114 7-limit temperaments

This is a list of linear temperaments with top complexity < 15, top
error < 15, and top badness < 100. I searched extensively without
adding to the list, which is probably complete. Most of the names are
old ones. In some cases I extended a 5-limit name to what seemed like
the appropriate 7-limit temperament, and in the case of The
Temperament Formerly Known as Duodecimal, am suggesting Waage or
Compton if one of these gentlemen invented it. There are a few new
names being suggested, none of which are yet etched in stone--not even
when the name is Bond, James Bond.

Number 1 Ennealimmal

[18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
TOP generators [133.3373752, 49.02398564]
bad: 4.918774 comp: 11.628267 err: .036377

Number 2 Meantone

[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
TOP generators [1201.698520, 504.1341314]
bad: 21.551439 comp: 3.562072 err: 1.698521

Number 3 Magic

[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
TOP generators [1201.276744, 380.7957184]
bad: 23.327687 comp: 4.274486 err: 1.276744

Number 4 Beep

[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
TOP generators [1194.642673, 254.8994697]
bad: 23.664749 comp: 1.292030 err: 14.176105

Number 5 Augmented

[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
TOP generators [399.9922103, 107.3111730]
bad: 27.081145 comp: 2.147741 err: 5.870879

Number 6 Pajara

[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
TOP generators [598.4467109, 106.5665459]
bad: 27.754421 comp: 2.988993 err: 3.106578

Number 7 Dominant Seventh

[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
TOP generators [1195.228951, 495.8810151]
bad: 28.744957 comp: 2.454561 err: 4.771049

Number 8 Schismic

[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
TOP generators [1200.760624, 498.1193303]
bad: 28.818558 comp: 5.618543 err: .912904

Number 9 Miracle

[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
TOP generators [1200.631014, 116.7206423]
bad: 29.119472 comp: 6.793166 err: .631014

Number 10 Orwell

[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
TOP generators [1199.532657, 271.4936472]
bad: 30.805067 comp: 5.706260 err: .946061

Number 11 Hemiwuerschmidt

[16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
TOP generators [1199.692003, 193.8224275]
bad: 31.386908 comp: 10.094876 err: .307997

Number 12 Catakleismic

[6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
TOP generators [1200.536355, 316.9063960]
bad: 32.938503 comp: 7.836558 err: .536356

Number 13 Father

[1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
TOP generators [1185.869125, 447.3863410]
bad: 33.256527 comp: 1.534101 err: 14.130876

Number 14 Blackwood

[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
TOP generators [239.1786927, 83.83059859]
bad: 34.210608 comp: 2.173813 err: 7.239629

Number 15 Semisixths

[7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
TOP generators [1198.389531, 443.1602931]
bad: 34.533812 comp: 4.630693 err: 1.610469

Number 16 Hemififths

[2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
TOP tuning [1199.700353, 1902.429930, 2785.617954, 3368.041901]
TOP generators [1199.700353, 351.3647888]
bad: 34.737019 comp: 10.766914 err: .299647

Number 17 Diminished

[4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
TOP generators [298.5321149, 101.4561401]
bad: 37.396767 comp: 2.523719 err: 5.871540

Number 18 Amity

[5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
TOP tuning [1199.723894, 1902.392618, 2786.717797, 3369.601033]
TOP generators [1199.723894, 339.3558130]
bad: 37.532790 comp: 11.659166 err: .276106

Number 19 Pelogic

[1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
TOP generators [1209.734056, 532.9412251]
bad: 39.824125 comp: 2.022675 err: 9.734056

Number 20 Parakleismic

[13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]]
TOP tuning [1199.738066, 1902.291445, 2786.921905, 3368.090564]
TOP generators [1199.738066, 315.1076065]
bad: 40.713036 comp: 12.467252 err: .261934

Number 21 {21/20, 28/27}

[1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
TOP generators [1214.253642, 509.4012304]
bad: 42.300772 comp: 1.722706 err: 14.253642

Number 22 Injera

[2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
TOP generators [600.8889070, 93.60982493]
bad: 42.529834 comp: 3.445412 err: 3.582707

Number 23 Dicot

[2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
TOP generators [1204.048159, 356.3998255]
bad: 42.920570 comp: 2.137243 err: 9.396316

Number 24 Hemifourths

[2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
TOP generators [1203.668841, 252.4803582]
bad: 43.552336 comp: 3.445412 err: 3.668842

Number 25 Waage? Compton? Duodecimal?

[0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
TOP generators [100.0514209, 16.55882096]
bad: 45.097159 comp: 8.548972 err: .617051

Number 26 Wizard

[12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
TOP generators [600.3197857, 216.7702531]
bad: 45.381303 comp: 8.423526 err: .639571

Number 27 Kleismic

[6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
TOP generators [1203.187309, 317.8344609]
bad: 45.676063 comp: 3.785579 err: 3.187309

Number 28 Negri

[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
TOP generators [1203.187309, 124.8419629]
bad: 46.125886 comp: 3.804173 err: 3.187309

Number 29 Nonkleismic

[10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
TOP generators [1198.828458, 309.8926610]
bad: 46.635848 comp: 6.309298 err: 1.171542

Number 30 Quartaminorthirds

[9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
TOP generators [1199.792743, 77.83315314]
bad: 47.721352 comp: 6.742251 err: 1.049791

Number 31 Tripletone

[3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
TOP generators [399.0200131, 92.45965769]
bad: 48.112067 comp: 4.045351 err: 2.939961

Number 32 Decimal

[4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
TOP generators [603.8288989, 250.6116362]
bad: 48.773723 comp: 2.523719 err: 7.657798

Number 33 {1029/1024, 4375/4374}

[12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]]
TOP tuning [1200.421488, 1901.286959, 2785.446889, 3367.642640]
TOP generators [600.2107440, 183.2944602]
bad: 50.004574 comp: 10.892116 err: .421488

Number 34 Superpythagorean

[1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
TOP generators [1197.596121, 489.4271829]
bad: 50.917015 comp: 4.602303 err: 2.403879

Number 35 Supermajor seconds

[3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
TOP generators [1201.698520, 232.5214630]
bad: 51.806440 comp: 5.522763 err: 1.698521

Number 36 Supersupermajor

[3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
TOP generators [1200.231587, 234.3804692]
bad: 52.638504 comp: 7.670504 err: .894655

Number 37 {6144/6125, 10976/10935} Hendecatonic?

[11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]]
TOP tuning [1199.662182, 1902.490429, 2787.098101, 3368.740066]
TOP generators [109.0601984, 48.46705632]
bad: 53.458690 comp: 12.579627 err: .337818

Number 38 {3136/3125, 5120/5103} Misty

[3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]]
TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021]
TOP generators [399.8871550, 96.94420930]
bad: 53.622498 comp: 12.585536 err: .338535

Number 39 {1728/1715, 4000/3993}

[11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
TOP generators [1199.083445, 45.17026643]
bad: 55.081549 comp: 7.752178 err: .916555

Number 40 {36/35, 160/147} Hystrix?

[3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]]
TOP tuning [1187.933715, 1892.564743, 2758.296667, 3402.700250]
TOP generators [1187.933715, 161.1008955]
bad: 55.952057 comp: 2.153383 err: 12.066285

Number 41 {28/27, 50/49}

[2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
TOP generators [595.7998193, 127.8698005]
bad: 56.092257 comp: 2.584059 err: 8.400361

Number 42 Porcupine

[3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
TOP generators [1196.905960, 162.3176609]
bad: 57.088650 comp: 4.295482 err: 3.094040

Number 43

[6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
TOP generators [598.4467109, 162.3159606]
bad: 57.621529 comp: 4.306766 err: 3.106578

Number 44 Octacot

[8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
TOP generators [1199.031259, 88.05739491]
bad: 58.217715 comp: 7.752178 err: .968741

Number 45 {25/24, 81/80} Jamesbond?

[0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
TOP generators [172.7759159, 86.69241190]
bad: 58.637859 comp: 2.493450 err: 9.431411

Number 46 Hemithirds

[15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
TOP tuning [1200.363229, 1901.194685, 2787.427555, 3367.479202]
TOP generators [1200.363229, 193.3505488]
bad: 60.573479 comp: 11.237086 err: .479706

Number 47

[12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]]
TOP tuning [1200.284965, 1901.503343, 2786.975381, 3369.219732]
TOP generators [600.1424823, 83.17776441]
bad: 61.101493 comp: 14.643003 err: .284965

Number 48 Flattone

[1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
TOP generators [1202.536419, 507.1379663]
bad: 61.126418 comp: 4.909123 err: 2.536420

Number 49 Diaschismic

[2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
TOP generators [599.3662015, 103.7870123]
bad: 61.527901 comp: 6.966993 err: 1.267597

Number 50 Superkleismic

[9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
TOP generators [1201.371918, 322.3731369]
bad: 62.364585 comp: 6.742251 err: 1.371918

Number 51

[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
TOP generators [1201.135545, 387.5841360]
bad: 62.703297 comp: 6.411729 err: 1.525246

Number 52 Tritonic

[5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]]
TOP tuning [1201.023211, 1900.333250, 2785.201472, 3365.953391]
TOP generators [1201.023211, 580.7519186]
bad: 63.536850 comp: 7.880073 err: 1.023211

Number 53

[1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]]
TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869]
TOP generators [1199.680495, 497.2520023]
bad: 64.536886 comp: 14.212326 err: .319505

Number 54

[6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
TOP generators [1202.659696, 82.97467050]
bad: 64.556006 comp: 4.306766 err: 3.480440

Number 55

[0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
TOP generators [99.80617249, 24.58395811]
bad: 65.630949 comp: 4.295482 err: 3.557008

Number 56

[2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
TOP generators [1204.567524, 355.9419091]
bad: 66.522610 comp: 2.696901 err: 9.146173

Number 57

[2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
TOP generators [1185.869125, 223.6931705]
bad: 66.774944 comp: 2.173813 err: 14.130876

Number 58

[5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
TOP generators [1194.335372, 99.13879319]
bad: 67.244049 comp: 3.445412 err: 5.664628

Number 59

[3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
TOP generators [1193.415676, 158.1468146]
bad: 67.670842 comp: 3.205865 err: 6.584324

Number 60

[3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
TOP generators [397.8052253, 76.63521863]
bad: 68.337269 comp: 3.221612 err: 6.584324

Number 61 Hemikleismic

[12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]]
TOP tuning [1199.411231, 1902.888178, 2785.151380, 3370.478790]
TOP generators [1199.411231, 158.5740148]
bad: 68.516458 comp: 10.787602 err: .588769

Number 62

[2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
TOP generators [592.7342285, 146.7842660]
bad: 68.668284 comp: 2.173813 err: 14.531543

Number 63

[8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]]
TOP tuning [1198.975478, 1900.576277, 2788.692580, 3365.949709]
TOP generators [1198.975478, 62.17183489]
bad: 68.767371 comp: 8.192765 err: 1.024522

Number 64

[3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
TOP generators [1202.900537, 570.4479508]
bad: 69.388565 comp: 4.891080 err: 2.900537

Number 65

[3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
TOP generators [1202.624742, 569.0491468]
bad: 70.105427 comp: 5.168119 err: 2.624742

Number 66

[17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
TOP tuning [1199.379215, 1900.971080, 2787.482526, 3370.568669]
TOP generators [1199.379215, 464.5804210]
bad: 71.416917 comp: 10.725806 err: .620785

Number 67

[11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]]
TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447]
TOP generators [1198.514750, 154.1766650]
bad: 71.539673 comp: 6.940227 err: 1.485250

Number 68

[3, -24, -1, -45, -10, 65] [[1, 1, 7, 3], [0, 3, -24, -1]]
TOP tuning [1200.486331, 1902.481504, 2787.442939, 3367.460603]
TOP generators [1200.486331, 233.9983907]
bad: 72.714599 comp: 12.227699 err: .486331

Number 69

[23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]]
TOP tuning [1199.671611, 1901.434518, 2786.108874, 3369.747810]
TOP generators [1199.671611, 386.7656515]
bad: 73.346343 comp: 14.944966 err: .328389

Number 70

[6, 29, -2, 32, -20, -86] [[1, 4, 14, 2], [0, -6, -29, 2]]
TOP tuning [1200.422358, 1901.285580, 2787.294397, 3367.645998]
TOP generators [1200.422357, 483.4006416]
bad: 73.516606 comp: 13.193267 err: .422358

Number 71

[7, -15, -16, -40, -45, 5] [[1, 5, -5, -5], [0, -7, 15, 16]]
TOP tuning [1200.210742, 1900.961474, 2784.858222, 3370.585685]
TOP generators [1200.210742, 585.7274621]
bad: 74.053446 comp: 10.869066 err: .626846

Number 72

[5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]]
TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008]
TOP generators [1192.540126, 139.5182509]
bad: 74.239244 comp: 3.154649 err: 7.459874

Number 73

[4, 21, -3, 24, -16, -66] [[1, 0, -6, 4], [0, 4, 21, -3]]
TOP tuning [1199.274449, 1901.646683, 2787.998389, 3370.862785]
TOP generators [1199.274449, 475.4116708]
bad: 74.381278 comp: 10.125066 err: .725551

Number 74

[3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
TOP generators [1195.486066, 559.3589487]
bad: 74.989802 comp: 4.075900 err: 4.513934

Number 75

[6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
TOP generators [399.8000105, 155.5708520]
bad: 76.576420 comp: 3.804173 err: 5.291448

Number 76

[13, 2, 30, -27, 11, 64] [[1, 6, 3, 13], [0, -13, -2, -30]]
TOP tuning [1200.672456, 1900.889183, 2786.148822, 3370.713730]
TOP generators [1200.672456, 407.9342733]
bad: 76.791305 comp: 10.686216 err: .672456

Number 77 Shrutar

[4, -8, 14, -22, 11, 55] [[2, 3, 5, 5], [0, 2, -4, 7]]
TOP tuning [1198.920873, 1903.665377, 2786.734051, 3365.796415]
TOP generators [599.4604367, 52.64203308]
bad: 76.825572 comp: 8.437555 err: 1.079127

Number 78

[12, 10, 25, -12, 6, 30] [[1, 6, 6, 12], [0, -12, -10, -25]]
TOP tuning [1199.028703, 1903.494472, 2785.274095, 3366.099130]
TOP generators [1199.028703, 440.8898120]
bad: 77.026097 comp: 8.905180 err: .971298

Number 79 Beatles

[2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
TOP generators [1197.104145, 354.7203384]
bad: 77.187771 comp: 5.162806 err: 2.895855

Number 80

[6, -12, 10, -33, -1, 57] [[2, 4, 3, 7], [0, -3, 6, -5]]
TOP tuning [1199.025947, 1903.033657, 2788.575394, 3371.560420]
TOP generators [599.5129735, 165.0060791]
bad: 78.320453 comp: 8.966980 err: .974054

Number 81

[4, 4, 0, -3, -11, -11] [[4, 6, 9, 11], [0, 1, 1, 0]]
TOP tuning [1212.384652, 1905.781495, 2815.069985, 3334.057793]
TOP generators [303.0961630, 63.74881402]
bad: 78.879803 comp: 2.523719 err: 12.384652

Number 82

[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
TOP generators [601.7004928, 230.8749260]
bad: 79.825592 comp: 4.619353 err: 3.740932

Number 83

[1, 6, 5, 7, 5, -5] [[1, 2, 5, 5], [0, -1, -6, -5]]
TOP tuning [1211.970043, 1882.982932, 2814.107292, 3355.064446]
TOP generators [1211.970043, 540.9571536]
bad: 79.928319 comp: 2.584059 err: 11.970043

Number 84 Squares

[4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
TOP generators [1201.698520, 426.4581630]
bad: 80.651668 comp: 6.890825 err: 1.698521

Number 85

[6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
TOP generators [199.0788921, 88.83392059]
bad: 80.672767 comp: 3.820609 err: 5.526647

Number 86

[7, 26, 25, 25, 20, -15] [[1, 5, 15, 15], [0, -7, -26, -25]]
TOP tuning [1199.352846, 1902.980716, 2784.811068, 3369.637284]
TOP generators [1199.352846, 584.8262161]
bad: 81.144087 comp: 11.197591 err: .647154

Number 87

[18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]]
TOP tuning [1200.448679, 1901.787880, 2785.271912, 3367.566305]
TOP generators [400.1495598, 83.18491309]
bad: 81.584166 comp: 13.484503 err: .448679

Number 88

[9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]
TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837]
TOP generators [1201.918557, 189.0109248]
bad: 81.594641 comp: 6.521440 err: 1.918557

Number 89

[1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
TOP generators [1195.155395, 496.2398890]
bad: 82.638059 comp: 4.075900 err: 4.974313

Number 90

[3, 7, -1, 4, -10, -22] [[1, 1, 1, 3], [0, 3, 7, -1]]
TOP tuning [1205.820043, 1890.417958, 2803.215176, 3389.260823]
TOP generators [1205.820043, 228.1993049]
bad: 82.914167 comp: 3.375022 err: 7.279064

Number 91

[6, 5, -31, -6, -66, -86] [[1, 0, 1, 11], [0, 6, 5, -31]]
TOP tuning [1199.976626, 1902.553087, 2785.437532, 3369.885264]
TOP generators [1199.976626, 317.0921813]
bad: 83.023430 comp: 14.832953 err: .377351

Number 92

[8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
TOP generators [599.2769413, 272.3123381]
bad: 83.268810 comp: 5.047438 err: 3.268439

Number 93

[4, 2, 9, -6, 3, 15] [[1, 3, 3, 6], [0, -4, -2, -9]]
TOP tuning [1208.170435, 1910.173796, 2767.342550, 3391.763218]
TOP generators [1208.170435, 428.5843770]
bad: 83.972208 comp: 3.205865 err: 8.170435

Number 94 Hexidecimal

[1, -3, 5, -7, 5, 20] [[1, 2, 1, 5], [0, -1, 3, -5]]
TOP tuning [1208.959294, 1887.754858, 2799.450479, 3393.977822]
TOP generators [1208.959293, 530.1637287]
bad: 84.341555 comp: 3.068202 err: 8.959294

Number 95

[6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
TOP generators [399.0200131, 46.12154491]
bad: 84.758945 comp: 5.369353 err: 2.939961

Number 96

[0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
TOP generators [99.83462277, 16.52294019]
bad: 85.896401 comp: 5.168119 err: 3.215955

Number 97

[11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]]
TOP tuning [1200.950404, 1901.347958, 2784.106944, 3366.157786]
TOP generators [1200.950404, 263.8594234]
bad: 85.962459 comp: 9.510433 err: .950404

Number 98 Slender

[13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]
TOP tuning [1200.337238, 1901.055858, 2784.996493, 3370.418508]
TOP generators [1200.337239, 38.43220154]
bad: 88.631905 comp: 12.499426 err: .567296

Number 99

[0, 5, 10, 8, 16, 9] [[5, 8, 12, 15], [0, 0, -1, -2]]
TOP tuning [1195.598382, 1912.957411, 2770.195472, 3388.313857]
TOP generators [239.1196765, 99.24064453]
bad: 89.758630 comp: 3.595867 err: 6.941749

Number 100

[1, -1, -5, -4, -11, -9] [[1, 2, 2, 1], [0, -1, 1, 5]]
TOP tuning [1185.210905, 1925.395162, 2815.448458, 3410.344145]
TOP generators [1185.210905, 445.0266480]
bad: 90.384580 comp: 2.472159 err: 14.789095

Number 101

[2, 8, -11, 8, -23, -48] [[1, 1, 0, 6], [0, 2, 8, -11]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3373.586984]
TOP generators [1201.698520, 348.7821945]
bad: 92.100337 comp: 7.363684 err: 1.698521

Number 102

[3, 12, 18, 12, 20, 8] [[3, 5, 8, 10], [0, -1, -4, -6]]
TOP tuning [1202.260038, 1898.372926, 2784.451552, 3375.170635]
TOP generators [400.7533459, 105.3938041]
bad: 92.910783 comp: 6.411729 err: 2.260038

Number 103

[4, -8, -20, -22, -43, -24] [[4, 6, 10, 13], [0, 1, -2, -5]]
TOP tuning [1199.003867, 1903.533834, 2787.453602, 3371.622404]
TOP generators [299.7509668, 105.0280329]
bad: 93.029698 comp: 9.663894 err: .996133

Number 104

[3, 0, -3, -7, -13, -7] [[3, 5, 7, 8], [0, -1, 0, 1]]
TOP tuning [1205.132027, 1884.438632, 2811.974729, 3337.800149]
TOP generators [401.7106756, 124.1147448]
bad: 94.336372 comp: 2.921642 err: 11.051598

Number 105

[4, 7, 2, 2, -8, -15] [[1, 2, 3, 3], [0, -4, -7, -2]]
TOP tuning [1190.204869, 1918.438775, 2762.165422, 3339.629125]
TOP generators [1190.204869, 115.4927407]
bad: 94.522719 comp: 3.014736 err: 10.400103

Number 106

[13, 19, 23, 0, 0, 0] [[1, 0, 0, 0], [0, 13, 19, 23]]
TOP tuning [1200.0, 1904.187463, 2783.043215, 3368.947050]
TOP generators [1200., 146.4759587]
bad: 94.757554 comp: 8.202087 err: 1.408527

Number 107

[2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]]
TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246]
TOP generators [603.2741324, 81.75384943]
bad: 94.764743 comp: 3.804173 err: 6.548265

Number 108

[2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]]
TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246]
TOP generators [603.2741324, 81.75384943]
bad: 94.764743 comp: 3.804173 err: 6.548265

Number 109

[1, -13, -2, -23, -6, 32] [[1, 2, -3, 2], [0, -1, 13, 2]]
TOP tuning [1197.567789, 1904.876372, 2780.666293, 3375.653987]
TOP generators [1197.567789, 490.2592046]
bad: 94.999539 comp: 6.249713 err: 2.432212

Number 110

[9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
TOP generators [133.0066710, 35.40561749]
bad: 95.729260 comp: 5.706260 err: 2.939961

Number 111

[5, 1, 9, -10, 0, 18] [[1, 0, 2, 0], [0, 5, 1, 9]]
TOP tuning [1193.274911, 1886.640142, 2763.877849, 3395.952256]
TOP generators [1193.274911, 377.3280283]
bad: 99.308041 comp: 3.205865 err: 9.662601

Number 112 Muggles

[5, 1, -7, -10, -25, -19] [[1, 0, 2, 5], [0, 5, 1, -7]]
TOP tuning [1203.148010, 1896.965522, 2785.689126, 3359.988323]
TOP generators [1203.148011, 379.3931044]
bad: 99.376477 comp: 5.618543 err: 3.148011

Number 113

[11, 6, 15, -16, -7, 18] [[1, 1, 2, 2], [0, 11, 6, 15]]
TOP tuning [1202.072164, 1905.239303, 2787.690040, 3363.008608]
TOP generators [1202.072164, 63.92428535]
bad: 99.809415 comp: 6.940227 err: 2.072164

Number 114

[1, -8, -26, -15, -44, -38] [[1, 2, -1, -8], [0, -1, 8, 26]]
TOP tuning [1199.424969, 1900.336158, 2788.685275, 3365.958541]
TOP generators [1199.424969, 498.5137806]
bad: 99.875385 comp: 9.888635 err: 1.021376

This list is attractive, but Meantone, Magic, Pajara, maybe
Injera to name a few are too low for my taste, if I'm reading
these errors right (they're weighted here, I take it).

If you could make this list finite with badness bounds only,
I'd be more impressed by claims that log-flat badness is
desirable (allows the comparison of ennealimmal with all
temperaments in a sense, not just the others on the list, or
whatever).

And I don't see how you figure schismic is less complex than
miracle in light of the maps given.

-Carl

>Number 1 Ennealimmal
>
>[18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
>TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
>TOP generators [133.3373752, 49.02398564]
>bad: 4.918774 comp: 11.628267 err: .036377
>
>
>Number 2 Meantone
>
>[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>TOP generators [1201.698520, 504.1341314]
>bad: 21.551439 comp: 3.562072 err: 1.698521
>
>
>Number 3 Magic
>
>[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>TOP generators [1201.276744, 380.7957184]
>bad: 23.327687 comp: 4.274486 err: 1.276744
>
>
>Number 4 Beep
>
>[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>TOP generators [1194.642673, 254.8994697]
>bad: 23.664749 comp: 1.292030 err: 14.176105
>
>
>Number 5 Augmented
>
>[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
>TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>TOP generators [399.9922103, 107.3111730]
>bad: 27.081145 comp: 2.147741 err: 5.870879
>
>
>Number 6 Pajara
>
>[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>TOP generators [598.4467109, 106.5665459]
>bad: 27.754421 comp: 2.988993 err: 3.106578
>
>
>Number 7 Dominant Seventh
>
>[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
>TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
>TOP generators [1195.228951, 495.8810151]
>bad: 28.744957 comp: 2.454561 err: 4.771049
>
>
>Number 8 Schismic
>
>[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>TOP generators [1200.760624, 498.1193303]
>bad: 28.818558 comp: 5.618543 err: .912904
>
>
>Number 9 Miracle
>
>[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>TOP generators [1200.631014, 116.7206423]
>bad: 29.119472 comp: 6.793166 err: .631014
>
>
>Number 10 Orwell
>
>[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
>TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
>TOP generators [1199.532657, 271.4936472]
>bad: 30.805067 comp: 5.706260 err: .946061

