User:Recentlymaterialized/Miscellaneous rank-2 temperaments: Difference between revisions

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== Whooshlet (441 & 3072) ==
== Whooshlet (441 & 3072) ==
''Main article: [[Whoosh]]''


[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]
[[Subgroup]]: [[2.3.5.7.13_subgroup|2.3.5.7.13]]
Line 1,021: Line 1,022:
! 5/1
! 5/1
! 7/1
! 7/1
! 11/1
! 13/1
! 13/1
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Revision as of 22:08, 6 January 2025

Senscape (27e & 87)

Subgroup: 2.3.5.7.11.13

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 400.3122Template:Cc, 220.5893Template:Cc 399.994Template:Cc, 220.6882Template:Cc 399.9947Template:Cc, 220.6886Template:Cc
Constrained ??? 1\3, 220.729Template:Cc 1\3, 220.6899Template:Cc 1\3, 220.6873Template:Cc
Destretched ??? 1\3, 220.4173Template:Cc 1\3, 220.6915Template:Cc 1\3, 220.6915Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.982Template:Cc 1903.4292Template:Cc 2786.1822Template:Cc 3365.482Template:Cc 4151.6942Template:Cc 4441.3165Template:Cc
Error -0.018Template:Cc +1.4742Template:Cc −0.1315Template:Cc −3.3439Template:Cc +0.3762Template:Cc +0.7889Template:Cc

Optimal ET sequence (TE): 27e, 60e, 87

Badness (Smith TE): 0.02560

Badness (Dirichlet): 1.05783

Tripierce (b30 & b39)

Subgroup: 3.5.7.13

Comma list: 245/243, 2197/2187

Mapping: [⟨3 1 7 7], ⟨0 4 -2 0]]

Generator preimages: 13/9, 13/7

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? ??? 634.5206Template:Cc, 1075.4459Template:Cc 634.5215Template:Cc, 1075.4475Template:Cc
Constrained ??? 1\b3, 1074.0886Template:Cc 1\b3, 1074.6634Template:Cc 1\b3, 1075.1282Template:Cc
Destretched ??? ??? 1\b3, 1074.5381Template:Cc 1\b3, 1074.5382Template:Cc

Prime harmonics (WE):

Prime 3/1 5/1 7/1 13/1
Tuning 1903.5617Template:Cc 2785.4124Template:Cc 3366.1979Template:Cc 4441.6439Template:Cc
Error +1.6067Template:Cc −0.9013Template:Cc −2.628Template:Cc +1.1162Template:Cc

Optimal ET sequence (TE): b9, b21, b30, b39, b69, b108, b177d

Badness (Smith TE): 0.00287

Badness (Dirichlet): 0.26872

Trisemipierce (b39 & b60)

Subgroup: 3.5.7.11.13

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 634.4577Template:Cc, 537.4447Template:Cc 634.5219Template:Cc, 537.7266Template:Cc 634.5229Template:Cc, 537.7274Template:Cc
Constrained ??? 1\b3, 537.0443Template:Cc 1\b3, 537.3589Template:Cc 1\b3, 537.7233Template:Cc
Destretched ??? 1\b3, 537.0443Template:Cc 1\b3, 537.2716Template:Cc 1\b3, 537.2716Template:Cc

Prime harmonics (WE):

Prime 3/1 5/1 7/1 11/1 13/1
Tuning 1903.5658Template:Cc 2785.4282Template:Cc 3366.2003Template:Cc 4151.2674Template:Cc 4441.6534Template:Cc
Error +1.6108Template:Cc −0.8855Template:Cc −2.6256Template:Cc −0.0505Template:Cc +1.1258Template:Cc

Optimal ET sequence (TE): b18, b21, b39, b99, b138, b177d

Badness (Smith TE): 0.00566

Badness (Dirichlet): 0.44749

Tridubhe (b69 & b78)

Subgroup: 3.5.7.13.17.19

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 634.4242Template:Cc, 220.2042Template:Cc 634.3781Template:Cc, 220.1438Template:Cc 634.3798Template:Cc, 220.1444Template:Cc
Constrained ??? 1\b3, 220.0518Template:Cc 1\b3, 220.0811Template:Cc 1\b3, 220.4087Template:Cc
Destretched ??? 1\b3, 220.0518Template:Cc 1\b3, 220.0074Template:Cc 1\b3, 220.0074Template:Cc

Prime harmonics (WE):

Prime 3/1 5/1 7/1 13/1 17/1 19/1
Tuning 1903.1344Template:Cc 2783.7095Template:Cc 3365.9811Template:Cc 4440.6468Template:Cc 4906.9877Template:Cc 5101.0782Template:Cc
Error +1.1794Template:Cc −2.6042Template:Cc −2.8448Template:Cc +0.1192Template:Cc +2.0323Template:Cc +3.5652Template:Cc

Optimal ET sequence (TE): b51, b60, b69, b147

Badness (Smith TE): 0.00523

Badness (Dirichlet): 0.48415

Trisemidubhe (b60 & b78)

