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Meantone family: Difference between revisions

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VisualWikitext
→Septimal meantone: +descriptions to each extension
→Septimal meantone: +CTWE/KE tunings; -diamond monotone and tradeoff (easily derivable and lack of significance)
Line 90: Line 90:
{{Multival|legend=1| 1 4 10 4 13 12 }}
{{Multival|legend=1| 1 4 10 4 13 12 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 696.9521
[[Optimal tuning]]:
* [[CTE]]: ~2 = 1\1, ~3/2 = 696.9521
* [[CTWE]]: ~2 = 1\1, ~3/2 = 696.6562


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }}
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
Line 101: Line 103:
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 700.000]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]


[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
[[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
Line 121: Line 121:
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.1676
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.1676
* CTWE: ~2 = 1\1, ~3/2 = 697.0315


Minimax tuning:  
Minimax tuning:  
Line 131: Line 133:
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]


Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
Algebraic generator: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.


{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b }}


Badness: 0.017027
Badness: 0.017027
Line 149: Line 150:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.8552
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.8552
* CTWE: ~2 = 1\1, ~3/2 = 696.7085


Minimax tuning:  
Minimax tuning:  
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: eigenmonzo (unchanged-interval) basis: 2.11/9
: eigenmonzo (unchanged-interval) basis: 2.11/9


Line 168: Line 171:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 | 0 1 4 10 18 15 26 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 | 0 1 4 10 18 15 26 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.6486
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.6486
* CTWE: ~2 = 1\1, ~3/2 = 696.4563


{{Optimal ET sequence|legend=1| 12fg, 19eg, 31, 50e }}
{{Optimal ET sequence|legend=1| 12fg, 19eg, 31, 50e }}
Line 181: Line 186:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 -40 | 0 1 4 10 18 15 26 28 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 -40 | 0 1 4 10 18 15 26 28 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.5551
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.5551
* CTWE: ~2 = 1\1, ~3/2 = 696.3526


{{Optimal ET sequence|legend=1| 12fghh, 19egh, 31, 50e }}
{{Optimal ET sequence|legend=1| 12fghh, 19egh, 31, 50e }}
Line 196: Line 203:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 | 0 1 4 10 18 15 7 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 | 0 1 4 10 18 15 7 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.0360
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.0360
* CTWE: ~2 = 1\1, ~3/2 = 696.6155


{{Optimal ET sequence|legend=1| 12f, 19eg, 31g }}
{{Optimal ET sequence|legend=1| 12f, 19eg, 31g }}
Line 209: Line 218:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 -10 | 0 1 4 10 18 15 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -7 -10 | 0 1 4 10 18 15 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2161
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.2161
* CTWE: ~2 = 1\1, ~3/2 = 696.6169


{{Optimal ET sequence|legend=1| 12f, 19egh, 31gh }}
{{Optimal ET sequence|legend=1| 12f, 19egh, 31gh }}
Line 224: Line 235:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9080
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.9080
* CTWE: ~2 = 1\1, ~3/2 = 696.9823


{{Optimal ET sequence|legend=1| 12f, 19e, 31 }}
{{Optimal ET sequence|legend=1| 12f, 31 }}


Badness: 0.019982
Badness: 0.019982
Line 237: Line 250:
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9308
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.9308
* CTWE: ~2 = 1\1, ~3/2 = 697.1061


{{Optimal ET sequence|legend=1| 12f, 19e, 31 }}
{{Optimal ET sequence|legend=1| 12f, 31 }}


Badness: 0.018047
Badness: 0.018047
Line 252: Line 267:
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2582
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.2582
* CTWE: ~2 = 1\1, ~3/2 = 697.2627


Minimax tuning:  
Minimax tuning:  
Line 261: Line 278:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 697.674]


{{Optimal ET sequence|legend=1| 12, 31, 43, 74 }}
{{Optimal ET sequence|legend=1| 12, 31, 43, 74 }}
Line 274: Line 290:
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.2996
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.2996
* CTWE: ~2 = 1\1, ~3/2 = 697.3303


{{Optimal ET sequence|legend=1| 12, 31, 43, 74g }}
{{Optimal ET sequence|legend=1| 12, 31, 43, 74g }}
Line 287: Line 305:
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.3271
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.3271
* CTWE: ~2 = 1\1, ~3/2 = 697.3736


{{Optimal ET sequence|legend=1| 12, 31, 43, 74gh }}
{{Optimal ET sequence|legend=1| 12, 31, 43, 74gh }}
Line 302: Line 322:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5155
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.5155
* CTWE: ~2 = 1\1, ~3/2 = 697.5241


