Yahoo Tuning Groups Ultimate Backup tuning-math 225/225 & 21/20 scales

By requiring that 225/224 be a comma and 21/20 a chroma, I got a number
of equations and inequalities which made it possible to directly search
the wedgies. This involved searching for 225/224 wedgies, of the form

[a, b, 2*b+2*a, d, 2*d+5*a, -2*d+5*b]

and such that |b| <= a. The form insures that 225/224 is a comma, and
the inequality insures that the size of scale with 21/20 as a chroma is
at least as large as the Graham complexity. I searched for a up to 31,
and filtered the result by requiring that the rms error be less than
15, and the geometric badness less than 20000. At the top of the list
is Orwell[18].

It should be noted this does not always work so easily; for instance
225/224 and 15/14 or 441/440 and 21/20 lead to problems. We are also
not guaranteed that everything on this list will actually work out
in practice; I haven't discussed the obstructions much but they do arise.

[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]

bad 1673.187049 comp 25.42062964 rms 2.589237496
graham 11 scale size 18 ratio 1.636364

[6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]

bad 1891.474472 comp 34.26986563 rms 1.610555448
graham 22 scale size 23 ratio 1.045455

[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]

bad 1935.541443 comp 21.62473825 rms 4.139050792
graham 12 scale size 16 ratio 1.333333

[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]

bad 2644.480844 comp 14.72969740 rms 12.18857055
graham 7 scale size 9 ratio 1.285714

[12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]

bad 3656.269932 comp 45.66691577 rms 1.753213831
graham 22 scale size 34 ratio 1.545455

[13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]

bad 3872.384714 comp 48.03151022 rms 1.678518039
graham 23 scale size 29 ratio 1.260870

[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]

bad 3960.691733 comp 33.57393404 rms 3.513715352
graham 18 scale size 25 ratio 1.388889

[13, 2, 30, -27, 11, 64] [[1, 6, 3, 13], [0, -13, -2, -30]]

bad 4674.708063 comp 55.18330983 rms 1.535108242
graham 30 scale size 41 ratio 1.366667

[11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]]

bad 5333.482178 comp 39.77267622 rms 3.371640164
graham 17 scale size 27 ratio 1.588235

[4, 4, 16, -3, 14, 26] [[4, 6, 9, 10], [0, 1, 1, 4]]

bad 5517.328547 comp 24.55359179 rms 9.151636960
graham 16 scale size 16 ratio 1.000000

[14, -6, 16, -42, -14, 54] [[2, 0, 6, 2], [0, 14, -6, 16]]

bad 6692.748195 comp 50.84125928 rms 2.589237496
graham 22 scale size 36 ratio 1.636364

[11, 6, 34, -16, 23, 62] [[1, 1, 2, 1], [0, 11, 6, 34]]

bad 7170.265679 comp 55.28147479 rms 2.346259298
graham 34 scale size 39 ratio 1.147059

[9, -7, 4, -32, -19, 29] [[1, 2, 2, 3], [0, -9, 7, -4]]

bad 7369.743470 comp 33.30471422 rms 6.644173248
graham 16 scale size 20 ratio 1.250000

[19, -5, 28, -52, -9, 79] [[1, -1, 3, -1], [0, 19, -5, 28]]

bad 7433.465748 comp 70.68042097 rms 1.487966281
graham 33 scale size 52 ratio 1.575758

[12, 10, 44, -12, 36, 74] [[2, 0, 2, -6], [0, 12, 10, 44]]

bad 7565.897894 comp 68.53973127 rms 1.610555448
graham 44 scale size 46 ratio 1.045455

[18, -9, 18, -56, -22, 67] [[9, 14, 21, 25], [0, 2, -1, 2]]

bad 7569.695667 comp 65.10901923 rms 1.785649073
graham 27 scale size 45 ratio 1.666667

[9, 5, 28, -13, 19, 51] [[1, 1, 2, 1], [0, 9, 5, 28]]

bad 7646.641388 comp 45.43499477 rms 3.704160176
graham 28 scale size 32 ratio 1.142857

[10, 2, 24, -20, 10, 50] [[2, 0, 4, -2], [0, 10, 2, 24]]

bad 7742.165768 comp 43.24947649 rms 4.139050792
graham 24 scale size 32 ratio 1.333333

[10, -10, 0, -39, -28, 28] [[10, 16, 23, 28], [0, -1, 1, 0]]

bad 7826.501965 comp 39.17133416 rms 5.100713978
graham 20 scale size 20 ratio 1.000000

[19, -17, 4, -71, -47, 57] [[1, -7, 10, 1], [0, 19, -17, 4]]

bad 7855.035658 comp 72.18391247 rms 1.507534728
graham 36 scale size 40 ratio 1.111111

[14, 6, 40, -23, 24, 76] [[2, 4, 5, 8], [0, -7, -3, -20]]

bad 8407.532757 comp 66.79114641 rms 1.884650276
graham 40 scale size 48 ratio 1.200000

