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Schismatic family: Difference between revisions

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The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and [[5/4]] is represented by a diminished fourth. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768.
The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and [[5/4]] is represented by a diminished fourth. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768.


== Schismatic aka Helmholtz ==
== Schismatic aka helmholtz ==
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Comma list]]: 32805/32768
[[Comma list]]: 32805/32768


[[Mapping]]: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}]
{{Mapping|legend=1| 1 0 15 | 0 1 -8 }}


: mapping generators: ~2, ~3
: mapping generators: ~2, ~3
Line 22: Line 22:
[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955]
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]
* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]


Line 60: Line 60:
[[Comma list]]: 225/224, 3125/3087
[[Comma list]]: 225/224, 3125/3087


[[Mapping]]: [{{val| 1 0 15 25 }}, {{val| 0 1 -8 -14 }}]
{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }}


: mapping generators: ~2, ~3
: mapping generators: ~2, ~3
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[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 1/15 0 -1/15 }}, {{monzo| 5/3 -8/15 0 8/15 }}, {{monzo| 5/3 -14/15 0 14/15 }}]
: {{monzo list| 1 0 0 0 | 5/3 1/15 0 -1/15 | 5/3 -8/15 0 8/15 | 5/3 -14/15 0 14/15 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 }}, {{monzo| 5/2 -1 0 1/2 }}, {{monzo| 25/8 -7/4 0 7/8 }}]
: {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 92: Line 92:
Comma list: 225/224, 385/384, 2200/2187
Comma list: 225/224, 385/384, 2200/2187


Mapping: [{{val| 1 0 15 25 -33 }}, {{val| 0 1 -8 -14 23 }}]
Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157
Line 98: Line 98:
Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: Eigenmonzo (unchanged-interval) basis: 2.9/7
: eigenmonzo (unchanged-interval) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
Line 114: Line 114:
Comma list: 225/224, 275/273, 325/324, 385/384
Comma list: 225/224, 275/273, 325/324, 385/384


Mapping: [{{val| 1 0 15 25 -33 -28 }}, {{val| 0 1 -8 -14 23 20 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113
Line 137: Line 137:
Comma list: 120/119, 154/153, 225/224, 273/272, 325/324
Comma list: 120/119, 154/153, 225/224, 273/272, 325/324


Mapping: [{{val| 1 0 15 25 -33 -28 -7 }}, {{val| 0 1 -8 -14 23 20 7 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092
Line 150: Line 150:
Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272
Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272


Mapping: [{{val| 1 0 15 25 -33 -28 -7 9 }}, {{val| 0 1 -8 -14 23 20 7 -3 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079
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Comma list: 225/224, 275/273, 325/324, 375/374, 385/384
Comma list: 225/224, 275/273, 325/324, 375/374, 385/384


Mapping: [{{val| 1 0 15 25 -33 -28 77 }}, {{val| 0 1 -8 -14 23 20 -46 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097
Line 176: Line 176:
Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374
Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374


Mapping: [{{val| 1 0 15 25 -33 -28 77 9 }}, {{val| 0 1 -8 -14 23 20 -46 -3 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 77 9 | 0 1 -8 -14 23 20 -46 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.098
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.098
Line 189: Line 189:
Comma list: 170/169, 225/224, 275/273, 325/324, 385/384
Comma list: 170/169, 225/224, 275/273, 325/324, 385/384


Mapping: [{{val| 1 0 15 25 -33 -28 -72 }}, {{val| 0 1 -8 -14 23 20 48 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144
Line 202: Line 202:
Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324
Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324


Mapping: [{{val| 1 0 15 25 -33 -28 -72 9 }}, {{val| 0 1 -8 -14 23 20 48 -3 }}]
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135
Line 215: Line 215:
Comma list: 100/99, 225/224, 245/242
Comma list: 100/99, 225/224, 245/242


Mapping: [{{val| 1 0 15 25 32 }}, {{val| 0 1 -8 -14 -18 }}]
Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321
Line 221: Line 221:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
: Eigenmonzo (unchanged-interval) basis: 2.11/9
: eigenmonzo (unchanged-interval) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
Line 237: Line 237:
Comma list: 100/99, 105/104, 196/195, 245/242
Comma list: 100/99, 105/104, 196/195, 245/242


Mapping: [{{val| 1 0 15 25 32 37 }}, {{val| 0 1 -8 -14 -18 -21 }}]
Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.559
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.559
Line 243: Line 243:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
: Eigenmonzo (unchanged-interval) basis: 2.13/9
: eigenmonzo (unchanged-interval) basis: 2.13/9


Tuning ranges:  
Tuning ranges:  
Line 260: Line 260:
Comma list: 100/99, 105/104, 120/119, 189/187, 196/195
Comma list: 100/99, 105/104, 120/119, 189/187, 196/195


Mapping: [{{val| 1 0 15 25 32 37 -7 }}, {{val| 0 1 -8 -14 -18 -21 7 }}]
Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312
Line 273: Line 273:
Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195
Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195


Mapping: [{{val| 1 0 15 25 32 37 -7 9 }}, {{val| 0 1 -8 -14 -18 -21 7 -3 }}]
Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357
Line 286: Line 286:
Comma list: 100/99, 105/104, 154/153, 170/169, 196/195
Comma list: 100/99, 105/104, 154/153, 170/169, 196/195


Mapping: [{{val| 1 0 15 25 32 37 58 }}, {{val| 0 1 -8 -14 -18 -21 -34 }}]
Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725
Line 299: Line 299:
Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189
Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189


Mapping: [{{val| 1 0 15 25 32 37 58 9 }}, {{val| 0 1 -8 -14 -18 -21 -34 -3 }}]
Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753
Line 312: Line 312:
Comma list: 85/84, 100/99, 105/104, 119/117, 221/220
Comma list: 85/84, 100/99, 105/104, 119/117, 221/220


Mapping: [{{val| 1 0 15 25 32 37 12 }}, {{val| 0 1 -8 -14 -18 -21 -5 }}]
Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717
Line 325: Line 325:
Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152
Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152