Gene, you rock!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> This is a list of linear temperaments with top complexity < 15, top
> error < 15, and top badness < 100. I searched extensively without
> adding to the list, which is probably complete. Most of the names
are
> old ones. In some cases I extended a 5-limit name to what seemed
like
> the appropriate 7-limit temperament, and in the case of The
> Temperament Formerly Known as Duodecimal, am suggesting Waage or
> Compton if one of these gentlemen invented it. There are a few new
> names being suggested, none of which are yet etched in stone--not
even
> when the name is Bond, James Bond.
>
> Number 1 Ennealimmal
>
> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> TOP generators [133.3373752, 49.02398564]
> bad: 4.918774 comp: 11.628267 err: .036377
>
>
> Number 2 Meantone
>
> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> TOP generators [1201.698520, 504.1341314]
> bad: 21.551439 comp: 3.562072 err: 1.698521
>
>
> Number 3 Magic
>
> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> TOP generators [1201.276744, 380.7957184]
> bad: 23.327687 comp: 4.274486 err: 1.276744
>
>
> Number 4 Beep
>
> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> TOP generators [1194.642673, 254.8994697]
> bad: 23.664749 comp: 1.292030 err: 14.176105
>
>
> Number 5 Augmented
>
> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> TOP generators [399.9922103, 107.3111730]
> bad: 27.081145 comp: 2.147741 err: 5.870879
>
>
> Number 6 Pajara
>
> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 106.5665459]
> bad: 27.754421 comp: 2.988993 err: 3.106578
>
>
> Number 7 Dominant Seventh
>
> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> TOP generators [1195.228951, 495.8810151]
> bad: 28.744957 comp: 2.454561 err: 4.771049
>
>
> Number 8 Schismic
>
> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> TOP generators [1200.760624, 498.1193303]
> bad: 28.818558 comp: 5.618543 err: .912904
>
>
> Number 9 Miracle
>
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014
>
>
> Number 10 Orwell
>
> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> TOP generators [1199.532657, 271.4936472]
> bad: 30.805067 comp: 5.706260 err: .946061
>
>
> Number 11 Hemiwuerschmidt
>
> [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> TOP generators [1199.692003, 193.8224275]
> bad: 31.386908 comp: 10.094876 err: .307997
>
>
> Number 12 Catakleismic
>
> [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
> TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
> TOP generators [1200.536355, 316.9063960]
> bad: 32.938503 comp: 7.836558 err: .536356
>
>
> Number 13 Father
>
> [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> TOP generators [1185.869125, 447.3863410]
> bad: 33.256527 comp: 1.534101 err: 14.130876
>
>
> Number 14 Blackwood
>
> [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
> TOP generators [239.1786927, 83.83059859]
> bad: 34.210608 comp: 2.173813 err: 7.239629
>
>
> Number 15 Semisixths
>
> [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
> TOP generators [1198.389531, 443.1602931]
> bad: 34.533812 comp: 4.630693 err: 1.610469
>
>
> Number 16 Hemififths
>
> [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
> TOP tuning [1199.700353, 1902.429930, 2785.617954, 3368.041901]
> TOP generators [1199.700353, 351.3647888]
> bad: 34.737019 comp: 10.766914 err: .299647
>
>
> Number 17 Diminished
>
> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> TOP generators [298.5321149, 101.4561401]
> bad: 37.396767 comp: 2.523719 err: 5.871540
>
>
> Number 18 Amity
>
> [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
> TOP tuning [1199.723894, 1902.392618, 2786.717797, 3369.601033]
> TOP generators [1199.723894, 339.3558130]
> bad: 37.532790 comp: 11.659166 err: .276106
>
>
> Number 19 Pelogic
>
> [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
> TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
> TOP generators [1209.734056, 532.9412251]
> bad: 39.824125 comp: 2.022675 err: 9.734056
>
>
> Number 20 Parakleismic
>
> [13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]]
> TOP tuning [1199.738066, 1902.291445, 2786.921905, 3368.090564]
> TOP generators [1199.738066, 315.1076065]
> bad: 40.713036 comp: 12.467252 err: .261934
>
>
> Number 21 {21/20, 28/27}
>
> [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
> TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
> TOP generators [1214.253642, 509.4012304]
> bad: 42.300772 comp: 1.722706 err: 14.253642
>
>
> Number 22 Injera
>
> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
> TOP generators [600.8889070, 93.60982493]
> bad: 42.529834 comp: 3.445412 err: 3.582707
>
>
> Number 23 Dicot
>
> [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
> TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
> TOP generators [1204.048159, 356.3998255]
> bad: 42.920570 comp: 2.137243 err: 9.396316
>
>
> Number 24 Hemifourths
>
> [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
> TOP generators [1203.668841, 252.4803582]
> bad: 43.552336 comp: 3.445412 err: 3.668842
>
>
> Number 25 Waage? Compton? Duodecimal?
>
> [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
> TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
> TOP generators [100.0514209, 16.55882096]
> bad: 45.097159 comp: 8.548972 err: .617051
>
>
> Number 26 Wizard
>
> [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
> TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
> TOP generators [600.3197857, 216.7702531]
> bad: 45.381303 comp: 8.423526 err: .639571
>
>
> Number 27 Kleismic
>
> [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
> TOP generators [1203.187309, 317.8344609]
> bad: 45.676063 comp: 3.785579 err: 3.187309
>
>
> Number 28 Negri
>
> [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
> TOP generators [1203.187309, 124.8419629]
> bad: 46.125886 comp: 3.804173 err: 3.187309
>
>
> Number 29 Nonkleismic
>
> [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
> TOP generators [1198.828458, 309.8926610]
> bad: 46.635848 comp: 6.309298 err: 1.171542
>
>
> Number 30 Quartaminorthirds
>
> [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
> TOP generators [1199.792743, 77.83315314]
> bad: 47.721352 comp: 6.742251 err: 1.049791
>
>
>
> Number 31 Tripletone
>
> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> TOP generators [399.0200131, 92.45965769]
> bad: 48.112067 comp: 4.045351 err: 2.939961
>
>
> Number 32 Decimal
>
> [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
> TOP generators [603.8288989, 250.6116362]
> bad: 48.773723 comp: 2.523719 err: 7.657798
>
>
> Number 33 {1029/1024, 4375/4374}
>
> [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]]
> TOP tuning [1200.421488, 1901.286959, 2785.446889, 3367.642640]
> TOP generators [600.2107440, 183.2944602]
> bad: 50.004574 comp: 10.892116 err: .421488
>
>
> Number 34 Superpythagorean
>
> [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
> TOP generators [1197.596121, 489.4271829]
> bad: 50.917015 comp: 4.602303 err: 2.403879
>
>
> Number 35 Supermajor seconds
>
> [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
> TOP generators [1201.698520, 232.5214630]
> bad: 51.806440 comp: 5.522763 err: 1.698521
>
>
> Number 36 Supersupermajor
>
> [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
> TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
> TOP generators [1200.231587, 234.3804692]
> bad: 52.638504 comp: 7.670504 err: .894655
>
>
> Number 37 {6144/6125, 10976/10935} Hendecatonic?
>
> [11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]]
> TOP tuning [1199.662182, 1902.490429, 2787.098101, 3368.740066]
> TOP generators [109.0601984, 48.46705632]
> bad: 53.458690 comp: 12.579627 err: .337818
>
>
> Number 38 {3136/3125, 5120/5103} Misty
>
> [3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]]
> TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021]
> TOP generators [399.8871550, 96.94420930]
> bad: 53.622498 comp: 12.585536 err: .338535
>
>
> Number 39 {1728/1715, 4000/3993}
>
> [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
> TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
> TOP generators [1199.083445, 45.17026643]
> bad: 55.081549 comp: 7.752178 err: .916555
>
>
> Number 40 {36/35, 160/147} Hystrix?
>
> [3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]]
> TOP tuning [1187.933715, 1892.564743, 2758.296667, 3402.700250]
> TOP generators [1187.933715, 161.1008955]
> bad: 55.952057 comp: 2.153383 err: 12.066285
>
>
> Number 41 {28/27, 50/49}
>
> [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
> TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
> TOP generators [595.7998193, 127.8698005]
> bad: 56.092257 comp: 2.584059 err: 8.400361
>
>
> Number 42 Porcupine
>
> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> TOP generators [1196.905960, 162.3176609]
> bad: 57.088650 comp: 4.295482 err: 3.094040
>
>
> Number 43
>
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 162.3159606]
> bad: 57.621529 comp: 4.306766 err: 3.106578
>
>
> Number 44 Octacot
>
> [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
> TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
> TOP generators [1199.031259, 88.05739491]
> bad: 58.217715 comp: 7.752178 err: .968741
>
>
> Number 45 {25/24, 81/80} Jamesbond?
>
> [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
> TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
> TOP generators [172.7759159, 86.69241190]
> bad: 58.637859 comp: 2.493450 err: 9.431411
>
>
> Number 46 Hemithirds
>
> [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
> TOP tuning [1200.363229, 1901.194685, 2787.427555, 3367.479202]
> TOP generators [1200.363229, 193.3505488]
> bad: 60.573479 comp: 11.237086 err: .479706
>
>
> Number 47
>
> [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]]
> TOP tuning [1200.284965, 1901.503343, 2786.975381, 3369.219732]
> TOP generators [600.1424823, 83.17776441]
> bad: 61.101493 comp: 14.643003 err: .284965
>
>
> Number 48 Flattone
>
> [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
> TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
> TOP generators [1202.536419, 507.1379663]
> bad: 61.126418 comp: 4.909123 err: 2.536420
>
>
> Number 49 Diaschismic
>
> [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
> TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
> TOP generators [599.3662015, 103.7870123]
> bad: 61.527901 comp: 6.966993 err: 1.267597
>
>
> Number 50 Superkleismic
>
> [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
> TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
> TOP generators [1201.371918, 322.3731369]
> bad: 62.364585 comp: 6.742251 err: 1.371918
>
>
> Number 51
>
> [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
> TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
> TOP generators [1201.135545, 387.5841360]
> bad: 62.703297 comp: 6.411729 err: 1.525246
>
>
> Number 52 Tritonic
>
> [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]]
> TOP tuning [1201.023211, 1900.333250, 2785.201472, 3365.953391]
> TOP generators [1201.023211, 580.7519186]
> bad: 63.536850 comp: 7.880073 err: 1.023211
>
>
> Number 53
>
> [1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]]
> TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869]
> TOP generators [1199.680495, 497.2520023]
> bad: 64.536886 comp: 14.212326 err: .319505
>
>
> Number 54
>
> [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
> TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
> TOP generators [1202.659696, 82.97467050]
> bad: 64.556006 comp: 4.306766 err: 3.480440
>
>
> Number 55
>
> [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
> TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
> TOP generators [99.80617249, 24.58395811]
> bad: 65.630949 comp: 4.295482 err: 3.557008
>
>
> Number 56
>
> [2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
> TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
> TOP generators [1204.567524, 355.9419091]
> bad: 66.522610 comp: 2.696901 err: 9.146173
>
>
> Number 57
>
> [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> TOP generators [1185.869125, 223.6931705]
> bad: 66.774944 comp: 2.173813 err: 14.130876
>
>
> Number 58
>
> [5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
> TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
> TOP generators [1194.335372, 99.13879319]
> bad: 67.244049 comp: 3.445412 err: 5.664628
>
>
> Number 59
>
> [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
> TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
> TOP generators [1193.415676, 158.1468146]
> bad: 67.670842 comp: 3.205865 err: 6.584324
>
>
> Number 60
>
> [3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
> TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
> TOP generators [397.8052253, 76.63521863]
> bad: 68.337269 comp: 3.221612 err: 6.584324
>
>
> Number 61 Hemikleismic
>
> [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]]
> TOP tuning [1199.411231, 1902.888178, 2785.151380, 3370.478790]
> TOP generators [1199.411231, 158.5740148]
> bad: 68.516458 comp: 10.787602 err: .588769
>
>
> Number 62
>
> [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> TOP generators [592.7342285, 146.7842660]
> bad: 68.668284 comp: 2.173813 err: 14.531543
>
>
> Number 63
>
> [8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]]
> TOP tuning [1198.975478, 1900.576277, 2788.692580, 3365.949709]
> TOP generators [1198.975478, 62.17183489]
> bad: 68.767371 comp: 8.192765 err: 1.024522
>
>
> Number 64
>
> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
> TOP generators [1202.900537, 570.4479508]
> bad: 69.388565 comp: 4.891080 err: 2.900537
>
>
> Number 65
>
> [3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
> TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
> TOP generators [1202.624742, 569.0491468]
> bad: 70.105427 comp: 5.168119 err: 2.624742
>
>
> Number 66
>
> [17, 6, 15, -30, -24, 18] [[1, -5, 0, -3], [0, 17, 6, 15]]
> TOP tuning [1199.379215, 1900.971080, 2787.482526, 3370.568669]
> TOP generators [1199.379215, 464.5804210]
> bad: 71.416917 comp: 10.725806 err: .620785
>
>
> Number 67
>
> [11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]]
> TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447]
> TOP generators [1198.514750, 154.1766650]
> bad: 71.539673 comp: 6.940227 err: 1.485250
>
>
> Number 68
>
> [3, -24, -1, -45, -10, 65] [[1, 1, 7, 3], [0, 3, -24, -1]]
> TOP tuning [1200.486331, 1902.481504, 2787.442939, 3367.460603]
> TOP generators [1200.486331, 233.9983907]
> bad: 72.714599 comp: 12.227699 err: .486331
>
>
> Number 69
>
> [23, -1, 13, -55, -44, 33] [[1, 9, 2, 7], [0, -23, 1, -13]]
> TOP tuning [1199.671611, 1901.434518, 2786.108874, 3369.747810]
> TOP generators [1199.671611, 386.7656515]
> bad: 73.346343 comp: 14.944966 err: .328389
>
>
> Number 70
>
> [6, 29, -2, 32, -20, -86] [[1, 4, 14, 2], [0, -6, -29, 2]]
> TOP tuning [1200.422358, 1901.285580, 2787.294397, 3367.645998]
> TOP generators [1200.422357, 483.4006416]
> bad: 73.516606 comp: 13.193267 err: .422358
>
>
> Number 71
>
> [7, -15, -16, -40, -45, 5] [[1, 5, -5, -5], [0, -7, 15, 16]]
> TOP tuning [1200.210742, 1900.961474, 2784.858222, 3370.585685]
> TOP generators [1200.210742, 585.7274621]
> bad: 74.053446 comp: 10.869066 err: .626846
>
>
> Number 72
>
> [5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]]
> TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008]
> TOP generators [1192.540126, 139.5182509]
> bad: 74.239244 comp: 3.154649 err: 7.459874
>
>
> Number 73
>
> [4, 21, -3, 24, -16, -66] [[1, 0, -6, 4], [0, 4, 21, -3]]
> TOP tuning [1199.274449, 1901.646683, 2787.998389, 3370.862785]
> TOP generators [1199.274449, 475.4116708]
> bad: 74.381278 comp: 10.125066 err: .725551
>
>
> Number 74
>
> [3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
> TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
> TOP generators [1195.486066, 559.3589487]
> bad: 74.989802 comp: 4.075900 err: 4.513934
>
>
> Number 75
>
> [6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
> TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
> TOP generators [399.8000105, 155.5708520]
> bad: 76.576420 comp: 3.804173 err: 5.291448
>
>
> Number 76
>
> [13, 2, 30, -27, 11, 64] [[1, 6, 3, 13], [0, -13, -2, -30]]
> TOP tuning [1200.672456, 1900.889183, 2786.148822, 3370.713730]
> TOP generators [1200.672456, 407.9342733]
> bad: 76.791305 comp: 10.686216 err: .672456
>
>
> Number 77 Shrutar
>
> [4, -8, 14, -22, 11, 55] [[2, 3, 5, 5], [0, 2, -4, 7]]
> TOP tuning [1198.920873, 1903.665377, 2786.734051, 3365.796415]
> TOP generators [599.4604367, 52.64203308]
> bad: 76.825572 comp: 8.437555 err: 1.079127
>
>
> Number 78
>
> [12, 10, 25, -12, 6, 30] [[1, 6, 6, 12], [0, -12, -10, -25]]
> TOP tuning [1199.028703, 1903.494472, 2785.274095, 3366.099130]
> TOP generators [1199.028703, 440.8898120]
> bad: 77.026097 comp: 8.905180 err: .971298
>
>
> Number 79 Beatles
>
> [2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
> TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
> TOP generators [1197.104145, 354.7203384]
> bad: 77.187771 comp: 5.162806 err: 2.895855
>
>
> Number 80
>
> [6, -12, 10, -33, -1, 57] [[2, 4, 3, 7], [0, -3, 6, -5]]
> TOP tuning [1199.025947, 1903.033657, 2788.575394, 3371.560420]
> TOP generators [599.5129735, 165.0060791]
> bad: 78.320453 comp: 8.966980 err: .974054
>
>
> Number 81
>
> [4, 4, 0, -3, -11, -11] [[4, 6, 9, 11], [0, 1, 1, 0]]
> TOP tuning [1212.384652, 1905.781495, 2815.069985, 3334.057793]
> TOP generators [303.0961630, 63.74881402]
> bad: 78.879803 comp: 2.523719 err: 12.384652
>
>
> Number 82
>
> [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
> TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
> TOP generators [601.7004928, 230.8749260]
> bad: 79.825592 comp: 4.619353 err: 3.740932
>
>
> Number 83
>
> [1, 6, 5, 7, 5, -5] [[1, 2, 5, 5], [0, -1, -6, -5]]
> TOP tuning [1211.970043, 1882.982932, 2814.107292, 3355.064446]
> TOP generators [1211.970043, 540.9571536]
> bad: 79.928319 comp: 2.584059 err: 11.970043
>
>
> Number 84 Squares
>
> [4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
> TOP generators [1201.698520, 426.4581630]
> bad: 80.651668 comp: 6.890825 err: 1.698521
>
>
> Number 85
>
> [6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
> TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
> TOP generators [199.0788921, 88.83392059]
> bad: 80.672767 comp: 3.820609 err: 5.526647
>
>
> Number 86
>
> [7, 26, 25, 25, 20, -15] [[1, 5, 15, 15], [0, -7, -26, -25]]
> TOP tuning [1199.352846, 1902.980716, 2784.811068, 3369.637284]
> TOP generators [1199.352846, 584.8262161]
> bad: 81.144087 comp: 11.197591 err: .647154
>
>
> Number 87
>
> [18, 15, -6, -18, -60, -56] [[3, 6, 8, 8], [0, -6, -5, 2]]
> TOP tuning [1200.448679, 1901.787880, 2785.271912, 3367.566305]
> TOP generators [400.1495598, 83.18491309]
> bad: 81.584166 comp: 13.484503 err: .448679
>
>
> Number 88
>
> [9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]
> TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837]
> TOP generators [1201.918557, 189.0109248]
> bad: 81.594641 comp: 6.521440 err: 1.918557
>
>
> Number 89
>
> [1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
> TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
> TOP generators [1195.155395, 496.2398890]
> bad: 82.638059 comp: 4.075900 err: 4.974313
>
>
> Number 90
>
> [3, 7, -1, 4, -10, -22] [[1, 1, 1, 3], [0, 3, 7, -1]]
> TOP tuning [1205.820043, 1890.417958, 2803.215176, 3389.260823]
> TOP generators [1205.820043, 228.1993049]
> bad: 82.914167 comp: 3.375022 err: 7.279064
>
>
>
> Number 91
>
> [6, 5, -31, -6, -66, -86] [[1, 0, 1, 11], [0, 6, 5, -31]]
> TOP tuning [1199.976626, 1902.553087, 2785.437532, 3369.885264]
> TOP generators [1199.976626, 317.0921813]
> bad: 83.023430 comp: 14.832953 err: .377351
>
>
> Number 92
>
> [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
> TOP generators [599.2769413, 272.3123381]
> bad: 83.268810 comp: 5.047438 err: 3.268439
>
>
> Number 93
>
> [4, 2, 9, -6, 3, 15] [[1, 3, 3, 6], [0, -4, -2, -9]]
> TOP tuning [1208.170435, 1910.173796, 2767.342550, 3391.763218]
> TOP generators [1208.170435, 428.5843770]
> bad: 83.972208 comp: 3.205865 err: 8.170435
>
>
> Number 94 Hexidecimal
>
> [1, -3, 5, -7, 5, 20] [[1, 2, 1, 5], [0, -1, 3, -5]]
> TOP tuning [1208.959294, 1887.754858, 2799.450479, 3393.977822]
> TOP generators [1208.959293, 530.1637287]
> bad: 84.341555 comp: 3.068202 err: 8.959294
>
>
> Number 95
>
> [6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
> TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
> TOP generators [399.0200131, 46.12154491]
> bad: 84.758945 comp: 5.369353 err: 2.939961
>
>
> Number 96
>
> [0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
> TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
> TOP generators [99.83462277, 16.52294019]
> bad: 85.896401 comp: 5.168119 err: 3.215955
>
>
> Number 97
>
> [11, -6, 10, -35, -15, 40] [[1, 4, 1, 5], [0, -11, 6, -10]]
> TOP tuning [1200.950404, 1901.347958, 2784.106944, 3366.157786]
> TOP generators [1200.950404, 263.8594234]
> bad: 85.962459 comp: 9.510433 err: .950404
>
>
> Number 98 Slender
>
> [13, -10, 6, -46, -27, 42] [[1, 2, 2, 3], [0, -13, 10, -6]]
> TOP tuning [1200.337238, 1901.055858, 2784.996493, 3370.418508]
> TOP generators [1200.337239, 38.43220154]
> bad: 88.631905 comp: 12.499426 err: .567296
>
>
> Number 99
>
> [0, 5, 10, 8, 16, 9] [[5, 8, 12, 15], [0, 0, -1, -2]]
> TOP tuning [1195.598382, 1912.957411, 2770.195472, 3388.313857]
> TOP generators [239.1196765, 99.24064453]
> bad: 89.758630 comp: 3.595867 err: 6.941749
>
>
> Number 100
>
> [1, -1, -5, -4, -11, -9] [[1, 2, 2, 1], [0, -1, 1, 5]]
> TOP tuning [1185.210905, 1925.395162, 2815.448458, 3410.344145]
> TOP generators [1185.210905, 445.0266480]
> bad: 90.384580 comp: 2.472159 err: 14.789095
>
>
> Number 101
>
> [2, 8, -11, 8, -23, -48] [[1, 1, 0, 6], [0, 2, 8, -11]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3373.586984]
> TOP generators [1201.698520, 348.7821945]
> bad: 92.100337 comp: 7.363684 err: 1.698521
>
>
> Number 102
>
> [3, 12, 18, 12, 20, 8] [[3, 5, 8, 10], [0, -1, -4, -6]]
> TOP tuning [1202.260038, 1898.372926, 2784.451552, 3375.170635]
> TOP generators [400.7533459, 105.3938041]
> bad: 92.910783 comp: 6.411729 err: 2.260038
>
>
> Number 103
>
> [4, -8, -20, -22, -43, -24] [[4, 6, 10, 13], [0, 1, -2, -5]]
> TOP tuning [1199.003867, 1903.533834, 2787.453602, 3371.622404]
> TOP generators [299.7509668, 105.0280329]
> bad: 93.029698 comp: 9.663894 err: .996133
>
>
> Number 104
>
> [3, 0, -3, -7, -13, -7] [[3, 5, 7, 8], [0, -1, 0, 1]]
> TOP tuning [1205.132027, 1884.438632, 2811.974729, 3337.800149]
> TOP generators [401.7106756, 124.1147448]
> bad: 94.336372 comp: 2.921642 err: 11.051598
>
>
> Number 105
>
> [4, 7, 2, 2, -8, -15] [[1, 2, 3, 3], [0, -4, -7, -2]]
> TOP tuning [1190.204869, 1918.438775, 2762.165422, 3339.629125]
> TOP generators [1190.204869, 115.4927407]
> bad: 94.522719 comp: 3.014736 err: 10.400103
>
>
>
> Number 106
>
> [13, 19, 23, 0, 0, 0] [[1, 0, 0, 0], [0, 13, 19, 23]]
> TOP tuning [1200.0, 1904.187463, 2783.043215, 3368.947050]
> TOP generators [1200., 146.4759587]
> bad: 94.757554 comp: 8.202087 err: 1.408527
>
>
> Number 107
>
> [2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]]
> TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246]
> TOP generators [603.2741324, 81.75384943]
> bad: 94.764743 comp: 3.804173 err: 6.548265
>
>
> Number 108
>
> [2, -6, -6, -14, -15, 3] [[2, 3, 5, 6], [0, 1, -3, -3]]
> TOP tuning [1206.548264, 1891.576247, 2771.109113, 3374.383246]
> TOP generators [603.2741324, 81.75384943]
> bad: 94.764743 comp: 3.804173 err: 6.548265
>
>
> Number 109
>
> [1, -13, -2, -23, -6, 32] [[1, 2, -3, 2], [0, -1, 13, 2]]
> TOP tuning [1197.567789, 1904.876372, 2780.666293, 3375.653987]
> TOP generators [1197.567789, 490.2592046]
> bad: 94.999539 comp: 6.249713 err: 2.432212
>
>
> Number 110
>
> [9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
> TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
> TOP generators [133.0066710, 35.40561749]
> bad: 95.729260 comp: 5.706260 err: 2.939961
>
>
> Number 111
>
> [5, 1, 9, -10, 0, 18] [[1, 0, 2, 0], [0, 5, 1, 9]]
> TOP tuning [1193.274911, 1886.640142, 2763.877849, 3395.952256]
> TOP generators [1193.274911, 377.3280283]
> bad: 99.308041 comp: 3.205865 err: 9.662601
>
>
> Number 112 Muggles
>
> [5, 1, -7, -10, -25, -19] [[1, 0, 2, 5], [0, 5, 1, -7]]
> TOP tuning [1203.148010, 1896.965522, 2785.689126, 3359.988323]
> TOP generators [1203.148011, 379.3931044]
> bad: 99.376477 comp: 5.618543 err: 3.148011
>
>
> Number 113
>
> [11, 6, 15, -16, -7, 18] [[1, 1, 2, 2], [0, 11, 6, 15]]
> TOP tuning [1202.072164, 1905.239303, 2787.690040, 3363.008608]
> TOP generators [1202.072164, 63.92428535]
> bad: 99.809415 comp: 6.940227 err: 2.072164
>
>
> Number 114
>
> [1, -8, -26, -15, -44, -38] [[1, 2, -1, -8], [0, -1, 8, 26]]
> TOP tuning [1199.424969, 1900.336158, 2788.685275, 3365.958541]
> TOP generators [1199.424969, 498.5137806]
> bad: 99.875385 comp: 9.888635 err: 1.021376

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> This list is attractive, but Meantone, Magic, Pajara, maybe
> Injera to name a few are too low for my taste, if I'm reading
> these errors right (they're weighted here, I take it).

I think log-flat badness has outlived its popularity :)

> And I don't see how you figure schismic is less complex than
> miracle in light of the maps given.

Probably the shortness of the fifths in the lattice wins it for
schismic . . .

>
> -Carl
>
> >Number 1 Ennealimmal
> >
> >[18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> >TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> >TOP generators [133.3373752, 49.02398564]
> >bad: 4.918774 comp: 11.628267 err: .036377
> >
> >
> >Number 2 Meantone
> >
> >[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> >TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> >TOP generators [1201.698520, 504.1341314]
> >bad: 21.551439 comp: 3.562072 err: 1.698521
> >
> >
> >Number 3 Magic
> >
> >[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> >TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> >TOP generators [1201.276744, 380.7957184]
> >bad: 23.327687 comp: 4.274486 err: 1.276744
> >
> >
> >Number 4 Beep
> >
> >[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> >TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> >TOP generators [1194.642673, 254.8994697]
> >bad: 23.664749 comp: 1.292030 err: 14.176105
> >
> >
> >Number 5 Augmented
> >
> >[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> >TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> >TOP generators [399.9922103, 107.3111730]
> >bad: 27.081145 comp: 2.147741 err: 5.870879
> >
> >
> >Number 6 Pajara
> >
> >[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> >TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> >TOP generators [598.4467109, 106.5665459]
> >bad: 27.754421 comp: 2.988993 err: 3.106578
> >
> >
> >Number 7 Dominant Seventh
> >
> >[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> >TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> >TOP generators [1195.228951, 495.8810151]
> >bad: 28.744957 comp: 2.454561 err: 4.771049
> >
> >
> >Number 8 Schismic
> >
> >[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> >TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> >TOP generators [1200.760624, 498.1193303]
> >bad: 28.818558 comp: 5.618543 err: .912904
> >
> >
> >Number 9 Miracle
> >
> >[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> >TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> >TOP generators [1200.631014, 116.7206423]
> >bad: 29.119472 comp: 6.793166 err: .631014
> >
> >
> >Number 10 Orwell
> >
> >[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> >TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> >TOP generators [1199.532657, 271.4936472]
> >bad: 30.805067 comp: 5.706260 err: .946061

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> This list is attractive, but Meantone, Magic, Pajara, maybe
> Injera to name a few are too low for my taste, if I'm reading
> these errors right (they're weighted here, I take it).
>
> If you could make this list finite with badness bounds only,
> I'd be more impressed by claims that log-flat badness is
> desirable (allows the comparison of ennealimmal with all
> temperaments in a sense, not just the others on the list, or
> whatever).

Log flat badness is deliberately designed not to be finite. and it
seems to me your objection is strange--do you think epimericity allows
comparison of one comma with another, while a log flat badness does not?

As for meantone, magic and pajara being too low, they are all near tht
top of the list. It would seem the list is doing exactly what you want
it to do.

You can make the list finite by bounding complexity, which is what
I've done.

> And I don't see how you figure schismic is less complex than
> miracle in light of the maps given.

Schismic gets to 3/2 in one generator step, and miracle takes six.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > This list is attractive, but Meantone, Magic, Pajara, maybe
> > Injera to name a few are too low for my taste, if I'm reading
> > these errors right (they're weighted here, I take it).
>
> I think log-flat badness has outlived its popularity :)

Not with me. However, an alternative which isn't simply ad-hoc
randomness would be nice.

>> This list is attractive, but Meantone, Magic, Pajara, maybe
>> Injera to name a few are too low for my taste, if I'm reading
>> these errors right (they're weighted here, I take it).
>>
>> If you could make this list finite with badness bounds only,
>> I'd be more impressed by claims that log-flat badness is
>> desirable (allows the comparison of ennealimmal with all
>> temperaments in a sense, not just the others on the list, or
>> whatever).
>
>Log flat badness is deliberately designed not to be finite. and it
>seems to me your objection is strange--do you think epimericity
>allows comparison of one comma with another, while a log flat badness
>does not?

What's the rub again? Within equally-sized complexity bins, log-flat
badness returns roughly the same number of temperaments? I guess
that makes sense.

>As for meantone, magic and pajara being too low, they are all near tht
>top of the list. It would seem the list is doing exactly what you want
>it to do.

I did say it was attractive...

>You can make the list finite by bounding complexity, which is what
>I've done.
>
>> And I don't see how you figure schismic is less complex than
>> miracle in light of the maps given.
>
>Schismic gets to 3/2 in one generator step, and miracle takes six.

What kind of complexity is this? Do you always use the same kind?
It seems you happily switch between geometric, weighted map-based,
and 3 other flavors when giving these lists. Providing a template
at the top of the lists showing units or something for each key
might help your readers.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> What kind of complexity is this?

It's the complexity which arises naturally out of the Tenney space
and dual val space point of view, as the norm on a bival. It
therefore gives more weight to lower primes such as 2 and 3 as
opposed to higher ones such as 5 and 7.

> Do you always use the same kind?

I wanted to do things from a TOP point of view, so I used something
consistent with that.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Number 8 Schismic
> >> >
> >> >[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> >> >TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> >> >TOP generators [1200.760624, 498.1193303]
> >> >bad: 28.818558 comp: 5.618543 err: .912904
> >> >
> >> >
> >> >Number 9 Miracle
> >> >
> >> >[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> >> >TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> >> >TOP generators [1200.631014, 116.7206423]
> >> >bad: 29.119472 comp: 6.793166 err: .631014
> //
> >> And I don't see how you figure schismic is less complex than
> >> miracle in light of the maps given.
> >
> >Probably the shortness of the fifths in the lattice wins it for
> >schismic . . .
>
> After I wrote that I reflected a bit on comma complexity vs. map
> complexity. Comma complexity gives you the number of notes you'd
> have to search to find the comma, on average (Kees points out that
> the symmetry of the lattice allows you to search 1/4 this numeber
> in the 5-limit, or something, but anyway...). Map complexity is
> the number of notes you need to complete the map *with contiguous
> chains of generators*.

Thus it will depend on the choice of generators. For so-called linear
temperaments, this is only made definite by fixing one of them to be
1/N octaves. For planar and higher-dimensional temperaments, the
choice is even more arbitrary. Comma complexity, or wedgie complexity
for higher codimensions, is well-defined, and is (according to Gene)
the natural generalization of the complexity measures we all agree on
for the simplest cases.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > What kind of complexity is this?
>
> It's the complexity which arises naturally out of the Tenney space
> and dual val space point of view, as the norm on a bival. It
> therefore gives more weight to lower primes such as 2 and 3 as
> opposed to higher ones such as 5 and 7.
>
> > Do you always use the same kind?
>
> I wanted to do things from a TOP point of view, so I used something
> consistent with that.

Carl, did you read the message that you got 3 copies of? This is what
Gene was addressing with his list.

>> Map complexity is
>> the number of notes you need to complete the map *with contiguous
>> chains of generators*.
>
>Thus it will depend on the choice of generators. For so-called linear
>temperaments, this is only made definite by fixing one of them to be
>1/N octaves.

I don't get thus. The map contains the consonances you want, and a
way to get the generators to hit them. Why does one of the generators
have to generate an octave all by itself?

>For planar and higher-dimensional temperaments, the
>choice is even more arbitrary. Comma complexity, or wedgie complexity
>for higher codimensions, is well-defined, and is (according to Gene)
>the natural generalization of the complexity measures we all agree on
>for the simplest cases.

I agree that comma complexity seems more desirable. I don't have
the commas Gene used for schismic and miracle handy (and I don't
know how to compute complexity from more than a single comma) but
what's all this talk about generator steps in a 3/2 then?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Map complexity is
> >> the number of notes you need to complete the map *with contiguous
> >> chains of generators*.
> >
> >Thus it will depend on the choice of generators. For so-called
linear
> >temperaments, this is only made definite by fixing one of them to
be
> >1/N octaves.
>
> I don't get thus. The map contains the consonances you want, and a
> way to get the generators to hit them. Why does one of the
generators
> have to generate an octave all by itself?