[9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]

bad 8811.637847 comp 33.78049801 rms 7.721906590
graham 16 scale size 25 ratio 1.562500

[18, 3, 42, -37, 16, 89] [[3, 5, 7, 9], [0, -6, -1, -14]]

bad 9988.496510 comp 76.80133416 rms 1.693411845
graham 42 scale size 57 ratio 1.357143

[10, -3, 14, -28, -6, 41] [[1, 6, 1, 9], [0, -10, 3, -14]]

bad 10168.63993 comp 36.93793274 rms 7.452769630
graham 17 scale size 27 ratio 1.588235

[20, -13, 14, -67, -34, 69] [[1, -1, 4, 1], [0, 20, -13, 14]]

bad 10378.94397 comp 72.69891283 rms 1.963800357
graham 33 scale size 47 ratio 1.424242

[17, -13, 8, -60, -35, 55] [[1, 2, 2, 3], [0, -17, 13, -8]]

bad 10577.11642 comp 62.75966800 rms 2.685381606
graham 30 scale size 38 ratio 1.266667

[8, -6, 4, -28, -16, 26] [[2, 4, 4, 6], [0, -8, 6, -4]]

bad 10577.92338 comp 29.45939480 rms 12.18857055
graham 14 scale size 18 ratio 1.285714

[15, -2, 26, -38, -1, 66] [[1, 4, 2, 7], [0, -15, 2, -26]]

bad 10687.83946 comp 57.60877001 rms 3.220421490
graham 28 scale size 43 ratio 1.535714

[20, -1, 38, -48, 4, 91] [[1, 8, 2, 15], [0, -20, 1, -38]]

bad 10785.36676 comp 78.79332776 rms 1.737224847
graham 39 scale size 59 ratio 1.512821

[17, -1, 32, -41, 3, 77] [[1, 7, 2, 13], [0, -17, 1, -32]]

bad 10962.29742 comp 66.78311603 rms 2.457922714
graham 33 scale size 50 ratio 1.515152

[7, 2, 18, -13, 9, 36] [[1, 4, 3, 9], [0, -7, -2, -18]]

bad 11045.68014 comp 31.38144021 rms 11.21622520
graham 18 scale size 23 ratio 1.277778

[19, 7, 52, -33, 29, 101] [[1, -2, 1, -7], [0, 19, 7, 52]]

bad 11567.12301 comp 88.27816455 rms 1.484290010
graham 52 scale size 64 ratio 1.230769

[25, -12, 26, -77, -29, 94] [[1, -4, 5, -3], [0, 25, -12, 26]]

bad 12185.51922 comp 90.49265062 rms 1.488049644
graham 38 scale size 63 ratio 1.657895

[15, -14, 2, -57, -39, 44] [[1, 3, 1, 3], [0, -15, 14, -2]]

bad 12223.03079 comp 57.58995263 rms 3.685407058
graham 29 scale size 31 ratio 1.068966

[8, 6, 28, -9, 22, 48] [[2, 5, 6, 12], [0, -4, -3, -14]]

bad 12236.41269 comp 44.04524617 rms 6.307482186
graham 28 scale size 30 ratio 1.071429

[10, 9, 38, -9, 32, 63] [[1, -1, 0, -7], [0, 10, 9, 38]]

bad 12949.78760 comp 58.80133944 rms 3.745313768
graham 38 scale size 39 ratio 1.026316

[16, -5, 22, -45, -10, 65] [[1, 9, 0, 13], [0, -16, 5, -22]]

bad 13459.64211 comp 58.97163144 rms 3.870323144
graham 27 scale size 43 ratio 1.592593

[24, -16, 16, -81, -42, 82] [[8, 12, 19, 22], [0, 3, -2, 2]]

bad 13518.53011 comp 87.38177875 rms 1.770466488
graham 40 scale size 56 ratio 1.400000

[25, -24, 2, -96, -67, 72] [[1, 4, 0, 3], [0, -25, 24, -2]]

bad 13886.41196 comp 96.72994445 rms 1.484117098
graham 49 scale size 51 ratio 1.040816

[15, 10, 50, -19, 37, 88] [[5, 7, 11, 11], [0, 3, 2, 10]]

bad 13983.81149 comp 79.58043642 rms 2.208070460
graham 50 scale size 55 ratio 1.100000

[11, 1, 24, -24, 7, 53] [[1, -2, 2, -5], [0, 11, 1, 24]]

bad 14084.32133 comp 45.56036466 rms 6.785182636
graham 24 scale size 34 ratio 1.416667

[24, -4, 40, -62, -4, 104] [[4, 2, 10, 4], [0, 12, -2, 20]]

bad 14625.07972 comp 91.33383152 rms 1.753213831
graham 44 scale size 68 ratio 1.545455

[7, 4, 22, -10, 15, 40] [[1, 1, 2, 1], [0, 7, 4, 22]]

bad 14634.49544 comp 35.59004417 rms 11.55368971
graham 22 scale size 25 ratio 1.136364