Mapping: [{{val| 1 0 15 25 32 37 12 9 }}, {{val| 0 1 -8 -14 -18 -21 -5 -3 }}]
Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716
Line 338: Line 338:
Comma list: 99/98, 176/175, 3125/3087
Comma list: 99/98, 176/175, 3125/3087


Mapping: [{{val| 1 0 15 25 51 }}, {{val| 0 1 -8 -14 -30 }}]
Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725
Line 344: Line 344:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: Eigenmonzo (unchanged-interval) basis: 2.11/9
: eigenmonzo (unchanged-interval) basis: 2.11/9


{{Optimal ET sequence|legend=1| 12, 41e, 53, 118d, 171de }}
{{Optimal ET sequence|legend=1| 12, 41e, 53, 118d, 171de }}
Line 355: Line 355:
Comma list: 99/98, 176/175, 275/273, 847/845
Comma list: 99/98, 176/175, 275/273, 847/845


Mapping: [{{val| 1 0 15 25 51 56 }}, {{val| 0 1 -8 -14 -30 -33 }}
Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747
Line 372: Line 372:
Comma list: 99/98, 120/119, 176/175, 275/273, 442/441
Comma list: 99/98, 120/119, 176/175, 275/273, 442/441


Mapping: [{{val| 1 0 15 25 51 56 -7 }}, {{val| 0 1 -8 -14 -30 -33 7 }}
Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680
Line 385: Line 385:
Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245
Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245


Mapping: [{{val| 1 0 15 25 51 56 -7 9 }}, {{val| 0 1 -8 -14 -30 -33 7 -3 }}
Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705
Line 398: Line 398:
Comma list: 121/120, 225/224, 3125/3087
Comma list: 121/120, 225/224, 3125/3087


Mapping: [{{val| 1 0 15 25 9 }}, {{val| 0 2 -16 -28 -7 }}]
Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }}


: mapping generators: ~2, ~110/63
: mapping generators: ~2, ~110/63
Line 413: Line 413:
Comma list: 121/120, 169/168, 225/224, 275/273
Comma list: 121/120, 169/168, 225/224, 275/273


Mapping: [{{val| 1 0 15 25 9 14 }}, {{val| 0 2 -16 -28 -7 -13 }}]
Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)
Line 428: Line 428:
Comma list: 225/224, 243/242, 3125/3087
Comma list: 225/224, 243/242, 3125/3087


Mapping: [{{val| 1 1 7 11 2 }}, {{val| 0 2 -16 -28 5 }}]
Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }}


: mapping generators: ~2, ~11/9
: mapping generators: ~2, ~11/9
Line 443: Line 443:
Comma list: 225/224, 243/242, 325/324, 640/637
Comma list: 225/224, 243/242, 325/324, 640/637


Mapping: [{{val| 1 1 7 11 2 -8 }}, {{val| 0 2 -16 -28 5 40 }}]
Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
Line 456: Line 456:
Comma list: 225/224, 1331/1323, 3125/3087
Comma list: 225/224, 1331/1323, 3125/3087


Mapping: [{{val| 1 2 -1 -3 0 }}, {{val| 0 -3 24 42 25 }}]
Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }}


: mapping generators: ~2, ~11/10
: mapping generators: ~2, ~11/10
Line 471: Line 471:
Comma list: 225/224, 275/273, 847/845, 1331/1323
Comma list: 225/224, 275/273, 847/845, 1331/1323


Mapping: [{{val| 1 2 -1 -3 0 -1 }}, {{val| 0 -3 24 42 25 34 }}]
Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963
Line 488: Line 488:
[[Comma list]]: 64/63, 360/343
[[Comma list]]: 64/63, 360/343


[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
{{Mapping|legend=1| 1 0 15 6 | 0 1 -8 -2 }}
 
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
Line 505: Line 503:
Comma list: 45/44, 64/63, 99/98
Comma list: 45/44, 64/63, 99/98


Mapping: [{{val| 1 0 15 6 13 }}, {{val| 0 1 -8 -2 -6 }}]
Mapping: {{mapping| 1 0 15 6 13 | 0 1 -8 -2 -6 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136
Line 522: Line 520:
[[Comma list]]: 4375/4374, 32805/32768
[[Comma list]]: 4375/4374, 32805/32768


[[Mapping]]: [{{val| 1 0 15 -59 }}, {{val| 0 1 -8 39 }}]
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }}
 
: mapping generators: ~2, ~3


{{Multival|legend=1| 1 -8 39 -15 59 113 }}
{{Multival|legend=1| 1 -8 39 -15 59 113 }}
Line 533: Line 529:
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 554: Line 550:
Comma list: 385/384, 3388/3375, 4375/4374
Comma list: 385/384, 3388/3375, 4375/4374


Mapping: [{{val| 1 0 15 -59 51 }}, {{val| 0 1 -8 39 -30 }}]
Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.722
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.722
Line 560: Line 556:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
: Eigenmonzo (unchanged-interval) basis: 2.11/7
: eigenmonzo (unchanged-interval) basis: 2.11/7


{{Optimal ET sequence|legend=1| 53, 118, 289e, 407de }}
{{Optimal ET sequence|legend=1| 53, 118, 289e, 407de }}
Line 571: Line 567:
Comma list: 352/351, 385/384, 625/624, 729/728
Comma list: 352/351, 385/384, 625/624, 729/728


Mapping: [{{val|1 0 15 -59 51 56}}, {{val|0 1 -8 39 -30 -33}}]
Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.745
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.745
Line 577: Line 573:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
: Eigenmonzo (unchanged-interval) basis: 2.13/7
: eigenmonzo (unchanged-interval) basis: 2.13/7


{{Optimal ET sequence|legend=1| 53, 118, 171e }}
{{Optimal ET sequence|legend=1| 53, 118, 171e }}
Line 588: Line 584:
Comma list: 352/351, 385/384, 561/560, 625/624, 729/728
Comma list: 352/351, 385/384, 561/560, 625/624, 729/728