Minimax tuning:  
Minimax tuning:  
Line 319: Line 341:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 | 0 1 4 10 18 27 38 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 | 0 1 4 10 18 27 38 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5076
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.5076
* CTWE: ~2 = 1\1, ~3/2 = 697.5113


{{Optimal ET sequence|legend=1| 12fg, 31fg, 43 }}
{{Optimal ET sequence|legend=1| 12fg, 31fg, 43 }}
Line 332: Line 356:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 -59 | 0 1 4 10 18 27 38 40 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 -59 | 0 1 4 10 18 27 38 40 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.4848
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.4848
* CTWE: ~2 = 1\1, ~3/2 = 697.4815


{{Optimal ET sequence|legend=1| 12fghh, 31fgh, 43 }}
{{Optimal ET sequence|legend=1| 12fghh, 31fgh, 43 }}
Line 345: Line 371:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 | 0 1 4 10 18 27 7 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 | 0 1 4 10 18 27 7 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.6098
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.6098
* CTWE: ~2 = 1\1, ~3/2 = 697.4333


{{Optimal ET sequence|legend=1| 12f, 31fg, 43g }}
{{Optimal ET sequence|legend=1| 12f, 31fg, 43g }}
Line 358: Line 386:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 -10 | 0 1 4 10 18 27 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -7 -10 | 0 1 4 10 18 27 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.7012
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.7012
* CTWE: ~2 = 1\1, ~3/2 = 697.4045


{{Optimal ET sequence|legend=1| 12f, 19effgh, 31fgh, 43gh }}
{{Optimal ET sequence|legend=1| 12f, 19effgh, 31fgh, 43gh }}
Line 371: Line 401:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5384
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.5384
* CTWE: ~2 = 1\1, ~3/2 = 697.6222


{{Optimal ET sequence|legend=1| 12f, 43 }}
{{Optimal ET sequence|legend=1| 12f, 43 }}
Line 384: Line 416:
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.5550
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.5550
* CTWE: ~2 = 1\1, ~3/2 = 697.6887


{{Optimal ET sequence|legend=1| 12f, 43 }}
{{Optimal ET sequence|legend=1| 12f, 43 }}
Line 399: Line 433:
: mapping generators: ~2, ~26/15
: mapping generators: ~2, ~26/15


Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 948.6109
Optimal tuning:
* CTE: ~2 = 1\1, ~26/15 = 948.6109
* CTWE: ~2 = 1\1, ~26/15 = 948.5065


{{Optimal ET sequence|legend=1| 19e, 43, 62 }}
{{Optimal ET sequence|legend=1| 19e, 43, 62 }}
Line 412: Line 448:
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}


Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 948.6173
Optimal tuning:
* CTE: ~2 = 1\1, ~26/15 = 948.6173
* CTWE: ~2 = 1\1, ~26/15 = 948.5169


{{Optimal ET sequence|legend=1| 19eg, 43, 62 }}
{{Optimal ET sequence|legend=1| 19eg, 43, 62 }}
Line 425: Line 463:
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}


Optimal tuning (CTE): ~2 = 1\1, ~19/11 = 948.6088
Optimal tuning:
* CTE: ~2 = 1\1, ~19/11 = 948.6088
* CTWE: ~2 = 1\1, ~19/11 = 948.5111


{{Optimal ET sequence|legend=1| 19egh, 43, 62 }}
{{Optimal ET sequence|legend=1| 19egh, 43, 62 }}
Line 440: Line 480:
: mapping generators: ~55/39, ~3
: mapping generators: ~55/39, ~3


Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 697.1678
Optimal tuning:
* CTE: ~55/39 = 1\2, ~3/2 = 697.1678
* CTWE: ~55/39 = 1\2, ~3/2 = 697.0545


{{Optimal ET sequence|legend=1| 12f, 38deefff, 50eff, 62, 136b }}
{{Optimal ET sequence|legend=1| 12f, 38deefff, 50eff, 62, 136b }}
Line 453: Line 495:
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 697.1740
Optimal tuning:
* CTE: ~17/12 = 1\2, ~3/2 = 697.1740
* CTWE: ~17/12 = 1\2, ~3/2 = 696.9858


{{Optimal ET sequence|legend=1| 12f, 50eff, 62, 136bg }}
{{Optimal ET sequence|legend=1| 12f, 50eff, 62, 136bg }}
Line 466: Line 510:
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 697.1871
Optimal tuning:
* CTE: ~17/12 = 1\2, ~3/2 = 697.1871
* CTWE: ~17/12 = 1\2, ~3/2 = 696.9638


{{Optimal ET sequence|legend=1| 12f, 50eff, 62 }}
{{Optimal ET sequence|legend=1| 12f, 50eff, 62 }}
Line 485: Line 531:
: mapping generator: ~2, ~3
: mapping generator: ~2, ~3


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.5311
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.5311
* CTWE: ~2 = 1\1, ~3/2 = 696.4509


Minimax tuning:  
Minimax tuning:  
Line 495: Line 543:
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]


Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.


