 394: Line 391:
{{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 401: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 166.046{{c}}
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


Minimax tuning:  
Minimax tuning:  
Line 413: Line 410:
{{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 422:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 165.185{{c}}
* [[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{{c}}, ~10/9 = 158.868{{c}}
* [[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]]:  
Line 436: Line 433:
{{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 443: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.768{{c}}
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.750{{c}}
* 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
 
== Hedgehog ==
{{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)). Its ploidacot is diploid omega-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
 
[[Comma list]]: 50/49, 245/243
 
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
 
: mapping generators: ~7/5, ~9/7
 
[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 600.000{{c}}, ~9/7 = 435.258{{c}}
: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
* [[POTE]]: ~7/5 = 600.000{{c}}, ~9/7 = 435.648{{c}}
: error map: {{val| 0.000 +4.989 -8.074 +9.414 }}
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
 
[[Badness]] (Smith): 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 tunings:
* CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.528{{c}}
* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.386{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
 
Badness (Smith): 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 tunings:
* CTE: ~7/5 = 600.000{{c}}, ~9/7 = 436.309{{c}}
* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.861{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
 
Badness (Smith): 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 tunings:
* CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.186{{c}}
* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 437.078{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 22f }}
 
Badness (Smith): 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 tunings:
* CTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.329{{c}}
* POTE: ~7/5 = 600.000{{c}}, ~9/7 = 435.425{{c}}
 
{{Optimal ET sequence|legend=0| 22 }}
 
Badness (Smith): 0.068406
 
; Music
* [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 ==
Line 553: Line 462:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 81.914{{c}}
* [[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{{c}}, ~21/20 = 82.505{{c}}
* [[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 567: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.802{{c}}
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.504{{c}}
* 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 582: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.912{{c}}
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.530{{c}}
* 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 597: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/20 = 82.034{{c}}
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 81.759{{c}}
* 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
Line 611: Line 520:


== 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.  
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.  


Line 617: Line 527:
[[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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~9/7 = 454.550{{c}}
* [[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{{c}}, ~9/7 = 454.448{{c}}
* [[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 636: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~9/7 = 454.505{{c}}
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 454.512{{c}}
* 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 651: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~13/10 = 454.480{{c}}
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* POTE: ~2 = 1200.000{{c}}, ~13/10 = 454.529{{c}}
* 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 ==
Line 673: Line 583:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.804{{c}}
* [[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{{c}}, ~36/35 = 54.384{{c}}
* [[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 687: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.702{{c}}
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.376{{c}}
* 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 702: 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 tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.575{{c}}
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.665{{c}}
* 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:Porcupine family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Porcupine family| ]] <!-- main article -->
[[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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