It doesn't, but that's how just about everyone has always described
just about all of them.

> >For planar and higher-dimensional temperaments, the
> >choice is even more arbitrary. Comma complexity, or wedgie
complexity
> >for higher codimensions, is well-defined, and is (according to
Gene)
> >the natural generalization of the complexity measures we all agree
on
> >for the simplest cases.
>
> I agree that comma complexity seems more desirable. I don't have
> the commas Gene used for schismic and miracle handy

He didn't (as he keeps insisting in another thread here now).

> (and I don't
> know how to compute complexity from more than a single comma)

You have to compute the wedgie, and then use the formula for wedgie
(top) complexity that he just posted.

> but
> what's all this talk about generator steps in a 3/2 then?

Complexity (if reasonably defined) is complexity, and the various
ways of looking at it are essentially equivalent.

>>>What kind of complexity is this?
>>
>>It's the complexity which arises naturally out of the Tenney space
>>and dual val space point of view, as the norm on a bival. It
>>therefore gives more weight to lower primes such as 2 and 3 as
>>opposed to higher ones such as 5 and 7.
>>
>>>Do you always use the same kind?
>>
>>I wanted to do things from a TOP point of view, so I used
>>something consistent with that.
>
>Carl, did you read the message that you got 3 copies of? This is
>what Gene was addressing with his list.

I was in some sort of trance when I read it, sorry.

>In the 3-limit, there's only one kind of regular TOP temperament:
>equal TOP temperament. For any instance of it, the complexity can
>be assessed by either
>
>() Measuring the Tenney harmonic distance of the commatic unison
>vector
>
>5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
>12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988
>
>() Calculating the number of notes per pure octave or 'tritave':
>
>5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
>.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
>12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
>.........TOP tritave = 1901 -> 19.01 notes per pure tritave.
>
>The latter results are precisely the former divided by 2: in
>particular, the base-2 Tenney harmonic distance gives 2 times the
>number of notes per tritave, and the base-3 Tenney harmonic distance
>gives 2 times the number of notes per octave. A funny 'switch' but
>agreement (up to a factor of exactly 2) nonetheless. In some way,
>both of these methods of course have to correspond to the same
>mathematical formula . . .

Ok, great!

>In the 5-limit, there are both 'linear' and equal TOP temperaments.
>For the 'linear' case, we can use the first method above (Tenney
>harmonic distance) to calculate complexity.

Did you repeat the above comparison for the two methods in the
5-limit?

>For the equal case, two
>commas are involved; if we delete the entries for prime p in the
>monzos for each of the commatic unison vectors and calculate the
>determinant of the remaining 2-by-2 matrix, we get the number of
>notes per tempered p; then we can use the usual TOP formula to get
>tempered p in terms of pure p and thus finally, the number of notes
>per pure p.

So this is a way to get map-complexity with two commas; what about
comma complexity with two commas?

>Note that there was no need to calculate the angle
>or 'straightness' of the commas; change the angles in your lattice
>and the number of notes the commas define remains the same, so
>angles can't really be relevant here.

Funny; when looking up TM stuff for Paul H. I ran across this same
reasoning...

////
>yes, a reduced basis will have good straightness, because the set of
>basis vectors is, in some sense, as short as possible. and, as we
>discussed before, shortness implies straightness. the "block" always
>has the same "area", so if the vectors are close to parallel,
>they'll have to be long to compensate. remember that whole confusing
>discussion? ///

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >>>What kind of complexity is this?
> >>
> >>It's the complexity which arises naturally out of the Tenney
space
> >>and dual val space point of view, as the norm on a bival. It
> >>therefore gives more weight to lower primes such as 2 and 3 as
> >>opposed to higher ones such as 5 and 7.
> >>
> >>>Do you always use the same kind?
> >>
> >>I wanted to do things from a TOP point of view, so I used
> >>something consistent with that.
> >
> >Carl, did you read the message that you got 3 copies of? This is
> >what Gene was addressing with his list.
>
> I was in some sort of trance when I read it, sorry.
>
> >In the 3-limit, there's only one kind of regular TOP temperament:
> >equal TOP temperament. For any instance of it, the complexity can
> >be assessed by either
> >
> >() Measuring the Tenney harmonic distance of the commatic unison
> >vector
> >
> >5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
> >12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988
> >
> >() Calculating the number of notes per pure octave or 'tritave':
> >
> >5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
> >.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
> >12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
> >.........TOP tritave = 1901 -> 19.01 notes per pure tritave.
> >
> >The latter results are precisely the former divided by 2: in
> >particular, the base-2 Tenney harmonic distance gives 2 times the
> >number of notes per tritave, and the base-3 Tenney harmonic
distance
> >gives 2 times the number of notes per octave. A funny 'switch' but
> >agreement (up to a factor of exactly 2) nonetheless. In some way,
> >both of these methods of course have to correspond to the same
> >mathematical formula . . .
>
> Ok, great!
>
> >In the 5-limit, there are both 'linear' and equal TOP
temperaments.
> >For the 'linear' case, we can use the first method above (Tenney
> >harmonic distance) to calculate complexity.
>
> Did you repeat the above comparison for the two methods in the
> 5-limit?

How can you? Linear temperaments and equal temperaments are different
entities in the 5-limit. However, I'd like to see Gene's general
formula and how it handles all these cases.

> >For the equal case, two
> >commas are involved; if we delete the entries for prime p in the
> >monzos for each of the commatic unison vectors and calculate the
> >determinant of the remaining 2-by-2 matrix, we get the number of
> >notes per tempered p; then we can use the usual TOP formula to get
> >tempered p in terms of pure p and thus finally, the number of
notes
> >per pure p.
>
> So this is a way to get map-complexity with two commas; what about
> comma complexity with two commas?

The determinant is an affine-geometric measure of "area" which of
course is invariant under change of comma basis, so it certainly
seems to represent the latter at least as clearly as it represents
the former.

> >Note that there was no need to calculate the angle
> >or 'straightness' of the commas; change the angles in your lattice
> >and the number of notes the commas define remains the same, so
> >angles can't really be relevant here.
>
> Funny; when looking up TM stuff for Paul H. I ran across this same
> reasoning...
>
> ////
> >yes, a reduced basis will have good straightness, because the set
of
> >basis vectors is, in some sense, as short as possible. and, as we
> >discussed before, shortness implies straightness. the "block"
always
> >has the same "area", so if the vectors are close to parallel,
> >they'll have to be long to compensate. remember that whole
confusing
> >discussion? ///
>
> -Carl

>> >In the 3-limit, there's only one kind of regular TOP temperament:
>> >equal TOP temperament. For any instance of it, the complexity can
>> >be assessed by either
>> >
>> >() Measuring the Tenney harmonic distance of the commatic unison
>> >vector
>> >
>> >5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
>> >12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988
>> >
>> >() Calculating the number of notes per pure octave or 'tritave':
>> >
>> >5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
>> >.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
>> >12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
>> >.........TOP tritave = 1901 -> 19.01 notes per pure tritave.
>> >
>> >The latter results are precisely the former divided by 2: in
>> >particular, the base-2 Tenney harmonic distance gives 2 times the
>> >number of notes per tritave, and the base-3 Tenney harmonic
>distance
>> >gives 2 times the number of notes per octave. A funny 'switch' but
>> >agreement (up to a factor of exactly 2) nonetheless. In some way,
>> >both of these methods of course have to correspond to the same
>> >mathematical formula . . .
>>
>> Ok, great!
>>
>> >In the 5-limit, there are both 'linear' and equal TOP
>> >temperaments.
>> >For the 'linear' case, we can use the first method above (Tenney
>> >harmonic distance) to calculate complexity.
>>
>> Did you repeat the above comparison for the two methods in the
>> 5-limit?
>
>How can you? Linear temperaments and equal temperaments are different
>entities in the 5-limit.

For linear temperaments can't you use both the map-based and
comma based approach, and see if the factor of 2 holds?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >In the 3-limit, there's only one kind of regular TOP
temperament:
> >> >equal TOP temperament. For any instance of it, the complexity
can
> >> >be assessed by either
> >> >
> >> >() Measuring the Tenney harmonic distance of the commatic
unison
> >> >vector
> >> >
> >> >5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
> >> >12-equal: log2(531441*524288) = 38.02, log3(531441*524288) =
23.988
> >> >
> >> >() Calculating the number of notes per pure octave or 'tritave':
> >> >
> >> >5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
> >> >.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
> >> >12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
> >> >.........TOP tritave = 1901 -> 19.01 notes per pure tritave.
> >> >
> >> >The latter results are precisely the former divided by 2: in
> >> >particular, the base-2 Tenney harmonic distance gives 2 times
the
> >> >number of notes per tritave, and the base-3 Tenney harmonic
> >distance
> >> >gives 2 times the number of notes per octave. A funny 'switch'
but
> >> >agreement (up to a factor of exactly 2) nonetheless. In some
way,
> >> >both of these methods of course have to correspond to the same
> >> >mathematical formula . . .
> >>
> >> Ok, great!
> >>
> >> >In the 5-limit, there are both 'linear' and equal TOP
> >> >temperaments.
> >> >For the 'linear' case, we can use the first method above (Tenney
> >> >harmonic distance) to calculate complexity.
> >>
> >> Did you repeat the above comparison for the two methods in the
> >> 5-limit?
> >
> >How can you? Linear temperaments and equal temperaments are
different
> >entities in the 5-limit.
>
> For linear temperaments can't you use both the map-based and
> comma based approach, and see if the factor of 2 holds?

What's the map-based approach, explicitly?

>> For linear temperaments can't you use both the map-based and
>> comma based approach, and see if the factor of 2 holds?
>
>What's the map-based approach, explicitly?

The minimum number of notes that completes the map.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> For linear temperaments can't you use both the map-based and
> >> comma based approach, and see if the factor of 2 holds?
> >
> >What's the map-based approach, explicitly?
>
> The minimum number of notes that completes the map.
>
> -Carl

How do you define 'completes the map'? And can you show how what I
did in the 3-limit satisfies this definition?

>> >> For linear temperaments can't you use both the map-based and
>> >> comma based approach, and see if the factor of 2 holds?
>> >
>> >What's the map-based approach, explicitly?
>>
>> The minimum number of notes that completes the map.
>>
>> -Carl
>
>How do you define 'completes the map'? And can you show how what I
>did in the 3-limit satisfies this definition?

For an et, the number of notes in an octave is guaranteed to
complete the map, but it may not be the minimum number of notes
to do so. So maybe the correct analog of what you did is the
number of tones in the Fokker block corresponding to the TM-reduced
basis of the temperament. But I'd be interested in the map thing.
It's just what we used to call Graham complexity.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> For linear temperaments can't you use both the map-based and
> >> >> comma based approach, and see if the factor of 2 holds?
> >> >
> >> >What's the map-based approach, explicitly?
> >>
> >> The minimum number of notes that completes the map.
> >>
> >> -Carl
> >
> >How do you define 'completes the map'? And can you show how what I
> >did in the 3-limit satisfies this definition?
>
> For an et, the number of notes in an octave is guaranteed to
> complete the map, but it may not be the minimum number of notes
> to do so. So maybe the correct analog of what you did is the
> number of tones in the Fokker block corresponding to the TM-reduced
> basis of the temperament.

. . . of the kernel of the temperament. Right, except this block only
exists in the case of equal temperament. Otherwise you have a "Fokker
strip", a "Fokker sheet", or what have you. In the co-dimension 1
case, comma complexity gives us the thickness of the Fokker sheet. In
the co-dimension 2 case, we need an affine-geometrical measure of the
cross-sectional area of the Fokker strip. Gene seems to be implying
that the norm of the wedgie gives this, but I'd love to see him (when
he has time) show how the cross-checking for the 3- and 5-limit cases
works out.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> For linear temperaments can't you use both the map-based and
> >> >> comma based approach, and see if the factor of 2 holds?
> >> >
> >> >What's the map-based approach, explicitly?
> >>
> >> The minimum number of notes that completes the map.
> >>
> >> -Carl
> >
> >How do you define 'completes the map'? And can you show how what I
> >did in the 3-limit satisfies this definition?
>
> For an et, the number of notes in an octave is guaranteed to
> complete the map, but it may not be the minimum number of notes
> to do so. So maybe the correct analog of what you did is the
> number of tones in the Fokker block corresponding to the TM-reduced
> basis of the temperament.

TM reduction is irrelevant; any other kernel basis will give the same
result.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

Gene seems to be implying
> that the norm of the wedgie gives this, but I'd love to see him (when
> he has time) show how the cross-checking for the 3- and 5-limit cases
> works out.

I'd need to know what you mean by cross-checking first.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> Gene seems to be implying
> > that the norm of the wedgie gives this, but I'd love to see him
(when
> > he has time) show how the cross-checking for the 3- and 5-limit
cases
> > works out.
>
> I'd need to know what you mean by cross-checking first.

The cross-checking that I showed in the 3-limit case (except I was
off by a factor of 2). In each limit and dimension, the complexity
measure should arise from a single general formula -- ||Wedgie||, I
suppose, but with a full elaboration for the grassmann-unaware --
and our paper should show how this reduces,

in the dimension-1 case, to the number of notes per log(frequency)
unit (assume we will also explain fokker determinants), and

in the codimension-1 case, to the length (scaled with a factor of
1/2 or however it works out) of the comma = the width of
the 'periodicity slice'.

On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith" <gwsmith@svpal.org>
wrote:

>Number 82
>
>[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
>TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
>TOP generators [601.7004928, 230.8749260]
>bad: 79.825592 comp: 4.619353 err: 3.740932

There are a number of interesting and potentially useful new tunings in
this list, but I'd like to draw attention to this one. I've been making
charts of the ET's produced by equal divisions of slightly stretched
octaves, from 1201 to 1205 cents, and noticed that the appearance of these
is quite different from the usual ones centered around the 12-19-22
triangle. Stretched octave ET's tend to cluster around 19, and the region
of meantone between 19 and 26 starts to get filled with new ET's. There's a
couple of new temperaments on the 1205 cent octave chart that don't show up
on the old familiar chart; one of these goes through 26 from 10-ET to
16-ET, and also includes 42, 68, 94, and 36-ET (plus some inconsistent
ones). When I plug 10 and 16 into the temperament finder, this is what I
end up with.

5/13, 229.4 cent generator

basis:
(0.5, 0.191135896755)

mapping by period and generator:
[(2, 0), (2, 3), (5, -1), (6, -1)]

mapping by steps:
[(16, 10), (25, 16), (37, 23), (45, 28)]

highest interval width: 4
complexity measure: 8 (10 for smallest MOS)
highest error: 0.014573 (17.488 cents)

The decatonic version of this scale seems to have some possibilities. I've
been playing around with it, originally in the 5-limit version with TOP
tuning [1203.571465, 1896.294363, 2778.021029, 3368.825906].

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > This list is attractive, but Meantone, Magic, Pajara, maybe
> > > Injera to name a few are too low for my taste, if I'm reading
> > > these errors right (they're weighted here, I take it).
> >
> > I think log-flat badness has outlived its popularity :)
>
> Not with me. However, an alternative which isn't simply ad-hoc
> randomness would be nice.

Thanks for the list Gene.

Gene or Paul,

Can one of you easily plot these 7-limit temperaments on an error vs.
complexity graph (log log or whatever seemed best with 5-limit) so we
can all think about what our subjective badness contours might look like.

Please label the points with the numbers-plus-names Gene gave them in
his list.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> >Number 82
> >
> >[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
> >TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
> >TOP generators [601.7004928, 230.8749260]
> >bad: 79.825592 comp: 4.619353 err: 3.740932

...

> When I plug 10 and 16 into the temperament finder, this is what I
> end up with.
>
> 5/13, 229.4 cent generator
>
> basis:
> (0.5, 0.191135896755)
>
> mapping by period and generator:
> [(2, 0), (2, 3), (5, -1), (6, -1)]
>
> mapping by steps:
> [(16, 10), (25, 16), (37, 23), (45, 28)]
>
> highest interval width: 4
> complexity measure: 8 (10 for smallest MOS)
> highest error: 0.014573 (17.488 cents)

This comparison of different outputs for the same temperament shows up
the need to correctly normalise the new weighted error and complexity
figures so they actually have units we can relate to. i.e. cents for
the error and gens per interval for the complexity.

This should be simple to do.

I think the correct normalisation of a weighted norm is the one where,
if every individual value happened to be X then the, the norm would
also be X, irrespective of the weights.

e.g. if the individual errors are E1, E2, ... En, and the respective
weights are W1, W2, ... Wn (all positive), I think the p-norm should
not be

[(|W1E1|**p + |W2E2|**p + ... |WnEn|**p)/n]**(1/p)

but instead

[(|W1E1|**p + |W2E2|**p + ... |WnEn|**p)/(W1**p + W2**p + ...
Wn**p)]**(1/p)

i.e. n is replaced by (W1**p + W2**p + ... Wn**p)

However it bothers me slightly that for minimax (p -> oo), this is
equivalent to

Max(|W1E1|, |W2E2|, ... |WnEn|)/Max(W1, W2, ... Wn)

It seems like I'd rather have

Max(|W1E1|, |W2E2|, ... |WnEn|)/Mean(W1, W2, ... Wn)

but I guess that would be inconsistent.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...>
wrote:
> > > > This list is attractive, but Meantone, Magic, Pajara, maybe
> > > > Injera to name a few are too low for my taste, if I'm reading
> > > > these errors right (they're weighted here, I take it).
> > >
> > > I think log-flat badness has outlived its popularity :)
> >
> > Not with me. However, an alternative which isn't simply ad-hoc
> > randomness would be nice.
>
> Thanks for the list Gene.
>
> Gene or Paul,
>
> Can one of you easily plot these 7-limit temperaments on an error
vs.
> complexity graph (log log or whatever seemed best with 5-limit) so
we
> can all think about what our subjective badness contours might look
like.

I could do that, but as this was done with a log-flat badness cutoff,
there will be a huge gaping hole in the graph. That's why I'm trying
to figure out the whole deal for myself, but no one's helping.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> > On Wed, 21 Jan 2004 09:08:14 -0000, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > >Number 82
> > >
> > >[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
> > >TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
> > >TOP generators [601.7004928, 230.8749260]
> > >bad: 79.825592 comp: 4.619353 err: 3.740932
>
> ...
>
> > When I plug 10 and 16 into the temperament finder, this is what I
> > end up with.
> >
> > 5/13, 229.4 cent generator
> >
> > basis:
> > (0.5, 0.191135896755)
> >
> > mapping by period and generator:
> > [(2, 0), (2, 3), (5, -1), (6, -1)]
> >
> > mapping by steps:
> > [(16, 10), (25, 16), (37, 23), (45, 28)]
> >
> > highest interval width: 4
> > complexity measure: 8 (10 for smallest MOS)
> > highest error: 0.014573 (17.488 cents)
>
> This comparison of different outputs for the same temperament shows
up
> the need to correctly normalise the new weighted error and
complexity
> figures so they actually have units we can relate to. i.e. cents for
> the error and gens per interval for the complexity.
>
> This should be simple to do.
>
> I think the correct normalisation of a weighted norm is the one
where,
> if every individual value happened to be X then the, the norm would
> also be X, irrespective of the weights.
>
> e.g. if the individual errors are E1, E2, ... En,

You realize that there are an infinite number of errors in the TOP
case.

>>Can one of you easily plot these 7-limit temperaments on an error
>>vs. complexity graph (log log or whatever seemed best with 5-limit)
>>so we can all think about what our subjective badness contours might
>>look like.
>
>I could do that, but as this was done with a log-flat badness cutoff,
>there will be a huge gaping hole in the graph. That's why I'm trying
>to figure out the whole deal for myself, but no one's helping.

Since I usually like the way you figure things out, I'll do whatever
I can to help, which may not be much. If there are any particular
msg. #s associated with this, I'll reread them. I didn't follow your
orthogonalization posts at all. :(

Part of the problem is these contours represent musical values, so
they're ultimately a matter of opinion. Log-flat badness has some
nice things going for it, I suppose. Back when I was coding it I
was asking things like, 'every time I double the number of commas
I search, how much of my top-10 list ought to change?'... without
much closure, I might add.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> I could do that, but as this was done with a log-flat badness cutoff,
> there will be a huge gaping hole in the graph. That's why I'm trying
> to figure out the whole deal for myself, but no one's helping.

Try asking for some specific form of help.

Since Herman has expressed his preferences as regards badness
functions, and his interest in "#82", I thought I'd cull the list of
114 by applying a more stringent cutoff of 1.355*comp + error <
10.71. This is an arbitrary choice among the linear functions of
complexity and error that could be chosen; it's chosen so that
Miracle, Blackwood, and Diaschismic make it in, but unfortunately
Waage does not. A slightly higher cutoff would take us outside Gene's
search range, but would probably still add temperaments of interest
to Herman. Anyway, here's the resulting list top-41 list:

Number 1 Meantone

[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
TOP generators [1201.698520, 504.1341314]
bad: 6.5251 comp: 3.562072 err: 1.698521

Number 2 Magic

[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
TOP generators [1201.276744, 380.7957184]
bad: 7.0687 comp: 4.274486 err: 1.276744

Number 3 Pajara

[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
TOP generators [598.4467109, 106.5665459]
bad: 7.1567 comp: 2.988993 err: 3.106578

Number 4 Semisixths

[7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
TOP generators [1198.389531, 443.1602931]
bad: 7.8851 comp: 4.630693 err: 1.610469

Number 5 Dominant Seventh

[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
TOP generators [1195.228951, 495.8810151]
bad: 8.0970 comp: 2.454561 err: 4.771049

Number 6 Injera

[2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
TOP generators [600.8889070, 93.60982493]
bad: 8.2512 comp: 3.445412 err: 3.582707

Number 7 Kleismic

[6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
TOP generators [1203.187309, 317.8344609]
bad: 8.3168 comp: 3.785579 err: 3.187309

Number 8 Hemifourths

[2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
TOP generators [1203.668841, 252.4803582]
bad: 8.3374 comp: 3.445412 err: 3.66884

Number 9 Negri

[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
TOP generators [1203.187309, 124.8419629]
bad: 8.3420 comp: 3.804173 err: 3.187309

Number 10 Tripletone

[3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
TOP generators [399.0200131, 92.45965769]
bad: 8.4214 comp: 4.045351 err: 2.939961

Number 11 Schismic

[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
TOP generators [1200.760624, 498.1193303]
bad: 8.5260 comp: 5.618543 err: .912904

Number 12 Superpythagorean

[1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
TOP generators [1197.596121, 489.4271829]
bad: 8.6400 comp: 4.602303 err: 2.403879

Number 13 Orwell

[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
TOP generators [1199.532657, 271.4936472]
bad: 8.6780 comp: 5.706260 err: .946061

Number 14 Augmented

[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
TOP generators [399.9922103, 107.3111730]
bad: 8.7811 comp: 2.147741 err: 5.870879

Number 15 Porcupine

[3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
TOP generators [1196.905960, 162.3176609]
bad: 8.9144 comp: 4.295482 err: 3.094040

Number 16

[6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
TOP generators [598.4467109, 162.3159606]
bad: 8.9422 comp: 4.306766 err: 3.106578

Number 17 Supermajor seconds

[3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
TOP generators [1201.698520, 232.5214630]
bad: 9.1819 comp: 5.522763 err: 1.698521

Number 18 Flattone

[1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
TOP generators [1202.536419, 507.1379663]
bad: 9.1883 comp: 4.909123 err: 2.536420

Number 19 Diminished

[4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
TOP generators [298.5321149, 101.4561401]
bad: 9.2912 comp: 2.523719 err: 5.871540

Number 20

[6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
TOP generators [1202.659696, 82.97467050]
bad: 9.3161 comp: 4.306766 err: 3.480440

Number 21

[0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
TOP generators [99.80617249, 24.58395811]
bad: 9.3774 comp: 4.295482 err: 3.557008

Number 22

[3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
TOP generators [1202.900537, 570.4479508]
bad: 9.5280 comp: 4.891080 err: 2.900537

Number 23

[3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
TOP generators [1202.624742, 569.0491468]
bad: 9.6275 comp: 5.168119 err: 2.624742

Number 24 Nonkleismic

[10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
TOP generators [1198.828458, 309.8926610]
bad: 9.7206 comp: 6.309298 err: 1.171542

Number 25 Miracle

[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
TOP generators [1200.631014, 116.7206423]
bad: 9.8358 comp: 6.793166 err: .631014

Number 26 Beatles

[2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
TOP generators [1197.104145, 354.7203384]
bad: 9.8915 comp: 5.162806 err: 2.895855

Number 27 -- formerly Number 82

[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
TOP generators [601.7004928, 230.8749260]
bad: 10.0002 comp: 4.619353 err: 3.740932

Number 28

[3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
TOP generators [1195.486066, 559.3589487]
bad: 10.0368 comp: 4.075900 err: 4.513934

Number 29

[8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
TOP generators [599.2769413, 272.3123381]
bad: 10.1077 comp: 5.047438 err: 3.268439

Number 30 Blackwood

[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
TOP generators [239.1786927, 83.83059859]
bad: 10.1851 comp: 2.173813 err: 7.239629

Number 31 Quartaminorthirds

[9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
TOP generators [1199.792743, 77.83315314]
bad: 10.1855 comp: 6.742251 err: 1.049791

Number 32

[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
TOP generators [1201.135545, 387.5841360]
bad: 10.2131 comp: 6.411729 err: 1.525246

Number 33

[6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
TOP generators [399.0200131, 46.12154491]
bad: 10.2154 comp: 5.369353 err: 2.939961

Number 34

[0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
TOP generators [99.83462277, 16.52294019]
bad: 10.2188 comp: 5.168119 err: 3.215955

Number 35

[5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
TOP generators [1194.335372, 99.13879319]
bad: 10.3332 comp: 3.445412 err: 5.664628

Number 36

[6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
TOP generators [399.8000105, 155.5708520]
bad: 10.4461 comp: 3.804173 err: 5.291448

Number 37

[1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
TOP generators [1195.155395, 496.2398890]
bad: 10.4972 comp: 4.075900 err: 4.974313

Number 38 Superkleismic

[9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
TOP generators [1201.371918, 322.3731369]
bad: 10.5077 comp: 6.742251 err: 1.371918

Number 39

[9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
TOP generators [133.0066710, 35.40561749]
bad: 10.6719 comp: 5.706260 err: 2.939961

Number 40

[6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
TOP generators [199.0788921, 88.83392059]
bad: 10.7036 comp: 3.820609 err: 5.526647

Number 41 Diaschismic

[2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
TOP generators [599.3662015, 103.7870123]
bad: 10.7079 comp: 6.966993 err: 1.267597

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Number 15 Porcupine
>
> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> TOP generators [1196.905960, 162.3176609]
> bad: 8.9144 comp: 4.295482 err: 3.094040
>
> Number 16
>
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 162.3159606]
> bad: 8.9422 comp: 4.306766 err: 3.106578

These two are related, and very close in terms of error and
complexity. Bi porcupine makes me think of porcupine sex, and I'm not
sure I want to, but some sort of porky name seems apt.

> Number 25 Miracle
>
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 9.8358 comp: 6.793166 err: .631014

This seems absurdly far down the list. I think mine was better.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > This comparison of different outputs for the same temperament shows
> up
> > the need to correctly normalise the new weighted error and
> complexity
> > figures so they actually have units we can relate to. i.e. cents for
> > the error and gens per interval for the complexity.
> >
> > This should be simple to do.
> >
> > I think the correct normalisation of a weighted norm is the one
> where,
> > if every individual value happened to be X then the, the norm would
> > also be X, irrespective of the weights.
> >
> > e.g. if the individual errors are E1, E2, ... En,
>
> You realize that there are an infinite number of errors in the TOP
> case.

No I didn't. How do you mean? Obviously you don't do an infinite
number of calculations. But even so, the normalisation factor may
still converge.

If it's a minimax type of thing then all we need is to divide the
current result by the maximum weight of any interval. Let me guess:
this is lg2(2) = 1 so there would be no change?

Is the weighting the same for the complexity? Minimax where gens per
interval is divided by lg2(product_complexity(interval))?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>Can one of you easily plot these 7-limit temperaments on an error
> >>vs. complexity graph (log log or whatever seemed best with 5-
limit)
> >>so we can all think about what our subjective badness contours
might
> >>look like.
> >
> >I could do that, but as this was done with a log-flat badness
cutoff,
> >there will be a huge gaping hole in the graph. That's why I'm
trying
> >to figure out the whole deal for myself, but no one's helping.
>
> Since I usually like the way you figure things out, I'll do whatever
> I can to help, which may not be much. If there are any particular
> msg. #s associated with this, I'll reread them. I didn't follow
your
> orthogonalization posts at all. :(
>
> Part of the problem is these contours represent musical values, so
> they're ultimately a matter of opinion.

Not the problem here, as Dave simply wanted a graph, for which the
best dataset would involve a single error cutoff and a single
complexity cutoff, both generous enough to satisfy everyone. Too big
a dataset to post to this list, but I hope to be doing this
eventually, starting with wedgies.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > I could do that, but as this was done with a log-flat badness
cutoff,
> > there will be a huge gaping hole in the graph. That's why I'm
trying
> > to figure out the whole deal for myself, but no one's helping.
>
> Try asking for some specific form of help.

I'm afraid I've done my best. Perhaps the way I'm seeing this stuff
bears no resemblance to the way anyone else is, but I do think it's
important that I understand it all myself, in at least one way.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > This comparison of different outputs for the same temperament
shows
> > up
> > > the need to correctly normalise the new weighted error and
> > complexity
> > > figures so they actually have units we can relate to. i.e.
cents for
> > > the error and gens per interval for the complexity.
> > >
> > > This should be simple to do.
> > >
> > > I think the correct normalisation of a weighted norm is the one
> > where,
> > > if every individual value happened to be X then the, the norm
would
> > > also be X, irrespective of the weights.
> > >
> > > e.g. if the individual errors are E1, E2, ... En,
> >
> > You realize that there are an infinite number of errors in the
TOP
> > case.
>
> No I didn't. How do you mean?

