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Sensipent family: Difference between revisions

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added pergens. fixed some errors. added the color name. corrected the mappings to match the 1st paragraph's description of the generator as well as all the optimal tunings. deleted the warning about mappings not matching since now they match. changed "1\1" to "1200¢" because "one edostep of 1edo" is a very unnatural, obtuse description of an octave, and also inaccurate because 1edo can be stretched or compressed, but these tunings require exactly 1200¢. changed 1\2 and 1\3 for the same reasons.
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Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125, seven make harmonic 6 and nine make harmonic 10.  
Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its [[pergen]] is (P8, ccP5/7) and its color name is Sepguti.  


The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These all use the same nominal generator as sensipent.  
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.  


Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  
Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  
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Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
'''IMPORTANT:''' Please be careful, when reading the mappings, to observe the orientation of the generator; most if not all mappings here use the generator that is closer to the octave than the unison, but show its tuning in terms of the generator that is closer to the unison than the octave.


== Sensipent ==
== Sensipent ==
Line 24: Line 22:
[[Comma list]]: 78732/78125
[[Comma list]]: 78732/78125


{{Mapping|legend=1| 1 6 8 | 0 -7 -9 }}
{{Mapping|legend=1| 1 -1 -1| 0 7 9 }}


: mapping generators: ~2, ~125/81
: mapping generators: ~2, ~162/125


[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~162/125 = 443.058
[[Optimal tuning]] ([[POTE]]): 2 = 1200¢, ~162/125 = 443.058


{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
Line 43: Line 41:
[[Comma list]]: 78732/78125, 961/960
[[Comma list]]: 78732/78125, 961/960


{{Mapping|legend=1| 1 6 8 10 | 0 -7 -9 -8}}
{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
 
: mapping generators: ~2, ~31/24


{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/28 = 443.050
[[Optimal tuning]] ([[CTE]]): 2 = 1200¢, ~31/24 = 443.050


Badness (Dirichlet): 0.243
Badness (Dirichlet): 0.243
Line 59: Line 59:
[[Comma list]]: 465/464, 576/575, 621/620, 729/725  
[[Comma list]]: 465/464, 576/575, 621/620, 729/725  


{{Mapping|legend=1| 1 6 8 2 20 10 | 0 -7 -9 4 -24 -8 }}
{{Mapping|legend=1| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}


{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 442.989
[[Optimal tuning]] ([[CTE]]): 2 = 1200¢, ~31/24 = 442.989


Badness (Dirichlet): 0.283
Badness (Dirichlet): 0.283
Line 74: Line 74:
[[Comma list]]: [[8019/8000]], [[16384/16335]]
[[Comma list]]: [[8019/8000]], [[16384/16335]]


{{Mapping|legend=1| 1 6 8 -6 | 0 -7 -9 15 }}
{{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
 
: mapping generators: ~2, ~128/99


{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~128/99 = 443.115
[[Optimal tuning]] ([[CTE]]): 2 = 1200¢, ~128/99 = 443.115


Badness (Dirichlet): 0.728
Badness (Dirichlet): 0.728
Line 89: Line 91:
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]


{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
: mapping generators: ~2, ~22/17


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~22/17 = 443.188
[[Optimal tuning]] ([[CTE]]): 2 = 1200¢, ~22/17 = 443.188


Badness (Dirichlet): 0.639
Badness (Dirichlet): 0.639
Line 102: Line 106:
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]]
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]]


{{Mapping|legend=1| 1 6 8 -6 -6 2 | 0 -7 -9 15 16 4 }}
{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.185
[[Optimal tuning]] ([[CTE]]): 2 = 1200¢, ~22/17 = 443.185


Badness (Dirichlet): 0.555
Badness (Dirichlet): 0.555
Line 117: Line 121:
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]], [[961/960]]
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]], [[961/960]]


{{Mapping|legend=1| 1 6 8 -6 -6 2 10 | 0 -7 -9 15 16 4 -8 }}
{{Mapping|legend=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.183
[[Optimal tuning]] ([[CTE]]): 2 = 1200¢, ~22/17 = 443.183


Badness (Dirichlet): 0.490
Badness (Dirichlet): 0.490
Line 135: Line 139:
[[Comma list]]: 126/125, 245/243
[[Comma list]]: 126/125, 245/243