Mapping: [{{val|1 0 15 -59 51 56 -91}}, {{val|0 1 -8 39 -30 -33 60}}]
Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.742
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.742
Line 594: Line 590:
Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
: Eigenmonzo (unchanged-interval) basis: 2.17/13
: eigenmonzo (unchanged-interval) basis: 2.17/13


{{Optimal ET sequence|legend=1| 53, 118, 171e, 289ef, 460eef }}
{{Optimal ET sequence|legend=1| 53, 118, 171e, 289ef, 460eef }}
Line 605: Line 601:
Comma list: 169/168, 325/324, 385/384, 3146/3125
Comma list: 169/168, 325/324, 385/384, 3146/3125


Mapping: [{{val| 1 0 15 -59 51 -28 }}, {{val| 0 1 -8 39 -30 20 }}]
Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Line 618: Line 614:
Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125
Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125


Mapping: [{{val| 1 0 15 -59 51 -28 -91 }}, {{val| 0 1 -8 39 -30 20 60 }}]
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.730
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.730
Line 631: Line 627:
Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625  
Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625  


Mapping: [{{val| 1 0 15 -59 51 -28 -91 9 }}, {{val| 0 1 -8 39 -30 20 60 -3 }}]
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.729
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.729
Line 646: Line 642:
Comma list: 540/539, 4375/4374, 32805/32768
Comma list: 540/539, 4375/4374, 32805/32768


Mapping: [{{val| 1 0 15 -59 135 }}, {{val| 0 1 -8 39 -83 }}]
Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.783
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.783
Line 652: Line 648:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: Eigenmonzo (unchanged-interval) basis: 2.11/7
: eigenmonzo (unchanged-interval) basis: 2.11/7


{{Optimal ET sequence|legend=1| 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee }}
{{Optimal ET sequence|legend=1| 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee }}
Line 663: Line 659:
Comma list: 540/539, 625/624, 729/728, 2200/2197
Comma list: 540/539, 625/624, 729/728, 2200/2197


Mapping: [{{val| 1 0 15 -59 135 56 }}, {{val| 0 1 -8 39 -83 -33 }}]
Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.784
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.784
Line 680: Line 676:
Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197
Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197


Mapping: [{{val| 1 0 15 -59 135 56 -91 }}, {{val| 0 1 -8 39 -83 -33 60 }}]
Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.777
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.777
Line 699: Line 695:
Comma list: 441/440, 4375/4374, 32805/32768
Comma list: 441/440, 4375/4374, 32805/32768


Mapping: [{{val| 1 0 15 -59 -136 }}, {{val| 0 1 -8 39 88 }}]
Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.724
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.724
Line 705: Line 701:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
: Eigenmonzo (unchanged-interval) basis: 2.11
: eigenmonzo (unchanged-interval) basis: 2.11


{{Optimal ET sequence|legend=1| 53e, 118, 289, 407d, 696d }}
{{Optimal ET sequence|legend=1| 53e, 118, 289, 407d, 696d }}
Line 716: Line 712:
Comma list: 441/440, 625/624, 729/728, 3584/3575
Comma list: 441/440, 625/624, 729/728, 3584/3575


Mapping: [{{val| 1 0 15 -59 -136 56 }}, {{val| 0 1 -8 39 88 -33 }}]
Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.738
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.738
Line 722: Line 718:
Minimax tuning:  
Minimax tuning:  
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: Eigenmonzo (unchanged-interval) basis: 2.13/11
: eigenmonzo (unchanged-interval) basis: 2.13/11


{{Optimal ET sequence|legend=1| 53e, 118, 171, 289f, 460ef }}
{{Optimal ET sequence|legend=1| 53e, 118, 171, 289f, 460ef }}
Line 733: Line 729:
Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873
Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873


Mapping: [{{val| 1 0 15 -59 -136 56 -91 }}, {{val| 0 1 -8 39 88 -33 60 }}]
Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740
Line 750: Line 746:
Comma list: 364/363, 441/440, 4375/4374, 32805/32768
Comma list: 364/363, 441/440, 4375/4374, 32805/32768


Mapping: [{{val| 1 0 15 -59 -136 -215 }}, {{val| 0 1 -8 39 88 138 }}]
Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Line 763: Line 759:
Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768
Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768


Mapping: [{{val| 1 0 15 -59 -136 -215 -91 }}, {{val| 0 1 -8 39 88 138 60 }}]
Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735
Line 778: Line 774:
Comma list: 3025/3024, 4375/4374, 32805/32768
Comma list: 3025/3024, 4375/4374, 32805/32768


Mapping: [{{val| 2 0 30 -118 -85 }}, {{val| 0 1 -8 39 29 }}]
Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }}


Mapping generators: ~99/70, ~3
: mapping generators: ~99/70, ~3


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757
Line 793: Line 789:
Comma list: 625/624, 729/728, 1575/1573, 4096/4095
Comma list: 625/624, 729/728, 1575/1573, 4096/4095


Mapping: [{{val| 2 0 30 -118 -85 112 }}, {{val| 0 1 -8 39 29 -33 }}]
Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }}


Mapping generators: ~99/70, ~3
Mapping generators: ~99/70, ~3
Line 808: Line 804:
Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873
Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873


Mapping: [{{val| 2 0 30 -118 -85 112 -182 }}, {{val| 0 1 -8 39 29 -33 60 }}]
Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.765
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.765
Line 821: Line 817:
Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768


Mapping: [{{val| 2 0 30 -118 -85 -243 }}, {{val| 0 1 -8 39 29 79 }}]
Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.769
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.769
Line 834: Line 830:
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768


Mapping: [{{val| 2 0 30 -118 -85 -243 -182 }}, {{val| 0 1 -8 39 29 79 60 }}]
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.764
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.764
Line 847: Line 843:
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864


Mapping: [{{val| 2 0 30 -118 -85 -243 -182 -169 }}, {{val| 0 1 -8 39 29 79 60 56 }}]
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.761
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.761
Line 860: Line 856:
Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768
Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768


