User:Recentlymaterialized/Miscellaneous rank-2 temperaments
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¢
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¢
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¢
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¢
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
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¢
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¢
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¢
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
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¢
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¢
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)
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¢
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
Quadrameantone (43 & 81)
Named for its meantone fifth at −4 generators, quadrameantone splits didacus's 14/5 into eight 25/22s. Due to the temperament's precision and complexity, the 3/1 from meantone is not formally included in the subgroup; if a more accurate 3/1 is desired, the temperament may be extended to the full 23-limit for a significant increase in scale size.
Comma list: 392/391, 476/475, 2300/2299, 3136/3125, 8096/8075
Mapping: [⟨1 12 27 21 18 23 39], ⟨0 -16 -40 -29 -23 -31 -57]]
Generator preimages: 2/1, 25/19
Norm Weil Tenney WE TE Optimized ??? 1199.6085¢, 725.5426¢ 1199.6487¢, 725.5704¢ 1199.6492¢, 725.5706¢ Constrained ??? 1\1, 725.8157¢ 1\1, 725.7958¢ 1\1, 725.8170¢ Destretched ??? 1\1, 725.7794¢ 1\1, 725.7828¢
Prime 2/1 5/1 7/1 11/1 17/1 19/1 23/1 Tuning 1199.6487¢ 2786.6589¢ 3367.701¢ 4151.0827¢ 4905.5587¢ 5099.2394¢ 5428.7895¢ Error −0.3513¢ +0.3452¢ −1.1249¢ −0.2353¢ +0.6033¢ +1.7264¢ +0.5152¢
Optimal ET sequence (TE): 38di, 43, 81, 124, 291ghi, 415ghhi
Badness (Smith TE): 0.00651
Badness (Dirichlet): 0.53307
2.5.7.11.13.17.19.23 subgroup
Subgroup: 2.5.7.11.13.17.19.23
Comma list: 392/391, 476/475, 1001/1000, 1496/1495, 3584/3575, 7744/7735
Mapping: [⟨1 12 27 21 -9 18 23 39], ⟨0 -16 -40 -29 21 -23 -31 -57]]
Generator preimages: 2/1, 25/19
Norm Weil Tenney WE TE Optimized ??? 1199.6083¢, 725.5424¢ 1199.6880¢, 725.5983¢ 1199.6884¢, 725.5986¢ Constrained ??? 1\1, 725.8157¢ 1\1, 725.7923¢ 1\1, 725.8129¢ Destretched ??? 1\1, 725.7793¢ 1\1, 725.7871¢
Prime 2/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1 Tuning 1199.6880¢ 2786.6822¢ 3367.6415¢ 4151.0954¢ 4440.3735¢ 4905.6216¢ 5099.2747¢ 5428.7253¢ Error −0.3120¢ +0.3684¢ −1.1844¢ −0.2225¢ −0.1542¢ +0.6662¢ +1.7617¢ +0.4509¢
Optimal ET sequence (TE): 43, 81, 124, 291ghi, 415ghhi
Badness (Smith TE): 0.00808
Badness (Dirichlet): 0.69675
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 208/207, 256/255, 364/363, 400/399, 442/441, 715/714, 1001/1000
Mapping: [⟨1 17 -4 -13 -8 12 -5 -8 -18], ⟨0 -39 16 40 29 -21 23 31 57]]
Generator preimages: 2/1, 21/16
Norm Weil Tenney WE TE Optimized ??? ??? 1199.3238¢, 473.9951¢ 1199.3250¢, 473.9956¢ Constrained ??? 1\1, 474.2641¢ 1\1, 474.2609¢ 1\1, 474.2484¢ Destretched ??? ??? 1\1, 474.2624¢
Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1 23/1 Tuning 1199.3238¢ 1902.6954¢ 2786.6265¢ 3368.5949¢ 4151.2677¢ 4437.9883¢ 4905.2684¢ 5099.2579¢ 5429.8927¢ Error −0.6762¢ +0.7404¢ +0.3128¢ −0.231¢ −0.0503¢ −2.5393¢ +0.313¢ +1.7449¢ +1.6183¢
Optimal ET sequence (TE): 43, 124, 167hi
Badness (Smith TE): 0.02501
Badness (Dirichlet): 1.79049
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¢
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¢
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¢
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
Hemilafa (130 & 181)
See also: 7-limit
Hemilafa was named by FloraC in 2022, after the 5-limit lafa temperament it splits.
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¢
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¢
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¢
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