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Sensipent family: Difference between revisions

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+ploidacots to accompany all the pergens
TallKite (talk | contribs)
autopatrolled, Sysops
3,990 edits
Changed 1\1 to 1200.000. Good point about significant digits. But optimal tunings routinely use slightly stretched/compressed octaves. "1200.000" unlike "1\1" makes it clear that the octave is exactly just. Granted, the reader could click on the link for POTE or CTE, read through several paragraphs, and find out that the octave is just, but why not communicate that fact directly? Likewise, 1\2 and 1\3 are changed to cents. (P.S. thanks for correcting mincents to mingen and other helpful changes)
Line 26: Line 26:
: mapping generators: ~2, ~162/125
: mapping generators: ~2, ~162/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.058
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 47: Line 47:
{{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, ~31/24 = 443.050
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050


Badness (Dirichlet): 0.243
Badness (Dirichlet): 0.243
Line 63: Line 63:
{{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 = 1200.000, ~31/24 = 442.989


Badness (Dirichlet): 0.283
Badness (Dirichlet): 0.283
Line 80: Line 80:
{{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 = 1200.000, ~128/99 = 443.115


Badness (Dirichlet): 0.728
Badness (Dirichlet): 0.728
Line 97: Line 97:
{{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 = 1200.000, ~22/17 = 443.188


Badness (Dirichlet): 0.639
Badness (Dirichlet): 0.639
Line 110: Line 110:
{{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 = 443.185
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185


Badness (Dirichlet): 0.555
Badness (Dirichlet): 0.555
Line 125: Line 125:
{{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 = 443.183
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.183


Badness (Dirichlet): 0.490
Badness (Dirichlet): 0.490
Line 146: Line 146:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 176: Line 176:
: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Line 192: Line 192:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2987
* CTE: ~2 = 1200.000, ~9/7 = 443.2987
* POTE: ~2 = 1\1, ~9/7 = 443.294
* POTE: ~2 = 1200.000, ~9/7 = 443.294


{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
Line 209: Line 209:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3658
* CTE: ~2 = 1200.000, ~9/7 = 443.3658
* POTE: ~2 = 1\1, ~9/7 = 443.321
* POTE: ~2 = 1200.000, ~9/7 = 443.321


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Line 224: Line 224:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3775
* CTE: ~2 = 1200.000, ~9/7 = 443.3775
* POTE: ~2 = 1\1, ~9/7 = 443.365
* POTE: ~2 = 1200.000, ~9/7 = 443.365


{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
Line 243: Line 243:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.4783
* CTE: ~2 = 1200.000, ~9/7 = 443.4783
* POTE: ~2 = 1\1, ~9/7 = 443.626
* POTE: ~2 = 1200.000, ~9/7 = 443.626


{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
Line 260: Line 260:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5075
* CTE: ~2 = 1200.000, ~9/7 = 443.5075
* POTE: ~2 = 1\1, ~9/7 = 443.559
* POTE: ~2 = 1200.000, ~9/7 = 443.559


{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
Line 275: Line 275:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5050
* CTE: ~2 = 1200.000, ~9/7 = 443.5050
* POTE: ~2 = 1\1, ~9/7 = 443.551
* POTE: ~2 = 1200.000, ~9/7 = 443.551


{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
Line 294: Line 294:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.1886
* CTE: ~2 = 1200.000, ~9/7 = 443.1886
* POTE: ~2 = 1\1, ~9/7 = 443.962
* POTE: ~2 = 1200.000, ~9/7 = 443.962


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
Line 311: Line 311:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2863
* CTE: ~2 = 1200.000, ~9/7 = 443.2863
* POTE: ~2 = 1\1, ~9/7 = 443.945
* POTE: ~2 = 1200.000, ~9/7 = 443.945


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
Line 328: Line 328:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7814
* CTE: ~2 = 1200.000, ~9/7 = 443.7814
* POTE: ~2 = 1\1, ~9/7 = 443.518
* POTE: ~2 = 1200.000, ~9/7 = 443.518