Mapping: [{{val| 4 0 60 -236 -170 -131 }}, {{val| 0 1 -8 39 29 23 }}]
Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }}


: mapping generators: ~208/175, ~3
: mapping generators: ~208/175, ~3
Line 877: Line 873:
[[Comma list]]: 126/125, 32805/32768
[[Comma list]]: 126/125, 32805/32768


[[Mapping]]: [{{val| 1 0 15 44 }}, {{val| 0 1 -8 -26 }}]
{{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }}


: mapping generators: ~2, ~3
: mapping generators: ~2, ~3
Line 886: Line 882:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] eigenmonzos (unchanged-intervals): 2, 7/6
* [[7-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.7/3
* [[9-odd-limit]] eigenmonzos (unchanged-intervals): 2, 9/7
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


{{Optimal ET sequence|legend=1| 12, 53d, 65, 77, 166c, 243c }}
{{Optimal ET sequence|legend=1| 12, 53d, 65, 77, 166c, 243c }}
Line 898: Line 894:
Comma list: 126/125, 176/175, 32805/32768
Comma list: 126/125, 176/175, 32805/32768


Mapping: [{{val| 1 0 15 44 70 }}, {{val| 0 1 -8 -26 -42 }}]
Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.172
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.172
Line 911: Line 907:
Comma list: 126/125, 176/175, 196/195, 5445/5408
Comma list: 126/125, 176/175, 196/195, 5445/5408


Mapping: [{{val| 1 0 15 44 70 75 }}, {{val| 0 1 -8 -26 -42 -45 }}]
Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.226
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.226
Line 924: Line 920:
Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873
Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873


Mapping: [{{val| 1 0 15 44 70 75 -7 }}, {{val| 0 1 -8 -26 -42 -45 7 }}]
Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.206
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.206
Line 937: Line 933:
Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323
Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323


Mapping: [{{val| 1 0 15 44 70 75 -7 9 }}, {{val| 0 1 -8 -26 -42 -45 7 -3 }}]
Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Line 950: Line 946:
Comma list: 126/125, 176/175, 729/728, 1287/1280
Comma list: 126/125, 176/175, 729/728, 1287/1280


Mapping: [{{val| 1 0 15 44 70 -47 }}, {{val| 0 1 -8 -26 -42 32 }}]
Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217
Line 963: Line 959:
Comma list: 126/125, 245/242, 896/891
Comma list: 126/125, 245/242, 896/891


Mapping: [{{val| 1 0 15 44 51 }}, {{val| 0 1 -8 -26 -30 }}]
Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.401
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.401
Line 976: Line 972:
Comma list: 126/125, 196/195, 245/242, 832/825
Comma list: 126/125, 196/195, 245/242, 832/825


Mapping: [{{val| 1 0 15 44 51 75 }}, {{val| 0 1 -8 -26 -30 -45 }}]
Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.348
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.348
Line 989: Line 985:
Comma list: 105/104, 126/125, 245/242, 352/351
Comma list: 105/104, 126/125, 245/242, 352/351


Mapping: [{{val| 1 0 15 44 51 56 }}, {{val| 0 1 -8 -26 -30 -33 }}]
Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.529
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.529
Line 1,002: Line 998:
[[Comma list]]: 3136/3125, 32805/32768
[[Comma list]]: 3136/3125, 32805/32768


[[Mapping]]: [{{val| 2 0 30 69 }}, {{val| 0 1 -8 -20 }}]
{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }}


: mapping generators: ~567/400, ~3
: mapping generators: ~567/400, ~3
Line 1,011: Line 1,007:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] eigenmonzos (unchanged-intervals): 2, 7/6
* [[7-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.7/3
* [[9-odd-limit]] eigenmonzos (unchanged-intervals): 2, 9/7
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


{{Optimal ET sequence|legend=1| 12, 106d, 118, 130, 248, 378, 508 }}
{{Optimal ET sequence|legend=1| 12, 106d, 118, 130, 248, 378, 508 }}
Line 1,023: Line 1,019:
Comma list: 441/440, 3136/3125, 8019/8000
Comma list: 441/440, 3136/3125, 8019/8000


Mapping: [{{val| 2 0 30 69 102 }}, {{val| 0 1 -8 -20 -30 }}]
Mapping: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.612
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.612
Line 1,036: Line 1,032:
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Comma list: 441/440, 729/728, 1001/1000, 3136/3125


Mapping: [{{val| 2 0 30 69 102 -75 }}, {{val| 0 1 -8 -20 -30 26 }}]
Mapping: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.590
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.590
Line 1,049: Line 1,045:
Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125


Mapping: [{{val| 2 0 30 69 102 -75 5 }}, {{val| 0 1 -8 -20 -30 26 1 }}]
Mapping: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.600
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.600
Line 1,062: Line 1,058:
Comma list: 351/350, 364/363, 441/440, 3136/3125
Comma list: 351/350, 364/363, 441/440, 3136/3125


Mapping: [{{val| 2 0 30 69 102 131 }}, {{val| 0 1 -8 -20 -30 -39 }}]
Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.565
Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.565
Line 1,075: Line 1,071:
Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125


Mapping: [{{val| 2 0 30 69 102 131 5 }}, {{val| 0 1 -8 -20 -30 -39 1 }}]
Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.595
Optimal tuning (POTE): ~55/39 = 1\2, ~3/2 = 701.595
Line 1,088: Line 1,084:
[[Comma list]]: 32805/32768, 1500625/1492992
[[Comma list]]: 32805/32768, 1500625/1492992


[[Mapping]]: [{{val| 2 1 22 -15 }}, {{val| 0 2 -16 19 }}]
{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }}


: mapping generators: ~1225/864, ~35/24
: mapping generators: ~1225/864, ~35/24
Line 1,105: Line 1,101:
Comma list: 385/384, 9801/9800, 14641/14580
Comma list: 385/384, 9801/9800, 14641/14580


