User:Recentlymaterialized/Miscellaneous rank-2 temperaments
Senscape (27e & 87)
Comma list: 196/195, 325/324, 441/440, 1001/1000
Mapping: [⟨3 2 2 4 1 10], ⟨0 5 9 8 17 2]]
Generator preimages: 63/50, 91/80
Norm Weil Tenney WE TE Optimized ??? 400.3122¢, 220.5893¢ 399.994¢, 220.6882¢ 399.9947¢, 220.6886¢ Constrained ??? 1\3, 220.729¢ 1\3, 220.6899¢ 1\3, 220.6873¢ Destretched ??? 1\3, 220.4173¢ 1\3, 220.6915¢ 1\3, 220.6915¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 Tuning 1199.982¢ 1903.4292¢ 2786.1822¢ 3365.482¢ 4151.6942¢ 4441.3165¢ Error -0.018¢ +1.4742¢ −0.1315¢ −3.3439¢ +0.3762¢ +0.7889¢
Optimal ET sequence (TE): 27e, 60e, 87
Badness (Smith TE): 0.02560
Badness (Dirichlet): 1.05783
Tripierce (b30 & b39)
Comma list: 245/243, 2197/2187
Mapping: [⟨3 1 7 7], ⟨0 4 -2 0]]
Generator preimages: 13/9, 13/7
Norm Weil Tenney WE TE Optimized ??? ??? 634.5206¢, 1075.4459¢ 634.5215¢, 1075.4475¢ Constrained ??? 1\b3, 1074.0886¢ 1\b3, 1074.6634¢ 1\b3, 1075.1282¢ Destretched ??? ??? 1\b3, 1074.5381¢ 1\b3, 1074.5382¢
Prime 3/1 5/1 7/1 13/1 Tuning 1903.5617¢ 2785.4124¢ 3366.1979¢ 4441.6439¢ Error +1.6067¢ −0.9013¢ −2.628¢ +1.1162¢
Optimal ET sequence (TE): b9, b21, b30, b39, b69, b108, b177d
Badness (Smith TE): 0.00287
Badness (Dirichlet): 0.26872
Trisemipierce (b39 & b60)
Comma list: 245/243, 847/845, 1575/1573
Mapping: [⟨3 1 7 4 7], ⟨0 4 -2 3 0]]
Generator preimages: 13/9, 15/11
Norm Weil Tenney WE TE Optimized ??? 634.4577¢, 537.4447¢ 634.5219¢, 537.7266¢ 634.5229¢, 537.7274¢ Constrained ??? 1\b3, 537.0443¢ 1\b3, 537.3589¢ 1\b3, 537.7233¢ Destretched ??? 1\b3, 537.0443¢ 1\b3, 537.2716¢ 1\b3, 537.2716¢
Prime 3/1 5/1 7/1 11/1 13/1 Tuning 1903.5658¢ 2785.4282¢ 3366.2003¢ 4151.2674¢ 4441.6534¢ Error +1.6108¢ −0.8855¢ −2.6256¢ −0.0505¢ +1.1258¢
Optimal ET sequence (TE): b18, b21, b39, b99, b138, b177d
Badness (Smith TE): 0.00566
Badness (Dirichlet): 0.44749
Tridubhe (b69 & b78)
Comma list: 245/243, 325/323, 665/663, 3213/3211
Mapping: [⟨3 3 6 7 6 7], ⟨0 4 -2 0 5 3]]
Generator preimages: 13/9, 17/15
Norm Weil Tenney WE TE Optimized ??? 634.4242¢, 220.2042¢ 634.3781¢, 220.1438¢ 634.3798¢, 220.1444¢ Constrained ??? 1\b3, 220.0518¢ 1\b3, 220.0811¢ 1\b3, 220.4087¢ Destretched ??? 1\b3, 220.0518¢ 1\b3, 220.0074¢ 1\b3, 220.0074¢
Prime 3/1 5/1 7/1 13/1 17/1 19/1 Tuning 1903.1344¢ 2783.7095¢ 3365.9811¢ 4440.6468¢ 4906.9877¢ 5101.0782¢ Error +1.1794¢ −2.6042¢ −2.8448¢ +0.1192¢ +2.0323¢ +3.5652¢
