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)
- See also: BPS III
Comma list: 245/243, 2197/2187
Mapping: [⟨3 1 7 7], ⟨0 2 -1 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)
- See also: BPS III - weak extension
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, 189/187)
Mapping: [⟨1 0 -32 -44 -58 (12 9) -73 -52 -47 -69], ⟨0 1 22 30 39 (-5 -3) 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: 12fikm, 43deffiiikkllmm, 55film, 67, 122(h)ilm
Badness (Smith TE): 0.01056 (0.01515)
Badness (Dirichlet): 1.07545 (1.77384)
Hemififive (68 & 94)
Comma list: 275/273, 325/324, 385/384, 1716/1715
Mapping: [⟨2 7 10 6 5 12], ⟨0 -10 -14 -1 5 -12]]
Generator preimages: 99/70, 8/7
Norm Weil Tenney WE TE Optimized ??? 599.9903 ¢, 229.6835 ¢ 600.0198 ¢, 229.7451 ¢ 600.0210 ¢, 229.7456 ¢ Constrained ??? 1\2, 229.6914 ¢ 1\2, 229.7361 ¢ 1\2, 229.7307 ¢ Destretched ??? 1\2, 229.6872 ¢ 1\2, 229.7375 ¢ 1\2, 229.7376 ¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 Tuning 1200.0396 ¢ 1902.6876 ¢ 2783.7665 ¢ 3370.3738 ¢ 4148.8246 ¢ 4443.2964 ¢ Error +0.0396 ¢ +0.7326 ¢ −2.5472 ¢ +1.5479 ¢ −2.4933 ¢ +2.7687 ¢
Optimal ET sequence (TE): 26, 68, 94
Badness (Smith TE): 0.03418
Badness (Dirichlet): 1.41232
19-limit
Comma list: 190/189, 275/273, 325/324, 385/384, 561/560, 595/594
Mapping: [⟨2 7 10 6 5 12 12 15], ⟨0 -10 -14 -1 5 -12 -10 -17]]
Generator preimages: 17/12, 8/7
Norm Weil Tenney WE TE Optimized ??? 599.9903 ¢, 229.6835 ¢ 600.0751 ¢, 229.7512 ¢ 600.0765 ¢, 229.7517 ¢ Constrained ??? 1\2, 229.6914 ¢ 1\2, 229.7168 ¢ 1\2, 229.6935 ¢ Destretched ??? 1\2, 229.6872 ¢ 1\2, 229.7224 ¢ 1\2, 229.7224 ¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 Tuning 1200.1502 ¢ 1903.0139 ¢ 2784.2344 ¢ 3370.6993 ¢ 4149.1313 ¢ 4443.8869 ¢ 4903.3893 ¢ 5095.3563 ¢ Error +0.1502 ¢ +1.0589 ¢ −2.0793 ¢ +1.8734 ¢ −2.1867 ¢ +3.3593 ¢ −1.5661 ¢ −2.1567 ¢
Optimal ET sequence (TE): 26, 68, 94
Badness (Smith TE): 0.01725
Badness (Dirichlet): 1.04955
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 190/189, 253/252, 275/273, 325/324, 385/384, 561/560, 595/594
Mapping: [⟨2 7 10 6 5 12 12 15 19], ⟨0 -10 -14 -1 5 -12 -10 -17 -26]]
Generator preimages: 17/12, 8/7
Norm Weil Tenney WE TE Optimized ??? 599.9903 ¢, 229.6835 ¢ 600.0744 ¢, 229.7474 ¢ 600.0758 ¢, 229.7480 ¢ Constrained ??? 1\2, 229.6914 ¢ 1\2, 229.7135 ¢ 1\2, 229.6908 ¢ Destretched ??? 1\2, 229.6872 ¢ 1\2, 229.7189 ¢ 1\2, 229.7190 ¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1 Tuning 1200.1488 ¢ 1903.0465 ¢ 2784.28 ¢ 3370.6991 ¢ 4149.1094 ¢ 4443.9237 ¢ 4903.4186 ¢ 5095.4097 ¢ 5427.9804 ¢ Error +0.1488 ¢ +1.0915 ¢ −2.0337 ¢ +1.8732 ¢ −2.2086 ¢ +3.3961 ¢ −1.5368 ¢ −2.1033 ¢ −0.2939 ¢
Optimal ET sequence (TE): 26i, 68, 94
Badness (Smith TE): 0.01458
Badness (Dirichlet): 1.04399
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