Mapping: [{{val| 2 1 22 -15 8 }}, {{val| 0 2 -16 19 -1 }}]
Mapping: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)
Line 1,118: Line 1,114:
Comma list: 352/351, 385/384, 729/728, 1575/1573
Comma list: 352/351, 385/384, 729/728, 1575/1573


Mapping: [{{val| 2 1 22 -15 8 15 }}, {{val| 0 2 -16 19 -1 -7 }}]
Mapping: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)
Line 1,131: Line 1,127:
Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
Comma list: 170/169, 289/288, 352/351, 385/384, 561/560


Mapping: [{{val| 2 1 22 -15 8 15 6 }}, {{val| 0 2 -16 19 -1 -7 2 }}]
Mapping: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)
Line 1,144: Line 1,140:
Comma list: 325/324, 385/384, 1573/1568, 14641/14580
Comma list: 325/324, 385/384, 1573/1568, 14641/14580


Mapping: [{{val| 2 1 22 -15 8 -36 }}, {{val| 0 2 -16 19 -1 40 }}]
Mapping: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)
Line 1,157: Line 1,153:
Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580


Mapping: [{{val| 2 1 22 -15 8 -36 6 }}, {{val| 0 2 -16 19 -1 40 2 }}]
Mapping: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)
Line 1,170: Line 1,166:
[[Comma list]]: 6144/6125, 19683/19600
[[Comma list]]: 6144/6125, 19683/19600


[[Mapping]]: [{{val| 1 0 15 -17 }}, {{val| 0 2 -16 25 }}]
{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }}


: mapping generators: ~2, ~140/81
: mapping generators: ~2, ~140/81
Line 1,187: Line 1,183:
Comma list: 540/539, 5632/5625, 8019/8000
Comma list: 540/539, 5632/5625, 8019/8000


Mapping: [{{val| 1 0 15 -17 51 }}, {{val| 0 2 -16 25 -60 }}]
Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }}


Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801
Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801
Line 1,200: Line 1,196:
Comma list: 351/350, 540/539, 676/675, 4096/4095
Comma list: 351/350, 540/539, 676/675, 4096/4095


Mapping: [{{val| 1 0 15 -17 51 14 }}, {{val| 0 2 -16 25 -60 -13 }}]
Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801
Line 1,213: Line 1,209:
Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095


Mapping: [{{val| 1 0 15 -17 51 14 -49 }}, {{val| 0 2 -16 25 -60 -13 67 }}]
Mapping: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810
Line 1,228: Line 1,224:
[[Comma list]]: 686/675, 32805/32768
[[Comma list]]: 686/675, 32805/32768


[[Mapping]]: [{{val| 1 2 -1 1 }}, {{val| 0 -3 24 13 }}]
{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }}


{{Multival|legend=1| 3 -24 -13 -45 -29 37 }}
{{Multival|legend=1| 3 -24 -13 -45 -29 37 }}
Line 1,243: Line 1,239:
Comma list: 245/242, 686/675, 896/891
Comma list: 245/242, 686/675, 896/891


Mapping: [{{val| 1 2 -1 1 0 }}, {{val| 0 -3 24 13 25 }}]
Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097
Line 1,256: Line 1,252:
Comma list: 91/90, 169/168, 245/242, 896/891
Comma list: 91/90, 169/168, 245/242, 896/891


Mapping: [{{val| 1 2 -1 1 0 3 }}, {{val| 0 -3 24 13 25 5 }}]
Mapping: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054
Line 1,271: Line 1,267:
[[Comma list]]: 32805/32768, 1071875/1062882
[[Comma list]]: 32805/32768, 1071875/1062882


[[Mapping]]: [{{val| 1 2 -1 10 }}, {{val| 0 -3 24 -52}}]
{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }}


{{Multival|legend=1| 3 -24 52 -45 74 188 }}
{{Multival|legend=1| 3 -24 52 -45 74 188 }}
Line 1,286: Line 1,282:
Comma list: 385/384, 4000/3993, 19712/19683
Comma list: 385/384, 4000/3993, 19712/19683


Mapping: [{{val| 1 2 -1 10 0}}, {{val| 0 -3 24 -52 25}}]
Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017
Line 1,299: Line 1,295:
Comma list: 325/324, 385/384, 1575/1573, 10985/10976
Comma list: 325/324, 385/384, 1575/1573, 10985/10976


Mapping: [{{val| 1 2 -1 10 0 12}}, {{val| 0 -3 24 -52 25 -60}}]
Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016
Line 1,312: Line 1,308:
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


Mapping: [{{val| 1 2 -1 10 0 12 -2}}, {{val| 0 -3 24 -52 25 -60 44}}]
Mapping: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012
Line 1,327: Line 1,323:
[[Comma list]]: 32805/32768, 244140625/243045684
[[Comma list]]: 32805/32768, 244140625/243045684


[[Mapping]]: [{{val| 1 2 -1 -12 }}, {{val| 0 -3 24 107 }}]
{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/567 = 166.0621
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~625/567 = 166.0621
Line 1,340: Line 1,336:
Comma list: 3025/3024, 4000/3993, 32805/32768
Comma list: 3025/3024, 4000/3993, 32805/32768


Mapping: [{{val| 1 2 -1 -12 0 }}, {{val| 0 -3 24 107 25 }}]
Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Line 1,353: Line 1,349:
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976


Mapping: [{{val| 1 2 -1 -12 0 -10 }}, {{val| 0 -3 24 107 25 99 }}]
Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628
Line 1,370: Line 1,366:
[[Comma list]]: 32805/32768, 118098/117649
[[Comma list]]: 32805/32768, 118098/117649


[[Mapping]]: [{{val| 2 1 22 2 }}, {{val| 0 3 -24 5 }}]
{{Mapping|legend=1| 2 1 22 2 | 0 3 -24 5 }}