{{Mapping|legend=1| 1 6 8 11 | 0 -7 -9 -13 }}
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}


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


{{Multival|legend=1| 7 9 13 -2 1 5 }}
{{Multival|legend=1| 7 9 13 -2 1 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
* [[CTE]]: ~2 = 1200¢, ~9/7 = 443.3166
* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
* [[POTE]]: ~2 = 1200¢, ~9/7 = 443.383


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 168: Line 172:
Comma list: 91/90, 126/125, 169/168
Comma list: 91/90, 126/125, 169/168


Sval mapping: {{mapping| 1 6 8 11 10 | 0 -7 -9 -13 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
 
Gencom mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 }}


: gencom: [2 14/9; 91/90 126/125 169/168]
: mapping generators: ~2, ~9/7


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
Optimal tuning (CTE): ~2 = 1200¢, ~9/7 = 443.4016


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Line 183: Line 185:
Comma list: 126/125, 245/243, 385/384
Comma list: 126/125, 245/243, 385/384


Mapping: {{mapping| 1 6 8 11 -6 | 0 -7 -9 -13 15 }}
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
 
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}
{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2987
* CTE: ~2 = 1200¢, ~9/7 = 443.2987
* POTE: ~2 = 1\1, ~9/7 = 443.294
* POTE: ~2 = 1200¢, ~9/7 = 443.294


{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
Line 200: Line 204:
Comma list: 91/90, 126/125, 169/168, 385/384
Comma list: 91/90, 126/125, 169/168, 385/384


Mapping: {{mapping| 1 6 8 11 -6 10 | 0 -7 -9 -13 15 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}


{{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}
{{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3658
* CTE: ~2 = 1200¢, ~9/7 = 443.3658
* POTE: ~2 = 1\1, ~9/7 = 443.321
* POTE: ~2 = 1200¢, ~9/7 = 443.321


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Line 217: Line 221:
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255


Mapping: {{mapping| 1 6 8 11 -6 10 -6 | 0 -7 -9 -13 15 -10 16 }}
Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3775
* CTE: ~2 = 1200¢, ~9/7 = 443.3775
* POTE: ~2 = 1\1, ~9/7 = 443.365
* POTE: ~2 = 1200¢, ~9/7 = 443.365


{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
Line 232: Line 236:
Comma list: 126/125, 176/175, 245/243
Comma list: 126/125, 176/175, 245/243


Mapping: {{mapping| 1 6 8 11 23 | 0 -7 -9 -13 -31 }}
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
 
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}
{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.4783
* CTE: ~2 = 1200¢, ~9/7 = 443.4783
* POTE: ~2 = 1\1, ~9/7 = 443.626
* POTE: ~2 = 1200¢, ~9/7 = 443.626


{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
Line 249: Line 255:
Comma list: 91/90, 126/125, 169/168, 352/351
Comma list: 91/90, 126/125, 169/168, 352/351


Mapping: {{mapping| 1 6 8 11 23 10 | 0 -7 -9 -13 -31 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}


{{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}
{{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5075
* CTE: ~2 = 1200¢, ~9/7 = 443.5075
* POTE: ~2 = 1\1, ~9/7 = 443.559
* POTE: ~2 = 1200¢, ~9/7 = 443.559


{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
Line 266: Line 272:
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


Mapping: {{mapping| 1 6 8 11 23 10 23 | 0 -7 -9 -13 -31 -10 -30 }}
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5050
* CTE: ~2 = 1200¢, ~9/7 = 443.5050
* POTE: ~2 = 1\1, ~9/7 = 443.551
* POTE: ~2 = 1200¢, ~9/7 = 443.551


{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
Line 281: Line 287:
Comma list: 56/55, 100/99, 245/243
Comma list: 56/55, 100/99, 245/243


Mapping: {{mapping| 1 6 8 11 6 | 0 -7 -9 -13 -4 }}
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
 
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}
{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.1886
* CTE: ~2 = 1200¢, ~9/7 = 443.1886
* POTE: ~2 = 1\1, ~9/7 = 443.962
* POTE: ~2 = 1200¢, ~9/7 = 443.962