{{Optimal ET sequence|legend=1| 12e, 19, 31, 81 }}
{{Optimal ET sequence|legend=1| 12e, 19, 31, 81, 112b }}


Badness: 0.021543
Badness: 0.021543
Line 514: Line 561:
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3563
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.3563
* CTWE: ~2 = 1\1, ~3/2 = 696.2347


Minimax tuning:  
Minimax tuning:  
Line 522: Line 571:
Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]


{{Optimal ET sequence|legend=1| 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=1| 19, 31, 50, 81 }}
Line 536: Line 584:
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3508
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.3508
* CTWE: ~2 = 1\1, ~3/2 = 696.2195


{{Optimal ET sequence|legend=1| 19g, 31, 50, 81, 131bd }}
{{Optimal ET sequence|legend=1| 19g, 31, 50, 81, 131bd }}
Line 549: Line 599:
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3471
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.3471
* CTWE: ~2 = 1\1, ~3/2 = 696.2137


{{Optimal ET sequence|legend=1| 19gh, 31, 50, 81 }}
{{Optimal ET sequence|legend=1| 19gh, 31, 50, 81 }}
Line 562: Line 614:
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4388
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.4388
* CTWE: ~2 = 1\1, ~3/2 = 696.4114


{{Optimal ET sequence|legend=1| 19, 31 }}
{{Optimal ET sequence|legend=1| 19, 31 }}
Line 575: Line 629:
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4838
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.4838
* CTWE: ~2 = 1\1, ~3/2 = 696.4973


{{Optimal ET sequence|legend=1| 12ef, 19, 31 }}
{{Optimal ET sequence|legend=1| 12ef, 19, 31 }}
Line 588: Line 644:
Mapping: {{mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }}
Mapping: {{mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2827
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.2827
* CTWE: ~2 = 1\1, ~3/2 = 696.4246


Minimax tuning:  
Minimax tuning:  
Line 605: Line 663:
Mapping: {{mapping| 1 0 -4 -13 24 10 12 | 0 1 4 10 -13 -4 -5 }}
Mapping: {{mapping| 1 0 -4 -13 24 10 12 | 0 1 4 10 -13 -4 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4069
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.4069
* CTWE: ~2 = 1\1, ~3/2 = 696.4129


{{Optimal ET sequence|legend=1| 12e, 19 }}
{{Optimal ET sequence|legend=1| 12e, 19 }}
Line 618: Line 678:
Mapping: {{mapping| 1 0 -4 -13 24 10 12 9 | 0 1 4 10 -13 -4 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 24 10 12 9 | 0 1 4 10 -13 -4 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4731
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.4731
* CTWE: ~2 = 1\1, ~3/2 = 696.4949


{{Optimal ET sequence|legend=1| 12e, 19 }}
{{Optimal ET sequence|legend=1| 12e, 19 }}
Line 631: Line 693:
Mapping: {{mapping| 1 0 -4 -13 24 10 -7 | 0 1 4 10 -13 -4 7 }}
Mapping: {{mapping| 1 0 -4 -13 24 10 -7 | 0 1 4 10 -13 -4 7 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.6614
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.6614
* CTWE: ~2 = 1\1, ~3/2 = 696.4462


{{Optimal ET sequence|legend=1| 12e, 19g, 31fg }}
{{Optimal ET sequence|legend=1| 12e, 19g, 31fg }}
Line 644: Line 708:
Mapping: {{mapping| 1 0 -4 -13 24 10 -7 -10 | 0 1 4 10 -13 -4 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 24 10 -7 -10 | 0 1 4 10 -13 -4 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 697.0160
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 697.0160
* CTWE: ~2 = 1\1, ~3/2 = 696.6337


{{Optimal ET sequence|legend=1| 12e, 19gh, 31fgh }}
{{Optimal ET sequence|legend=1| 12e, 19gh, 31fgh }}
Line 659: Line 725:
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.1527
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.1527
* CTWE: ~2 = 1\1, ~3/2 = 696.2083