Optimal ET sequence (TE): b51, b60, b69, b147
Badness (Smith TE): 0.00523
Badness (Dirichlet): 0.48415
Trisemidubhe (b60 & b78)
Comma list: 245/243, 325/323, 665/663, 847/845, 1575/1573
Mapping: [⟨6 2 14 8 14 7 11], ⟨0 4 -2 3 0 5 3]]
Generator preimages: 119/99, 15/11
Norm Weil Tenney WE TE Optimized ??? ??? 317.2028¢, 537.3966¢ 317.2037¢, 537.3981¢ Constrained ??? 1\b6, 537.0443¢ 1\b6, 537.1261¢ 1\b6, 537.5291¢ Destretched ??? ??? 1\b6, 1074.0806¢ 1\b6, 1074.0806¢
Prime 3/1 5/1 7/1 11/1 13/1 17/1 19/1 Tuning 1903.2168¢ 2783.9919¢ 3366.0459¢ 4149.8121¢ 4440.8391¢ 4907.4024¢ 5101.4204¢ Error +1.2618¢ −2.3218¢ −2.780¢ −1.5059¢ +0.3114¢ +2.447¢ +3.9074¢
Optimal ET sequence (TE): b18, b60, b78, b138, b216
Badness (Smith TE): 0.01046
Badness (Dirichlet): 0.95032
Badtone (31 & 67)
Comma list: 81/80, 144/143, 176/175, 196/195
Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]
Weil Tenney WE TE Optimized ??? 1200.4954¢, 232.6290¢ 1199.9347¢, 232.6275¢ 1199.9460¢, 232.6297¢ Constrained ??? 1\1, 232.6732¢ 1\1, 232.6392¢ 1\1, 232.6352¢ Destretched ??? 1\1, 232.5330¢ 1\1, 232.6402¢ 1\1, 232.6401¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 Tuning 1199.9347¢ 1897.817¢ 2791.5295¢ 3367.1765¢ 4150.4968¢ 4444.8759¢ Error −0.0653¢ −4.138¢ +5.2157¢ −1.6494¢ −0.8211¢ +4.3483¢
Optimal ET sequence (TE): 31, 67, 98
Badness (Smith TE): 0.03686
Badness (Dirichlet): 1.52300
Worsetone (12f & 67)
Lucas Chiaraviglio suggests the name mildtone.
Comma list: 81/80, 176/175, 196/195, 364/363
Mapping: [⟨1 0 -4 -32 -44 -58], ⟨0 1 4 22 30 39]]
Weil Tenney WE TE Optimized ??? 1199.6846¢, 1897.7889¢ 1199.7882¢, 1898.1434¢ 1199.8021¢, 1898.1654¢ Constrained ??? 1\1, 1898.2217¢ 1\1, 698.4715¢ 1\1, 698.4553¢ Destretched ??? 1\1, 1898.2878¢ 1\1, 1898.4785¢ 1\1, 1898.4785¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 Tuning 1199.7882¢ 1898.1434¢ 2793.4206¢ 3365.9316¢ 4153.6201¢ 4439.8756¢ Error −0.2118¢ −3.8116¢ +7.1069¢ −2.8943¢ +2.3022¢ −0.6521¢
Optimal ET sequence: 12f, 55f, 67
Badness (Smith TE): 0.04940
Badness (Dirichlet): 2.04145
2.3.7.11.13(.17.19).23.31.37.41 subgroup
Comma list: 287/286, 297/296, 364/363, 483/481, 621/616, 3627/3626, 4797/4774, (153/152, 273/272)
Mapping: [⟨1 0 -32 -44 -58 (-94 -97) -73 -52 -47 -69], ⟨0 1 22 30 39 (62 64) 49 36 33 47]]
Generator tunings (no-17 no-19):
Weil Tenney WE TE Optimized ??? 1201.1304¢, 1900.1656¢ 1200.9840¢, 1899.9400¢ 1200.9879¢, 1899.9461¢ Constrained ??? 1\1, 1898.6199¢ 1\1, 1898.4197¢ 1\1, 1898.4955¢ Destretched ??? 1\1, 1898.3773¢ 1\1, 1898.3833¢ 1\1, 1898.3833¢
