Xenharmonic Wiki

Porcupine family: Difference between revisions

← Older edit
VisualWikitext
Switch to Sintel's badness, WE & CWE tunings, per community consensus
m Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "
 
(15 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{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


Line 26: Line 19:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
: error map: {{val| 0.000 +5.859 -6.624 }}
: error map: {{val| 0.000 +5.859 -6.624 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.166{{c}}
: [[error map]]: {{val| 0.000 +5.547 -7.143 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.950{{c}}
: error map: {{val| 0.000 +6.194 -6.065 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 50: 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 64: 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:  
* WE: ~2 = 1200.3290{{c}}, ~10/9 = 164.1227{{c}}
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 163.9951{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.887{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.078{{c}} -->


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


Badness (Sintel): 0.303
Badness (Sintel): 0.303
==== Undecimation ====
Subgroup: 2.3.5.11.13
Comma list: 55/54, 100/99, 512/507
Subgroup-val mapping: {{mapping| 1 -1 -2 0 5 | 0 6 10 8 -3 }}
: mapping generators: ~2, ~88/65
Optimal tunings:
* WE: ~2 = 1199.4791{{c}}, ~88/65 = 517.9845{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/65 = 518.1740{{c}}
<!-- * 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 (Sintel): 1.21


== Septimal porcupine ==
== Septimal porcupine ==
Line 113: Line 84:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}
: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 163.203{{c}}
: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 162.880{{c}}
: error map: {{val| 0.000 +9.405 -0.714 +8.455 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 138: 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:  
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.105{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.747{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 165: 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:  
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.442{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.708{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 193: 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:  
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 162.636{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.277{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 206: 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 }}
Line 219: 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:  
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.378{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 162.482{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 240: 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:  
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 163.678{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 163.688{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 255: Line 212:


Badness (Sintel): 1.08
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 272: Line 247:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}
: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 161.306{{c}}
: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 159.691{{c}}
: error map: {{val| 0.000 +18.971 +15.229 -6.048 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 289: 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:  
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.365{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 159.807{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 309: 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:  
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 161.631{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.805{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 338: Line 305:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}
: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.391{{c}}
: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 164.412{{c}}
: error map: {{val| 0.000 +4.809 -8.375 +0.580 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 356: 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:  
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.321{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.552{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 377: 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:  
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.478{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 164.953{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
Line 406: Line 365:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}
: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.094{{c}}
: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.041{{c}}
: error map: {{val| 0.000 -0.077 -16.516 -10.299 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 424: 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:  
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.925{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.981{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 445: 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:  
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 166.046{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 165.974{{c}} -->


Minimax tuning:  
Minimax tuning:  
Line 475: Line 426:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}
: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~10/9 = 165.185{{c}}
: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~10/9 = 158.868{{c}}
: error map: {{val| 0.000 +21.442 +19.348 +72.306 }} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 493: 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:  
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 164.768{{c}}
* POTE: ~2 = 1200.000{{c}}, ~11/10 = 158.750{{c}} -->


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


Badness (Sintel): 0.886
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:
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}
<!-- * [[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]] (Sintel): 1.11
=== 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:
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}
<!-- * 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 (Sintel): 0.764
==== 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:
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}
<!-- * 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 (Sintel): 0.889
==== 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:
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}
<!-- * 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 (Sintel): 1.04
=== 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:
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
<!-- * 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 (Sintel): 2.26
; 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 621: Line 466:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}
: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/20 = 81.914{{c}}
: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/20 = 82.505{{c}}
: error map: {{val| 0.000 +3.012 -11.368 -16.342 }} -->


{{Optimal ET sequence|legend=1| 14c, 15, 29 }}
{{Optimal ET sequence|legend=1| 14c, 15, 29 }}
Line 635: 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:  
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.802{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.504{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 15, 29 }}
{{Optimal ET sequence|legend=0| 14c, 15, 29 }}
Line 652: 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:  
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 81.912{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 82.530{{c}} -->


{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}
{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}
Line 669: 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:  
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/20 = 82.034{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/20 = 81.759{{c}} -->


{{Optimal ET sequence|legend=0| 14c, 15 }}
{{Optimal ET sequence|legend=0| 14c, 15 }}
Line 685: 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 700: Line 536:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}
: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~14/9 = 745.450{{c}}
: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~14/9 = 745.552{{c}}
: error map: {{val| 0.000 +8.009 -3.040 -3.346 }} -->


{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
Line 714: Line 546:
Comma list: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675


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


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~14/9 = 745.495{{c}}
* POTE: ~2 = 1200.000{{c}}, ~14/9 = 745.488{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
Line 731: 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 -4 -7 -9 -4 -5 | 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:  
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~14/9 = 745.520{{c}}
* POTE: ~2 = 1200.000{{c}}, ~14/9 = 745.471{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
Line 759: Line 587:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}
: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.804{{c}}
: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~36/35 = 54.384{{c}}
: error map: {{val| 0.000 +8.585 -2.081 +13.636 }} -->


{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
Line 773: 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:  
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.702{{c}}
* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.376{{c}} -->


{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
Line 790: 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:  
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~36/35 = 54.575{{c}}
* POTE: ~2 = 1200.000{{c}}, ~36/35 = 54.665{{c}} -->


{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
Line 802: Line 622:
Badness (Sintel): 1.85
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]]
Retrieved from "https://en.xen.wiki/w/Porcupine_family"