Tuning ranges:  
Tuning ranges:  
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]


{{Optimal ET sequence|legend=1| 7d, 12, 19, 31e }}
{{Optimal ET sequence|legend=1| 7d, 12, 19, 31e }}
Line 677: Line 744:
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0983
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.0983
* CTWE: ~2 = 1\1, ~3/2 = 696.1241


{{Optimal ET sequence|legend=1| 7df, 12f, 19, 31e }}
{{Optimal ET sequence|legend=1| 7df, 12f, 19, 31e }}
Line 690: Line 759:
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2161
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.2161
* CTWE: ~2 = 1\1, ~3/2 = 696.4999


{{Optimal ET sequence|legend=1| 12f, 19, 31e }}
{{Optimal ET sequence|legend=1| 12f, 19, 31e }}
Line 703: Line 774:
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2774
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.2774
* CTWE: ~2 = 1\1, ~3/2 = 696.6370


{{Optimal ET sequence|legend=1| 12f, 19, 31e }}
{{Optimal ET sequence|legend=1| 12f, 19, 31e }}
Line 716: Line 789:
Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 | 0 1 4 10 6 15 7 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 | 0 1 4 10 6 15 7 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4501
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.4501
* CTWE: ~2 = 1\1, ~3/2 = 696.1417


{{Optimal ET sequence|legend=1| 7dfg, 12f, 19g }}
{{Optimal ET sequence|legend=1| 7dfg, 12f, 19g }}
Line 729: Line 804:
Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 -10 | 0 1 4 10 6 15 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -20 -7 -10 | 0 1 4 10 6 15 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.121
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.7925
* CTWE: ~2 = 1\1, ~3/2 = 696.3052


{{Optimal ET sequence|legend=1| 7dfgh, 12f, 19gh }}
{{Optimal ET sequence|legend=1| 7dfgh, 12f, 19gh }}
Line 742: Line 819:
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.7897
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 695.7897
* CTWE: ~2 = 1\1, ~3/2 = 695.2045


{{Optimal ET sequence|legend=1| 7d, 12, 19 }}
{{Optimal ET sequence|legend=1| 7d, 12, 19 }}
Line 755: Line 834:
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.0114
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.0114
* CTWE: ~2 = 1\1, ~3/2 = 695.8771


{{Optimal ET sequence|legend=1| 12, 19 }}
{{Optimal ET sequence|legend=1| 12, 19 }}
Line 768: Line 849:
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.1196
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.1196
* CTWE: ~2 = 1\1, ~3/2 = 696.1262


{{Optimal ET sequence|legend=1| 12, 19 }}
{{Optimal ET sequence|legend=1| 12, 19 }}
Line 872: Line 955:
Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 | 0 1 4 10 6 -4 7 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 | 0 1 4 10 6 -4 7 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.4125
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.4125
* CTWE: ~2 = 1\1, ~3/2 = 695.5584


{{Optimal ET sequence|legend=1| 7dg, 12, 19g }}
{{Optimal ET sequence|legend=1| 7dg, 12, 19g }}
Line 885: Line 970:
Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 -10 | 0 1 4 10 6 -4 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 -6 10 -7 -10 | 0 1 4 10 6 -4 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.9500
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.9500
* CTWE: ~2 = 1\1, ~3/2 = 695.9548


{{Optimal ET sequence|legend=1| 7dgh, 12, 19gh }}
{{Optimal ET sequence|legend=1| 7dgh, 12, 19gh }}
Line 898: Line 985:
Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 695.6202
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 695.6202
* CTWE: ~2 = 1\1, ~3/2 = 696.7474


{{Optimal ET sequence|legend=1| 7d, 12f, 19f }}
{{Optimal ET sequence|legend=1| 7d, 12f, 19f }}
Line 911: Line 1,000:
Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 | 0 1 4 10 6 3 7 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 | 0 1 4 10 6 3 7 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2789
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.2789
* CTWE: ~2 = 1\1, ~3/2 = 697.1744


{{Optimal ET sequence|legend=1| 7dg, 12f }}
{{Optimal ET sequence|legend=1| 7dg, 12f }}
Line 924: Line 1,015:
Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 -10 | 0 1 4 10 6 3 7 9 }}
Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 -10 | 0 1 4 10 6 3 7 9 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.8486
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.8486
* CTWE: ~2 = 1\1, ~3/2 = 697.7122