It's the minimax over *all* intervals.

> Obviously you don't do an infinite
> number of calculations.

No, you only need to set the primes, and the rest falls out correctly.

>But even so, the normalisation factor may
> still converge.

Yes, it may.

> If it's a minimax type of thing then all we need is to divide the
> current result by the maximum weight of any interval.

This doesn't seem right -- what's your reasoning?

> Let me guess:
> this is lg2(2) = 1 so there would be no change?

There is no need to use base 2 -- the result is the same regardless
of which base you use in the logarithms. The error is measured using
logarithms, but so is the complexity = log(n*d), so error divided by
complexity, which is what you're minimizing the maximum of, is
insensitive to choice of base.

> Is the weighting the same for the complexity? Minimax where gens per
> interval is divided by lg2(product_complexity(interval))?

No particular generator basis is assumed in the TOP complexity
calculations. Instead, it's a direct measure of how much the
tempering simplifies the lattice, and reduces (Gene seems to
imply/agree) to the number of notes per acoustic whatever in the case
of equal (1-dimensional) temperaments.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Number 15 Porcupine
> >
> > [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> > TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> > TOP generators [1196.905960, 162.3176609]
> > bad: 8.9144 comp: 4.295482 err: 3.094040
> >
> > Number 16
> >
> > [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> > TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> > TOP generators [598.4467109, 162.3159606]
> > bad: 8.9422 comp: 4.306766 err: 3.106578
>
> These two are related, and very close in terms of error and
> complexity. Bi porcupine makes me think of porcupine sex, and I'm
not
> sure I want to, but some sort of porky name seems apt.
>
> > Number 25 Miracle
> >
> > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > TOP generators [1200.631014, 116.7206423]
> > bad: 9.8358 comp: 6.793166 err: .631014
>
> This seems absurdly far down the list. I think mine was better.

I don't think this is necessarily too far down the list for every
conceivable user, including possibly Herman. As one would like all
the harmonies to come from a single generalized-diatonic scale, some
musicians may find the complexity of Miracle nearly prohibitive.
However, I was by no means suggesting a ranking like this for our
paper, and in fact I still believe we should avoid ranking
altogether, simply giving a "survey" (ordered by complexity, perhaps,
though almost certainly we would need to start with the most familiar
examples before generalizing) for the systems that fall beneath
whatever cutoff curve we choose to draw on the graph.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> It's the minimax over *all* intervals.

You mean the 1/log(product_complexity) weighted minimax over all
intervals in the prime limit.

> > Obviously you don't do an infinite
> > number of calculations.
>
> No, you only need to set the primes, and the rest falls out correctly.
>

Yes, It's very clever in that way. But still it flies in the face of
Partch's "observation one" (or whatever it is called) that as the
complexity of a frequency ratio increases it must be tuned more and
more accurately to be perceived as just (or something), until it
becomes too complex to hear that way even when tuned precisely.

The very fact that TOP cannot distinguish between a good 7-limit
temperament and a good 9-limit temperament (nor 8 or 10 limit) should
make one suspicious.

I suspect that while the results of tenney weighting are quite good
for 5-limit, and may be acceptable for 7-limit, we might find it
agreeing less and less with our subjective experience when we go to 11
and 13 limits. But then again, maybe not. I'll wait and see (or hear).

> >But even so, the normalisation factor may
> > still converge.
>
> Yes, it may.
>
> > If it's a minimax type of thing then all we need is to divide the
> > current result by the maximum weight of any interval.
>
> This doesn't seem right -- what's your reasoning?

I gave it in
/tuning-math/message/8876

It's what you get when you generalise the correctly normalised MAD
(p=1), RMS (p=2) etc. as p goes to infinity.

And in this application the interval with the maximum weight is the
one with the lowest product_complexity, namely the octave.

> > Let me guess:
> > this is lg2(2) = 1 so there would be no change?

I've just confirmed that for myself regarding errors (but not
complexity yet), by examining a few examples from Gene's list.
Although I should have given this maximum weight as 1/lg2(2).

> There is no need to use base 2 -- the result is the same regardless
> of which base you use in the logarithms. The error is measured using
> logarithms, but so is the complexity = log(n*d), so error divided by
> complexity, which is what you're minimizing the maximum of, is
> insensitive to choice of base.

Paul, that's clearly not what Gene has done in this list, otherwise
the error figures would be a factor of 1200 smaller than they are.
Gene has in fact already correctly normalised the errors and given
them in cents.

If you scan down the list
/tuning-math/message/8809
you will see that in most cases the error figure given for the
temperament happens to be the same as the cents error in the octave,
and is never less than it.

It's just that one now has to think like this:

If the minimax tenney-weighted error for the temperament is X cents,
then the errors in the following intervals could be as large as:

Interval Error(cents)
----------------------
1:2 X
2:3 2.6 X
4:5 4.3 X
5:6 4.9 X
4:7 4.8 X
5:7 5.1 X
6:7 5.4 X
4:9 5.2 X
5:9 5.5 X
7:9 6.0 X

where the multiplication factor is the base-2 log of the product of
the two sides of the ratio in lowest terms.

> > Is the weighting the same for the complexity? Minimax where gens per
> > interval is divided by lg2(product_complexity(interval))?
>
> No particular generator basis is assumed in the TOP complexity
> calculations. Instead, it's a direct measure of how much the
> tempering simplifies the lattice, and reduces (Gene seems to
> imply/agree) to the number of notes per acoustic whatever in the case
> of equal (1-dimensional) temperaments.

What's an "acoustic whatever"? Anything that relates to sound??? I'm
being a pedant here in case you didn't guess. :-)

OK. That sounds alright, but how do I relate it to upper limits (or
whatever) on the numbers of notes before I get a particular interval?
i.e. in a similar way to what I just did above with the errors.

>> Part of the problem is these contours represent musical values, so
>> they're ultimately a matter of opinion.
>
>Not the problem here, as Dave simply wanted a graph,

I was referring here to lists, not the graph. I was glad to see
Dave ask for that graph by the way, because I think it would help
clarify my perception of the lists I've seen.

-C.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
...
> > Number 25 Miracle
> >
> > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > TOP generators [1200.631014, 116.7206423]
> > bad: 9.8358 comp: 6.793166 err: .631014
>
> This seems absurdly far down the list. I think mine was better.

I agree with Gene here.

Paul has

k*comp + err < x

Isn't a log-flat badness cutoff equivalent to

k*log(comp) + log(err) < x

for some k and x?

If so, something in between might have the form

k*(comp**p) + err**p < x

where 0<p<1.

One might try

k*sqrt(comp) + sqrt(err) < x

for starters.

I'd really like to see these on a chart. Never mind about those Gene's
badness cutoff might have left out (for now). 114 seems like plenty to
choose from.

By the way, it seems that a useful rule of thumb is that the worst
error in the intervals we usually care about is about 5 times the
tenney-weighted minimax error. e.g. you can mentally round it to 2
significant digits, shift the decimal point to the right and divide by
2 to get a rough idea.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > It's the minimax over *all* intervals.
>
> You mean the 1/log(product_complexity) weighted minimax over all
> intervals in the prime limit.
>
> > > Obviously you don't do an infinite
> > > number of calculations.
> >
> > No, you only need to set the primes, and the rest falls out
correctly.
> >
>
> Yes, It's very clever in that way. But still it flies in the face of
> Partch's "observation one" (or whatever it is called) that as the
> complexity of a frequency ratio increases it must be tuned more and
> more accurately to be perceived as just (or something), until it
> becomes too complex to hear that way even when tuned precisely.

I don't think that's quite what Partch says. Manuel, at least, has
always insisted that simpler ratios need to be tuned more accurately,
and harmonic entropy and all the other discordance functions I've
seen show that the increase in discordance for a given amount of
mistuning is greatest for the simplest intervals.

> The very fact that TOP cannot distinguish between a good 7-limit
> temperament and a good 9-limit temperament (nor 8 or 10 limit)
should
> make one suspicious.

Such distinctions may be important for *scales*, but for
temperaments, I'm perfectly happy not to have to worry about them.
Any reasons I shouldn't be?

> I suspect that while the results of tenney weighting are quite good
> for 5-limit, and may be acceptable for 7-limit, we might find it
> agreeing less and less with our subjective experience when we go to
11
> and 13 limits. But then again, maybe not. I'll wait and see (or
hear).

Tenney weighting can be conceived of in other ways than you're
conceiving of it. For example, if you're looking at 13-limit, it
suffices to minimize the maximum weighted error of {13:8, 13:9,
13:10, 13:11, 13:12, 14:13} or any such lattice-spanning set of
intervals. Here the weights are all very close (13:8 gets 1.12 times
the weight of 14:13), *all* the ratios are ratios of 13 so simpler
intervals are not directly weighted *at all*, and yet the TOP result
will still be the same as if you just used the primes. I think TOP is
far more robust than you're giving it credit for.

> > There is no need to use base 2 -- the result is the same
regardless
> > of which base you use in the logarithms. The error is measured
using
> > logarithms, but so is the complexity = log(n*d), so error divided
by
> > complexity, which is what you're minimizing the maximum of, is
> > insensitive to choice of base.
>
> Paul, that's clearly not what Gene has done in this list, otherwise
> the error figures would be a factor of 1200 smaller than they are.

You mean 1200/log(2), right?

> Gene has in fact already correctly normalised the errors and given
> them in cents.

That's a weird interpretation of the units because it's really only
in cents for the octave, and often the octave doesn't even achieve
the reported error.

> It's just that one now has to think like this:

Yes.

> > > Is the weighting the same for the complexity? Minimax where
gens per
> > > interval is divided by lg2(product_complexity(interval))?
> >
> > No particular generator basis is assumed in the TOP complexity
> > calculations. Instead, it's a direct measure of how much the
> > tempering simplifies the lattice, and reduces (Gene seems to
> > imply/agree) to the number of notes per acoustic whatever in the
case
> > of equal (1-dimensional) temperaments.
>
> What's an "acoustic whatever"? Anything that relates to sound???
I'm
> being a pedant here in case you didn't guess. :-)

An interval that is fixed in (logarithmic) size, as measured for
example in cents.

> OK. That sounds alright, but how do I relate it to upper limits (or
> whatever) on the numbers of notes before I get a particular
>interval?

The word "before" implies some particular generating scheme. In fact
it seems to imply a *single* generator, which is not even possible
for planar, etc. temperaments. However, I wouldn't be surprised if it
is indeed possible to view TOP complexity this way, due to some
amazing theorem to be proved by Gene.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Yes, It's very clever in that way. But still it flies in the face of
> > Partch's "observation one" (or whatever it is called) that as the
> > complexity of a frequency ratio increases it must be tuned more and
> > more accurately to be perceived as just (or something), until it
> > becomes too complex to hear that way even when tuned precisely.
>
> I don't think that's quite what Partch says.

So what _does_ Partch say?

> Manuel, at least, has
> always insisted that simpler ratios need to be tuned more accurately,
> and harmonic entropy and all the other discordance functions I've
> seen show that the increase in discordance for a given amount of
> mistuning is greatest for the simplest intervals.

But surely it's obvious that beat rates go up as something like error
_times_ complexity, not down as error _divided_by_ complexity, even
though beat amplitudes do go down.

> > The very fact that TOP cannot distinguish between a good 7-limit
> > temperament and a good 9-limit temperament (nor 8 or 10 limit)
> should
> > make one suspicious.
>
> Such distinctions may be important for *scales*, but for
> temperaments, I'm perfectly happy not to have to worry about them.
> Any reasons I shouldn't be?

Sure. It is often possible to change generators slightly to reduce
errors in 7-odd-limit intervals at the expense of 9-odd-limit ones.
We've always accepted this in the past.

> > Paul, that's clearly not what Gene has done in this list, otherwise
> > the error figures would be a factor of 1200 smaller than they are.
>
> You mean 1200/log(2), right?

No. 1200.

Start with the error in ratio n:d as another frequency ratio R
(usually irrational). You claimed Gene was giving minimax of

log(R)/log(n*d)

which is the same as

lg2(R)/lg2(n*d)

in fact he is giving minimax of

1200*lg2(R)/lg2(n*d)

where 1200*lg2(R) is the interval's error in cents.

> > Gene has in fact already correctly normalised the errors and given
> > them in cents.
>
> That's a weird interpretation of the units because it's really only
> in cents for the octave, and often the octave doesn't even achieve
> the reported error.

But you agree it's a perfectly valid interpretation, due to the
obvious generalisation of the normalising of weighted MAD, RMS etc. to
the normalising of weighted minimax?

> > > > Is the weighting the same for the complexity? Minimax where
> gens per
> > > > interval is divided by lg2(product_complexity(interval))?
> > >
> > > No particular generator basis is assumed in the TOP complexity
> > > calculations. Instead, it's a direct measure of how much the
> > > tempering simplifies the lattice, and reduces (Gene seems to
> > > imply/agree) to the number of notes per acoustic [interval] in
> > > the case of equal (1-dimensional) temperaments.
...
> > OK. That sounds alright, but how do I relate it to upper limits (or
> > whatever) on the numbers of notes before I get a particular
> >interval?
>
> The word "before" implies some particular generating scheme.

Yes. Presumably any reasonable generating scheme should have some such
interprtetation in terms of the TOP complexity, otherwise what use is it?

> In fact
> it seems to imply a *single* generator, which is not even possible
> for planar, etc. temperaments.

I suspect one can still specify reasonable (maybe even optimal)
generating schemes with multiple generators. In fact since we're
tempering the octave, we apparently need to find such schemes already
for so-called linear temperaments.

Musicians will ultimately want finite scales (even finite compasses)
from these temperaments.

In any case, I'd be happy to hear _any_ such interpretation for this
new complexity measure, even if it only works for linear temperaments.

> However, I wouldn't be surprised if it
> is indeed possible to view TOP complexity this way, due to some
> amazing theorem to be proved by Gene.

I don't require any amazing theorems to be proved. I'd be happy if
such a concrete interpretation could just be shown to work for all
temperaments on Gene's latest list, or even just the first 10 or so.

If some complexity measure cannot be interpreted in concrete terms of
numbers of notes for particular intervals, then why should I trust it
as a means for comparing temperaments?

I'm sure it can be, I'm just dying to know how, or be given enough
information to figure it out for myself. Feel free to point me to
important posts I may have missed.

>> Manuel, at least, has
>> always insisted that simpler ratios need to be tuned more accurately,
>> and harmonic entropy and all the other discordance functions I've
>> seen show that the increase in discordance for a given amount of
>> mistuning is greatest for the simplest intervals.
>
>But surely it's obvious that beat rates go up as something like error
>_times_ complexity, not down as error _divided_by_ complexity, even
>though beat amplitudes do go down.

Without evaluating this, I'll claim that beat rates are not a good
indicator of discordance.

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > Yes, It's very clever in that way. But still it flies in the
face of
> > > Partch's "observation one" (or whatever it is called) that as
the
> > > complexity of a frequency ratio increases it must be tuned more
and
> > > more accurately to be perceived as just (or something), until it
> > > becomes too complex to hear that way even when tuned precisely.
> >
> > I don't think that's quite what Partch says.
>
> So what _does_ Partch say?

I'll have to go home and look.

> > Manuel, at least, has
> > always insisted that simpler ratios need to be tuned more
accurately,
> > and harmonic entropy and all the other discordance functions I've
> > seen show that the increase in discordance for a given amount of
> > mistuning is greatest for the simplest intervals.
>
> But surely it's obvious that beat rates go up as something like
error
> _times_ complexity, not down as error _divided_by_ complexity, even
> though beat amplitudes do go down.

Yes. But beating is virtually irrelevant for the perception of higher-
limit harmonies. Such harmonies, it seems to most people, *must* be
otonal in order to sound convincing at all, and this extreme otonal
bias indicates that virtual pitch and/or combinational tones are the
dominant factors -- beating awould predict equal or greater
consonance for the utonal harmonies, yet high-limit utonalities sound
very dissonant to many people.

> > > The very fact that TOP cannot distinguish between a good 7-limit
> > > temperament and a good 9-limit temperament (nor 8 or 10 limit)
> > should
> > > make one suspicious.
> >
> > Such distinctions may be important for *scales*, but for
> > temperaments, I'm perfectly happy not to have to worry about
them.
> > Any reasons I shouldn't be?
>
> Sure. It is often possible to change generators slightly to reduce
> errors in 7-odd-limit intervals at the expense of 9-odd-limit ones.
> We've always accepted this in the past.

Sure, but we were always assuming full octave-equivalence then, for
one thing.
>
> > > Gene has in fact already correctly normalised the errors and
given
> > > them in cents.
> >
> > That's a weird interpretation of the units because it's really
only
> > in cents for the octave, and often the octave doesn't even
achieve
> > the reported error.
>
> But you agree it's a perfectly valid interpretation, due to the
> obvious generalisation of the normalising of weighted MAD, RMS etc.
to
> the normalising of weighted minimax?

It's hard to say, because of course it's inconceivable for all of the
errors to be equal.

> > > > > Is the weighting the same for the complexity? Minimax where
> > gens per
> > > > > interval is divided by lg2(product_complexity(interval))?
> > > >
> > > > No particular generator basis is assumed in the TOP
complexity
> > > > calculations. Instead, it's a direct measure of how much the
> > > > tempering simplifies the lattice, and reduces (Gene seems to
> > > > imply/agree) to the number of notes per acoustic [interval]
in
> > > > the case of equal (1-dimensional) temperaments.
> ...
> > > OK. That sounds alright, but how do I relate it to upper limits
(or
> > > whatever) on the numbers of notes before I get a particular
> > >interval?
> >
> > The word "before" implies some particular generating scheme.
>
> Yes. Presumably any reasonable generating scheme should have some
such
> interprtetation in terms of the TOP complexity,

Possibly.

> otherwise what use is it?

See above. This seems like a more direct definition of 'complexity'
to me. If I take Gene's confirmation to heart, it's the affine-
geometrical size measure (length, area, volume, etc.) of the portion
of the lattice sufficient, under the relevant temperament, to
represent the entire lattice.

> > In fact
> > it seems to imply a *single* generator, which is not even
possible
> > for planar, etc. temperaments.
>
> I suspect one can still specify reasonable (maybe even optimal)
> generating schemes with multiple generators. In fact since we're
> tempering the octave, we apparently need to find such schemes
already
> for so-called linear temperaments.

I don't think we need to do any of this, nor should we want to for
the purposes of a paper brief enough to be published.

> In any case, I'd be happy to hear _any_ such interpretation for this
> new complexity measure,

See /tuning-math/message/8781 and
/tuning-math/message/8806 . . .

> even if it only works for linear >temperaments.

The whole point is that it works for equal, linear, planar . . . etc.

Oh, you said _such_ . . . Sorry. Well, I suspect that's still
possible, especially since Gene himself explained the lower
complexity of schismic vs. miracle in terms of generators-per-prime.

> I'd be happy if
> such a concrete interpretation could just be shown to work for all
> temperaments on Gene's latest list, or even just the first 10 or so.

Yes, I'm hoping Gene will help me to see this too. I've been groping
here for a better understanding of the math . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > But you agree it's a perfectly valid interpretation, due to the
> > obvious generalisation of the normalising of weighted MAD, RMS etc.
> to
> > the normalising of weighted minimax?
>
> It's hard to say, because of course it's inconceivable for all of the
> errors to be equal.

In our application perhaps, but not in general.

Here's another way to check the normalisation. Scaling all the weights
by the same factor (e.g. using a diffferent log base for the errors
and their weights) should not change the result.

Anyway, the main thing is that we're all happy with the scaling of the
error, irrespective of how we choose to interpret it.

> > Yes. Presumably any reasonable generating scheme should have some
> such
> > interprtetation in terms of the TOP complexity,
>
> Possibly.
>
> > otherwise what use is it?
>
> See above. This seems like a more direct definition of 'complexity'
> to me. If I take Gene's confirmation to heart, it's the affine-
> geometrical size measure (length, area, volume, etc.) of the portion
> of the lattice sufficient, under the relevant temperament, to
> represent the entire lattice.

That sounds pretty good, in a hand-waving sort of way, but when I try
to pin down what it actually means in terms of how big a scale we
might need to make good use of some temperament, my brain just keeps
sliding off it.

> > I suspect one can still specify reasonable (maybe even optimal)
> > generating schemes with multiple generators. In fact since we're
> > tempering the octave, we apparently need to find such schemes
> already
> > for so-called linear temperaments.
>
> I don't think we need to do any of this, nor should we want to for
> the purposes of a paper brief enough to be published.

So we're not goint to talk about how to obtain a finite scale from a
temperament. Now that the octave is just another generator (and the
pair of generators is no longer unique), don't we have some explaining
to do, about why we should iterate one particular generator modulo the
other?

> > In any case, I'd be happy to hear _any_ such interpretation for this
> > new complexity measure,
>
> See /tuning-math/message/8781 and
> /tuning-math/message/8806 . . .

Thanks. I may be some time ... :-)

> > even if it only works for linear >temperaments.
>
> The whole point is that it works for equal, linear, planar . . . etc.
>
> Oh, you said _such_ . . . Sorry. Well, I suspect that's still
> possible, especially since Gene himself explained the lower
> complexity of schismic vs. miracle in terms of generators-per-prime.

Yes. I meant "even if some particular _interpretation_ only works for
linear". I understand the _complexity_measure_ is intended to work for
all, and as such it sounds brilliant.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> One might try
>
> k*sqrt(comp) + sqrt(err) < x
>
> for starters.

All right, just for you . . . This time I'll insist Waage just makes
it in despite its high complexity. Then k=1.75 gives the most
inclusive cutoff which is still more stringent than Gene's, so
culling that list will work again.

Number 1 Meantone (Huygens)

[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
TOP generators [1201.698520, 504.1341314]
bad: 4.6061 comp: 3.562072 err: 1.698521

Number 2 Magic

[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
TOP generators [1201.276744, 380.7957184]
bad: 4.7480 comp: 4.274486 err: 1.276744

Number 3 Pajara

[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
TOP generators [598.4467109, 106.5665459]
bad: 4.7881 comp: 2.988993 err: 3.106578

Number 4 Dominant Seventh

[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
TOP generators [1195.228951, 495.8810151]
bad: 4.9260 comp: 2.454561 err: 4.771049

Number 5 Augmented

[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
TOP generators [399.9922103, 107.3111730]
bad: 4.9876 comp: 2.147741 err: 5.870879

Number 6 Semisixths

[7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
TOP generators [1198.389531, 443.1602931]
bad: 5.0349 comp: 4.630693 err: 1.610469

Number 7 Schismic

[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
TOP generators [1200.760624, 498.1193303]
bad: 5.1036 comp: 5.618543 err: .912904

Number 8 Injera

[2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
TOP generators [600.8889070, 93.60982493]
bad: 5.1411 comp: 3.445412 err: 3.582707

Number 9 Orwell

[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
TOP generators [1199.532657, 271.4936472]
bad: 5.1530 comp: 5.706260 err: .946061

Number 10 Hemifourths

[2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
TOP generators [1203.668841, 252.4803582]
bad: 5.1637 comp: 3.445412 err: 3.668842

Number 11 Kleismic

[6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
TOP generators [1203.187309, 317.8344609]
bad: 5.1902 comp: 3.785579 err: 3.187309

Number 12 Negri

[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
TOP generators [1203.187309, 124.8419629]
bad: 5.1986 comp: 3.804173 err: 3.187309

Number 13 Diminished

[4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
TOP generators [298.5321149, 101.4561401]
bad: 5.2032 comp: 2.523719 err: 5.871540

Number 14 Tripletone

[3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
TOP generators [399.0200131, 92.45965769]
bad: 5.2344 comp: 4.045351 err: 2.939961

Number 15 Blackwood

[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
TOP generators [239.1786927, 83.83059859]
bad: 5.2708 comp: 2.173813 err: 7.239629

Number 16 Superpythagorean

[1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
TOP generators [1197.596121, 489.4271829]
bad: 5.3047 comp: 4.602303 err: 2.403879

Number 17 Miracle

[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
TOP generators [1200.631014, 116.7206423]
bad: 5.3555 comp: 6.793166 err: .631014

Number 18 Porcupine

[3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
TOP generators [1196.905960, 162.3176609]
bad: 5.3860 comp: 4.295482 err: 3.094040

Number 19 Zeta Reticuli Trisex Porky

[6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
TOP generators [598.4467109, 162.3159606]
bad: 5.3943 comp: 4.306766 err: 3.106578

Number 20 Supermajor seconds

[3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
TOP generators [1201.698520, 232.5214630]
bad: 5.4159 comp: 5.522763 err: 1.698521

Number 21 Flattone

[1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
TOP generators [1202.536419, 507.1379663]
bad: 5.4700 comp: 4.909123 err: 2.536420

Number 22 Nonkleismic

[10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
TOP generators [1198.828458, 309.8926610]
bad: 5.4781 comp: 6.309298 err: 1.171542

Number 23

[6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
TOP generators [1202.659696, 82.97467050]
bad: 5.4973 comp: 4.306766 err: 3.480440

Number 24

[0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
TOP generators [99.80617249, 24.58395811]
bad: 5.5130 comp: 4.295482 err: 3.557008

Number 25 Decimal

[4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
TOP generators [603.8288989, 250.6116362]
bad: 5.5474 comp: 2.523719 err: 7.657798

Number 26 Quartaminorthirds

[9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
TOP generators [1199.792743, 77.83315314]
bad: 5.5686 comp: 6.742251 err: 1.049791

Number 27

[3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
TOP generators [1202.900537, 570.4479508]
bad: 5.5734 comp: 4.891080 err: 2.900537

Number 28

[3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
TOP generators [1202.624742, 569.0491468]
bad: 5.5985 comp: 5.168119 err: 2.624742

Number 29 Pelogic

[1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
TOP generators [1209.734056, 532.9412251]
bad: 5.6088 comp: 2.022675 err: 9.734056

Number 30 Dicot

[2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
TOP generators [1204.048159, 356.3998255]
bad: 5.6237 comp: 2.137243 err: 9.396316

Number 31

[5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
TOP generators [1194.335372, 99.13879319]
bad: 5.6284 comp: 3.445412 err: 5.664628

Number 32 Catakleismic

[6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
TOP generators [1200.536355, 316.9063960]
bad: 5.6313 comp: 7.836558 err: .536356

Number 33

[3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
TOP generators [1195.486066, 559.3589487]
bad: 5.6577 comp: 4.075900 err: 4.513934

Number 34

[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
TOP generators [1201.135545, 387.5841360]
bad: 5.6663 comp: 6.411729 err: 1.525246

Number 35 Beatles

[2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
TOP generators [1197.104145, 354.7203384]
bad: 5.6780 comp: 5.162806 err: 2.895855

Number 36 Formerly Number 82

[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
TOP generators [601.7004928, 230.8749260]
bad: 5.6954 comp: 4.619353 err: 3.740932

Number 37

[3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
TOP generators [1193.415676, 158.1468146]
bad: 5.6994 comp: 3.205865 err: 6.584324

Number 38

[3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
TOP generators [397.8052253, 76.63521863]
bad: 5.7070 comp: 3.221612 err: 6.584324

Number 39 {28/27, 50/49}

[2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
TOP generators [595.7998193, 127.8698005]
bad: 5.7115 comp: 2.584059 err: 8.400361

Number 40

[6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
TOP generators [399.8000105, 155.5708520]
bad: 5.7136 comp: 3.804173 err: 5.291448

Number 41 Superkleismic

[9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
TOP generators [1201.371918, 322.3731369]
bad: 5.7153 comp: 6.742251 err: 1.371918

Number 42

[8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
TOP generators [599.2769413, 272.3123381]
bad: 5.7395 comp: 5.047438 err: 3.268439

Number 43 Diaschismic

[2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
TOP generators [599.3662015, 103.7870123]
bad: 5.7450 comp: 6.966993 err: 1.267597

Number 44 Beep

[2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
TOP generators [1194.642673, 254.8994697]
bad: 5.7543 comp: 1.292030 err: 14.176105

Number 45

[1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
TOP generators [1195.155395, 496.2398890]
bad: 5.7634 comp: 4.075900 err: 4.974313

Number 46

[6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
TOP generators [399.0200131, 46.12154491]
bad: 5.7697 comp: 5.369353 err: 2.939961

Number 47

[6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
TOP generators [199.0788921, 88.83392059]
bad: 5.7715 comp: 3.820609 err: 5.526647

Number 48

[0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
TOP generators [99.83462277, 16.52294019]
bad: 5.7717 comp: 5.168119 err: 3.215955

Number 49 Supersupermajor

[3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
TOP generators [1200.231587, 234.3804692]
bad: 5.7926 comp: 7.670504 err: .894655

Number 50

[11, 13, 17, -5, -4, 3] [[1, 3, 4, 5], [0, -11, -13, -17]]
TOP tuning [1198.514750, 1899.600936, 2789.762356, 3371.570447]
TOP generators [1198.514750, 154.1766650]
bad: 5.8290 comp: 6.940227 err: 1.485250

Number 51 {1728/1715, 4000/3993}

[11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
TOP generators [1199.083445, 45.17026643]
bad: 5.8298 comp: 7.752178 err: .916555

Number 52 {25/24, 81/80} Jamesbond?

