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Starling temperaments: Difference between revisions

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autopatrolled, emailconfirmed
5,160 edits
"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 49: Line 49:
: [[Eigenmonzo]]s (unchanged-intervals): 2, 3
: [[Eigenmonzo]]s (unchanged-intervals): 2, 3


{{Val list|legend=1| 27, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 27, 31, 58, 89 }}


[[Badness]]: 0.027044
[[Badness]]: 0.027044
Line 62: Line 62:
POTE generator: ~6/5 = 310.144
POTE generator: ~6/5 = 310.144


Optimal GPV sequence: {{Val list| 27e, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 27e, 31, 58, 89 }}


Badness: 0.016842
Badness: 0.016842
Line 75: Line 75:
POTE generator: ~6/5 = 310.276
POTE generator: ~6/5 = 310.276


Optimal GPV sequence: {{Val list| 27e, 31, 58 }}
{{Optimal ET sequence|legend=1| 27e, 31, 58 }}


Badness: 0.017125
Badness: 0.017125
Line 88: Line 88:
POTE generator: ~6/5 = 310.381
POTE generator: ~6/5 = 310.381


Optimal GPV sequence: {{Val list| 27e, 31f, 58f }}
{{Optimal ET sequence|legend=1| 27e, 31f, 58f }}


Badness: 0.027568
Badness: 0.027568
Line 101: Line 101:
POTE generator: ~6/5 = 309.804
POTE generator: ~6/5 = 309.804


Optimal GPV sequence: {{Val list| 27eff, 31 }}
{{Optimal ET sequence|legend=1| 27eff, 31 }}


Badness: 0.029868
Badness: 0.029868
Line 114: Line 114:
POTE generator: ~6/5 = 309.737
POTE generator: ~6/5 = 309.737


Optimal GPV sequence: {{Val list| 27, 31 }}
{{Optimal ET sequence|legend=1| 27, 31 }}


Badness: 0.033434
Badness: 0.033434
Line 127: Line 127:
POTE generator: ~6/5 = 310.853
POTE generator: ~6/5 = 310.853


Optimal GPV sequence: {{Val list| 4, 23bc, 27e }}
{{Optimal ET sequence|legend=1| 4, 23bc, 27e }}


Badness: 0.048687
Badness: 0.048687
Line 146: Line 146:
[[POTE generator]]: ~25/24 = 78.039
[[POTE generator]]: ~25/24 = 78.039


{{Val list|legend=1| 15, 31, 46, 77, 123 }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}


[[Badness]]: 0.122765
[[Badness]]: 0.122765
Line 171: Line 171:
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  
[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.  


{{Val list|legend=1| 15, 31, 46, 77, 185, 262cd }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185, 262cd }}


[[Badness]]: 0.031056
[[Badness]]: 0.031056
Line 193: Line 193:
Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.


Optimal GPV sequence: {{Val list| 15, 31, 46, 77, 262cdee, 339cdeee }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}


Badness: 0.016687
Badness: 0.016687
Line 206: Line 206:
POTE generator: ~21/20 = 78.219
POTE generator: ~21/20 = 78.219


Optimal GPV sequence: {{Val list| 15, 31f, 46 }}
{{Optimal ET sequence|legend=1| 15, 31f, 46 }}


Badness: 0.023461
Badness: 0.023461
Line 219: Line 219:
POTE generator: ~21/20 = 77.709
POTE generator: ~21/20 = 77.709


Optimal GPV sequence: {{Val list| 15, 31, 77ff, 108eff, 139efff }}
{{Optimal ET sequence|legend=1| 15, 31, 77ff, 108eff, 139efff }}


Badness: 0.021328
Badness: 0.021328
Line 232: Line 232:
POTE generator: ~21/20 = 77.958
POTE generator: ~21/20 = 77.958


Optimal GPV sequence: {{Val list| 15f, 31, 46, 77 }}
{{Optimal ET sequence|legend=1| 15f, 31, 46, 77 }}


Badness: 0.020665
Badness: 0.020665
Line 245: Line 245:
POTE generator: ~21/20 = 78.003
POTE generator: ~21/20 = 78.003