{{Optimal ET sequence|legend=1| 7dgh, 12f }}
{{Optimal ET sequence|legend=1| 7dgh, 12f }}
Line 939: Line 1,032:
Mapping: {{mapping| 1 0 -4 -13 5 | 0 1 4 10 -1 }}
Mapping: {{mapping| 1 0 -4 -13 5 | 0 1 4 10 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.7022
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.7022
* CTWE: ~2 = 1\1, ~3/2 = 695.1805


{{Optimal ET sequence|legend=1| 7d, 12e, 19e }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e }}
Line 952: Line 1,047:
Mapping: {{mapping| 1 0 -4 -13 5 10 | 0 1 4 10 -1 -4 }}
Mapping: {{mapping| 1 0 -4 -13 5 10 | 0 1 4 10 -1 -4 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.2408
Optimal tuning:
* CTE: ~2 = 1\1, ~3/2 = 696.2408
* CTWE: ~2 = 1\1, ~3/2 = 695.0144


{{Optimal ET sequence|legend=1| 7d, 12e, 19e }}
{{Optimal ET sequence|legend=1| 7d, 12e, 19e }}
Line 965: Line 1,062:
Mapping: {{mapping| 1 1 0 -3 2 4 | 0 2 8 20 5 -1 }}
Mapping: {{mapping| 1 1 0 -3 2 4 | 0 2 8 20 5 -1 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/9 = 348.5444
Optimal tuning:
* CTE: ~2 = 1\1, ~11/9 = 348.5444
* CTWE: ~2 = 1\1, ~3/2 = 348.5009


{{Optimal ET sequence|legend=1| 7d, 24d, 31 }}
{{Optimal ET sequence|legend=1| 7d, 24d, 31 }}
Line 982: Line 1,081:
: mapping generators: ~63/44, ~3
: mapping generators: ~63/44, ~3


Optimal tuning (CTE): ~63/44 = 1\2, ~3/2 = 696.5199
Optimal tuning:
* CTE: ~63/44 = 1\2, ~3/2 = 696.5199
* CTWE: ~63/44 = 1\2, ~3/2 = 696.1908


{{Optimal ET sequence|legend=1| 12, 26de, 38d, 50 }}
{{Optimal ET sequence|legend=1| 12, 26de, 38d, 50 }}
Line 995: Line 1,096:
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}


Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 696.3410
Optimal tuning:
* CTE: ~55/39 = 1\2, ~3/2 = 696.3410
* CTWE: ~55/39 = 1\2, ~3/2 = 696.0066


{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
Line 1,008: Line 1,111:
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 696.3526
Optimal tuning:
* CTE: ~17/12 = 1\2, ~3/2 = 696.3526
* CTWE: ~17/12 = 1\2, ~3/2 = 695.9317


{{Optimal ET sequence|legend=1| 12f, 38df, 50 }}
{{Optimal ET sequence|legend=1| 12f, 38df, 50 }}
Line 1,021: Line 1,126:
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}


Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 696.3837
Optimal tuning:
* CTE: ~17/12 = 1\2, ~3/2 = 696.3837
* CTWE: ~17/12 = 1\2, ~3/2 = 695.8947


{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
{{Optimal ET sequence|legend=1| 12f, 26deff, 38df, 50 }}
Line 1,034: Line 1,141:
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.7074
: mapping generators: ~2, ~11/10
 
Optimal tuning:
* CTE: ~2 = 1\1, ~11/10 = 167.7074
* CTWE: ~2 = 1\1, ~11/10 = 167.7749


{{Optimal ET sequence|legend=1|7d, 36d, 43, 50, 93 }}
{{Optimal ET sequence|legend=1| 7d, 36d, 43, 50, 93 }}


Badness: 0.050729
Badness: 0.050729
Line 1,047: Line 1,158:
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.7121
Optimal tuning:
* CTE: ~2 = 1\1, ~11/10 = 167.7121
* CTWE: ~2 = 1\1, ~11/10 = 167.7728


{{Optimal ET sequence|legend=1|7d, 36d, 43, 50, 93 }}
{{Optimal ET sequence|legend=1| 7d, 43, 50, 93 }}


Badness: 0.035445
Badness: 0.035445
Line 1,060: Line 1,173:
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}


Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 167.7047
Optimal tuning:
* CTE: ~2 = 1\1, ~11/10 = 167.7047
* CTWE: ~2 = 1\1, ~11/10 = 167.7682


{{Optimal ET sequence|legend=1|7dg, 36dg, 43, 50, 93 }}
{{Optimal ET sequence|legend=1| 7dg, 43, 50, 93 }}


Badness: 0.025221
Badness: 0.025221
Retrieved from "https://en.xen.wiki/w/Meantone_family"