[0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
TOP generators [172.7759159, 86.69241190]
bad: 5.8344 comp: 2.493450 err: 9.431411

Number 53

[5, 3, 7, -7, -3, 8] [[1, 1, 2, 2], [0, 5, 3, 7]]
TOP tuning [1192.540126, 1890.131381, 2803.635005, 3361.708008]
TOP generators [1192.540126, 139.5182509]
bad: 5.8395 comp: 3.154649 err: 7.459874

Number 54

[9, -2, 14, -24, -3, 38] [[1, 3, 2, 5], [0, -9, 2, -14]]
TOP tuning [1201.918556, 1904.657347, 2781.858962, 3363.439837]
TOP generators [1201.918557, 189.0109248]
bad: 5.8541 comp: 6.521440 err: 1.918557

Number 55 Octacot

[8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
TOP generators [1199.031259, 88.05739491]
bad: 5.8567 comp: 7.752178 err: .968741

Number 56 Wizard

[12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
TOP generators [600.3197857, 216.7702531]
bad: 5.8788 comp: 8.423526 err: .639571

Number 57

[9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
TOP generators [133.0066710, 35.40561749]
bad: 5.8950 comp: 5.706260 err: 2.939961

Number 58 Squares

[4, 16, 9, 16, 3, -24] [[1, 3, 8, 6], [0, -4, -16, -9]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.067656]
TOP generators [1201.698520, 426.4581630]
bad: 5.8971 comp: 6.890825 err: 1.698521

Number 59

[2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
TOP generators [1204.567524, 355.9419091]
bad: 5.8982 comp: 2.696901 err: 9.146173

Number 60 Waage

[0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
TOP generators [100.0514209, 16.55882096]
bad: 5.9023 comp: 8.548972 err: .617051

> I'd really like to see these on a chart.

I think Gene may be using the wrong norm to get his complexity
values. I'll wait until I'm sure they're right or corrected.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> That sounds pretty good, in a hand-waving sort of way, but when I
try
> to pin down what it actually means in terms of how big a scale we
> might need to make good use of some temperament, my brain just keeps
> sliding off it.

Might that be, in a way, a necessary result of dropping octave-
equivalence?

> > > I suspect one can still specify reasonable (maybe even optimal)
> > > generating schemes with multiple generators. In fact since we're
> > > tempering the octave, we apparently need to find such schemes
> > already
> > > for so-called linear temperaments.
> >
> > I don't think we need to do any of this, nor should we want to
for
> > the purposes of a paper brief enough to be published.
>
> So we're not goint to talk about how to obtain a finite scale from a
> temperament. Now that the octave is just another generator (and the
> pair of generators is no longer unique), don't we have some
explaining
> to do, about why we should iterate one particular generator modulo
the
> other?

For linear temperaments, I agree that we may still want to give the
period/generator specification, but there are many ways to justify
the attention on this. For example, though we didn't assume octave-
equivalence, we may want to assume that the musician will generally
be most interested in subsets of the temperament that repeat at the
(tempered) octave.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > One might try
> >
> > k*sqrt(comp) + sqrt(err) < x
> >
> > for starters.
>
> All right, just for you . . . This time I'll insist Waage just makes
> it in despite its high complexity. Then k=1.75 gives the most
> inclusive cutoff which is still more stringent than Gene's, so
> culling that list will work again.
...

Well I wouldn't want as many as sixty. I think about half that would
be plenty.

> I think Gene may be using the wrong norm to get his complexity
> values. I'll wait until I'm sure they're right or corrected.

Well something's wrong. Whether its the badness functions or only the
complexity I don't know. But Diaschismic shouldn't be so far down. I
don't think Miracle should be so far down either. Sure it gets a hit
for having 6 gens to the fifth, but not that much of a hit I would think.

And where's Shrutar?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > That sounds pretty good, in a hand-waving sort of way, but when I
> try
> > to pin down what it actually means in terms of how big a scale we
> > might need to make good use of some temperament, my brain just keeps
> > sliding off it.
>
> Might that be, in a way, a necessary result of dropping octave-
> equivalence?

It might be, but in any case I'd still like to get on top of it.

Without octave equivalence we can still make it finite by limiting it
to the range of human hearing, say 10 octaves.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> No particular generator basis is assumed in the TOP complexity
> calculations. Instead, it's a direct measure of how much the
> tempering simplifies the lattice, and reduces (Gene seems to
> imply/agree) to the number of notes per acoustic whatever in the
case
> of equal (1-dimensional) temperaments.

I thought equal temperaments were "0-dimensional"?

The complexity measure for an equal temperament pretty much reduces
to the division of the octave. You could take the maximum of
|(mapping of p)/log2(p)| if you wanted to be precise about it.

>For linear temperaments, I agree that we may still want to give the
>period/generator specification, but there are many ways to justify
>the attention on this. For example, though we didn't assume octave-
>equivalence, we may want to assume that the musician will generally
>be most interested in subsets of the temperament that repeat at the
>(tempered) octave.

As long as the temperament maps the octave, will it not be periodic
at said octave?

-Carl

>So we're not goint to talk about how to obtain a finite scale from a
>temperament. Now that the octave is just another generator (and the
>pair of generators is no longer unique), don't we have some explaining
>to do, about why we should iterate one particular generator modulo the
>other?

No generator is privileged in the mod relationship (any may be taken
as mod any other). And since the pair of generators was never unique,
we have no more explaining to do now than we ever did. Though I am
interested in any relationship between these two ways of looking at
temperament.

>> > even if it only works for linear temperaments.
>>
>> The whole point is that it works for equal, linear, planar . . . etc.
>>
>> Oh, you said _such_ . . . Sorry. Well, I suspect that's still
>> possible, especially since Gene himself explained the lower
>> complexity of schismic vs. miracle in terms of generators-per-prime.
>
>Yes. I meant "even if some particular _interpretation_ only works for
>linear". I understand the _complexity_measure_ is intended to work for
>all, and as such it sounds brilliant.

Graham saw fit to report the size of the smallest MOS, which seems
founded on the noble goal of trying to measure complexity in terms of
number of notes, across temperaments. However it seems likely that
size of smallest MOS is just an artifact of the particular choice of
generators, which we now know is not unique. I've suggested
consonance/notes graphs, but again: how to choose the notes? I've
also suggested number of notes in a corresponding Fokker block, but
I'll have to defer to Paul on the status of that suggestion.

-Carl

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > That sounds pretty good, in a hand-waving sort of way, but when
I
> > try
> > > to pin down what it actually means in terms of how big a scale
we
> > > might need to make good use of some temperament, my brain just
keeps
> > > sliding off it.
> >
> > Might that be, in a way, a necessary result of dropping octave-
> > equivalence?
>
> It might be, but in any case I'd still like to get on top of it.
>
> Without octave equivalence we can still make it finite by limiting
it
> to the range of human hearing, say 10 octaves.

Sure, but then we won't have a clear choice of generator even for
linear temperaments. I think *scale* (finite non-ET pitch set)
questions would require a separate paper, beginning from periodicity
blocks . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > No particular generator basis is assumed in the TOP complexity
> > calculations. Instead, it's a direct measure of how much the
> > tempering simplifies the lattice, and reduces (Gene seems to
> > imply/agree) to the number of notes per acoustic whatever in the
> case
> > of equal (1-dimensional) temperaments.
>
> I thought equal temperaments were "0-dimensional"?

We've always said linear temperaments were actually 2-dimensional, so
equal temperaments are 1-dimensional, and are generated by their
step. Remember we've dropped octave-equivalence to get TOP.

> The complexity measure for an equal temperament pretty much reduces
> to the division of the octave.

More precisely, to the division of the *acoustical* octave, or any
other fixed value in cents, apparently . . .

> You could take the maximum of
> |(mapping of p)/log2(p)| if you wanted to be precise about it.

Could you take another look at the "Attn: Gene 2" post and explain
what's going on there, mathematically? I didn't use the maximum but
that was only 3-limit . . .

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >For linear temperaments, I agree that we may still want to give
the
> >period/generator specification, but there are many ways to justify
> >the attention on this. For example, though we didn't assume octave-
> >equivalence, we may want to assume that the musician will
generally
> >be most interested in subsets of the temperament that repeat at
the
> >(tempered) octave.
>
> As long as the temperament maps the octave, will it not be periodic
> at said octave?
>
> -Carl

Unless it's an equal temperament, it will have an infinite density of
notes, periodic at any conceivable interval in the temperament.
Whether the *finite subsets* you choose repeat at the octave or not,
though, is up to you.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >So we're not goint to talk about how to obtain a finite scale from
a
> >temperament. Now that the octave is just another generator (and the
> >pair of generators is no longer unique), don't we have some
explaining
> >to do, about why we should iterate one particular generator modulo
the
> >other?
>
> No generator is privileged in the mod relationship (any may be taken
> as mod any other). And since the pair of generators was never
unique,
> we have no more explaining to do now than we ever did.

Well, it was a lot more unique before, since one of the generators
was totally unique -- the period -- and the other could be defined
uniquely within any given half-period range (usually taken to be 0 to
1/2).

> I've
> also suggested number of notes in a corresponding Fokker block, but
> I'll have to defer to Paul on the status of that suggestion.

Again, this is exactly what I've been discussing in this thread and
in "Attn: Gene 2", except that you have to replace "Fokker block"
with "Fokker strip", "Fokker slice", or whatever is appropriate
(block only appropriate for ET). In the latter cases, the number of
notes is infinite, but the "size" of the relevant construct in the
lattice is still ultra-meaningful, and what I've been itching about
here.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > One might try
> > >
> > > k*sqrt(comp) + sqrt(err) < x
> > >
> > > for starters.
> >
> > All right, just for you . . . This time I'll insist Waage just
makes
> > it in despite its high complexity. Then k=1.75 gives the most
> > inclusive cutoff which is still more stringent than Gene's, so
> > culling that list will work again.
> ...
>
> Well I wouldn't want as many as sixty. I think about half that would
> be plenty.

Of course our paper should have even fewer. I'd like to see Injera
but I won't insist on it. But I was just playing along for the
purposes of this thread.

> > I think Gene may be using the wrong norm to get his complexity
> > values. I'll wait until I'm sure they're right or corrected.
>
> Well something's wrong. Whether its the badness functions or only
the
> complexity I don't know.

Would you take a closer look, then? I think it's important to rethink
the problem each time, to allow old prejudices a chance of dissolving.

> But Diaschismic shouldn't be so far down.

For 7-limit? This is only one of at least 3 possible 7-limit
mappings, so it's weird to even use that name.

> And where's Shrutar?

It fell off the end, with a badness of 6.1221 in this formulation.
Doesn't shine as a 7-limit linear temperament . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> I think Gene may be using the wrong norm to get his complexity
> values. I'll wait until I'm sure they're right or corrected.

What's the defintion of "right"?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> Well something's wrong. Whether its the badness functions or only
the
> complexity I don't know. But Diaschismic shouldn't be so far down. I
> don't think Miracle should be so far down either. Sure it gets a hit
> for having 6 gens to the fifth, but not that much of a hit I would
think.

It gets an even harder hit for making the major sixth so complex,
which seems fair enough.

> And where's Shrutar?

Far down the list somewhere.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> We've always said linear temperaments were actually 2-dimensional,
so
> equal temperaments are 1-dimensional, and are generated by their
> step. Remember we've dropped octave-equivalence to get TOP.

You boiled me in oil and rendered me down for lard when I first got
here for wanting things this way, and now you claim we've always said
it? :)

> Could you take another look at the "Attn: Gene 2" post and explain
> what's going on there, mathematically? I didn't use the maximum but
> that was only 3-limit . . .

I could but it might be one of those posts where I don't know what
you are really asking for.

>> > No particular generator basis is assumed in the TOP complexity
>> > calculations. Instead, it's a direct measure of how much the
>> > tempering simplifies the lattice, and reduces (Gene seems to
>> > imply/agree) to the number of notes per acoustic whatever in the
>> > case of equal (1-dimensional) temperaments.
>>
>> I thought equal temperaments were "0-dimensional"?
>
>We've always said linear temperaments were actually 2-dimensional,
>so equal temperaments are 1-dimensional, and are generated by their
>step. Remember we've dropped octave-equivalence to get TOP.

Are you suggesting that TOP "linear" temperaments have a greater
dimensionality than old-style "linear" temperaments?

-Carl

>> I've also suggested number of notes in a corresponding Fokker block,
>> but I'll have to defer to Paul on the status of that suggestion.
>
>Again, this is exactly what I've been discussing in this thread and
>in "Attn: Gene 2", except that you have to replace "Fokker block"
>with "Fokker strip", "Fokker slice", or whatever is appropriate
>(block only appropriate for ET). In the latter cases, the number of
>notes is infinite, but the "size" of the relevant construct in the
>lattice is still ultra-meaningful, and what I've been itching about
>here.

Ok.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> > No particular generator basis is assumed in the TOP complexity
> >> > calculations. Instead, it's a direct measure of how much the
> >> > tempering simplifies the lattice, and reduces (Gene seems to
> >> > imply/agree) to the number of notes per acoustic whatever in
the
> >> > case of equal (1-dimensional) temperaments.
> >>
> >> I thought equal temperaments were "0-dimensional"?
> >
> >We've always said linear temperaments were actually 2-dimensional,
> >so equal temperaments are 1-dimensional, and are generated by
their
> >step. Remember we've dropped octave-equivalence to get TOP.
>
> Are you suggesting that TOP "linear" temperaments have a greater
> dimensionality than old-style "linear" temperaments?
>
> -Carl

If we assume total octave-equivalence, and just deal in pitch
classes, then the dimensionality of everything basically goes down by
one. But so far, the mathematics of temperament (for example,
detecting torsion) seems to be a lot more straightforward if we don't
assume octave-equivalence, and just treat 2 like any other prime
throughout.

>> Are you suggesting that TOP "linear" temperaments have a greater
>> dimensionality than old-style "linear" temperaments?
>
>If we assume total octave-equivalence, and just deal in pitch
>classes, then the dimensionality of everything basically goes down by
>one. But so far, the mathematics of temperament (for example,
>detecting torsion) seems to be a lot more straightforward if we don't
>assume octave-equivalence, and just treat 2 like any other prime
>throughout.

Agreed but this doesn't seem to address the question. A TOP
"linear" temperament requires the same number of commas as an
old-style "linear" temperament... oh, but the TOP lattice has
an extra dimension. Ok.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:

> > > I think Gene may be using the wrong norm to get his complexity
> > > values. I'll wait until I'm sure they're right or corrected.

Are you sure yet?

> > Well something's wrong. Whether its the badness functions or only
> the
> > complexity I don't know.
>
> Would you take a closer look, then?

This would be a lot easier with an error versus complexity plot.

>I think it's important to rethink
> the problem each time, to allow old prejudices a chance of dissolving.
>

Yes. For example, the tempering of the octave may have reduced the
worst error significantly in some cases. And I may be getting used to
the idea of using the maximum function (inf-norm) for the complexity.

But it's possible that the combination of using maximum _and_ the new
weighting for complexity is too much. i.e. taking it too far from my
(most people's?) subjective judgement of complexity. But then again,
simply reducing k (the penalty for complexity relative to error) in
some badness function might restore sanity for me.

> > But Diaschismic shouldn't be so far down.
>
> For 7-limit? This is only one of at least 3 possible 7-limit
> mappings, so it's weird to even use that name.

Oops. Yeah. I was thinking of the most obvious one

pajara <0 1, -2 -2]

which _is_ appropriately high on the list, but I was looking at

15-limit diaschismic <0 1, -2 -8 ...]

which I agree should be well down the 7-limit list.

Then there's

"56-ET" diaschismic <0 1, -2 9]

and

shrutar <0 1, -2 3.5]

> > And where's Shrutar?
>
> It fell off the end, with a badness of 6.1221 in this formulation.
> Doesn't shine as a 7-limit linear temperament . . .

But wasn't that it's whole raison d'etre?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
>
> > > > I think Gene may be using the wrong norm to get his
complexity
> > > > values. I'll wait until I'm sure they're right or corrected.
>
> Are you sure yet?

No; but it did seem Gene was suggesting that at least one other
complexity formulation would result if we treated the wedgies as
multimonzos instead of as multivals. I'm hoping to get a clear
intuitive, geometrical, and musical understanding of this stuff,
which I was itching for with my "Attn: Gene 2" post, which I'm still
hoping will get some (attention). It may merely require a clear view
of what these mathematical quantities (wedgies?) are saying in the 3-
limit and 5-limit.

> > > Well something's wrong. Whether its the badness functions or
only
> > the
> > > complexity I don't know.
> >
> > Would you take a closer look, then?
>
> This would be a lot easier with an error versus complexity plot.

I'm afraid my arm is up for the task of cutting and pasting all of
Gene's numbers and names that this would require. If someone puts the
relevant information in a table, then anyone could make the required
plot, even you, with Excel.

> >I think it's important to rethink
> > the problem each time, to allow old prejudices a chance of
dissolving.
> >
>
> Yes. For example, the tempering of the octave may have reduced the
> worst error significantly in some cases. And I may be getting used
to
> the idea of using the maximum function (inf-norm) for the
complexity.

Well, for 3-limit and 5-limit complexity, I've been using the
weighted L_1 norm of the monzo, i.e., the Tenney harmonic distance of
the comma. So don't get *too* used to it. :)

> But it's possible that the combination of using maximum _and_ the
new
> weighting for complexity is too much. i.e. taking it too far from my
> (most people's?) subjective judgement of complexity.

Not unreasonable. I look forward to the day when this is all crystal
clear and unsubjective.

> But then again,
> simply reducing k (the penalty for complexity relative to error) in
> some badness function might restore sanity for me.

Sure. Of course, I still hope we can do away with badness and simply
choose by looking at the graph. We should choose some cutoff that
corresponds to a wide swath of empty space in the graph, so that
modest changes in the cutoff do not affect which temperaments make it
in and which don't.

> > > But Diaschismic shouldn't be so far down.
> >
> > For 7-limit? This is only one of at least 3 possible 7-limit
> > mappings, so it's weird to even use that name.
>
> Oops. Yeah. I was thinking of the most obvious one
>
> pajara <0 1, -2 -2]
>
> which _is_ appropriately high on the list, but I was looking at
>
> 15-limit diaschismic <0 1, -2 -8 ...]
>
> which I agree should be well down the 7-limit list.
>
> Then there's
>
> "56-ET" diaschismic <0 1, -2 9]
>
> and
>
> shrutar <0 1, -2 3.5]
>
> > > And where's Shrutar?
> >
> > It fell off the end, with a badness of 6.1221 in this
formulation.
> > Doesn't shine as a 7-limit linear temperament . . .
>
> But wasn't that it's whole raison d'etre?

11-limit, I thought . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > We've always said linear temperaments were actually 2-dimensional,
> so
> > equal temperaments are 1-dimensional, and are generated by their
> > step. Remember we've dropped octave-equivalence to get TOP.
>
> You boiled me in oil and rendered me down for lard when I first got
> here for wanting things this way, and now you claim we've always said
> it? :)

Hee hee. :-)

What I remember we gave you a hard time about, was not that linear
temperaments are 2-dimensional without octave-equivalence, but that
you wanted to call them "planar" (which would have been too confusing
a departure from the historical usage). We wanted "Linear temperament"
to be the constant name of the musical object which remains
essentially the same while its mathematical models vary in dimensionality.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > > wrote:
> >
> > > > > I think Gene may be using the wrong norm to get his
> complexity
> > > > > values. I'll wait until I'm sure they're right or corrected.
> >
> > Are you sure yet?
>
> No; but it did seem Gene was suggesting that at least one other
> complexity formulation would result if we treated the wedgies as
> multimonzos instead of as multivals. I'm hoping to get a clear
> intuitive, geometrical, and musical understanding of this stuff,
> which I was itching for with my "Attn: Gene 2" post, which I'm still
> hoping will get some (attention). It may merely require a clear view
> of what these mathematical quantities (wedgies?) are saying in the 3-
> limit and 5-limit.

If Gene could show how to obtain this complexity figure from a
period-generator prime-mapping in such a way that you get the same
result no matter how you factor the generators (octave, tritave etc.),
this would be a big help to my understanding.

>
> > > > Well something's wrong. Whether its the badness functions or
> only
> > > the
> > > > complexity I don't know.
> > >
> > > Would you take a closer look, then?
> >
> > This would be a lot easier with an error versus complexity plot.
>
> I'm afraid my arm is up for the task of cutting and pasting all of
> Gene's numbers and names that this would require. If someone puts the
> relevant information in a table, then anyone could make the required
> plot, even you, with Excel.

Gene,

Can you please supply your latest list of 114 as a tab-delimited
table, one line per temperament, with the first row being the column
headings?

> > But it's possible that the combination of using maximum _and_ the
> new
> > weighting for complexity is too much. i.e. taking it too far from my
> > (most people's?) subjective judgement of complexity.
>
> Not unreasonable. I look forward to the day when this is all crystal
> clear and unsubjective.

How could it ever be so, except in a statistical sense, by average the
subjectivity of a lot of humans. It is after all a human perceptual or
cognitive property we're trying to model.

What we're really looking for is something that's mathematically
simple and yet close enough to represent the typical human experience.
Trouble is, not many humans have much experience with many linear
temperaments, including us.

> > But then again,
> > simply reducing k (the penalty for complexity relative to error) in
> > some badness function might restore sanity for me.
>
> Sure. Of course, I still hope we can do away with badness and simply
> choose by looking at the graph. We should choose some cutoff that
> corresponds to a wide swath of empty space in the graph, so that
> modest changes in the cutoff do not affect which temperaments make it
> in and which don't.

Yes. I like that idea too. But by "a wide swath" don't you mean one
that it's easy to put a simple smooth curve thru? And you must have
some general idea of which way this intergalactic moat must curve.

> > > > And where's Shrutar?
> > >
> > > It fell off the end, with a badness of 6.1221 in this
> formulation.
> > > Doesn't shine as a 7-limit linear temperament . . .
> >
> > But wasn't that it's whole raison d'etre?
>
> 11-limit, I thought . . .

You're probably right.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > > We've always said linear temperaments were actually 2-
dimensional,
> > so
> > > equal temperaments are 1-dimensional, and are generated by
their
> > > step. Remember we've dropped octave-equivalence to get TOP.
> >
> > You boiled me in oil and rendered me down for lard when I first
got
> > here for wanting things this way, and now you claim we've always
said
> > it? :)
>
> Hee hee. :-)
>
> What I remember we gave you a hard time about, was not that linear
> temperaments are 2-dimensional without octave-equivalence, but that
> you wanted to call them "planar" (which would have been too
confusing
> a departure from the historical usage). We wanted "Linear
temperament"
> to be the constant name of the musical object which remains
> essentially the same while its mathematical models vary in
>dimensionality.

Unfortunately for us, 'linear temperament' has probably never
referred to a multiple-chains-per-octave system (like pajara,
diminished, augmented, ennealimmal . . .) before we started using it
that way, and some of the original users of the term (say, Erv
Wilson) might be rather upset with this slight generalization.

(Now hiding head under sand while Gene throws a fit :)

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Dave Keenan"
<d.keenan@b...>
> > > > wrote:
> > >
> > > > > > I think Gene may be using the wrong norm to get his
> > complexity
> > > > > > values. I'll wait until I'm sure they're right or
corrected.
> > >
> > > Are you sure yet?
> >
> > No; but it did seem Gene was suggesting that at least one other
> > complexity formulation would result if we treated the wedgies as
> > multimonzos instead of as multivals. I'm hoping to get a clear
> > intuitive, geometrical, and musical understanding of this stuff,
> > which I was itching for with my "Attn: Gene 2" post, which I'm
still
> > hoping will get some (attention). It may merely require a clear
view
> > of what these mathematical quantities (wedgies?) are saying in
the 3-
> > limit and 5-limit.
>
> If Gene could show how to obtain this complexity figure from a
> period-generator prime-mapping in such a way that you get the same
> result no matter how you factor the generators (octave, tritave
etc.),
> this would be a big help to my understanding.

Where did he fall short?

>
> > > But it's possible that the combination of using maximum _and_
the
> > new
> > > weighting for complexity is too much. i.e. taking it too far
from my
> > > (most people's?) subjective judgement of complexity.
> >
> > Not unreasonable. I look forward to the day when this is all
crystal
> > clear and unsubjective.
>
> How could it ever be so, except in a statistical sense, by average
the
> subjectivity of a lot of humans. It is after all a human perceptual
or
> cognitive property we're trying to model.

With the complexity measure, I'm hoping it will be an affine-
geometrical property (in the Tenney lattice), purely mathematical but
as unobjectionable as the 3-limit cases I already illustrated.

> What we're really looking for is something that's mathematically
> simple and yet close enough to represent the typical human
experience.
> Trouble is, not many humans have much experience with many linear
> temperaments, including us.

OK, so I meant unsubjective to me. :)

> > > But then again,
> > > simply reducing k (the penalty for complexity relative to
error) in
> > > some badness function might restore sanity for me.
> >
> > Sure. Of course, I still hope we can do away with badness and
simply
> > choose by looking at the graph. We should choose some cutoff that
> > corresponds to a wide swath of empty space in the graph, so that
> > modest changes in the cutoff do not affect which temperaments
make it
> > in and which don't.
>
> Yes. I like that idea too. But by "a wide swath" don't you mean one
> that it's easy to put a simple smooth curve thru?

Yes.

> And you must have
> some general idea of which way this intergalactic moat must curve.

True. Even though it's out-Keenaning Keenan with respect to Smith, I
still think a straight line -- if not a *convex* curve, perish the
thought -- makes some sense. Both error and complexity are things
that we typically judge and compare in a *linear* fashion, so
performing various operations on them seems arbitrary at best. At
least, it seems that if there's zero error, doubling the complexity
should double the badness; and if there's zero complexity, doubling
the error should double the badness.

> > > > > And where's Shrutar?
> > > >
> > > > It fell off the end, with a badness of 6.1221 in this
> > formulation.
> > > > Doesn't shine as a 7-limit linear temperament . . .
> > >
> > > But wasn't that it's whole raison d'etre?
> >
> > 11-limit, I thought . . .
>
> You're probably right.

Well, you came up with the final tuning, so you should know :) I
recall you tempering out 896:891 (though I think this was replaced
with 176:175 in Gene's TM reduction), so it would seem to be 11-
limit . . .

>> What I remember we gave you a hard time about, was not that linear
>> temperaments are 2-dimensional without octave-equivalence, but that
>> you wanted to call them "planar" (which would have been too
>> confusing a departure from the historical usage). We wanted "Linear
>> temperament" to be the constant name of the musical object which
>> remains essentially the same while its mathematical
>> models vary in dimensionality.
>
>Unfortunately for us, 'linear temperament' has probably never
>referred to a multiple-chains-per-octave system (like pajara,
>diminished, augmented, ennealimmal . . .) before we started using it
>that way, and some of the original users of the term (say, Erv
>Wilson) might be rather upset with this slight generalization.

I can't remember Erv ever using the term, and if he had, I can't
imagine him getting upset (by any stretch of the word) over something
like this!

-Carl

>True. Even though it's out-Keenaning Keenan with respect to Smith, I
>still think a straight line -- if not a *convex* curve, perish the
>thought -- makes some sense. Both error and complexity are things
>that we typically judge and compare in a *linear* fashion, so
>performing various operations on them seems arbitrary at best. At
>least, it seems that if there's zero error, doubling the complexity
>should double the badness; and if there's zero complexity, doubling
>the error should double the badness.

John deLaubenfels seemed to feel that error is perceived quadratically,
and this is the way he implemented error pain at one point in his
software.

Complexity, I should think, should definitely be punished more than
1:1. Using 8 notes instead of 7 notes would seem to demand more
than eight 7ths the mental energy.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What I remember we gave you a hard time about, was not that
linear
> >> temperaments are 2-dimensional without octave-equivalence, but
that
> >> you wanted to call them "planar" (which would have been too
> >> confusing a departure from the historical usage). We
wanted "Linear
> >> temperament" to be the constant name of the musical object which
> >> remains essentially the same while its mathematical
> >> models vary in dimensionality.
> >
> >Unfortunately for us, 'linear temperament' has probably never
> >referred to a multiple-chains-per-octave system (like pajara,
> >diminished, augmented, ennealimmal . . .) before we started using
it
> >that way, and some of the original users of the term (say, Erv
> >Wilson) might be rather upset with this slight generalization.
>
> I can't remember Erv ever using the term, and if he had, I can't
> imagine him getting upset (by any stretch of the word) over
something
> like this!
>
> -Carl

Well, a quick look shows that he used the terms "linear
mapping", "linear scale", "linear notation", etc., in a way that
almost certainly assumes *one* and only one chain of octave-
equivalent pitch-classes. On page 5 of

http://www.anaphoria.com/xen3a.PDF

he mentions linear scales/notations of 5, 7, 8, 9, 11, 12, and 13
elements, generated by all the possible generators between 1/2 and
1/3 octave, and goes on to say that he has yet to consider linear
systems which would be generated by a half-fifth or half-fourth, but
the idea of *multiple* chains would not seem to fit into his rubric
here. By the way, some of the simplest multiple-chain systems,
augmented and diaschsimic/pajara, would immediately yield
the "missing" numbers in the list -- 6 and 10, respectively.

Given the clarification that I got on Wilson's MOS concept from
Daniel, Kraig, and others recently, which prompted me to start using
the "DE" terminology, and given his writings, I suspect (very
strongly) that his linear temperament concept is similar.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >True. Even though it's out-Keenaning Keenan with respect to Smith,
I
> >still think a straight line -- if not a *convex* curve, perish the
> >thought -- makes some sense. Both error and complexity are things
> >that we typically judge and compare in a *linear* fashion, so
> >performing various operations on them seems arbitrary at best. At
> >least, it seems that if there's zero error, doubling the
complexity
> >should double the badness; and if there's zero complexity,
doubling
> >the error should double the badness.
>
> John deLaubenfels seemed to feel that error is perceived
>quadratically,
> and this is the way he implemented error pain at one point

At every point.

> in his
> software.

That's mainly because the quadratic function was by far the easiest
one to implement in software!

But I am not at all adverse to assuming a quadratic penatly on error.

> Complexity, I should think, should definitely be punished more than
> 1:1. Using 8 notes instead of 7 notes would seem to demand more
> than eight 7ths the mental energy.