Optimal GPV sequence: {{Val list| 15f, 31, 46, 77, 123e, 200ceg }}
{{Optimal ET sequence|legend=1| 15f, 31, 46, 77, 123e, 200ceg }}


Badness: 0.016768
Badness: 0.016768
Line 258: Line 258:
POTE generator: ~21/20 = 77.839
POTE generator: ~21/20 = 77.839


Optimal GPV sequence: {{Val list| 16, 30, 46, 62, 108ef }}
{{Optimal ET sequence|legend=1| 16, 30, 46, 62, 108ef }}


Badness: 0.032749
Badness: 0.032749
Line 271: Line 271:
POTE generator: ~40/39 = 39.044
POTE generator: ~40/39 = 39.044


Optimal GPV sequence: {{Val list| 30, 31, 61, 92f, 123f }}
{{Optimal ET sequence|legend=1| 30, 31, 61, 92f, 123f }}


Badness: 0.047059
Badness: 0.047059
Line 284: Line 284:
POTE generator: ~45/44 = 38.921
POTE generator: ~45/44 = 38.921


Optimal GPV sequence: {{Val list| 31, 92e, 123, 154, 185 }}
{{Optimal ET sequence|legend=1| 31, 92e, 123, 154, 185 }}


Badness: 0.061275
Badness: 0.061275
Line 297: Line 297:
POTE generator: ~45/44 = 38.948
POTE generator: ~45/44 = 38.948


Optimal GPV sequence: {{Val list| 31, 92e, 123, 154 }}
{{Optimal ET sequence|legend=1| 31, 92e, 123, 154 }}


Badness: 0.057919
Badness: 0.057919
Line 310: Line 310:
POTE generator: ~40/39 = 38.993
POTE generator: ~40/39 = 38.993


Optimal GPV sequence: {{Val list| 31, 92ef, 123f }}
{{Optimal ET sequence|legend=1| 31, 92ef, 123f }}


Badness: 0.057931
Badness: 0.057931
Line 339: Line 339:
: [[Eigenmonzo]]s (unchanged-intervals): 2, 3
: [[Eigenmonzo]]s (unchanged-intervals): 2, 3


{{Val list|legend=1| 8d, 23d, 31, 101, 132c, 163c }}
{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}


[[Badness]]: 0.050389
[[Badness]]: 0.050389
Line 361: Line 361:
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.
Algebraic generator: positive root of 15''x''<sup>2</sup> - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.


Optimal GPV sequence: {{Val list| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}
{{Optimal ET sequence|legend=1| 8d, 23de, 31, 101, 132ce, 163ce, 194cee }}


Badness: 0.025621
Badness: 0.025621
Line 374: Line 374:
POTE generator: ~11/10 = 154.478
POTE generator: ~11/10 = 154.478


Optimal GPV sequence: {{Val list| 8d, 23de, 31, 70f, 101ff }}
{{Optimal ET sequence|legend=1| 8d, 23de, 31, 70f, 101ff }}


Badness: 0.023323
Badness: 0.023323
Line 392: Line 392:
[[POTE generator]]: ~6/5 = 311.679
[[POTE generator]]: ~6/5 = 311.679


{{Val list|legend=1| 27, 50, 77 }}
{{Optimal ET sequence|legend=1| 27, 50, 77 }}


[[Badness]]: 0.073509
[[Badness]]: 0.073509
Line 405: Line 405:
POTE generator: ~6/5 = 311.587
POTE generator: ~6/5 = 311.587


Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}


Badness: 0.056915
Badness: 0.056915
Line 418: Line 418:
POTE generator: ~6/5 = 311.591
POTE generator: ~6/5 = 311.591


Optimal GPV sequence: {{Val list| 27e, 77, 104c, 181c }}
{{Optimal ET sequence|legend=1| 27e, 77, 104c, 181c }}


Badness: 0.035582
Badness: 0.035582
Line 433: Line 433:
[[POTE generator]]: ~6/5 = 312.602
[[POTE generator]]: ~6/5 = 312.602


{{Val list|legend=1| 42, 46, 96d, 142d, 238dd }}
{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}