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
Line 298: Line 306:
Comma list: 56/55, 78/77, 91/90, 100/99
Comma list: 56/55, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 6 8 11 6 10 | 0 -7 -9 -13 -4 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}


{{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}
{{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2863
* CTE: ~2 = 1200¢, ~9/7 = 443.2863
* POTE: ~2 = 1\1, ~9/7 = 443.945
* POTE: ~2 = 1200¢, ~9/7 = 443.945


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
Line 315: Line 323:
Comma list: 55/54, 77/75, 99/98
Comma list: 55/54, 77/75, 99/98


Mapping: {{mapping| 1 6 8 11 11 | 0 -7 -9 -13 -12 }}
Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
 
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7814
* CTE: ~2 = 1200¢, ~9/7 = 443.7814
* POTE: ~2 = 1\1, ~9/7 = 443.518
* POTE: ~2 = 1200¢, ~9/7 = 443.518


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
Line 330: Line 340:
Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 55/54, 66/65, 77/75, 143/140


Mapping: {{mapping| 1 6 8 11 11 11 | 0 -7 -9 -13 -12 -11 }}
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7877
* CTE: ~2 = 1200¢, ~9/7 = 443.7877
* POTE: ~2 = 1\1, ~9/7 = 443.506
* POTE: ~2 = 1200¢, ~9/7 = 443.506


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
Line 340: Line 350:
Badness: 0.023258
Badness: 0.023258


=== Bisensi ===
=== Bisensi (P8/2, ccP5/7) ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 126/125, 245/243
Comma list: 121/120, 126/125, 245/243


Mapping: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
Pergen mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8}}


: mapping generators: ~99/70, ~9/7
Mincents mapping: [⟨2 5 7 9 9], ⟨0 -7 -9 -13 -8]]
: mapping generators: ~99/70, ~11/10
: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~11/10 = 156.6312
* CTE: ~99/70 = 600¢, ~9/7 = 443.3688, ~11/10 = 156.6312
* POTE: ~99/70 = 1\2, ~11/10 = 156.692
* POTE: ~99/70 = 600¢, ~9/7 = 443.308, ~11/10 = 156.692


{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
Line 362: Line 374:
Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 121/120, 126/125, 169/168


Mapping: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
Pergen mapping: {{mapping| 2 -2 -2 -4 1 0| 0 7 9 13 8 10}}
: mapping generators: ~99/70, ~9/7
 
Mincents mapping: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
 
: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~11/10 = 156.5584
* CTE: ~55/39 = 600¢, ~9/7 = 443.4416, ~11/10 = 156.5584
* POTE: ~55/39 = 1\2, ~11/10 = 156.725
* POTE: ~55/39 = 600¢, ~9/7 = 443.275, ~11/10 = 156.725


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
Line 377: Line 394:
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168


Mapping: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
Pergen mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
 
: mapping generators: ~99/70, ~9/7
 
Mincents mapping:  {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
 
: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~11/10 = 156.5534
* CTE: ~17/12 = 600¢, ~9/7 = 443.4466, ~11/10 = 156.5534


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
Line 386: Line 409:
Badness: 0.0188
Badness: 0.0188


=== Hemisensi ===
=== Hemisensi (P8, ccP5/14) ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 243/242, 245/242
Comma list: 126/125, 243/242, 245/242


Mapping: {{mapping| 1 13 17 24 32 | 0 -14 -18 -26 -35 }}
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}


: mapping generators: ~2, ~44/25
: mapping generators: ~2, ~25/22


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.5981
* CTE: ~2 = 1200¢, ~25/22 = 221.5981
* POTE: ~2 = 1\1, ~25/22 = 221.605
* POTE: ~2 = 1200¢, ~25/22 = 221.605


{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
Line 408: Line 431:
Comma list: 91/90, 126/125, 169/168, 243/242
Comma list: 91/90, 126/125, 169/168, 243/242


Mapping: {{mapping| 1 13 17 24 32 30 | 0 -14 -18 -26 -35 -30 }}
Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.6333
* CTE: ~2 = 1200¢, ~25/22 = 221.6333
* POTE: ~2 = 1\1, ~25/22 = 221.556
* POTE: ~2 = 1200¢, ~25/22 = 221.556