We could use a quadratic penalty on the complexity too. But now we're
talking about *convex* badness contours, while Dave and especially
Gene were proposing *concave* ones (Dave suggested using k*sqrt
(error) + sqrt(complexity)). I think the difficulty with convex
badness contours is that they imply a rather sudden cutoff for some
maximum error and some maximum complexity. This is, indeed, probably
a very accurate reflection of what's relevant for a particular
musician working in a particular style. Against this, though, is the
idea that different musicians may have different needs as regards
lowness of complexity and lowness of error, so superimposing their
individually-convex badness criteria could lead to a 'global' badness
criterion that isn't convex. Taking this idea to its logical extreme
leads to things like log-flat badness.

>> >Unfortunately for us, 'linear temperament' has probably never
>> >referred to a multiple-chains-per-octave system (like pajara,
>> >diminished, augmented, ennealimmal . . .) before we started using
>> >it that way, and some of the original users of the term (say, Erv
>> >Wilson) might be rather upset with this slight generalization.
>>
>> I can't remember Erv ever using the term, and if he had, I can't
>> imagine him getting upset (by any stretch of the word) over
>> something like this!
>
>Well, a quick look shows that he used the terms "linear
>mapping", "linear scale", "linear notation", etc., in a way that
>almost certainly assumes *one* and only one chain of octave-
>equivalent pitch-classes.

But not a chain of one and only one generator.

>he mentions linear scales/notations of 5, 7, 8, 9, 11, 12, and 13
>elements, generated by all the possible generators between 1/2 and
>1/3 octave, and goes on to say that he has yet to consider linear
>systems which would be generated by a half-fifth or half-fourth, but
>the idea of *multiple* chains would not seem to fit into his rubric
>here.

But he doesn't think of these as temperaments.

>Given the clarification that I got on Wilson's MOS concept from
>Daniel, Kraig, and others recently, which prompted me to start using
>the "DE" terminology, and given his writings, I suspect (very
>strongly) that his linear temperament concept is similar.

His concept of temperament is that "I would almost never do it".

In any case, if you haven't met Erv, you might not realize that
he is about as non-committal on terminology and "suspicious of
language" as it's possible to get.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Unfortunately for us, 'linear temperament' has probably never
> >> >referred to a multiple-chains-per-octave system (like pajara,
> >> >diminished, augmented, ennealimmal . . .) before we started
using
> >> >it that way, and some of the original users of the term (say,
Erv
> >> >Wilson) might be rather upset with this slight generalization.
> >>
> >> I can't remember Erv ever using the term, and if he had, I can't
> >> imagine him getting upset (by any stretch of the word) over
> >> something like this!
> >
> >Well, a quick look shows that he used the terms "linear
> >mapping", "linear scale", "linear notation", etc., in a way that
> >almost certainly assumes *one* and only one chain of octave-
> >equivalent pitch-classes.
>
> But not a chain of one and only one generator.

Huh? How not??

> >he mentions linear scales/notations of 5, 7, 8, 9, 11, 12, and 13
> >elements, generated by all the possible generators between 1/2 and
> >1/3 octave, and goes on to say that he has yet to consider linear
> >systems which would be generated by a half-fifth or half-fourth,
but
> >the idea of *multiple* chains would not seem to fit into his
rubric
> >here.
>
> But he doesn't think of these as temperaments.

When he does talk about some of these as temperaments (or the related
just intonation structures with an undistributed commatic unison
vector), though, they're all single-chain.

>> But not a chain of one and only one generator.
>
>Huh? How not??

Because he doesn't temper, the generator varies in size
depending on where you are in the map.

>When he does talk about some of these as temperaments (or the related
>just intonation structures with an undistributed commatic unison
>vector), though, they're all single-chain.

..of a particular generator in scale steps, not interval size.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> But not a chain of one and only one generator.
> >
> >Huh? How not??
>
> Because he doesn't temper, the generator varies in size
> depending on where you are in the map.

That's what I meant by an undistributed commatic unison vector. But
there's still one and only one chain, in contradistinction with
pajara, augmented, diminished, ennealimmal, etc. . . . which was my
point.

> >When he does talk about some of these as temperaments (or the
related
> >just intonation structures with an undistributed commatic unison
> >vector), though, they're all single-chain.
>
> ..of a particular generator in scale steps, not interval size.

Yes.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What I remember we gave you a hard time about, was not that
linear
> >> temperaments are 2-dimensional without octave-equivalence, but
that
> >> you wanted to call them "planar" (which would have been too
> >> confusing a departure from the historical usage). We
wanted "Linear
> >> temperament" to be the constant name of the musical object which
> >> remains essentially the same while its mathematical
> >> models vary in dimensionality.
> >
> >Unfortunately for us, 'linear temperament' has probably never
> >referred to a multiple-chains-per-octave system (like pajara,
> >diminished, augmented, ennealimmal . . .) before we started using
it
> >that way, and some of the original users of the term (say, Erv
> >Wilson) might be rather upset with this slight generalization.
>
> I can't remember Erv ever using the term,

How about the first line of

http://www.anaphoria.com/xen2.PDF

?

>> >> But not a chain of one and only one generator.
>> >
>> >Huh? How not??
>>
>> Because he doesn't temper, the generator varies in size
>> depending on where you are in the map.
>
>That's what I meant by an undistributed commatic unison vector. But
>there's still one and only one chain, in contradistinction with
>pajara, augmented, diminished, ennealimmal, etc. . . . which was my
>point.

Yes, it's long been agreed that these have historically been missed.

-Carl

>But I am not at all adverse to assuming a quadratic penatly on error.
>
>> Complexity, I should think, should definitely be punished more than
>> 1:1. Using 8 notes instead of 7 notes would seem to demand more
>> than eight 7ths the mental energy.
>
>We could use a quadratic penalty on the complexity too.

If we square both terms, doesn't this give the same ranking?

In March '02 I wrote, "I think I'd rather have a smooth pain function,
like ms, and a stronger exponent on complexity."

>But now we're talking about *convex* badness contours, while Dave
>and especially Gene were proposing *concave* ones (Dave suggested
>using k*sqrt (error) + sqrt(complexity)). I think the difficulty with
>convex badness contours is that they imply a rather sudden cutoff
>for some maximum error and some maximum complexity. This is, indeed,
>probably a very accurate reflection of what's relevant for a
>particular musician working in a particular style. Against this,
>though, is the idea that different musicians may have different needs
>as regards lowness of complexity and lowness of error, so
>superimposing their individually-convex badness criteria could lead
>to a 'global' badness criterion that isn't convex. Taking this idea
>to its logical extreme leads to things like log-flat badness.

Right. I can live with log-flat badness.

By the way, when doing ms error, if an error is less than a cent
it will get *smaller* when squared. Do you see this as a good
thing, should we be ceilinging these to 1 before squarring, or...?

-Carl

>> I can't remember Erv ever using the term,
>
>How about the first line of
>
>http://www.anaphoria.com/xen2.PDF

I didn't remember it. :)

He seems to be arguing strongly here for temperament, but he
either changed his mind before I met with him or he was using
the term here to mean 'mapping to an MOS' rather than actual
temperament. You can see from his later stuff that he lists
both pitches when they coincide on a single key, and when I
asked him about how he'd tune that, he said one or the other
could be used in both contexts (a wolfish sort of temperament!)
or an off-keyboard switch could be used to select between them,
but never would he use a single averaged pitch.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >But I am not at all adverse to assuming a quadratic penatly on
error.
> >
> >> Complexity, I should think, should definitely be punished more
than
> >> 1:1. Using 8 notes instead of 7 notes would seem to demand more
> >> than eight 7ths the mental energy.
> >
> >We could use a quadratic penalty on the complexity too.
>
> If we square both terms, doesn't this give the same ranking?

No, because following Dave, we're adding the terms, not multiplying
them. Dave restated Gene's product as a sum of logs.

> In March '02 I wrote, "I think I'd rather have a smooth pain
function,
> like ms, and a stronger exponent on complexity."

In response to what?

> >But now we're talking about *convex* badness contours, while Dave
> >and especially Gene were proposing *concave* ones (Dave suggested
> >using k*sqrt (error) + sqrt(complexity)). I think the difficulty
with
> >convex badness contours is that they imply a rather sudden cutoff
> >for some maximum error and some maximum complexity. This is,
indeed,
> >probably a very accurate reflection of what's relevant for a
> >particular musician working in a particular style. Against this,
> >though, is the idea that different musicians may have different
needs
> >as regards lowness of complexity and lowness of error, so
> >superimposing their individually-convex badness criteria could lead
> >to a 'global' badness criterion that isn't convex. Taking this idea
> >to its logical extreme leads to things like log-flat badness.
>
> Right. I can live with log-flat badness.

Yecchhh . . .

> By the way, when doing ms error, if an error is less than a cent
> it will get *smaller* when squared.

No, you can't compare cents to cents-squared. These quantities do not
have the same dimension.

> Do you see this as a good
> thing, should we be ceilinging these to 1 before squarring, or...?

To 1 cent? Definitely not -- there's no justification for treating 1
cent as a special error size.

Besides, the errors Gene gave are only in units of cents if you're
looking at the error of the octave -- other intervals have different
units, since it's minimax Tenney-weighed error we're looking at.

>> >> Complexity, I should think, should definitely be punished more
>> >> than 1:1. Using 8 notes instead of 7 notes would seem to demand
>> >> more than eight 7ths the mental energy.
>> >
>> >We could use a quadratic penalty on the complexity too.
>>
>> If we square both terms, doesn't this give the same ranking?
>
>No, because following Dave, we're adding the terms, not multiplying
>them. Dave restated Gene's product as a sum of logs.

Oh. Why do that?

>> In March '02 I wrote, "I think I'd rather have a smooth pain
>> function, like ms, and a stronger exponent on complexity."
>
>In response to what?

Dunno, but by "stronger" I meant "stronger than whatever we use
on error". And "exponent" maybe shouldn't be taken literally...
I just meant to say that I'm willing to accept lots of error for
a small savings in notes.

>> By the way, when doing ms error, if an error is less than a cent
>> it will get *smaller* when squared.
>
>No, you can't compare cents to cents-squared. These quantities do not
>have the same dimension.
>
>> Do you see this as a good
>> thing, should we be ceilinging these to 1 before squarring, or...?
>
>To 1 cent? Definitely not -- there's no justification for treating 1
>cent as a special error size.
>
>Besides, the errors Gene gave are only in units of cents if you're
>looking at the error of the octave -- other intervals have different
>units, since it's minimax Tenney-weighed error we're looking at.

This was a more general question. When calculating the rms error of
an equal temperament, as we used to do, we just allow things less
than 1 to get smaller and influence the mean? It seems one is special
whether or not we do anything.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> Complexity, I should think, should definitely be punished
more
> >> >> than 1:1. Using 8 notes instead of 7 notes would seem to
demand
> >> >> more than eight 7ths the mental energy.
> >> >
> >> >We could use a quadratic penalty on the complexity too.
> >>
> >> If we square both terms, doesn't this give the same ranking?
> >
> >No, because following Dave, we're adding the terms, not
multiplying
> >them. Dave restated Gene's product as a sum of logs.
>
> Oh. Why do that?

So that he could understand Gene's badness and my linear badness in
the same form, and propose a compromise.

> >> In March '02 I wrote, "I think I'd rather have a smooth pain
> >> function, like ms, and a stronger exponent on complexity."
> >
> >In response to what?
>
> Dunno, but by "stronger" I meant "stronger than whatever we use
> on error". And "exponent" maybe shouldn't be taken literally...
> I just meant to say that I'm willing to accept lots of error for
> a small savings in notes.

Well, according to log-flat badness, this will only happen at very
low complexity values. At high complexity values, log-flat badness is
essentially "flat" in error.

> >> By the way, when doing ms error, if an error is less than a cent
> >> it will get *smaller* when squared.
> >
> >No, you can't compare cents to cents-squared. These quantities do
not
> >have the same dimension.
> >
> >> Do you see this as a good
> >> thing, should we be ceilinging these to 1 before squarring,
or...?
> >
> >To 1 cent? Definitely not -- there's no justification for treating
1
> >cent as a special error size.
> >
> >Besides, the errors Gene gave are only in units of cents if you're
> >looking at the error of the octave -- other intervals have
different
> >units, since it's minimax Tenney-weighed error we're looking at.
>
> This was a more general question.

Agreed -- in fact, rms is used in all kinds of fields, including
experimental error analysis, electrical engineering, and acoustics.

> When calculating the rms error of
> an equal temperament, as we used to do, we just allow things less
> than 1 to get smaller

Again, they don't *really* get smaller, because they don't have the
same units.

> and influence the mean?

Yes.

> It seems one is special
> whether or not we do anything.

Incorrect. If you change to a different system of units (say,
millicents) so that nothing's smaller than 1, perform the rms
calculation, and then change back to the original units, you get the
same answer. Try it!

>> >> >We could use a quadratic penalty on the complexity too.
>> >>
>> >> If we square both terms, doesn't this give the same ranking?
>> >
>> >No, because following Dave, we're adding the terms, not
>> >multiplying them. Dave restated Gene's product as a sum of logs.
>>
>> Oh. Why do that?
>
>So that he could understand Gene's badness and my linear badness in
>the same form, and propose a compromise.

Ah. Is yours the one from the Attn: Gene post? It involves taking
determinants, which I haven't fully learned how to do yet.

>> >> In March '02 I wrote, "I think I'd rather have a smooth pain
>> >> function, like ms, and a stronger exponent on complexity."
>> >
>> >In response to what?
>>
>> Dunno, but by "stronger" I meant "stronger than whatever we use
>> on error". And "exponent" maybe shouldn't be taken literally...
>> I just meant to say that I'm willing to accept lots of error for
>> a small savings in notes.
>
>Well, according to log-flat badness, this will only happen at very
>low complexity values. At high complexity values, log-flat badness is
>essentially "flat" in error.

That's good to know, but the above is just my value judgement, and
as you point out log-flat badness frees us from those, in a sense.

>> It seems one is special
>> whether or not we do anything.
>
>Incorrect. If you change to a different system of units (say,
>millicents) so that nothing's smaller than 1, perform the rms
>calculation, and then change back to the original units, you get the
>same answer. Try it!

Of course you're right. Thanks.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Ah. Is yours the one from the Attn: Gene post? It involves taking
> determinants, which I haven't fully learned how to do yet.

You stick it into your computer, and press the button. :)

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >We could use a quadratic penalty on the complexity too.
> >> >>
> >> >> If we square both terms, doesn't this give the same ranking?
> >> >
> >> >No, because following Dave, we're adding the terms, not
> >> >multiplying them. Dave restated Gene's product as a sum of logs.
> >>
> >> Oh. Why do that?
> >
> >So that he could understand Gene's badness and my linear badness
in
> >the same form, and propose a compromise.
>
> Ah. Is yours the one from the Attn: Gene post?

No, it was the toy "Hermanic" example.

> That's good to know, but the above is just my value judgement, and
> as you point out log-flat badness frees us from those, in a sense.

But it results in an infinite number of temperaments, or none at all,
depending on what level of badness you use as your cutoff.

>> >So that he could understand Gene's badness and my linear badness
>> >in the same form, and propose a compromise.
>>
>> Ah. Is yours the one from the Attn: Gene post?
>
>No, it was the toy "Hermanic" example.

This, I guess:

"I thought I'd cull the list of 114 by applying a more stringent
cutoff of 1.355*comp + error < 10.71. This is an arbitrary choice
among the linear functions of complexity and error that could be
chosen"

You don't say what kind of comp and error you're using.

>> That's good to know, but the above is just my value judgement, and
>> as you point out log-flat badness frees us from those, in a sense.
>
>But it results in an infinite number of temperaments, or none at all,
>depending on what level of badness you use as your cutoff.

...as I was trying to complain recently, when I said I'd be a lot
more impressed if it didn't need cutoffs.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >So that he could understand Gene's badness and my linear badness
> >> >in the same form, and propose a compromise.
> >>
> >> Ah. Is yours the one from the Attn: Gene post?
> >
> >No, it was the toy "Hermanic" example.
>
> This, I guess:
>
> "I thought I'd cull the list of 114 by applying a more stringent
> cutoff of 1.355*comp + error < 10.71. This is an arbitrary choice
> among the linear functions of complexity and error that could be
> chosen"
>
> You don't say what kind of comp and error you're using.

Copied from Gene's "114".

> >> That's good to know, but the above is just my value judgement,
and
> >> as you point out log-flat badness frees us from those, in a
sense.
> >
> >But it results in an infinite number of temperaments, or none at
all,
> >depending on what level of badness you use as your cutoff.
>
> ...as I was trying to complain recently, when I said I'd be a lot
> more impressed if it didn't need cutoffs.

Well you always have at least one cutoff -- namely, the maximum
allowed value of the badness function itself -- and of the various
badness functions that have been proposed, a few (including log-flat)
require one or two additional cutoffs to yield a finite list. A
straight line doesn't.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> Yes. I like that idea too. But by "a wide swath" don't you mean one
> that it's easy to put a simple smooth curve thru?

Right.

> And you must have
> some general idea of which way this intergalactic moat must curve.

In the 5-limit linear case, it would be really easy to do this if we
didn't want to go out to the complexity of schismic and
kleismic/hanson (the only argument that would arise would be whether
the father and beep couple should be in or out, leading to a grand
total of 11 or 9 5-limit LTs). Unfortunately, the low error of
schismic has proved tantalizing enough for a few musicians to
construct instruments capable of playing the large extended scales
that its approximations require. If we consider such complexity
justifiable, it seems we should be interested in 15 to 17 5-limit
LTs, or 17 to 19 if we include father and beep. The couple residing
in "the middle of the road" is 2187;2048 and 3126;2916. With Herman,
we could split the difference and select only the better of the pair,
2187;2048 (Dave, have you *heard* Blackwood's 21-equal suite?) . . .
I don't think anyone's talked about the 20480;19683 system. But if
schismic, and certainly if semisixths, is not too complex to be a
useful alternative to strict JI, why shouldn't this system merit some
attention from musicians too? I don't think near-JI triads sound
enough better than chords in this system (which are purer than those
of augmented, porcupine, or diminished) to merit a much higher
allowed complexity to generate them linearly.

A problem with our plan to have versions of these badness curves for
sets of temperaments of different dimensions is that moving to a
higher-dimensional tuning system would theoretically increase the
badness by an infinite factor. But in practice you never use an
infinite swath of the lattice so eventually any complex enough 5-
limit linear temperament becomes indistinguishable from a planar
system.

I'm speaking like you now, Dave! ;)

>In the 5-limit linear case, it would be really easy to do this if we
>didn't want to go out to the complexity of schismic and
>kleismic/hanson

While I think it would be nice to name this after Larry Hanson (and
I certainly agreeable to the idea), my preference is to keep kleismic,
since it tells the name of the comma involved, and has a fairly-well
established body of use. What say everybody?

-Carl

This time I'll try L_1 (multimonzo interpretation?) instead of
L_infinity (multival interpretation?) to get complexity from the
wedgie. Let's see how it affects the rankings -- we don't need to
worry about scaling because Gene's badness measure is
multiplicative . . .

The top 10 get re-ordered as follows, though this is probably not the
new top 10 overall . . .

1.
> Number 1 Ennealimmal
>
> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> TOP generators [133.3373752, 49.02398564]
> bad: 4.918774 comp: 11.628267 err: .036377

39.8287 -> bad = 57.7058

2.
> Number 2 Meantone
>
> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> TOP generators [1201.698520, 504.1341314]
> bad: 21.551439 comp: 3.562072 err: 1.698521

11.7652 -> bad = 235.1092

3.
> Number 9 Miracle
>
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014

21.1019 --> bad = 280.9843

4.
> Number 7 Dominant Seventh
>
> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> TOP generators [1195.228951, 495.8810151]
> bad: 28.744957 comp: 2.454561 err: 4.771049

7.9560 -> bad = 301.9952

5.
> Number 3 Magic
>
> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> TOP generators [1201.276744, 380.7957184]
> bad: 23.327687 comp: 4.274486 err: 1.276744

15.5360 -> bad = 308.1642

6.
> Number 4 Beep
>
> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> TOP generators [1194.642673, 254.8994697]
> bad: 23.664749 comp: 1.292030 err: 14.176105

4.7295 -> bad = 317.0935

7.
> Number 6 Pajara
>
> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 106.5665459]
> bad: 27.754421 comp: 2.988993 err: 3.106578

10.4021 -> bad = 336.1437

8.
> Number 10 Orwell
>
> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> TOP generators [1199.532657, 271.4936472]
> bad: 30.805067 comp: 5.706260 err: .946061

19.9797 -> bad = 377.6573

9.
> Number 8 Schismic
>
> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> TOP generators [1200.760624, 498.1193303]
> bad: 28.818558 comp: 5.618543 err: .912904

20.2918 --> bad = 375.8947

10.
> Number 5 Augmented
>
> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> TOP generators [399.9922103, 107.3111730]
> bad: 27.081145 comp: 2.147741 err: 5.870879

8.3046 -> bad = 404.8933

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >In the 5-limit linear case, it would be really easy to do this if
we
> >didn't want to go out to the complexity of schismic and
> >kleismic/hanson
>
> While I think it would be nice to name this after Larry Hanson (and
> I certainly agreeable to the idea), my preference is to keep
kleismic,
> since it tells the name of the comma involved,

A rare feature, and Pythagorean doesn't eat the Pythagorean comma,
for example . . .

> and has a fairly-well
> established body of use. What say everybody?
>
> -Carl

Fairly-well established body of use?

>> and has a fairly-well
>> established body of use. What say everybody?
>
>Fairly-well established body of use?

Yes.

-Carl

oops -- I may have done that all wrong. The scaling factors for the
elements of the wedgie, the ones that you divide by to calculate the
multival norm -- do you have to *multiply* by them when you calculate
the multimonzo norm?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> This time I'll try L_1 (multimonzo interpretation?) instead of
> L_infinity (multival interpretation?) to get complexity from the
> wedgie. Let's see how it affects the rankings -- we don't need to
> worry about scaling because Gene's badness measure is
> multiplicative . . .
>
> The top 10 get re-ordered as follows, though this is probably not
the
> new top 10 overall . . .
>
> 1.
> > Number 1 Ennealimmal
> >
> > [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> > TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> > TOP generators [133.3373752, 49.02398564]
> > bad: 4.918774 comp: 11.628267 err: .036377
>
> 39.8287 -> bad = 57.7058
>
> 2.
> > Number 2 Meantone
> >
> > [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> > TOP generators [1201.698520, 504.1341314]
> > bad: 21.551439 comp: 3.562072 err: 1.698521
>
> 11.7652 -> bad = 235.1092
>
> 3.
> > Number 9 Miracle
> >
> > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > TOP generators [1200.631014, 116.7206423]
> > bad: 29.119472 comp: 6.793166 err: .631014
>
> 21.1019 --> bad = 280.9843
>
> 4.
> > Number 7 Dominant Seventh
> >
> > [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> > TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> > TOP generators [1195.228951, 495.8810151]
> > bad: 28.744957 comp: 2.454561 err: 4.771049
>
> 7.9560 -> bad = 301.9952
>
> 5.
> > Number 3 Magic
> >
> > [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> > TOP generators [1201.276744, 380.7957184]
> > bad: 23.327687 comp: 4.274486 err: 1.276744
>
> 15.5360 -> bad = 308.1642
>
> 6.
> > Number 4 Beep
> >
> > [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> > TOP generators [1194.642673, 254.8994697]
> > bad: 23.664749 comp: 1.292030 err: 14.176105
>
> 4.7295 -> bad = 317.0935
>
> 7.
> > Number 6 Pajara
> >
> > [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> > TOP generators [598.4467109, 106.5665459]
> > bad: 27.754421 comp: 2.988993 err: 3.106578
>
> 10.4021 -> bad = 336.1437
>
> 8.
> > Number 10 Orwell
> >
> > [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> > TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> > TOP generators [1199.532657, 271.4936472]
> > bad: 30.805067 comp: 5.706260 err: .946061
>
> 19.9797 -> bad = 377.6573
>
> 9.
> > Number 8 Schismic
> >
> > [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> > TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> > TOP generators [1200.760624, 498.1193303]
> > bad: 28.818558 comp: 5.618543 err: .912904
>
> 20.2918 --> bad = 375.8947
>
> 10.
> > Number 5 Augmented
> >
> > [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> > TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> > TOP generators [399.9922103, 107.3111730]
> > bad: 27.081145 comp: 2.147741 err: 5.870879
>
> 8.3046 -> bad = 404.8933

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> While I think it would be nice to name this after Larry Hanson (and
> I certainly agreeable to the idea), my preference is to keep
kleismic,
> since it tells the name of the comma involved, and has a fairly-well
> established body of use. What say everybody?

I like kleismic, but which version of kleismic did Hanson like?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> oops -- I may have done that all wrong. The scaling factors for the
> elements of the wedgie, the ones that you divide by to calculate
the
> multival norm -- do you have to *multiply* by them when you
calculate
> the multimonzo norm?

You rescale the monzos by multiplying, but it all comes out the same
up to a constant factor, so the real difference is L1 vs L_inf.
There's something to be said for L1 in that comparison.

>> While I think it would be nice to name this after Larry Hanson (and
>> I certainly agreeable to the idea), my preference is to keep
>> kleismic, since it tells the name of the comma involved, and has a
>> fairly-well established body of use. What say everybody?
>
>I like kleismic, but which version of kleismic did Hanson like?

AFAIK, mainly the 53-note version.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > While I think it would be nice to name this after Larry Hanson
(and
> > I certainly agreeable to the idea), my preference is to keep
> kleismic,
> > since it tells the name of the comma involved, and has a fairly-
well
> > established body of use. What say everybody?
>
> I like kleismic, but which version of kleismic did Hanson like?

5-limit -- 19-, 34-, 53-, and 72-equal.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> 5-limit -- 19-, 34-, 53-, and 72-equal.

Any music available?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > 5-limit -- 19-, 34-, 53-, and 72-equal.
>
> Any music available?

Neil Haverstick used Hanson's 34-equal guitars on his CDs . . .

I'm trying to re-rank your top 10 using multimonzo L_1 norm (see next
post). Could you do this too to provide an independent check?

This time, I'll multiply, instead of dividing, the elements of the
wedgie by the relevant "unit areas" . . . still using L_1 . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> This time I'll try L_1 (multimonzo interpretation?) instead of
> L_infinity (multival interpretation?) to get complexity from the
> wedgie. Let's see how it affects the rankings -- we don't need to
> worry about scaling because Gene's badness measure is
> multiplicative . . .
>
> The top 10 get re-ordered as follows, though this is probably not
the
> new top 10 overall . . .

1.
> 1.
> > Number 1 Ennealimmal
> >
> > [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> > TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> > TOP generators [133.3373752, 49.02398564]
> > bad: 4.918774 comp: 11.628267 err: .036377
>
> 39.8287 -> bad = 57.7058

464.95 -> bad = 7864

2.
> 7.
> > Number 6 Pajara
> >
> > [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> > TOP generators [598.4467109, 106.5665459]
> > bad: 27.754421 comp: 2.988993 err: 3.106578
>
> 10.4021 -> bad = 336.1437

130.6 -> bad = 52987

3.
> 3.
> > Number 9 Miracle
> >
> > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > TOP generators [1200.631014, 116.7206423]
> > bad: 29.119472 comp: 6.793166 err: .631014
>
> 21.1019 --> bad = 280.9843

310.15 -> bad = 60699

4.
> 2.
> > Number 2 Meantone
> >
> > [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> > TOP generators [1201.698520, 504.1341314]
> > bad: 21.551439 comp: 3.562072 err: 1.698521
>
> 11.7652 -> bad = 235.1092

189.73 -> bad = 61144

5.
> 6.
> > Number 4 Beep
> >
> > [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> > TOP generators [1194.642673, 254.8994697]
> > bad: 23.664749 comp: 1.292030 err: 14.176105
>
> 4.7295 -> bad = 317.0935

69.852 -> bad = 69170

6.
> 9.
> > Number 8 Schismic
> >
> > [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> > TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> > TOP generators [1200.760624, 498.1193303]
> > bad: 28.818558 comp: 5.618543 err: .912904
>
> 20.2918 --> bad = 375.8947

291.09 -> bad = 77353

7.
> 5.
> > Number 3 Magic
> >
> > [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> > TOP generators [1201.276744, 380.7957184]
> > bad: 23.327687 comp: 4.274486 err: 1.276744
>
> 15.5360 -> bad = 308.1642

265.95 -> bad = 90301

8.
> 8.
> > Number 10 Orwell
> >
> > [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> > TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> > TOP generators [1199.532657, 271.4936472]
> > bad: 30.805067 comp: 5.706260 err: .946061
>
> 19.9797 -> bad = 377.6573

324.9486 -> bad = 99896

9.
> 10.
> > Number 5 Augmented
> >
> > [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> > TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> > TOP generators [399.9922103, 107.3111730]
> > bad: 27.081145 comp: 2.147741 err: 5.870879
>
> 8.3046 -> bad = 404.8933

143.07 -> bad = 1.2017e+005

10.
> 4.
> > Number 7 Dominant Seventh
> >
> > [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> > TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> > TOP generators [1195.228951, 495.8810151]
> > bad: 28.744957 comp: 2.454561 err: 4.771049
>
> 7.9560 -> bad = 301.9952

162.2 -> bad = 1.2552e+005

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > oops -- I may have done that all wrong. The scaling factors for
the
> > elements of the wedgie, the ones that you divide by to calculate
> the
> > multival norm -- do you have to *multiply* by them when you
> calculate
> > the multimonzo norm?
>
> You rescale the monzos by multiplying, but it all comes out the
same
> up to a constant factor,

I don't think so! The element that you were formerly multiplying by
the largest factor, you're now dividing by the largest factor, and
vice versa!

Whoops -- I forgot that the order of the elements changes too!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > oops -- I may have done that all wrong. The scaling factors for
> the
> > > elements of the wedgie, the ones that you divide by to
calculate
> > the
> > > multival norm -- do you have to *multiply* by them when you
> > calculate
> > > the multimonzo norm?
> >
> > You rescale the monzos by multiplying, but it all comes out the
> same
> > up to a constant factor,
>
> I don't think so! The element that you were formerly multiplying by
> the largest factor, you're now dividing by the largest factor, and
> vice versa!