[[Badness]]: 0.078049
[[Badness]]: 0.078049
Line 446: Line 446:
POTE generator: ~6/5 = 312.601
POTE generator: ~6/5 = 312.601


Optimal GPV sequence: {{Val list| 42, 46, 96d, 142d, 238dd }}
{{Optimal ET sequence|legend=1| 42, 46, 96d, 142d, 238dd }}


Badness: 0.044499
Badness: 0.044499
Line 459: Line 459:
POTE generator: ~6/5 = 312.564
POTE generator: ~6/5 = 312.564


Optimal GPV sequence: {{Val list| 42, 46, 96d, 238ddf }}
{{Optimal ET sequence|legend=1| 42, 46, 96d, 238ddf }}


Badness: 0.029693
Badness: 0.029693
Line 476: Line 476:
[[POTE generator]]: ~8/7 = 222.797
[[POTE generator]]: ~8/7 = 222.797


{{Val list|legend=1| 16, 27, 43, 70, 167ccdd }}
{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}


[[Badness]]: 0.087500
[[Badness]]: 0.087500
Line 489: Line 489:
POTE generator: ~8/7 = 222.898
POTE generator: ~8/7 = 222.898


Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e }}
{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e }}


Badness: 0.043336
Badness: 0.043336
Line 502: Line 502:
POTE generator: ~8/7 = 222.961
POTE generator: ~8/7 = 222.961


Optimal GPV sequence: {{Val list| 16, 27e, 43, 70e, 113cdee }}
{{Optimal ET sequence|legend=1| 16, 27e, 43, 70e, 113cdee }}


Badness: 0.028920
Badness: 0.028920
Line 519: Line 519:
[[POTE generator]]: ~175/128 = 558.605
[[POTE generator]]: ~175/128 = 558.605


{{Val list|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=1| 15, 43, 58 }}


[[Badness]]: 0.088441
[[Badness]]: 0.088441
Line 532: Line 532:
POTE generator: ~11/8 = 558.620
POTE generator: ~11/8 = 558.620


Optimal GPV sequence: {{Val list| 15, 43, 58 }}
{{Optimal ET sequence|legend=1| 15, 43, 58 }}


Badness: 0.033078
Badness: 0.033078
Line 545: Line 545:
POTE generator: ~11/8 = 558.589
POTE generator: ~11/8 = 558.589


Optimal GPV sequence: {{Val list| 15, 43, 58 }}
{{Optimal ET sequence|legend=1| 15, 43, 58 }}


Badness: 0.022838
Badness: 0.022838
Line 558: Line 558:
POTE generator: ~11/8 = 558.509
POTE generator: ~11/8 = 558.509


Optimal GPV sequence: {{Val list| 15, 43, 58 }}
{{Optimal ET sequence|legend=1| 15, 43, 58 }}


Badness: 0.022293
Badness: 0.022293
Line 571: Line 571:
POTE generator: ~11/8 = 558.504
POTE generator: ~11/8 = 558.504


Optimal GPV sequence: {{Val list| 15, 43, 58h }}
{{Optimal ET sequence|legend=1| 15, 43, 58h }}


Badness: 0.018938
Badness: 0.018938
Line 584: Line 584:
POTE generator: ~11/8 = 558.522
POTE generator: ~11/8 = 558.522


Optimal GPV sequence: {{Val list| 15, 43, 58hi }}
{{Optimal ET sequence|legend=1| 15, 43, 58hi }}


Badness: 0.016581
Badness: 0.016581
Line 599: Line 599:
POTE generator: ~11/8 = 558.520
POTE generator: ~11/8 = 558.520


Optimal GPV sequence: {{Val list| 15, 43, 58hi }}
{{Optimal ET sequence|legend=1| 15, 43, 58hi }}


Badness: 0.013762
Badness: 0.013762
Line 616: Line 616:
[[POTE generator]]: ~135/98 = 541.828
[[POTE generator]]: ~135/98 = 541.828


{{Val list|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}
{{Optimal ET sequence|legend=1| 11cd, 20cd, 31, 206bcd, 237bcd, 268bcd, 299bcd, 330bbcd }}