Subgroup: 3.5.7.11.13.17.19

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? ??? 317.2028Template:Cc, 537.3966Template:Cc 317.2037Template:Cc, 537.3981Template:Cc
Constrained ??? 1\b6, 537.0443Template:Cc 1\b6, 537.1261Template:Cc 1\b6, 537.5291Template:Cc
Destretched ??? ??? 1\b6, 1074.0806Template:Cc 1\b6, 1074.0806Template:Cc

Prime harmonics (WE):

Prime 3/1 5/1 7/1 11/1 13/1 17/1 19/1
Tuning 1903.2168Template:Cc 2783.9919Template:Cc 3366.0459Template:Cc 4149.8121Template:Cc 4440.8391Template:Cc 4907.4024Template:Cc 5101.4204Template:Cc
Error +1.2618Template:Cc −2.3218Template:Cc −2.780Template:Cc −1.5059Template:Cc +0.3114Template:Cc +2.447Template:Cc +3.9074Template:Cc

Optimal ET sequence (TE): b18, b60, b78, b138, b216

Badness (Smith TE): 0.01046

Badness (Dirichlet): 0.95032

Badtone (31 & 67)

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]

Generator preimages: 2/1, 8/7

Generator tunings:

Weil Tenney WE TE
Optimized ??? 1200.4954Template:Cc, 232.6290Template:Cc 1199.9347Template:Cc, 232.6275Template:Cc 1199.9460Template:Cc, 232.6297Template:Cc
Constrained ??? 1\1, 232.6732Template:Cc 1\1, 232.6392Template:Cc 1\1, 232.6352Template:Cc
Destretched ??? 1\1, 232.5330Template:Cc 1\1, 232.6402Template:Cc 1\1, 232.6401Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.9347Template:Cc 1897.817Template:Cc 2791.5295Template:Cc 3367.1765Template:Cc 4150.4968Template:Cc 4444.8759Template:Cc
Error −0.0653Template:Cc −4.138Template:Cc +5.2157Template:Cc −1.6494Template:Cc −0.8211Template:Cc +4.3483Template:Cc

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.

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 176/175, 196/195, 364/363

Mapping: [⟨1 0 -4 -32 -44 -58], ⟨0 1 4 22 30 39]]

Generator preimages: 2/1, 3/1

Generator tunings:

Weil Tenney WE TE
Optimized ??? 1199.6846Template:Cc, 1897.7889Template:Cc 1199.7882Template:Cc, 1898.1434Template:Cc 1199.8021Template:Cc, 1898.1654Template:Cc
Constrained ??? 1\1, 1898.2217Template:Cc 1\1, 698.4715Template:Cc 1\1, 698.4553Template:Cc
Destretched ??? 1\1, 1898.2878Template:Cc 1\1, 1898.4785Template:Cc 1\1, 1898.4785Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.7882Template:Cc 1898.1434Template:Cc 2793.4206Template:Cc 3365.9316Template:Cc 4153.6201Template:Cc 4439.8756Template:Cc
Error −0.2118Template:Cc −3.8116Template:Cc +7.1069Template:Cc −2.8943Template:Cc +2.3022Template:Cc −0.6521Template:Cc

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 preimages: 2/1, 3/1

Generator tunings (no-17 no-19):

Weil Tenney WE TE
Optimized ??? 1201.1304Template:Cc, 1900.1656Template:Cc 1200.9840Template:Cc, 1899.9400Template:Cc 1200.9879Template:Cc, 1899.9461Template:Cc
Constrained ??? 1\1, 1898.6199Template:Cc 1\1, 1898.4197Template:Cc 1\1, 1898.4955Template:Cc
Destretched ??? 1\1, 1898.3773Template:Cc 1\1, 1898.3833Template:Cc 1\1, 1898.3833Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 7/1 11/1 13/1 23/1 31/1 37/1 41/1
Tuning 1200.984Template:Cc 1899.940Template:Cc 3367.1912Template:Cc 4154.903Template:Cc 4440.5866Template:Cc 5425.2263Template:Cc 5946.6708Template:Cc 6251.7709Template:Cc 6429.2824Template:Cc
Error +0.984Template:Cc −2.015Template:Cc −1.6347Template:Cc +3.585Template:Cc +0.059Template:Cc −3.0481Template:Cc +1.6352Template:Cc +0.4269Template:Cc +0.220Template:Cc

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)

Subgroup: 2.3.5.7.13(.23)

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 171.4255Template:Cc, 231.2847Template:Cc 171.4244Template:Cc, 231.2902Template:Cc 171.4244Template:Cc, 231.2902Template:Cc
Constrained ??? 1\7, 231.2969Template:Cc 1\7, 231.2963Template:Cc 1\7, 231.2991Template:Cc
Destretched ??? 1\7, 231.2888Template:Cc 1\7, 231.2943Template:Cc 1\7, 231.2943Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 13/1 23/1
Tuning 1199.971Template:Cc 1902.0144Template:Cc 2786.3108Template:Cc 3368.6227Template:Cc 4440.6893Template:Cc 5428.3714Template:Cc
Error −0.029Template:Cc +0.0594Template:Cc −0.0029Template:Cc −0.2032Template:Cc +0.1616Template:Cc +0.097Template:Cc