{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
Line 423: Line 446:
[[Comma list]]: 225/224, 78732/78125
[[Comma list]]: 225/224, 78732/78125


{{Mapping|legend=1| 1 6 8 23 | 0 -7 -9 -32 }}
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
 
: mapping generators: ~2, ~162/125


{{Multival|legend=1| 7 9 32 -2 31 49 }}
{{Multival|legend=1| 7 9 32 -2 31 49 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
[[Optimal tuning]] ([[POTE]]): ~2 = 1200¢, ~162/125 = 442.755


{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
Line 438: Line 463:
[[Comma list]]: 5120/5103, 78732/78125
[[Comma list]]: 5120/5103, 78732/78125


{{Mapping|legend=1| 1 6 8 -18 | 0 -7 -9 33 }}
{{Mapping|legend=1| 1 -1 -1 15| 0 7 9 -33 }}
 
: mapping generators: ~2, ~162/125


{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
[[Optimal tuning]] ([[POTE]]): ~2 = 1200¢, ~162/125 = 443.289


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
Line 453: Line 480:
Comma list: 176/175, 1331/1323, 5120/5103
Comma list: 176/175, 1331/1323, 5120/5103


Mapping: {{mapping| 1 6 8 -18 -6 | 0 -7 -9 33 15 }}
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15}}
 
: mapping generators: ~2, ~128/99


Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
Optimal tuning (POTE): ~2 = 1200¢, ~128/99 = 443.274


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
Line 466: Line 495:
Comma list: 176/175, 351/350, 847/845, 1331/1323
Comma list: 176/175, 351/350, 847/845, 1331/1323


Mapping: {{mapping| 1 6 8 -18 -6 -19 | 0 -7 -9 33 15 36 }}
Mapping: {{mapping| 1 -1 -1 15 9 17| 0 7 9 -33 -15 -36}}
: mapping generators: ~2, ~84/65


Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
Optimal tuning (POTE): ~2 = 1200¢, ~84/65 = 443.270


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
Line 479: Line 509:
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714


Mapping: {{mapping| 1 6 8 -18 -6 -19 -6 | 0 -7 -9 33 15 36 16 }}
Mapping: {{mapping| 1 -1 -1 15 9 17 10| 0 7 9 -33 -15 -36 -16}}
: mapping generators: ~2, ~22/17


Optimal tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
Optimal tuning (POTE): ~2 = 1200¢, ~22/17 = 443.270


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
Line 487: Line 518:
Badness: 0.018105
Badness: 0.018105


== Bison ==
== Bison (P8/2, ccP5/7) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 78732/78125
[[Comma list]]: 6144/6125, 78732/78125


{{Mapping|legend=1| 2 5 7 3 | 0 -7 -9 10 }}
Pergen {{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10}}
 
: mapping generators: ~567/400, ~162/125
 
Mincents {{Mapping|legend=1| 2 5 7 3 | 0 -7 -9 10 }}


: mapping generators: ~567/400, ~35/32
: mapping generators: ~567/400, ~35/32
Line 498: Line 533:
{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}


[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~35/32 = 156.925
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600¢, ~162/125 = 443.075, ~35/32 = 156.925


{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
Line 511: Line 546:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
Pergen {{Mapping|legend=1| 2 -2 -2 13 18| 0 7 9 -10 -15}}
: mapping generators: ~567/400, ~162/125
 
Mincents mapping: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/32 = 156.883
: mapping generators: ~567/400, ~35/32
 
Optimal tuning (POTE): ~99/70 = 600¢, ~162/125 = 443.117, ~35/32 = 156.883


{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
Line 524: Line 564:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


Mapping: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
Pergen {{Mapping|legend=1| 2 -2 -2 13 18 17| 0 7 9 -10 -15 -13}}
: mapping generators: ~55/39, ~162/125
 
Mincents mapping: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
 
: mapping generators: ~55/39, ~35/32


Optimal tuning (POTE): ~55/39 = 1\2, ~35/32 = 156.904
Optimal tuning (POTE): ~55/39 = 600¢, ~162/125 = 443.096, ~35/32 = 156.904


{{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
Line 532: Line 577:
Badness: 0.023504
Badness: 0.023504