Prime 2/1 3/1 7/1 11/1 13/1 23/1 31/1 37/1 41/1 Tuning 1200.984¢ 1899.940¢ 3367.1912¢ 4154.903¢ 4440.5866¢ 5425.2263¢ 5946.6708¢ 6251.7709¢ 6429.2824¢ Error +0.984¢ −2.015¢ −1.6347¢ +3.585¢ +0.059¢ −3.0481¢ +1.6352¢ +0.4269¢ +0.220¢
Optimal ET sequence: 12f(ghh)ikm, 43deff(ggghhh)iiikkllmm, 55f(gh)ilm, 67p, 122(g)ilm
Badness (Smith TE): 0.01056 (0.01156)
Badness (Dirichlet): 1.07545 (1.35270)
Cyanide (140 & 441)
Comma list: 2401/2400, 4096/4095, (35000/34983), 140625/140608
Mapping: [⟨7 3 23 21 34 (60)], ⟨0 6 -5 -1 -6 (-21)]]
Generator preimages: 3675/3328, 8/7
Norm Weil Tenney WE TE Optimized ??? 171.4255¢, 231.2847¢ 171.4244¢, 231.2902¢ 171.4244¢, 231.2902¢ Constrained ??? 1\7, 231.2969¢ 1\7, 231.2963¢ 1\7, 231.2991¢ Destretched ??? 1\7, 231.2888¢ 1\7, 231.2943¢ 1\7, 231.2943¢
Prime 2/1 3/1 5/1 7/1 13/1 23/1 Tuning 1199.971¢ 1902.0144¢ 2786.3108¢ 3368.6227¢ 4440.6893¢ 5428.3714¢ Error −0.029¢ +0.0594¢ −0.0029¢ −0.2032¢ +0.1616¢ +0.097¢
Optimal ET sequence (TE): 21, 119, 140, 301, 441 (add-23: 21ii, 140, 301, 441, 3227f)
Badness (Smith TE): 0.01465 (0.01044)
Badness (Dirichlet): 0.50663 (0.49340)
13-limit
Comma list: 1288/1287, 2080/2079, 2401/2400, 4096/4095, 35000/34983
Mapping: [⟨7 3 23 21 62 34 (60)], ⟨0 6 -5 -1 -28 -6 (-21)]]
Generator preimages: 875/792, 8/7
Norm Weil Tenney WE TE Optimized ??? 171.4227, 231.3059¢ 171.4242, 231.3085¢ 171.4242, 231.3085¢ Constrained ??? 1\7, 231.3138¢ 1\7, 231.3149¢ 1\7, 231.3180¢ Destretched ??? 1\7, 231.3138¢ 1\7, 231.3144¢ 1\7, 231.3144¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 23/1 Tuning 1199.9697¢ 1902.1239¢ 2786.2149¢ 3368.6006¢ 4151.6642¢ 4440.5731¢ 5427.9754¢ Error −0.0303¢ +0.1689¢ −0.0988¢ −0.2253¢ +0.3463¢ +0.0454¢ −0.2989¢
Optimal ET sequence (TE): 140, 301e, 441, 581, 1022, 1603e (same sequence for add-23)
Badness (Smith TE): 0.02947 (0.01904)
Badness (Dirichlet): 1.21770 (1.01017)
581 & 742 (weak extension)
Comma list: 2401/2400, 3136/3135, 4096/4095, 5985/5984, 12376/12375, 14875/14872
Mapping: [⟨7 15 13 19 24 22 31 28], ⟨0 -18 15 3 1 18 -11 8]] (23/1: [⟨18], ⟨63]])
Generator preimages: 85/77, 143/140
Generator tunings (add-23):
Norm Weil Tenney WE TE Optimized ??? 171.4266¢, 37.1864¢ 171.4244¢, 37.1884¢ 171.4244¢, 37.1884¢ Constrained ??? 1\7, 37.1867¢ 1\7, 37.1893¢ 1\7, 37.1893¢ Destretched ??? 1\7, 37.1868¢ 1\7, 37.1893¢ 1\7, 37.1893¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 Tuning ( ¢) 1199.97¢ 1901.97¢ 2786.34¢ 3368.63¢ 4151.37¢ 4440.73¢ 4905.08¢ 5097.39¢ Error −0.0291¢ +0.0195¢ +0.0302¢ −0.1966¢ +0.0566¢ +0.2014¢ +0.1289¢ −0.1218¢
