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| Line 40: |
Line 40: |
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| Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek. | | Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek. |
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| == Sensi ==
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| {{Main| Sensi }}
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| {{See also| Sensipent family #Sensi }}
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| Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 19 & 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 11, 19 and 27 are available.
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| === Septimal sensi ===
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| [[Subgroup]]: 2.3.5.7
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|
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| [[Comma list]]: 126/125, 245/243
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| {{Mapping|legend=1| 1 6 8 11 | 0 -7 -9 -13 }}
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| : mapping generators: ~2, ~14/9
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| {{Multival|legend=1| 7 9 13 -2 1 5 }}
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| [[Optimal tuning]]s:
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| * [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
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| * [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
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| [[Minimax tuning]]:
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| * [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
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| : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7
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| * [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
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| : [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
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| [[Tuning ranges of regular temperaments|Tuning ranges]]:
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| * 7-odd-limit [[diamond monotone]]: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
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| * 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
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| * 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
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| * 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]
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| [[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.
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| {{Optimal ET sequence|legend=1| 19, 27, 46 }}
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| [[Badness]]: 0.025622
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| ==== 2.3.5.7.13 subgroup (sensation) ====
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| Subgroup: 2.3.5.7.13
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|
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| Comma list: 91/90, 126/125, 169/168
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| Sval mapping: {{mapping| 1 6 8 11 10 | 0 -7 -9 -13 -10 }}
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| Gencom mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 }}
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| : gencom: [2 14/9; 91/90 126/125 169/168]
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| Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
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| {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
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| === Sensor ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 126/125, 245/243, 385/384
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| Mapping: {{mapping| 1 6 8 11 -6 | 0 -7 -9 -13 15 }}
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| {{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.2987
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| * POTE: ~2 = 1\1, ~9/7 = 443.294
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| {{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
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| Badness: 0.037942
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 91/90, 126/125, 169/168, 385/384
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| Mapping: {{mapping| 1 6 8 11 -6 10 | 0 -7 -9 -13 15 -10 }}
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| {{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.3658
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| * POTE: ~2 = 1\1, ~9/7 = 443.321
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| {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
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| Badness: 0.025575
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| ==== 17-limit ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
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| Mapping: {{mapping| 1 6 8 11 -6 10 -6 | 0 -7 -9 -13 15 -10 16 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.3775
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| * POTE: ~2 = 1\1, ~9/7 = 443.365
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| {{Optimal ET sequence|legend=1| 19, 27, 46 }}
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| Badness: 0.022908
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| === Sensus ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 126/125, 176/175, 245/243
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| Mapping: {{mapping| 1 6 8 11 23 | 0 -7 -9 -13 -31 }}
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| {{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.4783
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| * POTE: ~2 = 1\1, ~9/7 = 443.626
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| {{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
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| Badness: 0.029486
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|
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 91/90, 126/125, 169/168, 352/351
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| Mapping: {{mapping| 1 6 8 11 23 10 | 0 -7 -9 -13 -31 -10 }}
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| {{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.5075
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| * POTE: ~2 = 1\1, ~9/7 = 443.559
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| {{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
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| Badness: 0.020789
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| ==== 17-limit ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
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| Mapping: {{mapping| 1 6 8 11 23 10 23 | 0 -7 -9 -13 -31 -10 -30 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.5050
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| * POTE: ~2 = 1\1, ~9/7 = 443.551
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| {{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
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| Badness: 0.016238
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| === Sensis ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 56/55, 100/99, 245/243
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| Mapping: {{mapping| 1 6 8 11 6 | 0 -7 -9 -13 -4 }}
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| {{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.1886
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| * POTE: ~2 = 1\1, ~9/7 = 443.962
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| {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
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| Badness: 0.028680
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 56/55, 78/77, 91/90, 100/99
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| Mapping: {{mapping| 1 6 8 11 6 10 | 0 -7 -9 -13 -4 -10 }}
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| {{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.2863
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| * POTE: ~2 = 1\1, ~9/7 = 443.945
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| {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
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| Badness: 0.020017
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| === Sensa ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 55/54, 77/75, 99/98
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| Mapping: {{mapping| 1 6 8 11 11 | 0 -7 -9 -13 -12 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.7814
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| * POTE: ~2 = 1\1, ~9/7 = 443.518
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| {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
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| Badness: 0.036835
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|
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 55/54, 66/65, 77/75, 143/140
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| Mapping: {{mapping| 1 6 8 11 11 11 | 0 -7 -9 -13 -12 -11 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~9/7 = 443.7877
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| * POTE: ~2 = 1\1, ~9/7 = 443.506
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| {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
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| Badness: 0.023258
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| === Bisensi ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 121/120, 126/125, 245/243
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| Mapping: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
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| : mapping generators: ~99/70, ~11/10
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| Optimal tunings:
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| * CTE: ~99/70 = 1\2, ~11/10 = 156.6312
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| * POTE: ~99/70 = 1\2, ~11/10 = 156.692
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| {{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
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| Badness: 0.041723
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|
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 91/90, 121/120, 126/125, 169/168
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| Mapping: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
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| Optimal tunings:
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| * CTE: ~55/39 = 1\2, ~11/10 = 156.5584
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| * POTE: ~55/39 = 1\2, ~11/10 = 156.725
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| {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
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| Badness: 0.026339
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| ==== 17-limit ====
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| Subgroup: 2.3.5.7.11.13.17
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| Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
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| Mapping: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
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| Optimal tunings:
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| * CTE: ~17/12 = 1\2, ~11/10 = 156.5534
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| {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
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| Badness: 0.0188
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| === Hemisensi ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 126/125, 243/242, 245/242
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| Mapping: {{mapping| 1 13 17 24 32 | 0 -14 -18 -26 -35 }}
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| : mapping generators: ~2, ~44/25
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~25/22 = 221.5981
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| * POTE: ~2 = 1\1, ~25/22 = 221.605
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| {{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
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| Badness: 0.048714
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|
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| ==== 13-limit ====
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| Subgroup: 2.3.5.7.11.13
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|
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| Comma list: 91/90, 126/125, 169/168, 243/242
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| Mapping: {{mapping| 1 13 17 24 32 30 | 0 -14 -18 -26 -35 -30 }}
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| Optimal tunings:
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| * CTE: ~2 = 1\1, ~25/22 = 221.6333
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| * POTE: ~2 = 1\1, ~25/22 = 221.556
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| {{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
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| Badness: 0.033016
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| == Cohemiripple == | | == Cohemiripple == |