Mapping generators: ~343/243, ~9/7
: mapping generators: ~343/243, ~9/7


{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
{{Multival|legend=1| 6 -48 10 -90 -1 158 }}
Line 1,387: Line 1,383:
Comma list: 540/539, 4000/3993, 32805/32768
Comma list: 540/539, 4000/3993, 32805/32768


Mapping: [{{val| 2 1 22 2 25 }}, {{val| 0 3 -24 5 -25 }}]
Mapping: {{mapping| 2 1 22 2 25 | 0 3 -24 5 -25 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Line 1,400: Line 1,396:
Comma list: 540/539, 729/728, 1575/1573, 4096/4095
Comma list: 540/539, 729/728, 1575/1573, 4096/4095


Mapping: [{{val| 2 1 22 2 25 -2 }}, {{val| 0 3 -24 5 -25 13 }}]
Mapping: {{mapping| 2 1 22 2 25 -2 | 0 3 -24 5 -25 13 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911
Line 1,413: Line 1,409:
[[Comma list]]: 32805/32768, 250047/250000
[[Comma list]]: 32805/32768, 250047/250000


[[Mapping]]: [{{val| 3 0 45 94 }}, {{val| 0 1 -8 -18 }}]
{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }}


: mapping generators: ~63/50, ~3
: mapping generators: ~63/50, ~3
Line 1,436: Line 1,432:
Comma list: 441/440, 4375/4356, 32805/32768
Comma list: 441/440, 4375/4356, 32805/32768


Mapping: [{{val| 3 0 45 94 134 }}, {{val| 0 1 -8 -18 -26 }}]
Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824
Line 1,449: Line 1,445:
Comma list: 364/363, 441/440, 625/624, 13720/13689
Comma list: 364/363, 441/440, 625/624, 13720/13689


Mapping: [{{val| 3 0 45 94 134 168 }}, {{val| 0 1 -8 -18 -26 -33 }}]
Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821
Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821
Line 1,462: Line 1,458:
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619


Mapping: [{{val| 3 0 45 94 134 168 -2 }}, {{val| 0 1 -8 -18 -26 -33 3 }}]
Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810
Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810
Line 1,475: Line 1,471:
Comma list: 540/539, 32805/32768, 137781/137500
Comma list: 540/539, 32805/32768, 137781/137500


Mapping: [{{val| 3 0 45 94 -137 }}, {{val| 0 1 -8 -18 31 }}]
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685
Line 1,488: Line 1,484:
Comma list: 540/539, 729/728, 4096/4095, 31250/31213
Comma list: 540/539, 729/728, 4096/4095, 31250/31213


Mapping: [{{val| 3 0 45 94 -137 -103 }}, {{val| 0 1 -8 -18 31 24 }}]
Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689
Line 1,501: Line 1,497:
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095


Mapping: [{{val| 3 0 45 94 -137 -103 -2 }}, {{val| 0 1 -8 -18 31 24 3 }}]
Mapping: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688
Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688
Line 1,516: Line 1,512:
Comma list: 9801/9800, 32805/32768, 151263/151250
Comma list: 9801/9800, 32805/32768, 151263/151250


Mapping: [{{val| 6 0 90 188 287 }}, {{val| 0 1 -8 -18 -28 }}]
Mapping: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }}


Mapping generators: ~55/49, ~3
: mapping generators: ~55/49, ~3


Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460
Line 1,531: Line 1,527:
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375


Mapping: [{{val| 6 0 90 188 287 355 }}, {{val| 0 1 -8 -18 -28 -35 }}]
Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }}


Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256
Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256
Line 1,546: Line 1,542:
Comma list: 3025/3024, 32805/32768, 102487/102400
Comma list: 3025/3024, 32805/32768, 102487/102400


Mapping: [{{val| 3 0 45 94 8 }}, {{val| 0 2 -16 -36 1 }}]
Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }}


: mapping generators: ~63/50, ~693/400
: mapping generators: ~63/50, ~693/400
Line 1,561: Line 1,557:
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712


Mapping: [{{val| 3 0 45 94 8 42 }}, {{val| 0 2 -16 -36 1 -13 }}]
Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)
Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)
Line 1,574: Line 1,570:
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264


Mapping: [{{val| 3 0 45 94 8 42 -2 }}, {{val| 0 2 -16 -36 1 -13 6 }}]
Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }}


Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)
Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)
Line 1,587: Line 1,583:
[[Comma list]]: 32805/32768, 367653125/362797056
[[Comma list]]: 32805/32768, 367653125/362797056


[[Mapping]]: [{{val| 3 0 45 -32 }}, {{val| 0 2 -16 17 }}]
{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }}


: mapping generators: ~1536/1225, ~34300/19683
: mapping generators: ~1536/1225, ~34300/19683
Line 1,602: Line 1,598:
Comma list: 385/384, 14700/14641, 19712/19683
Comma list: 385/384, 14700/14641, 19712/19683


Mapping: [{{val| 3 0 45 -32 8 }}, {{val| 0 2 -16 17 1 }}]
Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }}


Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9629
Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9629
Line 1,615: Line 1,611:
[[Comma list]]: 2401/2400, 32805/32768
[[Comma list]]: 2401/2400, 32805/32768


[[Mapping]]: [{{val| 1 1 7 5 }}, {{val| 0 4 -32 -15 }}]
{{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }}


: mapping generators: ~2, ~448/405
: mapping generators: ~2, ~448/405
Line 1,636: Line 1,632:
Comma list: 243/242, 441/440, 16384/16335
Comma list: 243/242, 441/440, 16384/16335


Mapping: [{{val| 1 1 7 5 2 }}, {{val| 0 4 -32 -15 10 }}]
Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
Line 1,649: Line 1,645:
Comma list: 243/242, 364/363, 441/440, 3584/3575
Comma list: 243/242, 364/363, 441/440, 3584/3575


Mapping: [{{val| 1 1 7 5 2 -2 }}, {{val| 0 4 -32 -15 10 39 }}]
Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409
Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409
Line 1,662: Line 1,658:
Comma list: 2401/2400, 9801/9800, 32805/32768
Comma list: 2401/2400, 9801/9800, 32805/32768