== Subpental ==
== Subpental (P8, c<sup>4</sup>P4/14) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 538: Line 583:


{{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}
{{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}
: mapping generators: ~2, ~56/45


{{Multival|legend=1| 14 18 45 -4 32 54 }}
{{Multival|legend=1| 14 18 45 -4 32 54 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
[[Optimal tuning]] ([[POTE]]): ~2 = 1200¢, ~56/45 = 378.467


{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
Line 554: Line 601:
Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}
Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
Optimal tuning (POTE): ~2 = 1200¢, ~56/45 = 378.440


{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
Line 567: Line 614:
Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}
Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
Optimal tuning (POTE): ~2 = 1200¢, ~56/45 = 378.437


{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
Line 573: Line 620:
Badness: 0.023940
Badness: 0.023940


== Heinz ==
== Heinz (P8, c<sup>9</sup>P5/21) ==
A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (= 103 - 46).
A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (= 103 - 46).


Line 580: Line 627:
[[Comma list]]: 1029/1024, 78732/78125
[[Comma list]]: 1029/1024, 78732/78125


{{Mapping|legend=1| 1 13 17 -1 | 0 -21 -27 7 }}
{{Mapping|legend=1| 1 -8 -10 6| 0 21 27 -7 }}


: mapping generators: ~2, ~35/24
: mapping generators: ~2, ~48/35


{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815
[[Optimal tuning]] ([[POTE]]): ~2 = 1200¢, ~48/35 = 546.815


{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
Line 597: Line 644:
Comma list: 385/384, 441/440, 78732/78125
Comma list: 385/384, 441/440, 78732/78125


Mapping: {{mapping| 1 13 17 -1 4 | 0 -21 -27 7 -1 }}
{{Mapping|legend=1| 1 -8 -10 6 3| 0 21 27 -7 1}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
: mapping generators: ~2, ~11/8
 
Optimal tuning (POTE): ~2 = 1200¢, ~11/8 = 547.631


{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
Line 610: Line 659:
Comma list: 351/350, 385/384, 441/440, 847/845
Comma list: 351/350, 385/384, 441/440, 847/845


Mapping: {{mapping| 1 13 17 -1 4 -5 | 0 -21 -27 7 -1 16 }}
{{Mapping|legend=1| 1 -8 -10 6 3 11| 0 21 27 -7 1 -16}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.629
Optimal tuning (POTE): ~2 = 1200¢, ~11/8 = 547.629


{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
Line 623: Line 672:
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845


Mapping: {{mapping| 1 13 17 -1 4 -5 3 | 0 -21 -27 7 -1 16 2 }}
{{Mapping|legend=1| 1 -8 -10 6 3 11 5| 0 21 27 -7 1 -16 -2}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.635
Optimal tuning (POTE): ~2 = 1200¢, ~11/8 = 547.635


{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
Line 636: Line 685:
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968


Mapping: {{mapping| 1 13 17 -1 4 -5 3 -5 | 0 -21 -27 7 -1 16 2 17 }}
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 12| 0 21 27 -7 1 -16 -2 -17}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.614
Optimal tuning (POTE): ~2 = 1200¢, ~11/8 = 547.614


{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
Line 644: Line 693:
Badness: 0.019005
Badness: 0.019005


== Trisensory ==
== Trisensory (P8/3, M6/21) ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 650: Line 699:


{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
: mapping generators: ~63/50, ~36/35


{{Multival|legend=1| 21 27 12 -6 -40 -48 }}
{{Multival|legend=1| 21 27 12 -6 -40 -48 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400¢, ~36/35 = 43.147


{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
Line 666: Line 717:
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.292
Optimal tuning (POTE): ~63/50 = 400¢, ~36/35 = 43.292


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 679: Line 730:
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
: mapping generators: ~49/39, ~36/35
 
Optimal tuning (POTE): ~49/39 = 400¢, ~36/35 = 43.288


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 692: Line 745:
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.276
Optimal tuning (POTE): ~49/39 = 400¢, ~36/35 = 43.276


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Line 705: Line 758:
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.292
Optimal tuning (POTE): ~49/39 = 400¢, ~36/35 = 43.292


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Retrieved from "https://en.xen.wiki/w/Sensipent_family"