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

Subgroup: 2.3.5.7.11.13(.23)

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 171.4227, 231.3059Template:Cc 171.4242, 231.3085Template:Cc 171.4242, 231.3085Template:Cc
Constrained ??? 1\7, 231.3138Template:Cc 1\7, 231.3149Template:Cc 1\7, 231.3180Template:Cc
Destretched ??? 1\7, 231.3138Template:Cc 1\7, 231.3144Template:Cc 1\7, 231.3144Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 23/1
Tuning 1199.9697Template:Cc 1902.1239Template:Cc 2786.2149Template:Cc 3368.6006Template:Cc 4151.6642Template:Cc 4440.5731Template:Cc 5427.9754Template:Cc
Error −0.0303Template:Cc +0.1689Template:Cc −0.0988Template:Cc −0.2253Template:Cc +0.3463Template:Cc +0.0454Template:Cc −0.2989Template:Cc

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)

Subgroup: 2.3.5.7.11.13.17.19

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 ??? 171.4266Template:Cc, 37.1864Template:Cc 171.4244Template:Cc, 37.1884Template:Cc 171.4244Template:Cc, 37.1884Template:Cc
Constrained ??? 1\7, 37.1867Template:Cc 1\7, 37.1893Template:Cc 1\7, 37.1893Template:Cc
Destretched ??? 1\7, 37.1868Template:Cc 1\7, 37.1893Template:Cc 1\7, 37.1893Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1 17/1 19/1
Tuning ( ¢) 1199.97Template:Cc 1901.97Template:Cc 2786.34Template:Cc 3368.63Template:Cc 4151.37Template:Cc 4440.73Template:Cc 4905.08Template:Cc 5097.39Template:Cc
Error −0.0291Template:Cc +0.0195Template:Cc +0.0302Template:Cc −0.1966Template:Cc +0.0566Template:Cc +0.2014Template:Cc +0.1289Template:Cc −0.1218Template:Cc

Optimal ET sequence (TE): 161, 420, 581, 742, 1323

Badness (Smith TE): 0.01260

Badness (Dirichlet): 0.76654

130 & 181

See also: 7-limit

Subgroup: 2.3.5.7.13

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 1199.983Template:Cc, 470.7428Template:Cc 1199.9727Template:Cc, 470.7388Template:Cc 1199.9727Template:Cc, 470.7388Template:Cc
Constrained ??? 1\1, 470.7495Template:Cc 1\1, 470.7495Template:Cc 1\1, 470.7493Template:Cc
Destretched ??? 1\1, 470.7495Template:Cc 1\1, 470.7495Template:Cc 1\1, 470.7495Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 13/1
Tuning 1199.9727Template:Cc 1901.9687Template:Cc 2786.4055Template:Cc 3368.6607Template:Cc 4440.6683Template:Cc
Error −0.0273Template:Cc +0.0137Template:Cc +0.0918Template:Cc −0.1652Template:Cc +0.1407Template:Cc

Optimal ET sequence (TE): 51, 79c, 130, 311, 441, 1453, 1894, 2335

Badness (Smith TE): 0.01381

Badness (Dirichlet): 0.47741

13-limit

Subgroup: 2.3.5.7.11.13

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 1199.9305Template:Cc, 470.7167Template:Cc 1199.9386Template:Cc, 470.7221Template:Cc 1199.9387Template:Cc, 470.7221Template:Cc
Constrained ??? 1\1, 470.7411Template:Cc 1\1, 470.746Template:Cc 1\1, 470.745Template:Cc
Destretched ??? 1\1, 470.744Template:Cc 1\1, 470.7462Template:Cc 1\1, 470.7461Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 11/1 13/1
Tuning 1199.9386Template:Cc 1901.9946Template:Cc 2786.1202Template:Cc 3368.482Template:Cc 4152.0197Template:Cc 4440.6721Template:Cc
Error −0.0614Template:Cc +0.0396Template:Cc −0.1936Template:Cc −0.3439Template:Cc +0.7017Template:Cc +0.1445Template:Cc

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

Subgroup: 2.3.5.7.13

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

Generator tunings:

Norm Weil Tenney WE TE
Optimized ??? 400.0024Template:Cc, 160.5488Template:Cc 400.0034Template:Cc, 160.5495Template:Cc 400.0034Template:Cc, 160.5495Template:Cc
Constrained ??? 1\, 160.5478Template:Cc 1\, 160.5480Template:Cc 1\, 160.5476Template:Cc
Destretched ??? 1\3, 160.5478Template:Cc 1\3, 160.5481Template:Cc 1\3, 160.5481Template:Cc

Prime harmonics (WE):

Prime 2/1 3/1 5/1 7/1 13/1
Tuning 1200.0102Template:Cc 1901.9291Template:Cc 2786.3213Template:Cc 3368.7981Template:Cc 4440.575Template:Cc
Error +0.0102Template:Cc −0.0259Template:Cc +0.0076Template:Cc −0.0278Template:Cc +0.0473Template:Cc

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


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