Mapping: [{{val| 2 2 14 10 23 }}, {{val| 0 4 -32 -15 -55 }}]
Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }}


: mapping generators: ~99/70, ~448/405
: mapping generators: ~99/70, ~448/405
Line 1,679: Line 1,675:
[[Comma list]]: 32805/32768, 9765625/9680832
[[Comma list]]: 32805/32768, 9765625/9680832


[[Mapping]]: [{{val| 1 2 -1 -4 }}, {{val| 0 -5 40 82 }}]
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }}


{{Multival|legend=1| 5 -40 -82 -75 -144 -78 }}
{{Multival|legend=1| 5 -40 -82 -75 -144 -78 }}
Line 1,694: Line 1,690:
Comma list: 1375/1372, 4375/4356, 32805/32768
Comma list: 1375/1372, 4375/4356, 32805/32768


Mapping: [{{val| 1 2 -1 -4 -7 }}, {{val| 0 -5 40 82 126 }}]
Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616
Line 1,707: Line 1,703:
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647


Mapping: [{{val| 1 2 -1 -4 -7 -9 }}, {{val| 0 -5 40 82 126 153 }}]
Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612
Line 1,720: Line 1,716:
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619


Mapping: [{{val| 1 2 -1 -4 -7 -9 5 }}, {{val| 0 -5 40 82 126 153 -11 }}]
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612
Line 1,733: Line 1,729:
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971


Mapping: [{{val| 1 2 -1 -4 -7 -9 5 4 }}, {{val| 0 -5 40 82 126 153 -11 3 }}]
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615
Line 1,748: Line 1,744:
[[Comma list]]: 32805/32768, 49009212/48828125
[[Comma list]]: 32805/32768, 49009212/48828125


[[Mapping]]: [{{val| 1 2 -1 -5 }}, {{val| 0 -5 40 94 }}]
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }}


{{Multival|legend=1| 5 -40 -94 -75 -163 -106 }}
{{Multival|legend=1| 5 -40 -94 -75 -163 -106 }}
Line 1,763: Line 1,759:
Comma list: 441/440, 32805/32768, 1953125/1951488
Comma list: 441/440, 32805/32768, 1953125/1951488


Mapping: [{{val| 1 2 -1 -5 -8 }}, {{val| 0 -5 40 94 138 }}]
Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653
Line 1,776: Line 1,772:
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Comma list: 364/363, 441/440, 32805/32768, 109512/109375


Mapping: [{{val| 1 2 -1 -5 -8 -11 }}, {{val| 0 -5 40 94 138 177 }}]
Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }}


Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658
Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658
Line 1,789: Line 1,785:
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768


Mapping: [{{val| 1 2 -1 -5 -8 -11 5 }}, {{val| 0 -5 40 94 138 177 -11 }}]
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656
Line 1,802: Line 1,798:
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859


Mapping: [{{val| 1 2 -1 -5 -8 -11 5 4 }}, {{val| 0 -5 40 94 138 177 -11 3 }}]
Mapping: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659
Line 1,815: Line 1,811:
Comma list: 5632/5625, 8019/8000, 151263/151250
Comma list: 5632/5625, 8019/8000, 151263/151250


Mapping: [{{val| 1 2 -1 -5 -9 }}, {{val| 0 -5 40 94 150 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }}


Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671
Line 1,828: Line 1,824:
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000


Mapping: [{{val| 1 2 -1 -5 -9 -11 }}, {{val| 0 -5 40 94 150 177 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }}


Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661
Line 1,841: Line 1,837:
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750


Mapping: [{{val| 1 2 -1 -5 -9 -11 5 }}, {{val| 0 -5 40 94 150 177 -11 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665
Line 1,854: Line 1,850:
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700


Mapping: [{{val| 1 2 -1 -5 -9 -11 5 4 }}, {{val| 0 -5 40 94 150 177 -11 3 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.668
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.668
Line 1,867: Line 1,863:
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436


Mapping: [{{val| 1 2 -1 -5 -9 14 }}, {{val| 0 -5 40 94 150 -124 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }}


Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.672
Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.672
Line 1,880: Line 1,876:
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157


Mapping: [{{val| 1 2 -1 -5 -9 14 5 }}, {{val| 0 -5 40 94 150 -124 -11 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.671
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.671
Line 1,893: Line 1,889:
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137


Mapping: [{{val| 1 2 -1 -5 -9 14 5 4 }}, {{val| 0 -5 40 94 150 -124 -11 3 }}]
Mapping: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.672
Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.672
Line 1,908: Line 1,904:
[[Comma list]]: 32805/32768, 235298/234375
[[Comma list]]: 32805/32768, 235298/234375


[[Mapping]]: [{{val| 1 2 -1 -1 }}, {{val| 0 -6 48 55 }}]
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }}


: mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20
Line 1,925: Line 1,921:
Comma list: 441/440, 4000/3993, 235298/234375
Comma list: 441/440, 4000/3993, 235298/234375


Mapping: [{{val| 1 2 -1 -1 0 }}, {{val| 0 -6 48 55 50 }}]
Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
Line 1,938: Line 1,934:
Comma list: 364/363, 441/440, 676/675, 10985/10976
Comma list: 364/363, 441/440, 676/675, 10985/10976


Mapping: [{{val| 1 2 -1 -1 0 1 }}, {{val| 0 -6 48 55 50 39 }}]
Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049
Line 1,953: Line 1,949:
[[Comma list]]: 32805/32768, 829440/823543
[[Comma list]]: 32805/32768, 829440/823543


[[Mapping]]: [{{val| 1 3 -9 2 }}, {{val| 0 -7 -56 4 }}]
{{Mapping|legend=1| 1 3 -9 2 | 0 -7 -56 4 }}


{{Multival|legend=1| 7 56 -4 231 -26 -76 }}
{{Multival|legend=1| 7 56 -4 231 -26 -76 }}
Line 1,968: Line 1,964:
Comma list: 540/539, 15488/15435, 32805/32768
Comma list: 540/539, 15488/15435, 32805/32768