Here's the calculation for Pajara, the way I now understand it should
be done:

take the two unison vectors

|1 0 2 -2>
|6 -2 0 -1>

Now find the determinant, and the "area" it represents, in each of
the basis planes:

|1 0| = -2*(e23) -> 2*lg2(3) = 3.1699
|6 -2|

|1 2| = -12*(e25) -> 12*lg2(5) = 27.863
|6 0|

|1 -2| = 11*(e27) -> 11*lg2(7) = 30.881
|6 -1|

|0 2| = 4*(e35) -> 4*lg2(3)*lg2(5) = 14.721
|-2 0|

|0 -2| = -4*(e37) => 4*lg2(3)*lg2(7) = 17.798
|-2 -1|

|2 -2| = -2*(e57) => 2*lg2(5)*lg2(7) = 13.037
|0 -1|

The sum is 107.47.

So the below was wrong. I forgot that you reverse the order of the
elements to convert a multival wedgie into a multimonzo wedgie! Doing
so would, indeed, give the same rankings as my original L_1
calculation. But that's gotta be the right norm. The Tenney lattice
is set up to measure complexity, and the norm we always associate
with it is the L_1 norm. Isn't that right? The L_1 norm on the monzo
is what I've been using all along to calculate complexity for the
codimension-1 case, in my graphs and in the "Attn: Gene 2" post . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> This time, I'll multiply, instead of dividing, the elements of the
> wedgie by the relevant "unit areas" . . . still using L_1 . . .
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > This time I'll try L_1 (multimonzo interpretation?) instead of
> > L_infinity (multival interpretation?) to get complexity from the
> > wedgie. Let's see how it affects the rankings -- we don't need to
> > worry about scaling because Gene's badness measure is
> > multiplicative . . .
> >
> > The top 10 get re-ordered as follows, though this is probably not
> the
> > new top 10 overall . . .
>
> 1.
> > 1.
> > > Number 1 Ennealimmal
> > >
> > > [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> > > TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> > > TOP generators [133.3373752, 49.02398564]
> > > bad: 4.918774 comp: 11.628267 err: .036377
> >
> > 39.8287 -> bad = 57.7058
>
> 464.95 -> bad = 7864
>
> 2.
> > 7.
> > > Number 6 Pajara
> > >
> > > [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> > > TOP generators [598.4467109, 106.5665459]
> > > bad: 27.754421 comp: 2.988993 err: 3.106578
> >
> > 10.4021 -> bad = 336.1437
>
> 130.6 -> bad = 52987
>
> 3.
> > 3.
> > > Number 9 Miracle
> > >
> > > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > > TOP generators [1200.631014, 116.7206423]
> > > bad: 29.119472 comp: 6.793166 err: .631014
> >
> > 21.1019 --> bad = 280.9843
>
> 310.15 -> bad = 60699
>
> 4.
> > 2.
> > > Number 2 Meantone
> > >
> > > [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> > > TOP generators [1201.698520, 504.1341314]
> > > bad: 21.551439 comp: 3.562072 err: 1.698521
> >
> > 11.7652 -> bad = 235.1092
>
> 189.73 -> bad = 61144
>
> 5.
> > 6.
> > > Number 4 Beep
> > >
> > > [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> > > TOP generators [1194.642673, 254.8994697]
> > > bad: 23.664749 comp: 1.292030 err: 14.176105
> >
> > 4.7295 -> bad = 317.0935
>
> 69.852 -> bad = 69170
>
> 6.
> > 9.
> > > Number 8 Schismic
> > >
> > > [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> > > TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> > > TOP generators [1200.760624, 498.1193303]
> > > bad: 28.818558 comp: 5.618543 err: .912904
> >
> > 20.2918 --> bad = 375.8947
>
> 291.09 -> bad = 77353
>
> 7.
> > 5.
> > > Number 3 Magic
> > >
> > > [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> > > TOP generators [1201.276744, 380.7957184]
> > > bad: 23.327687 comp: 4.274486 err: 1.276744
> >
> > 15.5360 -> bad = 308.1642
>
> 265.95 -> bad = 90301
>
> 8.
> > 8.
> > > Number 10 Orwell
> > >
> > > [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> > > TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> > > TOP generators [1199.532657, 271.4936472]
> > > bad: 30.805067 comp: 5.706260 err: .946061
> >
> > 19.9797 -> bad = 377.6573
>
> 324.9486 -> bad = 99896
>
> 9.
> > 10.
> > > Number 5 Augmented
> > >
> > > [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> > > TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> > > TOP generators [399.9922103, 107.3111730]
> > > bad: 27.081145 comp: 2.147741 err: 5.870879
> >
> > 8.3046 -> bad = 404.8933
>
> 143.07 -> bad = 1.2017e+005
>
> 10.
> > 4.
> > > Number 7 Dominant Seventh
> > >
> > > [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> > > TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> > > TOP generators [1195.228951, 495.8810151]
> > > bad: 28.744957 comp: 2.454561 err: 4.771049
> >
> > 7.9560 -> bad = 301.9952
>
> 162.2 -> bad = 1.2552e+005

I re-ranked Gene's top 64 using L_1 and got the following top 32.
Anything missing?

1.
> Ennealimmal
>
> [18, 27, 18, 1, -22, -34] [[9, 15, 22, 26], [0, -2, -3, -2]]
> TOP tuning [1200.036377, 1902.012656, 2786.350297, 3368.723784]
> TOP generators [133.3373752, 49.02398564]
> bad: 4.918774 comp: 11.628267 err: .036377

39.8287 -> bad = 57.7058

2.
> Meantone (Huygens)
>
> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> TOP generators [1201.698520, 504.1341314]
> bad: 21.551439 comp: 3.562072 err: 1.698521

11.7652 -> bad = 235.1092

3.
> Miracle
>
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 29.119472 comp: 6.793166 err: .631014

21.1019 --> bad = 280.9843

4.
> Hemiwuerschmidt
>
> [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> TOP generators [1199.692003, 193.8224275]
> bad: 31.386908 comp: 10.094876 err: .307997

31.212 -> bad = 300.04

5.
> Dominant Seventh
>
> [1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
> TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
> TOP generators [1195.228951, 495.8810151]
> bad: 28.744957 comp: 2.454561 err: 4.771049

7.9560 -> bad = 301.9952

6.
> Blackwood
>
> [0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
> TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
> TOP generators [239.1786927, 83.83059859]
> bad: 34.210608 comp: 2.173813 err: 7.239629

6.4749 -> bad = 303.52

7.
> Magic
>
> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> TOP generators [1201.276744, 380.7957184]
> bad: 23.327687 comp: 4.274486 err: 1.276744

15.5360 -> bad = 308.1642

8.
> Beep
>
> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> TOP generators [1194.642673, 254.8994697]
> bad: 23.664749 comp: 1.292030 err: 14.176105

4.7295 -> bad = 317.0935

9.
> Pajara
>
> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 106.5665459]
> bad: 27.754421 comp: 2.988993 err: 3.106578

10.4021 -> bad = 336.1437

10.
> Semisixths
>
> [7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
> TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
> TOP generators [1198.389531, 443.1602931]
> bad: 34.533812 comp: 4.630693 err: 1.610469

14.459 -> bad = 336.67

11.
> Catakleismic
>
> [6, 5, 22, -6, 18, 37] [[1, 0, 1, -3], [0, 6, 5, 22]]
> TOP tuning [1200.536356, 1901.438376, 2785.068335, 3370.331646]
> TOP generators [1200.536355, 316.9063960]
> bad: 32.938503 comp: 7.836558 err: .536356

25.127 -> bad = 338.65

12.
> Diminished
>
> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> TOP generators [298.5321149, 101.4561401]
> bad: 37.396767 comp: 2.523719 err: 5.871540

7.917 -> bad = 368.02

13.
> Schismic
>
> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
> TOP generators [1200.760624, 498.1193303]
> bad: 28.818558 comp: 5.618543 err: .912904

20.2918 --> bad = 375.8947

14.
> Orwell
>
> [7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
> TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
> TOP generators [1199.532657, 271.4936472]
> bad: 30.805067 comp: 5.706260 err: .946061

19.9797 -> bad = 377.6573

15.
> Hemififths
>
> [2, 25, 13, 35, 15, -40] [[1, 1, -5, -1], [0, 2, 25, 13]]
> TOP tuning [1199.700353, 1902.429930, 2785.617954, 3368.041901]
> TOP generators [1199.700353, 351.3647888]
> bad: 34.737019 comp: 10.766914 err: .299647

35.677 -> bad = 381.41

16.
> Father
>
> [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> TOP generators [1185.869125, 447.3863410]
> bad: 33.256527 comp: 1.534101 err: 14.130876

5.2007 -> bad = 382.2

17.
> Amity
>
> [5, 13, -17, 9, -41, -76] [[1, 3, 6, -2], [0, -5, -13, 17]]
> TOP tuning [1199.723894, 1902.392618, 2786.717797, 3369.601033]
> TOP generators [1199.723894, 339.3558130]
> bad: 37.532790 comp: 11.659166 err: .276106

38.128 -> bad = 401.39

18.
> Augmented
>
> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
> TOP generators [399.9922103, 107.3111730]
> bad: 27.081145 comp: 2.147741 err: 5.870879

8.3046 -> bad = 404.8933

19.
> Parakleismic
>
> [13, 14, 35, -8, 19, 42] [[1, 5, 6, 12], [0, -13, -14, -35]]
> TOP tuning [1199.738066, 1902.291445, 2786.921905, 3368.090564]
> TOP generators [1199.738066, 315.1076065]
> bad: 40.713036 comp: 12.467252 err: .261934

39.586 -> bad = 410.46

20.
> Tripletone
>
> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> TOP generators [399.0200131, 92.45965769]
> bad: 48.112067 comp: 4.045351 err: 2.939961

12.125 -> bad = 432.24

21.
> {21/20, 28/27}
>
> [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
> TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
> TOP generators [1214.253642, 509.4012304]
> bad: 42.300772 comp: 1.722706 err: 14.253642

5.5723 -> bad = 442.58

22.
> Decimal
>
> [4, 2, 2, -6, -8, -1] [[2, 4, 5, 6], [0, -2, -1, -1]]
> TOP tuning [1207.657798, 1914.092323, 2768.532858, 3372.361757]
> TOP generators [603.8288989, 250.6116362]
> bad: 48.773723 comp: 2.523719 err: 7.657798

7.6792 -> bad = 451.58

23.
> Hemifourths
>
> [2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
> TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
> TOP generators [1203.668841, 252.4803582]
> bad: 43.552336 comp: 3.445412 err: 3.668842

11.204 -> bad = 460.59

24.
> Negri
>
> [4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
> TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
> TOP generators [1203.187309, 124.8419629]
> bad: 46.125886 comp: 3.804173 err: 3.187309

12.125 -> bad = 468.55

25.
> Nonkleismic
>
> [10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
> TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
> TOP generators [1198.828458, 309.8926610]
> bad: 46.635848 comp: 6.309298 err: 1.171542

20.326 -> bad = 484

26.
> Kleismic
>
> [6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
> TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
> TOP generators [1203.187309, 317.8344609]
> bad: 45.676063 comp: 3.785579 err: 3.187309

12.409 -> bad = 490.77

27.
> Dicot
>
> [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
> TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
> TOP generators [1204.048159, 356.3998255]
> bad: 42.920570 comp: 2.137243 err: 9.396316

7.2314 -> bad = 491.37

28.
> Superpythagorean
>
> [1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
> TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
> TOP generators [1197.596121, 489.4271829]
> bad: 50.917015 comp: 4.602303 err: 2.403879

14.431 -> bad = 500.61

29.
> Injera
>
> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
> TOP generators [600.8889070, 93.60982493]
> bad: 42.529834 comp: 3.445412 err: 3.582707

11.918 -> bad = 508.85

30.
> {25/24, 81/80} Jamesbond?
>
> [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
> TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
> TOP generators [172.7759159, 86.69241190]
> bad: 58.637859 comp: 2.493450 err: 9.431411

7.4202 -> bad = 519.28

31.
> Quartaminorthirds
>
> [9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
> TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
> TOP generators [1199.792743, 77.83315314]
> bad: 47.721352 comp: 6.742251 err: 1.049791

22.397 -> bad = 526.59

32.
> Pelogic
>
> [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
> TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
> TOP generators [1209.734056, 532.9412251]
> bad: 39.824125 comp: 2.022675 err: 9.734056

7.426 -> bad = 536.78

**********************************************************************

> Number 43
>
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 162.3159606]
> bad: 57.621529 comp: 4.306766 err: 3.106578

13.19 -> bad = 540.44

> Number 36 Supersupermajor
>
> [3, 17, -1, 20, -10, -50] [[1, 1, -1, 3], [0, 3, 17, -1]]
> TOP tuning [1200.231588, 1903.372996, 2784.236389, 3366.314293]
> TOP generators [1200.231587, 234.3804692]
> bad: 52.638504 comp: 7.670504 err: .894655

24.923 -> bad = 555.72

> Number 47
>
> [12, 34, 20, 26, -2, -49] [[2, 4, 7, 7], [0, -6, -17, -10]]
> TOP tuning [1200.284965, 1901.503343, 2786.975381, 3369.219732]
> TOP generators [600.1424823, 83.17776441]
> bad: 61.101493 comp: 14.643003 err: .284965

44.37 -> bad = 561

> Number 46 Hemithirds
>
> [15, -2, -5, -38, -50, -6] [[1, 4, 2, 2], [0, -15, 2, 5]]
> TOP tuning [1200.363229, 1901.194685, 2787.427555, 3367.479202]
> TOP generators [1200.363229, 193.3505488]
> bad: 60.573479 comp: 11.237086 err: .479706

34.589 -> bad = 573.94

> Number 44 Octacot
>
> [8, 18, 11, 10, -5, -25] [[1, 1, 1, 2], [0, 8, 18, 11]]
> TOP tuning [1199.031259, 1903.490418, 2784.064367, 3366.693863]
> TOP generators [1199.031259, 88.05739491]
> bad: 58.217715 comp: 7.752178 err: .968741

24.394 -> bad = 576.47

> Number 35 Supermajor seconds
>
> [3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
> TOP generators [1201.698520, 232.5214630]
> bad: 51.806440 comp: 5.522763 err: 1.698521

18.448 -> bad = 578.06

> Number 25 Waage? Compton? Duodecimal?
>
> [0, 12, 24, 19, 38, 22] [[12, 19, 28, 34], [0, 0, -1, -2]]
> TOP tuning [1200.617051, 1900.976998, 2785.844725, 3370.558188]
> TOP generators [100.0514209, 16.55882096]
> bad: 45.097159 comp: 8.548972 err: .617051

30.795 -> bad = 585.17

> Number 55
>
> [0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
> TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
> TOP generators [99.80617249, 24.58395811]
> bad: 65.630949 comp: 4.295482 err: 3.557008

12.84 -> bad = 586.43

> Number 48 Flattone
>
> [1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
> TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
> TOP generators [1202.536419, 507.1379663]
> bad: 61.126418 comp: 4.909123 err: 2.536420

15.376 -> bad = 599.67

> Number 38 {3136/3125, 5120/5103} Misty
>
> [3, -12, -30, -26, -56, -36] [[3, 5, 6, 6], [0, -1, 4, 10]]
> TOP tuning [1199.661465, 1902.491566, 2787.099767, 3368.765021]
> TOP generators [399.8871550, 96.94420930]
> bad: 53.622498 comp: 12.585536 err: .338535

42.92 -> bad = 623.63

> Number 41 {28/27, 50/49}
>
> [2, 6, 6, 5, 4, -3] [[2, 3, 4, 5], [0, 1, 3, 3]]
> TOP tuning [1191.599639, 1915.269258, 2766.808679, 3362.608498]
> TOP generators [595.7998193, 127.8698005]
> bad: 56.092257 comp: 2.584059 err: 8.400361

8.701 -> bad = 635.97

> Number 63
>
> [8, 13, 23, 2, 14, 17] [[1, 2, 3, 4], [0, -8, -13, -23]]
> TOP tuning [1198.975478, 1900.576277, 2788.692580, 3365.949709]
> TOP generators [1198.975478, 62.17183489]
> bad: 68.767371 comp: 8.192765 err: 1.024522

25.137 -> bad = 647.35

> Number 49 Diaschismic
>
> [2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
> TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
> TOP generators [599.3662015, 103.7870123]
> bad: 61.527901 comp: 6.966993 err: 1.267597

22.629 -> bad = 649.07

> Number 57
>
> [2, -2, 1, -8, -4, 8] [[1, 2, 2, 3], [0, -2, 2, -1]]
> TOP tuning [1185.869125, 1924.351909, 2819.124589, 3333.914203]
> TOP generators [1185.869125, 223.6931705]
> bad: 66.774944 comp: 2.173813 err: 14.130876

6.7795 -> bad = 649.47

> Number 59
>
> [3, 5, 9, 1, 6, 7] [[1, 2, 3, 4], [0, -3, -5, -9]]
> TOP tuning [1193.415676, 1912.390908, 2789.512955, 3350.341372]
> TOP generators [1193.415676, 158.1468146]
> bad: 67.670842 comp: 3.205865 err: 6.584324

9.9461 -> bad = 651.35

> Number 56
>
> [2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
> TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
> TOP generators [1204.567524, 355.9419091]
> bad: 66.522610 comp: 2.696901 err: 9.146173

8.4704 -> bad = 656.21

> Number 26 Wizard
>
> [12, -2, 20, -31, -2, 52] [[2, 1, 5, 2], [0, 6, -1, 10]]
> TOP tuning [1200.639571, 1900.941305, 2784.828674, 3368.342104]
> TOP generators [600.3197857, 216.7702531]
> bad: 45.381303 comp: 8.423526 err: .639571

32.407 -> bad = 671.69

> Number 37 {6144/6125, 10976/10935} Hendecatonic?
>
> [11, -11, 22, -43, 4, 82] [[11, 17, 26, 30], [0, 1, -1, 2]]
> TOP tuning [1199.662182, 1902.490429, 2787.098101, 3368.740066]
> TOP generators [109.0601984, 48.46705632]
> bad: 53.458690 comp: 12.579627 err: .337818

44.677 -> bad = 674.3

> Number 42 Porcupine
>
> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> TOP generators [1196.905960, 162.3176609]
> bad: 57.088650 comp: 4.295482 err: 3.094040

14.796 -> bad = 677.35

> Number 33 {1029/1024, 4375/4374}
>
> [12, 22, -4, 7, -40, -71] [[2, 5, 8, 5], [0, -6, -11, 2]]
> TOP tuning [1200.421488, 1901.286959, 2785.446889, 3367.642640]
> TOP generators [600.2107440, 183.2944602]
> bad: 50.004574 comp: 10.892116 err: .421488

40.255 -> bad = 683

> Number 39 {1728/1715, 4000/3993}
>
> [11, 18, 5, 3, -23, -39] [[1, 2, 3, 3], [0, -11, -18, -5]]
> TOP tuning [1199.083445, 1901.293958, 2784.185538, 3371.399002]
> TOP generators [1199.083445, 45.17026643]
> bad: 55.081549 comp: 7.752178 err: .916555

28.441 -> bad = 741.38

> Number 62
>
> [2, -2, -2, -8, -9, 1] [[2, 3, 5, 6], [0, 1, -1, -1]]
> TOP tuning [1185.468457, 1924.986952, 2816.886876, 3409.621105]
> TOP generators [592.7342285, 146.7842660]
> bad: 68.668284 comp: 2.173813 err: 14.531543

7.1855 -> bad = 750.29

> Number 40 {36/35, 160/147} Hystrix?
>
> [3, 5, 1, 1, -7, -12] [[1, 2, 3, 3], [0, -3, -5, -1]]
> TOP tuning [1187.933715, 1892.564743, 2758.296667, 3402.700250]
> TOP generators [1187.933715, 161.1008955]
> bad: 55.952057 comp: 2.153383 err: 12.066285

8.0882 -> bad = 789.37

> Number 61 Hemikleismic
>
> [12, 10, -9, -12, -48, -49] [[1, 0, 1, 4], [0, 12, 10, -9]]
> TOP tuning [1199.411231, 1902.888178, 2785.151380, 3370.478790]
> TOP generators [1199.411231, 158.5740148]
> bad: 68.516458 comp: 10.787602 err: .588769

36.649 -> bad = 790.81

> Number 52 Tritonic
>
> [5, -11, -12, -29, -33, 3] [[1, 4, -3, -3], [0, -5, 11, 12]]
> TOP tuning [1201.023211, 1900.333250, 2785.201472, 3365.953391]
> TOP generators [1201.023211, 580.7519186]
> bad: 63.536850 comp: 7.880073 err: 1.023211

27.923 -> bad = 797.81

> Number 50 Superkleismic
>
> [9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
> TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
> TOP generators [1201.371918, 322.3731369]
> bad: 62.364585 comp: 6.742251 err: 1.371918

24.524 -> bad = 825.11

> Number 54
>
> [6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
> TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
> TOP generators [1202.659696, 82.97467050]
> bad: 64.556006 comp: 4.306766 err: 3.480440

15.623 -> bad = 849.49

> Number 53
>
> [1, 33, 27, 50, 40, -30] [[1, 2, 16, 14], [0, -1, -33, -27]]
> TOP tuning [1199.680495, 1902.108988, 2785.571846, 3369.722869]
> TOP generators [1199.680495, 497.2520023]
> bad: 64.536886 comp: 14.212326 err: .319505

51.639 -> bad = 851.99

> Number 51
>
> [8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
> TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
> TOP generators [1201.135545, 387.5841360]
> bad: 62.703297 comp: 6.411729 err: 1.525246

23.841 -> bad = 866.91

> Number 60
>
> [3, 0, 9, -7, 6, 21] [[3, 5, 7, 9], [0, -1, 0, -3]]
> TOP tuning [1193.415676, 1912.390908, 2784.636577, 3350.341372]
> TOP generators [397.8052253, 76.63521863]
> bad: 68.337269 comp: 3.221612 err: 6.584324

11.571 -> bad = 881.53

> Number 64
>
> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
> TOP generators [1202.900537, 570.4479508]
> bad: 69.388565 comp: 4.891080 err: 2.900537

17.521 -> bad = 890.45

> Number 58
>
> [5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
> TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
> TOP generators [1194.335372, 99.13879319]
> bad: 67.244049 comp: 3.445412 err: 5.664628

12.818 -> bad = 930.67

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > You rescale the monzos by multiplying, but it all comes out the
> same
> > up to a constant factor,
>
> I don't think so! The element that you were formerly multiplying by
> the largest factor, you're now dividing by the largest factor, and
> vice versa!

If you mulitply by p5p7 and then divide by p2p3p5p7, you get 1/p2p3,
and so forth.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> So the below was wrong. I forgot that you reverse the order of the
> elements to convert a multival wedgie into a multimonzo wedgie!
Doing
> so would, indeed, give the same rankings as my original L_1
> calculation. But that's gotta be the right norm. The Tenney lattice
> is set up to measure complexity, and the norm we always associate
> with it is the L_1 norm. Isn't that right? The L_1 norm on the
monzo
> is what I've been using all along to calculate complexity for the
> codimension-1 case, in my graphs and in the "Attn: Gene 2"
post . . .

To me it seemed there was a reasonable case for either norm, which
means you could argue you could use any Lp norm also, since they lie
between L1 and L_inf. Do you need me to redo the calculation using
the L1 norm?

I think it would be a good idea to stick to the normalization by
dividing, since for higher limits a linear temperament wedgie is
still a bival, so it's easier.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > So the below was wrong. I forgot that you reverse the order of
the
> > elements to convert a multival wedgie into a multimonzo wedgie!
> Doing
> > so would, indeed, give the same rankings as my original L_1
> > calculation. But that's gotta be the right norm. The Tenney
lattice
> > is set up to measure complexity, and the norm we always associate
> > with it is the L_1 norm. Isn't that right? The L_1 norm on the
> monzo
> > is what I've been using all along to calculate complexity for the
> > codimension-1 case, in my graphs and in the "Attn: Gene 2"
> post . . .
>
> To me it seemed there was a reasonable case for either norm,

What's the case for L_inf norm? It doesn't seem to agree with the
purpose we're using the Tenney lattice in the first place . . .

>Do you need me to redo the calculation using
> the L1 norm?

That would be excellent, and then I could make a graph for Dave . . .

> I think it would be a good idea to stick to the normalization by
> dividing, since for higher limits a linear temperament wedgie is
> still a bival, so it's easier.

OK . . .

Wedgie norm for 12-equal:

Take the two unison vectors

|7 0 -3>
|-4 4 -1>

Now find the determinant, and the "area" it represents, in each of
the basis planes:

|7 0| = 28*(e23) -> 28/lg2(5) = 12.059
|-4 4|

|7 -3| = -19*(e25) -> 19/lg2(3) = 11.988
|-4 -1|

|0 -3| = 12*(e35) -> 12 = 12
|4 -1|

sum = 36.047

If I just use the maximum (L_inf = 12.059) as a measure of notes per
acoustical octave, then I "predict" tempered octaves of 1194.1 cents.
If I use the sum (L_1), dividing by the "mystery constant" 3,
I "predict" tempered octaves of 1198.4 cents. Neither one is the TOP
value . . . :( . . . but what sorts of error criteria, if any, *do*
they optimize?

So the cross-checking I found for the 3-limit case in "Attn: Gene 2"
/tuning-math/message/8799
doesn't seem to work in the 5-limit ET case for either the L_1 or
L_inf norms.

However, if I just add the largest and smallest values above:

28/lg2(5)+19/lg2(3)

I do predict the correct tempered octave (aside from a factor of 2),

1197.67406985219 cents.

So what sort of norm, if any, did I use to calculate complexity this
time? It's related to how we temper for TOP . . .

On Sat, 31 Jan 2004 08:59:44 -0000, "Paul Erlich" <perlich@aya.yale.edu>
wrote:

>In the 5-limit linear case, it would be really easy to do this if we
>didn't want to go out to the complexity of schismic and
>kleismic/hanson (the only argument that would arise would be whether
>the father and beep couple should be in or out, leading to a grand
>total of 11 or 9 5-limit LTs).

Father is really only useful up to 8 notes, and beep up to 9, but father
could potentially be of interest as a "warping" of traditional melody and
harmony, since tempering out semitones drastically changes the effect of
both melodies and harmonic progressions. As a "temperament" in the sense of
approximating JI, it isn't worth considering. Beep could theoretically be
of similar use as a "warping" of Bohlen-Pierce-type harmony, since 27/25 is
the BP-scale's equivalent of a semitone, but in other respects it seems
less interesting than other 5-limit tunings.

> Unfortunately, the low error of
>schismic has proved tantalizing enough for a few musicians to
>construct instruments capable of playing the large extended scales
>that its approximations require. If we consider such complexity
>justifiable, it seems we should be interested in 15 to 17 5-limit
>LTs, or 17 to 19 if we include father and beep.

Schismic and kleismic/hanson start being useful (barely) around 12 notes,
but the tiny size of schismic steps beyond 12 notes is a drawback until you
get to around 41 notes when the steps are a bit more evenly spaced.
Kleismic[15] and kleismic[19] are usable and have reasonably sized steps.
So the complexity of kleismic in a musically useful sense isn't really
comparable to schismic; this is one thing that the horagrams are useful
for.

> The couple residing
>in "the middle of the road" is 2187;2048 and 3126;2916. With Herman,
>we could split the difference and select only the better of the pair,
>2187;2048 (Dave, have you *heard* Blackwood's 21-equal suite?) . . .

Star Wars fans will recognize 2187 as Princess Leia's cell number, if
that's of any help in assigning a name to this temperament. The TOP tuning
for these:

[-11 7] (2187;2048) [1205.145343, 1893.799825]
[-2 -6 5] (3125;2916) [1205.183460, 1910.170591, 2774.278093]

Certainly, [-11 7] is useful as a 3-limit temperament, and this might be a
useful way to think of 7-ET in Thai and Burmese music. Since it's basically
a 3-limit temperament like Blackwood or Aristoxenan, its 5:1 can be tuned
just, resulting in 10.3 cent flat major thirds (slightly better than the
13.7 cent sharp thirds of 21-ET).

>I don't think anyone's talked about the 20480;19683 system. But if
>schismic, and certainly if semisixths, is not too complex to be a
>useful alternative to strict JI, why shouldn't this system merit some
>attention from musicians too? I don't think near-JI triads sound
>enough better than chords in this system (which are purer than those
>of augmented, porcupine, or diminished) to merit a much higher
>allowed complexity to generate them linearly.

This is [12 -9 1], TOP tuning: [1197.596121, 1905.765059, 2780.732080]. It
goes through 22 and 27 on the temperament chart, between porcupine and
diaschismic. The tiny steps are just barely big enough to be perceived as
melodic steps rather than commas, around the size of 22-ET steps. It might
be worth looking into.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >In the 5-limit linear case, it would be really easy to do this if we
> >didn't want to go out to the complexity of schismic and
> >kleismic/hanson
>
> While I think it would be nice to name this after Larry Hanson (and
> I certainly agreeable to the idea), my preference is to keep kleismic,
> since it tells the name of the comma involved, and has a fairly-well
> established body of use. What say everybody?
>
> -Carl

I have to agree with Carl. While Hanson may well be a perfectly
appropriate name for it, when a name has been as extensively used as
"kleismic" (even if only in these archives), you'd not only need to
show what's right about the new name, but also what's wrong about the
old name.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> That would be excellent, and then I could make a graph for
Dave . . .

Do I need to push any farther towards either less accurate or more
complex temperaments to make everyone happy?

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:

> So the complexity of kleismic in a musically useful sense isn't
really
> comparable to schismic; this is one thing that the horagrams are
useful
> for.