[[Badness]]: 0.099801
[[Badness]]: 0.099801
Line 629: Line 629:
POTE generator: ~15/11 = 541.772
POTE generator: ~15/11 = 541.772


Optimal GPV sequence: {{Val list| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}
{{Optimal ET sequence|legend=1| 11cdee, 20cde, 31, 144cd, 175cd, 206bcde, 237bcde }}


Badness: 0.042719
Badness: 0.042719
Line 642: Line 642:
POTE generator: ~15/11 = 541.778
POTE generator: ~15/11 = 541.778


Optimal GPV sequence: {{Val list| 11cdeef, 20cdef, 31 }}
{{Optimal ET sequence|legend=1| 11cdeef, 20cdef, 31 }}


Badness: 0.037849
Badness: 0.037849
Line 657: Line 657:
[[POTE generator]]: ~35/27 = 455.445
[[POTE generator]]: ~35/27 = 455.445


{{Val list|legend=1| 50, 58, 108, 166c, 408ccc }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}


[[Badness]]: 0.097786
[[Badness]]: 0.097786
Line 670: Line 670:
POTE generator: ~35/27 = 455.373
POTE generator: ~35/27 = 455.373


Optimal GPV sequence: {{Val list| 50, 58, 108, 166ce, 224cee }}
{{Optimal ET sequence|legend=1| 50, 58, 108, 166ce, 224cee }}


Badness: 0.041190
Badness: 0.041190
Line 683: Line 683:
POTE generator: ~35/27 = 455.347
POTE generator: ~35/27 = 455.347


Optimal GPV sequence: {{Val list| 50, 58, 166cef, 224ceeff }}
{{Optimal ET sequence|legend=1| 50, 58, 166cef, 224ceeff }}


Badness: 0.023877
Badness: 0.023877
Line 704: Line 704:
[[POTE generator]]: ~35/24 = 657.818
[[POTE generator]]: ~35/24 = 657.818


{{Val list|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}
{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}


[[Badness]]: 0.101191
[[Badness]]: 0.101191
Line 717: Line 717:
POTE generator: ~16/11 = 657.923
POTE generator: ~16/11 = 657.923


Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}


Badness: 0.067291
Badness: 0.067291
Line 730: Line 730:
POTE generator: ~16/11 = 657.854
POTE generator: ~16/11 = 657.854


Optimal GPV sequence: {{Val list| 11b, 20b, 31 }}
{{Optimal ET sequence|legend=1| 11b, 20b, 31 }}


=== Marrakesh ===
=== Marrakesh ===
Line 741: Line 741:
POTE generator: ~22/15 = 657.791
POTE generator: ~22/15 = 657.791


Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c }}
{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c }}


Badness: 0.040539
Badness: 0.040539
Line 754: Line 754:
POTE generator: ~22/15 = 657.756
POTE generator: ~22/15 = 657.756


Optimal GPV sequence: {{Val list| 31, 73, 104c, 135c, 239ccf }}
{{Optimal ET sequence|legend=1| 31, 73, 104c, 135c, 239ccf }}


Badness: 0.040774
Badness: 0.040774
Line 767: Line 767:
POTE generator: ~22/15 = 657.700
POTE generator: ~22/15 = 657.700


Optimal GPV sequence: {{Val list| 31, 104cff, 135cff }}
{{Optimal ET sequence|legend=1| 31, 104cff, 135cff }}


Badness: 0.041395
Badness: 0.041395
Line 784: Line 784:
[[POTE generator]]: ~5/4 = 391.094
[[POTE generator]]: ~5/4 = 391.094


{{Val list|legend=1| 43, 46, 89, 135c, 359cc }}
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}


[[Badness]]: 0.110873
[[Badness]]: 0.110873
Line 797: Line 797:
POTE generator: ~5/4 = 391.075
POTE generator: ~5/4 = 391.075


Optimal GPV sequence: {{Val list| 43, 46, 89, 135c, 224c }}
{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 224c }}


Badness: 0.043438
Badness: 0.043438
Line 810: Line 810:
POTE generator: ~5/4 = 391.073
POTE generator: ~5/4 = 391.073