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
Line 343: Line 343:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7877
* CTE: ~2 = 1200.000, ~9/7 = 443.7877
* POTE: ~2 = 1\1, ~9/7 = 443.506
* POTE: ~2 = 1200.000, ~9/7 = 443.506


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
Line 364: Line 364:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~9/7 = 443.3688 (~11/10 = 156.6312)
* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
* POTE: ~99/70 = 1\2, ~9/7 = 443.308 (~11/10 = 156.692)
* POTE: ~99/70 = 600.000, ~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 383: Line 383:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~9/7 = 443.4416, ~11/10 = 156.5584
* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
* POTE: ~55/39 = 1\2, ~9/7 = 443.275, ~11/10 = 156.725
* POTE: ~55/39 = 600.000, ~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 402: Line 402:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~9/7 = 443.4466 (~11/10 = 156.5534)
* CTE: ~17/12 = 600.000, ~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 420: Line 420:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.5981
* CTE: ~2 = 1200.000, ~25/22 = 221.5981
* POTE: ~2 = 1\1, ~25/22 = 221.605
* POTE: ~2 = 1200.000, ~25/22 = 221.605


{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
Line 435: Line 435:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.6333
* CTE: ~2 = 1200.000, ~25/22 = 221.6333
* POTE: ~2 = 1\1, ~25/22 = 221.556
* POTE: ~2 = 1200.000, ~25/22 = 221.556


{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
Line 453: Line 453:
{{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 = 1200.000, ~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 470: Line 470:
{{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 = 1200.000, ~162/125 = 443.289


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
Line 485: Line 485:
: mapping generators: ~2, ~128/99
: mapping generators: ~2, ~128/99


Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
Optimal tuning (POTE): ~2 = 1200.000, ~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 500: Line 500:
: mapping generators: ~2, ~84/65
: mapping generators: ~2, ~84/65


Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
Optimal tuning (POTE): ~2 = 1200.000, ~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 515: Line 515:
: mapping generators: ~2, ~22/17
: mapping generators: ~2, ~22/17


Optimal tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
Optimal tuning (POTE): ~2 = 1200.000, ~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 536: Line 536:
{{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, ~162/125 = 443.075 (~35/32 = 156.925)
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~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 553: Line 553:
:: mapping generators: ~567/400, ~35/32
:: mapping generators: ~567/400, ~35/32


Optimal tuning (POTE): ~99/70 = 1\2, ~162/125 = 443.117 (~35/32 = 156.883)
Optimal tuning (POTE): ~99/70 = 600.000, ~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 570: Line 570:
:: mapping generators: ~55/39, ~35/32
:: mapping generators: ~55/39, ~35/32


Optimal tuning (POTE): ~55/39 = 1\2, ~162/125 = 443.096 (~35/32 = 156.904)
Optimal tuning (POTE): ~55/39 = 600.000, ~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 589: Line 589:
{{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 = 1200.000, ~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 602: Line 602:
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 = 1200.000, ~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 615: Line 615:
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 = 1200.000, ~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 634: Line 634:
{{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 = 1200.000, ~48/35 = 546.815


{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
Line 649: Line 649:
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
Optimal tuning (POTE): ~2 = 1200.000, ~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 662: Line 662:
{{Mapping|legend=1| 1 -8 -10 6 3 11 | 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 = 1200.000, ~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 675: Line 675:
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 | 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 = 1200.000, ~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 688: Line 688:
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 12 | 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 = 1200.000, ~11/8 = 547.614


{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
Line 707: Line 707:
{{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 = 400.000, ~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 720: Line 720:
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 = 400.000, ~36/35 = 43.292


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 735: Line 735:
: mapping generators: ~49/39, ~36/35
: mapping generators: ~49/39, ~36/35


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 748: Line 748:
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 = 400.000, ~36/35 = 43.276


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Line 761: Line 761:
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 = 400.000, ~36/35 = 43.292


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
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