The horagrams assume distributionally even, octave-repeating scales;
our complexity measures don't.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > That would be excellent, and then I could make a graph for
> Dave . . .
>
> Do I need to push any farther towards either less accurate or more
> complex temperaments to make everyone happy?

No; I'm thinking that, if anything, it's near the *center* where log-
flat badness may suffer from some paucity, for our exploratory
purposes.

Please remember to post the results as a delimited table, with one
temperament per row. That way, most of us will be able to graph them
easily.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:

> Schismic and kleismic/hanson start being useful (barely) around 12
notes,
> but the tiny size of schismic steps beyond 12 notes is a drawback
until you
> get to around 41 notes when the steps are a bit more evenly spaced.

Others may feel differently. Schismic-17 is a favorite of Wilson and
others and closely resembles the medieval Arabic system; Helmholtz
and Groven used 24 and 36 notes, respectively. Justin White seemed to
be most interested in the 29-note version.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > Yes. I like that idea too. But by "a wide swath" don't you mean one
> > that it's easy to put a simple smooth curve thru?
>
> Right.
>
> > And you must have
> > some general idea of which way this intergalactic moat must curve.
>
> In the 5-limit linear case, it would be really easy to do this if we
> didn't want to go out to the complexity of schismic and
> kleismic/hanson (the only argument that would arise would be whether
> the father and beep couple should be in or out, leading to a grand
> total of 11 or 9 5-limit LTs).

There's no doubt in my mind that father and beep should be out. I
notice Herman agrees that these are not of interest as approximations
of 5-limit JI.

And yes, I can see what you mean, when I look at
/tuning-math/files/Erlich/dave3.gif
(I've decided I like to see both axes linear)
There's a wide moat cutting off these 9.

> Unfortunately, the low error of
> schismic has proved tantalizing enough for a few musicians to
> construct instruments capable of playing the large extended scales
> that its approximations require.

Yes. Schismic and kleismic must be in.

> If we consider such complexity
> justifiable, it seems we should be interested in 15 to 17 5-limit
> LTs,

> or 17 to 19 if we include father and beep. The couple residing
> in "the middle of the road" is 2187;2048 and 3126;2916. With Herman,
> we could split the difference and select only the better of the pair,
> 2187;2048

But they are so close in both error and complexity. If our error and
complexity measures are any good at all then it's either both or none.

(Dave, have you *heard* Blackwood's 21-equal suite?) . . .

No I haven't. But hearing it wouldn't tell me if its complexity was
objectionable. I'd have to think about composing in it or playing it
for that. I suppose the fact that Blackwood did so, suggests it's OK,
but only if it was actually that 5-limit linear temperament that
Blackwood used, and not some other or more general way of viewing of
21-ET.

And I'll take your word that it sounds ok, but are you certain that
its consonances only depend on the actual 2187;2048 linear mapping?

> I don't think anyone's talked about the 20480;19683 system. But if
> schismic, and certainly if semisixths, is not too complex to be a
> useful alternative to strict JI, why shouldn't this system merit some
> attention from musicians too?

That's easy to answer. It's error is way larger than either schismic
or kleismic, and I wouldn't mind omitting semisixths. Has anyone
expressed a strong desire to include it (78125;78732).

> I don't think near-JI [e.g. schismic] triads sound
> enough better than chords in this system (which are purer than those
> of augmented, porcupine, or diminished) to merit a much higher
> allowed complexity to generate them linearly.

They are only slightly purer than augmented, and about 50% worse than
meantone. This certainly doesn't allow 20480;19683 "in" when it has
complexity near that of schismic.

First you say it would be nice not to have to admit anything as
complex as schismic. Then you say schismic has to be admitted because
people have found schismic useful because of its very low error. Then
you say we should admit stuff as complex as schismic but with far
greater errors. Doesn't seem consistent to me.

I think I know what you're trying to do. I can see that the only
really wide moats/channels happen to depart from the X axis in a
direction that is almost due north and go quite a long way in that
direction before curving west. But I don't think this is acceptable
from a psychological standpoint. Straight lines I could definitely
live with. Convex curves (as viewed from the side opposite the origin)
I find more difficult to accept. The farthest I'd be willing to go in
the convex direction would be k*error^2 + comp^2 < x and the curve
should only just admit 24;25 and 128;135 (neutral thirds and pelogic)
and only just admit schismic (32768;32805), and possibly just admit
Negri (16384;16875).

However, I can see that if we plot that curve then if there's anything
that's only just outside it we'll feel obliged to adjust the curve to
include it, and repeat this until there are none "just outside the
curve", and then I can see that we will probably end up including
semisixths and the "middle of the road pair" (actually large error
_and_ high complexity) 2187;2048 and 3126;2916.

I could live with these 18. Although I note that a straight line can
be drawn that includes all but these last two, and a convex curve with
powers intermediate between 1 and 2 could do so more cleanly.

On the dave3.gif plot, the formula would be something like

(5000*error)^2 + comp^2 < 35^2

> A problem with our plan to have versions of these badness curves for
> sets of temperaments of different dimensions is that moving to a
> higher-dimensional tuning system would theoretically increase the
> badness by an infinite factor.

I don't understand?

> But in practice you never use an
> infinite swath of the lattice so eventually any complex enough 5-
> limit linear temperament becomes indistinguishable from a planar
> system.
>
> I'm speaking like you now, Dave! ;)

In the above paragraph? In what way? I don't follow.

Certainly you are now agreeing that log-flat badness with additional
cutoffs on error and complexity is not the best way to choose a
reduced list. What made you change you mind? Was it seeing them plotted?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > That would be excellent, and then I could make a graph for
> > Dave . . .
> >
> > Do I need to push any farther towards either less accurate or more
> > complex temperaments to make everyone happy?
>
> No; I'm thinking that, if anything, it's near the *center* where log-
> flat badness may suffer from some paucity, for our exploratory
> purposes.

Agreed.

> Please remember to post the results as a delimited table, with one
> temperament per row. That way, most of us will be able to graph them
> easily.

Agreed.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > Yes. I like that idea too. But by "a wide swath" don't you mean
one
> > > that it's easy to put a simple smooth curve thru?
> >
> > Right.
> >
> > > And you must have
> > > some general idea of which way this intergalactic moat must
curve.
> >
> > In the 5-limit linear case, it would be really easy to do this if
we
> > didn't want to go out to the complexity of schismic and
> > kleismic/hanson (the only argument that would arise would be
whether
> > the father and beep couple should be in or out, leading to a
grand
> > total of 11 or 9 5-limit LTs).
>
> There's no doubt in my mind that father and beep should be out. I
> notice Herman agrees that these are not of interest as
approximations
> of 5-limit JI.
>
> And yes, I can see what you mean, when I look at
> /tuning-math/files/Erlich/dave3.gif
> (I've decided I like to see both axes linear)
> There's a wide moat cutting off these 9.
>
> > Unfortunately, the low error of
> > schismic has proved tantalizing enough for a few musicians to
> > construct instruments capable of playing the large extended
scales
> > that its approximations require.
>
> Yes. Schismic and kleismic must be in.
>
> > If we consider such complexity
> > justifiable, it seems we should be interested in 15 to 17 5-limit
> > LTs,
>
> > or 17 to 19 if we include father and beep. The couple residing
> > in "the middle of the road" is 2187;2048 and 3126;2916. With
Herman,
> > we could split the difference and select only the better of the
pair,
> > 2187;2048
>
> But they are so close in both error and complexity. If our error and
> complexity measures are any good at all then it's either both or
none.

*** But Dave, the pair is stradding the smooth curve that passes
through the center of the moat everywhere else. How is one to decide?
I prefer this arbitrariness for just 2 temperaments when it makes the
rest so unarbitrary . . .

> (Dave, have you *heard* Blackwood's 21-equal suite?) . . .
>
> No I haven't. But hearing it wouldn't tell me if its complexity was
> objectionable.

I think hearing it makes a *great* case that it's complexity is
unobjectionable.

I'd have to think about composing in it or playing it
> for that. I suppose the fact that Blackwood did so, suggests it's
OK,
> but only if it was actually that 5-limit linear temperament that
> Blackwood used, and not some other or more general way of viewing of
> 21-ET.

He modulates his 'diatonic' music around a circle of 7 fifths -- thus
availing himself of 2048;2187. He might not have used its "native
scale", but yes, he used the temperament.

> And I'll take your word that it sounds ok, but are you certain that
> its consonances only depend on the actual 2187;2048 linear mapping?

In my view, linear mappings depend on consonances, not vice
versa . . .

Anyway, see above. The diatonic scale is so unusual in 21-equal that
(or since) the syntonic comma is *negative*, so the usual diatonic
play has to deal with this particular "quirk", certainly not
tempering it out . . .

> > I don't think anyone's talked about the 20480;19683 system. But
if
> > schismic, and certainly if semisixths, is not too complex to be a
> > useful alternative to strict JI, why shouldn't this system merit
some
> > attention from musicians too?
>
> That's easy to answer. It's error is way larger than either schismic
> or kleismic, and I wouldn't mind omitting semisixths. Has anyone
> expressed a strong desire to include it (78125;78732).

But the moat!

> > I don't think near-JI [e.g. schismic] triads sound
> > enough better than chords in this system (which are purer than
those
> > of augmented, porcupine, or diminished) to merit a much higher
> > allowed complexity to generate them linearly.
>
> They are only slightly purer than augmented, and about 50% worse
than
> meantone. This certainly doesn't allow 20480;19683 "in" when it has
> complexity near that of schismic.

I'm arguing that, along this particular line of thinking, complexity
does one thing to music, and error another, but there's no urgent
reason more of one should limit your tolerance for the other . . .

> First you say it would be nice not to have to admit anything as
> complex as schismic. Then you say schismic has to be admitted
because
> people have found schismic useful because of its very low error.
Then
> you say we should admit stuff as complex as schismic but with far
> greater errors. Doesn't seem consistent to me.

I drew a moat, stepped beyond it, and drew a new moat to accomodate
the expansion.

> I think I know what you're trying to do. I can see that the only
> really wide moats/channels happen to depart from the X axis in a
> direction that is almost due north and go quite a long way in that
> direction before curving west. But I don't think this is acceptable
> from a psychological standpoint. Straight lines I could definitely
> live with. Convex curves (as viewed from the side opposite the
origin)
> I find more difficult to accept.

See above . . .

> > A problem with our plan to have versions of these badness curves
for
> > sets of temperaments of different dimensions is that moving to a
> > higher-dimensional tuning system would theoretically increase the
> > badness by an infinite factor.
>
> I don't understand?

What's the area of a volume?

> > But in practice you never use an
> > infinite swath of the lattice so eventually any complex enough 5-
> > limit linear temperament becomes indistinguishable from a planar
> > system.
> >
> > I'm speaking like you now, Dave! ;)
>
> In the above paragraph? In what way? I don't follow.

It's this:

> Certainly you are now agreeing that log-flat badness with additional
> cutoffs on error and complexity is not the best way to choose a
> reduced list.

That's what I meant.

> What made you change you mind? Was it seeing them plotted?

I have thousands of your posts swimming around my head at all
times :) The real reason -- because I want them to get used for
***music*** (which is why speaking to people like herman is
useful . . .;)

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:

> > > oThe couple residing
> > > in "the middle of the road" is 2187;2048 and 3126;2916. With
> Herman,
> > > we could split the difference and select only the better of the
> pair,
> > > 2187;2048
> >
> > But they are so close in both error and complexity. If our error and
> > complexity measures are any good at all then it's either both or
> none.
>
> *** But Dave, the pair is stradding the smooth curve that passes
> through the center of the moat everywhere else.

Which smooth curve? Which moat? I note that there are no islands (or
stepping stones) in moats. :-)

> How is one to decide?
>
> I prefer this arbitrariness for just 2 temperaments when it makes the
> rest so unarbitrary . . .

Can you easily re-plot dave3.jpg showing a quarter-ellipse

(k*err)^2 + comp^2 = x^2

that passes thru 24;25 (neutral thirds) and 78125;78732 (semisixths)?

This is not intended to represent the centre of the moat but one edge
of it.

> I'm arguing that, along this particular line of thinking, complexity
> does one thing to music, and error another, but there's no urgent
> reason more of one should limit your tolerance for the other . . .

I disagree. I think people will only tolerate more complexity if it
gives them less errolr and vice versa. You seem to be arguing for a
rectangular region where

max(k*err, comp) < x

But I'm willing to move far enough in this direction to admit of
elliptical regions.

They would address Carl's points about a doubling of complexity being
more than twice as inconvenient, and JdL using error-squared for
mistuning-pain.

> The real reason -- because I want them to get used for
> ***music*** (which is why speaking to people like herman is
> useful . . .;)

Amen.

There's something VERY CREEPY about my complexity values. I'm going
to have to accept this as *the* correct scaling for complexity (I'm
already convinced this is the correct formulation too, i.e. L_1 norm,
for the time being) . . .

> 2.
> > Meantone (Huygens)
> >
> > [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> > TOP generators [1201.698520, 504.1341314]
> > bad: 21.551439 comp: 3.562072 err: 1.698521
>
> 11.7652 -> bad = 235.1092

11.7652 -> 12 -? chromatic scale

> 3.
> > Miracle
> >
> > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > TOP generators [1200.631014, 116.7206423]
> > bad: 29.119472 comp: 6.793166 err: .631014
>
> 21.1019 --> bad = 280.9843

21.1019 --> 21 -? blackjack scale

> 7.
> > Magic
> >
> > [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> > TOP generators [1201.276744, 380.7957184]
> > bad: 23.327687 comp: 4.274486 err: 1.276744
>
> 15.5360 -> bad = 308.1642

15.5360 --? 16 -? magic-16 MOS scale

> 8.
> > Beep
> >
> > [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> > TOP generators [1194.642673, 254.8994697]
> > bad: 23.664749 comp: 1.292030 err: 14.176105
>
> 4.7295 -> bad = 317.0935

4.7295 --> 5 -? beep-5 MOS scale

> 9.
> > Pajara
> >
> > [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> > TOP generators [598.4467109, 106.5665459]
> > bad: 27.754421 comp: 2.988993 err: 3.106578
>
> 10.4021 -> bad = 336.1437

10.4021 --> 10 -? symmetrical or omnitetrachordal decatonic scale

> 12.
> > Diminished
> >
> > [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> > TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> > TOP generators [298.5321149, 101.4561401]
> > bad: 37.396767 comp: 2.523719 err: 5.871540
>
> 7.917 -> bad = 368.02

7.917 --> 8 -? octatonic scale

> 16.
> > Father
> >
> > [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> > TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> > TOP generators [1185.869125, 447.3863410]
> > bad: 33.256527 comp: 1.534101 err: 14.130876
>
> 5.2007 -> bad = 382.2

5.2007 --> 5 -? father-5 MOS scale

> 20.
> > Tripletone
> >
> > [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> > TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> > TOP generators [399.0200131, 92.45965769]
> > bad: 48.112067 comp: 4.045351 err: 2.939961
>
> 12.125 -> bad = 432.24

12.125 --> 12 -? 12-note DE with "augmented" symmetry

> 21.
> > {21/20, 28/27}
> >
> > [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
> > TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
> > TOP generators [1214.253642, 509.4012304]
> > bad: 42.300772 comp: 1.722706 err: 14.253642
>
> 5.5723 -> bad = 442.58

5.5723 --> 5 -? 5-note MOS

> 27.
> > Dicot
> >
> > [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
> > TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
> > TOP generators [1204.048159, 356.3998255]
> > bad: 42.920570 comp: 2.137243 err: 9.396316
>
> 7.2314 -> bad = 491.37

7.2314 --> 7 -? Mohajira

> 29.
> > Injera
> >
> > [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
> > TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
> > TOP generators [600.8889070, 93.60982493]
> > bad: 42.529834 comp: 3.445412 err: 3.582707
>
> 11.918 -> bad = 508.85

11.918 --> 12 -? 12-note Injera DE or omnitetrachordal scale

> 30.
> > {25/24, 81/80} Jamesbond?
> >
> > [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
> > TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
> > TOP generators [172.7759159, 86.69241190]
> > bad: 58.637859 comp: 2.493450 err: 9.431411
>
> 7.4202 -> bad = 519.28

7.4202 -> 7 -? 7-note equal-tempered scale

> 32.
> > Pelogic
> >
> > [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
> > TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
> > TOP generators [1209.734056, 532.9412251]
> > bad: 39.824125 comp: 2.022675 err: 9.734056
>
> 7.426 -> bad = 536.78

7.426 --> 7 --> 7-note "P.E.Logic Diatonic" scale

Creepy, isn't it?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> Can you easily re-plot dave3.jpg showing a quarter-ellipse
>
> (k*err)^2 + comp^2 = x^2
>
> that passes thru 24;25 (neutral thirds) and 78125;78732
(semisixths)?
>
> This is not intended to represent the centre of the moat but one
edge
> of it.

Tough assignment . . . you?

> > I'm arguing that, along this particular line of thinking,
complexity
> > does one thing to music, and error another, but there's no urgent
> > reason more of one should limit your tolerance for the other . . .
>
> I disagree. I think people will only tolerate more complexity if it
> gives them less errolr and vice versa.

It'll generally be different people, though. For a given individual,
very low error doesn't make complexity any easier to tolerate than
merely tolerable error. Nevertheless, I think we should use something
close to a straight line, slightly convex or concave to best fit
a "moat".

Sorry I skipped this:

4.
> Hemiwuerschmidt
>
> [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> TOP generators [1199.692003, 193.8224275]
> bad: 31.386908 comp: 10.094876 err: .307997

31.212 -> bad = 300.04

31.212 --> 31 -? 31-note proper MOS

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> There's something VERY CREEPY about my complexity values. I'm going
> to have to accept this as *the* correct scaling for complexity (I'm
> already convinced this is the correct formulation too, i.e. L_1
norm,
> for the time being) . . .
>
> > 2.
> > > Meantone (Huygens)
> > >
> > > [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
> > > TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
> > > TOP generators [1201.698520, 504.1341314]
> > > bad: 21.551439 comp: 3.562072 err: 1.698521
> >
> > 11.7652 -> bad = 235.1092
>
> 11.7652 -> 12 -? chromatic scale
>
> > 3.
> > > Miracle
> > >
> > > [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> > > TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> > > TOP generators [1200.631014, 116.7206423]
> > > bad: 29.119472 comp: 6.793166 err: .631014
> >
> > 21.1019 --> bad = 280.9843
>
> 21.1019 --> 21 -? blackjack scale
>
> > 7.
> > > Magic
> > >
> > > [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
> > > TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
> > > TOP generators [1201.276744, 380.7957184]
> > > bad: 23.327687 comp: 4.274486 err: 1.276744
> >
> > 15.5360 -> bad = 308.1642
>
> 15.5360 --? 16 -? magic-16 MOS scale
>
> > 8.
> > > Beep
> > >
> > > [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
> > > TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
> > > TOP generators [1194.642673, 254.8994697]
> > > bad: 23.664749 comp: 1.292030 err: 14.176105
> >
> > 4.7295 -> bad = 317.0935
>
> 4.7295 --> 5 -? beep-5 MOS scale
>
> > 9.
> > > Pajara
> > >
> > > [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
> > > TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
> > > TOP generators [598.4467109, 106.5665459]
> > > bad: 27.754421 comp: 2.988993 err: 3.106578
> >
> > 10.4021 -> bad = 336.1437
>
> 10.4021 --> 10 -? symmetrical or omnitetrachordal decatonic scale
>
> > 12.
> > > Diminished
> > >
> > > [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
> > > TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
> > > TOP generators [298.5321149, 101.4561401]
> > > bad: 37.396767 comp: 2.523719 err: 5.871540
> >
> > 7.917 -> bad = 368.02
>
> 7.917 --> 8 -? octatonic scale
>
> > 16.
> > > Father
> > >
> > > [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
> > > TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
> > > TOP generators [1185.869125, 447.3863410]
> > > bad: 33.256527 comp: 1.534101 err: 14.130876
> >
> > 5.2007 -> bad = 382.2
>
> 5.2007 --> 5 -? father-5 MOS scale
>
> > 20.
> > > Tripletone
> > >
> > > [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> > > TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
> > > TOP generators [399.0200131, 92.45965769]
> > > bad: 48.112067 comp: 4.045351 err: 2.939961
> >
> > 12.125 -> bad = 432.24
>
> 12.125 --> 12 -? 12-note DE with "augmented" symmetry
>
> > 21.
> > > {21/20, 28/27}
> > >
> > > [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
> > > TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
> > > TOP generators [1214.253642, 509.4012304]
> > > bad: 42.300772 comp: 1.722706 err: 14.253642
> >
> > 5.5723 -> bad = 442.58
>
> 5.5723 --> 5 -? 5-note MOS
>
> > 27.
> > > Dicot
> > >
> > > [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
> > > TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
> > > TOP generators [1204.048159, 356.3998255]
> > > bad: 42.920570 comp: 2.137243 err: 9.396316
> >
> > 7.2314 -> bad = 491.37
>
> 7.2314 --> 7 -? Mohajira
>
> > 29.
> > > Injera
> > >
> > > [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
> > > TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
> > > TOP generators [600.8889070, 93.60982493]
> > > bad: 42.529834 comp: 3.445412 err: 3.582707
> >
> > 11.918 -> bad = 508.85
>
> 11.918 --> 12 -? 12-note Injera DE or omnitetrachordal scale
>
> > 30.
> > > {25/24, 81/80} Jamesbond?
> > >
> > > [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
> > > TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
> > > TOP generators [172.7759159, 86.69241190]
> > > bad: 58.637859 comp: 2.493450 err: 9.431411
> >
> > 7.4202 -> bad = 519.28
>
> 7.4202 -> 7 -? 7-note equal-tempered scale
>
> > 32.
> > > Pelogic
> > >
> > > [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
> > > TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
> > > TOP generators [1209.734056, 532.9412251]
> > > bad: 39.824125 comp: 2.022675 err: 9.734056
> >
> > 7.426 -> bad = 536.78
>
> 7.426 --> 7 --> 7-note "P.E.Logic Diatonic" scale
>
> Creepy, isn't it?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> There's something VERY CREEPY about my complexity values. I'm going
> to have to accept this as *the* correct scaling for complexity (I'm
> already convinced this is the correct formulation too, i.e. L_1 norm,
> for the time being) . . .
...
> Creepy, isn't it?

Woah! Yes.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > Can you easily re-plot dave3.jpg showing a quarter-ellipse
> >
> > (k*err)^2 + comp^2 = x^2
> >
> > that passes thru 24;25 (neutral thirds) and 78125;78732
> (semisixths)?
> >
> > This is not intended to represent the centre of the moat but one
> edge
> > of it.
>
> Tough assignment . . . you?

I don't have the data to plot them all. If you just want me to give
you k and x I'll only need the error and complexity figures for these
two temperaments.

Are they still the same as used on dave3.jpg, or are you about to
change them?

> > I disagree. I think people will only tolerate more complexity if it
> > gives them less errolr and vice versa.
>
> It'll generally be different people, though. For a given individual,
> very low error doesn't make complexity any easier to tolerate than
> merely tolerable error.

Very true. So that makes the quadratic (elliptical) case look good

> Nevertheless, I think we should use something
> close to a straight line, slightly convex or concave to best fit
> a "moat".

Sounds good to me. So we can parameterise it as

(err/max_err)^p + (comp/max_comp)^p < 1, where 0.5<=p<=2.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > There's something VERY CREEPY about my complexity values. I'm
going
> > to have to accept this as *the* correct scaling for complexity
(I'm
> > already convinced this is the correct formulation too, i.e. L_1
norm,
> > for the time being) . . .
> ...
> > Creepy, isn't it?
>
> Woah! Yes.

So when I explained this as the number of notes in the bivector, I
must have been onto something. It's interesting how the *proper*
scales are "attractors" for this measure, but improper DEs don't seem
to be at all -- horagrams about to be uploaded . . .

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > Can you easily re-plot dave3.jpg showing a quarter-ellipse
> > >
> > > (k*err)^2 + comp^2 = x^2
> > >
> > > that passes thru 24;25 (neutral thirds) and 78125;78732
> > (semisixths)?
> > >
> > > This is not intended to represent the centre of the moat but
one
> > edge
> > > of it.
> >
> > Tough assignment . . . you?
>
> I don't have the data to plot them all. If you just want me to give
> you k and x I'll only need the error and complexity figures for
these
> two temperaments.
>
> Are they still the same as used on dave3.jpg,

Yes.

> or are you about to
> change them?

No. You should know how to calculate them by now: log(n/d)*log(n*d)
and log(n*d) respectively.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Please remember to post the results as a delimited table, with one
> temperament per row. That way, most of us will be able to graph them
> easily.

You'd better give a sample of what you want; I'll see if Maple can
accomodate you.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > There's something VERY CREEPY about my complexity values. I'm
> going
> > > to have to accept this as *the* correct scaling for complexity
> (I'm
> > > already convinced this is the correct formulation too, i.e. L_1
> norm,
> > > for the time being) . . .
> > ...
> > > Creepy, isn't it?
> >
> > Woah! Yes.
>
> So when I explained this as the number of notes in the bivector, I
> must have been onto something. It's interesting how the *proper*
> scales are "attractors" for this measure, but improper DEs don't
seem
> to be at all

Except Blackjack . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Please remember to post the results as a delimited table, with
one
> > temperament per row. That way, most of us will be able to graph
them
> > easily.
>
> You'd better give a sample of what you want; I'll see if Maple can
> accomodate you.

[[w, e, d, g, i, e] error] works fine for me, I can calculate the
complexity . . .

[[6, -7, -2, -25, -20, 15] .630134]
[[1, 4, 10, 4, 13, 12] 1.698521]
[[5, 1, 12, -10, 5, 25] 1.276744]
[[2, -4, -4, -11, -12, 2] 3.106578]
[[4, 4, 4, -3, -5, -2] 5.871540]

>I'm arguing that, along this particular line of thinking, complexity
>does one thing to music, and error another, but there's no urgent
>reason more of one should limit your tolerance for the other . . .

Taking this to its logical extreme, wouldn't we abandon badness
alltogether?

-Carl

>There's something VERY CREEPY about my complexity values.

Wow dude. What sort of DES are these? Not the smallest
apparently.

-Carl

>> > TOP generators [1201.698520, 504.1341314]

So how are these generators being chosen? Hermite? I confess
I don't know how to 'refactor' a generator basis.

-Carl

Herman:
>>Schismic and kleismic/hanson start being useful (barely) around 12 > > notes,
> >>but the tiny size of schismic steps beyond 12 notes is a drawback > > until you
> >>get to around 41 notes when the steps are a bit more evenly spaced.

Paul E:
> Others may feel differently. Schismic-17 is a favorite of Wilson and > others and closely resembles the medieval Arabic system; Helmholtz > and Groven used 24 and 36 notes, respectively. Justin White seemed to > be most interested in the 29-note version.

I played a lot with the 29 note scale, and it worked fine. It's nice to have some small intervals to throw in when you want them. It's a serious contender because it's still based on fifths, so it's familiar and can work with simple adaptations of regular notation. But that also makes it easy to find and so overrated.

It should be included in a 9-limit, or weighted 7-limit, list anyway.

Graham

Paul Erlich wrote:

> I don't think that's quite what Partch says. Manuel, at least, has > always insisted that simpler ratios need to be tuned more accurately, > and harmonic entropy and all the other discordance functions I've > seen show that the increase in discordance for a given amount of > mistuning is greatest for the simplest intervals.

Did you ever track down what Partch said?

Harmonic entropy can obviously be used to prove whatever you like. It also shows that the troughs get narrower the more complex the limit, so it takes a smaller mistuning before the putative ratio becomes irrelevant.

It also shows that, if all intervals are equally mistuned, the more complex ones will have the highest entropy. So they're the ones for which the mistuning is most problematic, and where you should start for optimization.

> Such distinctions may be important for *scales*, but for > temperaments, I'm perfectly happy not to have to worry about them. > Any reasons I shouldn't be?

You're using temperaments to construct scales, aren't you? If you don't want more than 18 notes in your scale, miracle is a contender in the 7-limit but not the 9-limit. And if you don't want errors more than 6 cents, you can use meantone in the 7-limit but not the 9-limit. There's no point in using intervals that are uselessly complex or inaccurate so you need to know whether you want the wider 9-limit when choosing the temperament.

> Tenney weighting can be conceived of in other ways than you're > conceiving of it. For example, if you're looking at 13-limit, it > suffices to minimize the maximum weighted error of {13:8, 13:9, > 13:10, 13:11, 13:12, 14:13} or any such lattice-spanning set of > intervals. Here the weights are all very close (13:8 gets 1.12 times > the weight of 14:13), *all* the ratios are ratios of 13 so simpler > intervals are not directly weighted *at all*, and yet the TOP result > will still be the same as if you just used the primes. I think TOP is > far more robust than you're giving it credit for.

It's really an average over all odd-limit minimaxes. And the higher you get probably the less difference it makes -- but then the harder the consonances will be to hear anyway. For the special case of 7 vs 9 limit, which is the most important, it seems to make quite a difference.

Once you've established that the 9-limit intervals are playable and audible, it may make sense to weight the simple ones higher because you expect to use them more. You could even generate the weights statistically from the score. But there are usually so few usable temperaments in any situation, you may as well consider each one individually and subjectively.

Oh, yes, I think the 9-limit calculation can be done by giving 3 a weight of a half. That places 9 on an equal footing with 5 and 7, and I think it works better than vaguely talking about the number of consonances. After all, how do you share a comma between 3:2 and 9:8? I still don't know how the 15-limit would work.

I'm expecting the limit of this calculation as the odd limit tends to infinity will be the same as this Kees metric. And as the integer limit goes to infinity, it'll probably give the Tenney metric. But as the integers don't get much beyond 10, infinity isn't really an important consideration.

Not that it does much harm either, because the minimax always depends on the most complex intervals, which will have roughly equal weighting. The same as octave specific metrics give roughly the same results as odd-limit style octave equivalent ones if you allow for octave stretching.

Graham