Optimal ET sequence (TE): 161, 420, 581, 742, 1323
Badness (Smith TE): 0.01260
Badness (Dirichlet): 0.76654
130 & 181
See also: 7-limit
Comma list: 2401/2400, 4096/4095, 492128/492075
Mapping: [⟨1 11 -22 -7 19], ⟨0 -24 62 25 -39]]
Generator preimages: 2/1, 21/16
Norm Weil Tenney WE TE Optimized ??? 1199.983¢, 470.7428¢ 1199.9727¢, 470.7388¢ 1199.9727¢, 470.7388¢ Constrained ??? 1\1, 470.7495¢ 1\1, 470.7495¢ 1\1, 470.7493¢ Destretched ??? 1\1, 470.7495¢ 1\1, 470.7495¢ 1\1, 470.7495¢
Prime 2/1 3/1 5/1 7/1 13/1 Tuning 1199.9727¢ 1901.9687¢ 2786.4055¢ 3368.6607¢ 4440.6683¢ Error −0.0273¢ +0.0137¢ +0.0918¢ −0.1652¢ +0.1407¢
Optimal ET sequence (TE): 51, 79c, 130, 311, 441, 1453, 1894, 2335
Badness (Smith TE): 0.01381
Badness (Dirichlet): 0.47741
13-limit
Comma list: 1575/1573, 2080/2079, 4096/4095, 4459/4455
Mapping: [⟨1 11 -22 -7 -24 19], ⟨0 -24 62 25 70 -39]]
Generator preimages: 2/1, 21/16
Norm Weil Tenney WE TE Optimized ??? 1199.9305¢, 470.7167¢ 1199.9386¢, 470.7221¢ 1199.9387¢, 470.7221¢ Constrained ??? 1\1, 470.7411¢ 1\1, 470.746¢ 1\1, 470.745¢ Destretched ??? 1\1, 470.744¢ 1\1, 470.7462¢ 1\1, 470.7461¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 Tuning 1199.9386¢ 1901.9946¢ 2786.1202¢ 3368.482¢ 4152.0197¢ 4440.6721¢ Error −0.0614¢ +0.0396¢ −0.1936¢ −0.3439¢ +0.7017¢ +0.1445¢
Optimal ET sequence (TE): 51, 130, 311, 441, 752e, 1193e
Badness (Smith TE): 0.02172
Badness (Dirichlet): 0.89734
Whooshlet (441 & 3072)
Main article: Whoosh
Comma list: 140625/140608, 250047/250000, 1990656/1990625 (9801/9800 can be added if the full 13-limit is desired)
Mapping: [⟨3 18 17 2 40], ⟨0 -33 -25 16 -72]]
Generator preimages: 63/50, 192/175
Norm Weil Tenney WE TE Optimized ??? 400.0024¢, 160.5488¢ 400.0034¢, 160.5495¢ 400.0034¢, 160.5495¢ Constrained ??? 1\, 160.5478¢ 1\, 160.5480¢ 1\, 160.5476¢ Destretched ??? 1\3, 160.5478¢ 1\3, 160.5481¢ 1\3, 160.5481¢
Prime 2/1 3/1 5/1 7/1 13/1 Tuning 1200.0102¢ 1901.9291¢ 2786.3213¢ 3368.7981¢ 4440.575¢ Error +0.0102¢ −0.0259¢ +0.0076¢ −0.0278¢ +0.0473¢
Optimal ET sequence (TE): 15, 411f, 426, 441, 1308, 1749, 2190, 6129, 8319, 10509 (full 13-limit: 426, 882, 1308, 2190, 5688, 7878, 17946bde)
Badness (Smith TE): 0.0142139
Badness (Dirichlet): 0.4914805