Optimal GPV sequence: {{Val list| 43, 46, 89, 135cf, 224cf }}
{{Optimal ET sequence|legend=1| 43, 46, 89, 135cf, 224cf }}


Badness: 0.030666
Badness: 0.030666
Line 827: Line 827:
[[POTE generator]]: ~343/270 = 446.568
[[POTE generator]]: ~343/270 = 446.568


{{Val list|legend=1| 8d, 35, 43 }}
{{Optimal ET sequence|legend=1| 8d, 35, 43 }}


[[Badness]]: 0.148531
[[Badness]]: 0.148531
Line 840: Line 840:
POTE generator: ~72/55 = 446.616
POTE generator: ~72/55 = 446.616


Optimal GPV sequence: {{Val list| 8d, 35, 43 }}
{{Optimal ET sequence|legend=1| 8d, 35, 43 }}


Badness: 0.059449
Badness: 0.059449
Line 853: Line 853:
POTE generator: ~13/10 = 446.598
POTE generator: ~13/10 = 446.598


Optimal GPV sequence: {{Val list| 8d, 35f, 43 }}
{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}


Badness: 0.035258
Badness: 0.035258
Line 866: Line 866:
POTE generator: ~13/10 = 446.631
POTE generator: ~13/10 = 446.631


Optimal GPV sequence: {{Val list| 8d, 35f, 43 }}
{{Optimal ET sequence|legend=1| 8d, 35f, 43 }}


Badness: 0.025907
Badness: 0.025907
Line 883: Line 883:
[[POTE generator]]: ~3/2 = 701.244
[[POTE generator]]: ~3/2 = 701.244


{{Val list|legend=1| 27, 81, 108, 135c, 243c }}
{{Optimal ET sequence|legend=1| 27, 81, 108, 135c, 243c }}


[[Badness]]: 0.173308
[[Badness]]: 0.173308
Line 896: Line 896:
POTE generator: ~3/2 = 700.001
POTE generator: ~3/2 = 700.001


Optimal GPV sequence: {{Val list| 27e, 81, 108 }}
{{Optimal ET sequence|legend=1| 27e, 81, 108 }}


Badness: 0.078060
Badness: 0.078060
Line 909: Line 909:
POTE generator: ~3/2 = 700.867
POTE generator: ~3/2 = 700.867


Optimal GPV sequence: {{Val list| 27e, 81, 108, 243ceef }}
{{Optimal ET sequence|legend=1| 27e, 81, 108, 243ceef }}


Badness: 0.057145
Badness: 0.057145
Line 922: Line 922:
POTE generator: ~3/2 = 700.397
POTE generator: ~3/2 = 700.397


Optimal GPV sequence: {{Val list| 27eg, 81, 108g }}
{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}


Badness: 0.042106
Badness: 0.042106
Line 935: Line 935:
POTE generator: ~3/2 = 700.429
POTE generator: ~3/2 = 700.429


Optimal GPV sequence: {{Val list| 27eg, 81, 108g }}
{{Optimal ET sequence|legend=1| 27eg, 81, 108g }}


Badness: 0.030415
Badness: 0.030415
Line 948: Line 948:
POTE generator: ~3/2 = 701.595
POTE generator: ~3/2 = 701.595


Optimal GPV sequence: {{Val list| 27eg, 81gg, 108, 135ce }}
{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ce }}


Badness: 0.047163
Badness: 0.047163
Line 961: Line 961:
POTE generator: ~3/2 = 701.673
POTE generator: ~3/2 = 701.673


Optimal GPV sequence: {{Val list| 27eg, 81gg, 108, 135ceh }}
{{Optimal ET sequence|legend=1| 27eg, 81gg, 108, 135ceh }}


Badness: 0.034176
Badness: 0.034176
Line 974: Line 974:
POTE generator: ~3/2 = 699.179
POTE generator: ~3/2 = 699.179


Optimal GPV sequence: {{Val list| 27e, 54bdef, 81f, 108f }}
{{Optimal ET sequence|legend=1| 27e, 54bdef, 81f, 108f }}


Badness: 0.052732
Badness: 0.052732
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