[26, -8, 36, -73, -16, 106] [[2, 2, 5, 4], [0, 13, -4, 18]]

bad 14810.48305 comp 95.90825535 rms 1.610116282
graham 44 scale size 70 ratio 1.590909

[25, 0, 50, -58, 9, 116] [[25, 40, 58, 71], [0, -1, 0, -2]]

bad 14998.42416 comp 100.1661880 rms 1.494869699
graham 50 scale size 75 ratio 1.500000

[21, -9, 24, -63, -21, 81] [[3, 0, 9, 3], [0, 21, -9, 24]]

bad 15058.68343 comp 76.26188891 rms 2.589237496
graham 33 scale size 54 ratio 1.636364

[26, -20, 12, -92, -54, 84] [[2, 4, 4, 6], [0, -26, 20, -12]]

bad 15489.53885 comp 96.06302042 rms 1.678518039
graham 46 scale size 58 ratio 1.260870

[16, 2, 36, -34, 12, 78] [[2, -2, 4, -6], [0, 16, 2, 36]]

bad 15842.76693 comp 67.14786808 rms 3.513715352
graham 36 scale size 50 ratio 1.388889

[11, -11, 0, -43, -31, 31] [[11, 17, 26, 31], [0, 1, -1, 0]]

bad 15984.88332 comp 43.08846759 rms 8.609687186
graham 22 scale size 22 ratio 1.000000

[20, 11, 62, -29, 42, 113] [[1, 1, 2, 1], [0, 20, 11, 62]]

bad 16004.08424 comp 100.7161993 rms 1.577728140
graham 62 scale size 71 ratio 1.145161

[17, 11, 56, -22, 41, 99] [[1, -2, 0, -9], [0, 17, 11, 56]]

bad 16052.65711 comp 89.40526263 rms 2.008263812
graham 56 scale size 62 ratio 1.107143

[21, 3, 48, -44, 17, 103] [[3, 5, 7, 9], [0, -7, -1, -16]]

bad 16311.24569 comp 88.75350222 rms 2.070694980
graham 48 scale size 66 ratio 1.375000

[16, 7, 46, -26, 28, 87] [[1, -6, -1, -19], [0, 16, 7, 46]]

bad 16435.37138 comp 76.63496479 rms 2.798501846
graham 46 scale size 55 ratio 1.195652

[16, -10, 12, -53, -26, 56] [[2, 1, 6, 4], [0, 8, -5, 6]]

bad 16488.92261 comp 58.03978434 rms 4.894864788
graham 26 scale size 38 ratio 1.461538

[15, -9, 12, -49, -23, 53] [[3, 3, 8, 7], [0, 5, -3, 4]]

bad 16785.97003 comp 54.32725934 rms 5.687361716
graham 24 scale size 36 ratio 1.500000

[18, 15, 66, -18, 54, 111] [[3, 0, 3, -9], [0, 18, 15, 66]]

bad 17023.27025 comp 102.8095969 rms 1.610555448
graham 66 scale size 69 ratio 1.045455

[23, -8, 30, -66, -17, 92] [[1, -9, 6, -11], [0, 23, -8, 30]]

bad 17130.95974 comp 84.30066077 rms 2.410569942
graham 38 scale size 61 ratio 1.605263

[15, 3, 36, -30, 15, 75] [[3, 0, 6, -3], [0, 15, 3, 36]]

bad 17419.87298 comp 64.87421474 rms 4.139050792
graham 36 scale size 48 ratio 1.333333

[21, -21, 0, -82, -59, 59] [[21, 33, 49, 59], [0, 1, -1, 0]]

bad 17502.75864 comp 82.25980175 rms 2.586611398
graham 42 scale size 42 ratio 1.000000

[23, -20, 6, -85, -55, 70] [[1, 10, -5, 5], [0, -23, 20, -6]]

bad 17672.69851 comp 86.82372813 rms 2.344369208
graham 43 scale size 49 ratio 1.139535

[26, 4, 60, -54, 22, 128] [[2, 12, 6, 26], [0, -26, -4, -60]]

bad 18698.83223 comp 110.3666196 rms 1.535108242
graham 60 scale size 82 ratio 1.366667

[14, -1, 26, -34, 2, 63] [[1, 6, 2, 11], [0, -14, 1, -26]]

bad 18845.84375 comp 54.77717330 rms 6.280820060
graham 27 scale size 41 ratio 1.518519

[12, -7, 10, -39, -18, 43] [[1, -3, 5, -1], [0, 12, -7, 10]]

bad 19574.18705 comp 43.42843741 rms 10.37851763
graham 19 scale size 29 ratio 1.526316

[12, 5, 34, -20, 20, 65] [[1, -4, 0, -13], [0, 12, 5, 34]]

bad 19840.40051 comp 56.94920411 rms 6.117516042
graham 34 scale size 41 ratio 1.205882

225/225 & 21/20 scales

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>