Mapping: [{{val| 1 3 -9 2 -2 }}, {{val| 0 -7 -56 4 27 }}]
Mapping: {{mapping| 1 3 -9 2 -2 | 0 -7 -56 4 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616
Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616
Line 1,981: Line 1,977:
Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Comma list: 540/539, 729/728, 1573/1568, 4096/4095


Mapping: [{{val| 1 3 -9 2 -2 13 }}, {{val| 0 -7 -56 4 27 -46 }}]
Mapping: {{mapping| 1 3 -9 2 -2 13 | 0 -7 -56 4 27 -46 }}


Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610
Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610
Line 1,996: Line 1,992:
[[Comma list]]: 32805/32768, 420175/419904
[[Comma list]]: 32805/32768, 420175/419904


[[Mapping]]: [{{val| 1 1 7 0 }}, {{val| 0 5 -40 24 }}]
{{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }}


: mapping generators: ~2, ~243/224
: mapping generators: ~2, ~243/224
Line 2,013: Line 2,009:
Comma list: 385/384, 1331/1323, 19712/19683
Comma list: 385/384, 1331/1323, 19712/19683


Mapping: [{{val| 1 1 7 0 1 }}, {{val| 0 5 -40 24 21 }}]
Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }}


Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365
Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365
Line 2,026: Line 2,022:
Comma list: 352/351, 385/384, 729/728, 1331/1323
Comma list: 352/351, 385/384, 729/728, 1331/1323


Mapping: [{{val| 1 1 7 0 1 3 }}, {{val| 0 5 -40 24 21 6 }}]
Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363
Line 2,039: Line 2,035:
[[Comma list]]: 16875/16807, 32805/32768
[[Comma list]]: 16875/16807, 32805/32768


[[Mapping]]: [{{val| 1 0 15 12 }}, {{val| 0 5 -40 -29 }}]
{{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }}


: mapping generators: ~2, ~56/45
: mapping generators: ~2, ~56/45
Line 2,056: Line 2,052:
Comma list: 540/539, 1375/1372, 32805/32768
Comma list: 540/539, 1375/1372, 32805/32768


Mapping: [{{val| 1 0 15 12 -7 }}, {{val| 0 5 -40 -29 33 }}]
Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
Line 2,069: Line 2,065:
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
Comma list: 540/539, 729/728, 1375/1372, 4096/4095


Mapping: [{{val| 1 0 15 12 -7 -15 }}, {{val| 0 5 -40 -29 33 59 }}]
Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
Line 2,084: Line 2,080:
[[Comma list]]: 32805/32768, 390625/388962
[[Comma list]]: 32805/32768, 390625/388962


[[Mapping]]: [{{val| 4 0 60 119 }}, {{val| 0 1 -8 -17 }}]
{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }}


: mapping generators: ~25/21, ~3
: mapping generators: ~25/21, ~3
Line 2,101: Line 2,097:
Comma list: 1375/1372, 6250/6237, 32805/32768
Comma list: 1375/1372, 6250/6237, 32805/32768


Mapping: [{{val| 4 0 60 119 185 }}, {{val| 0 1 -8 -17 -27 }}]
Mapping: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176
Line 2,114: Line 2,110:
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647


Mapping: [{{val| 4 0 60 119 185 224 }}, {{val| 0 1 -8 -17 -27 -33 }}]
Mapping: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }}


Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158
Line 2,129: Line 2,125:
[[Comma list]]: 32805/32768, 516560652/514714375
[[Comma list]]: 32805/32768, 516560652/514714375


[[Mapping]]: [{{val| 7 0 105 -56 }}, {{val| 0 1 -8 7 }}]
{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }}


: mapping generators: ~8575/7776, ~3
: mapping generators: ~8575/7776, ~3
Line 2,146: Line 2,142:
Comma list: 3025/3024, 24057/24010, 32805/32768
Comma list: 3025/3024, 24057/24010, 32805/32768


Mapping: [{{val| 7 0 105 -56 -120 }}, {{val| 0 1 -8 7 13 }}]
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }}


Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.719
Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.719
Line 2,159: Line 2,155:
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024


Mapping: [{{val| 7 0 105 -56 -120 37 }}, {{val| 0 1 -8 7 13 -1 }}]
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }}


Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.724
Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.724
Line 2,174: Line 2,170:
[[Comma list]]: 32805/32768, 2259436291848/2251875390625
[[Comma list]]: 32805/32768, 2259436291848/2251875390625


[[Mapping]]: [{{val| 8 0 120 -117 }}, {{val| 0 1 -8 11 }}]
{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }}


: mapping generators: ~42875/39366, ~3
: mapping generators: ~42875/39366, ~3
Line 2,191: Line 2,187:
Comma list: 9801/9800, 32805/32768, 46656/46585
Comma list: 9801/9800, 32805/32768, 46656/46585


Mapping: [{{val| 8 0 120 -117 15 }}, {{val| 0 1 -8 11 1 }}]
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.713
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.713
Line 2,204: Line 2,200:
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655


Mapping: [{{val| 8 0 120 -117 15 93 }}, {{val| 0 1 -8 11 1 -5 }}]
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }}


Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.725
Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.725
Line 2,219: Line 2,215:
[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}
[[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }}


[[Mapping]]: [{{val| 1 1 7 6 }}, {{val| 0 13 -104 -71 }}]
{{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }}


: mapping generators: ~2, ~1323/1280
: mapping generators: ~2, ~1323/1280
Line 2,234: Line 2,230:
Comma list: 441/440, 32805/32768, 55296000/55240493
Comma list: 441/440, 32805/32768, 55296000/55240493


Mapping: [{{val| 1 1 7 6 4 }}, {{val| 0 13 -104 -71 -12 }}]
Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }}


Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744
Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744
Line 2,243: Line 2,239:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Schismatic]]
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Schismatic family| ]] <!-- main article -->
[[Category:Schismatic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Schismatic_family"