I took the same 15 7-limit intervals as before, and this time took

sets of three. After eliminating the ones with linear dependence or

torsion, I obtained the following, which lists the validity measure,

the intervals, and the dual val:

[1.448457604, [4375/4374, 2401/2400, 6144/6125],

99*a+157*b+230*c+278*d]

[1.509442953, [4375/4374, 2401/2400, 3136/3125],

99*a+157*b+230*c+278*d]

[1.487593558, [4375/4374, 2401/2400, 1029/1024],

114*b+72*a+167*c+202*d]

[1.247713548, [4375/4374, 2401/2400, 1728/1715], 43*b+27*a+63*c+76*d]

[1.261826622, [4375/4374, 2401/2400, 126/125], 43*b+27*a+63*c+76*d]

[1.269638870, [4375/4374, 2401/2400, 245/243], 43*b+27*a+63*c+76*d]

[1.800467556, [4375/4374, 2401/2400, 81/80], 71*b+45*a+104*c+126*d]

[1.455372564, [4375/4374, 2401/2400, 50/49], 28*b+18*a+41*c+50*d]

[1.466326603, [4375/4374, 2401/2400, 49/48], 28*b+18*a+41*c+50*d]

[1.296080214, [4375/4374, 2401/2400, 36/35], 22*c+26*d+15*b+9*a]

[3.790362186, [4375/4374, 2401/2400, 28/27], 85*c+102*d+58*b+36*a]

[1.968212894, [4375/4374, 6144/6125, 3136/3125],

99*a+157*b+230*c+278*d]

[1.457979089, [4375/4374, 6144/6125, 1029/1024],

73*b+46*a+129*d+107*c]

[1.779804660, [4375/4374, 6144/6125, 1728/1715],

84*b+53*a+123*c+149*d]

[1.681227707, [4375/4374, 6144/6125, 126/125], 73*b+46*a+129*d+107*c]

[1.698999914, [4375/4374, 6144/6125, 245/243], 73*b+46*a+129*d+107*c]

[1.116544333, [4375/4374, 6144/6125, 81/80], 11*b+7*a+16*c+20*d]

[1.142843389, [4375/4374, 6144/6125, 64/63], 11*b+7*a+16*c+20*d]

[4.102976382, [4375/4374, 6144/6125, 50/49], 169*d+139*c+95*b+60*a]

[2.544310889, [4375/4374, 6144/6125, 49/48], 62*b+39*a+91*c+109*d]

[3.561534162, [4375/4374, 6144/6125, 28/27], 89*d+51*b+32*a+75*c]

[2.763160012, [4375/4374, 3136/3125, 1029/1024],

331*d+187*b+118*a+274*c]

[2.468277477, [4375/4374, 3136/3125, 1728/1715],

127*b+80*a+186*c+225*d]

[1.249197248, [4375/4374, 3136/3125, 126/125], 19*a+30*b+44*c+53*d]

[1.254634753, [4375/4374, 3136/3125, 245/243], 19*a+30*b+44*c+53*d]

[3.283589564, [4375/4374, 3136/3125, 64/63], 142*c+97*b+61*a+172*d]

[2.733807197, [4375/4374, 3136/3125, 50/49], 42*a+67*b+98*c+119*d]

[2.083409549, [4375/4374, 3136/3125, 36/35], 54*c+37*b+23*a+66*d]

[1.174027616, [4375/4374, 3136/3125, 28/27], 10*c+13*d+4*a+7*b]

[1.951242152, [4375/4374, 3136/3125, 25/24], 23*b+15*a+34*c+40*d]

[1.389479277, [4375/4374, 1029/1024, 1728/1715], 60*c+73*d+41*b+26*a]

[1.824149922, [4375/4374, 1029/1024, 126/125], 73*b+46*a+129*d+107*c]

[1.843432955, [4375/4374, 1029/1024, 245/243], 73*b+46*a+129*d+107*c]

[1.577077172, [4375/4374, 1029/1024, 81/80], 60*c+73*d+41*b+26*a]

[1.517338321, [4375/4374, 1029/1024, 64/63], 47*c+56*d+32*b+20*a]

[1.672594979, [4375/4374, 1029/1024, 49/48], 47*c+56*d+32*b+20*a]

[1.220941678, [4375/4374, 1029/1024, 36/35], 9*b+6*a+13*c+17*d]

[4.347012006, [4375/4374, 1029/1024, 25/24], 90*d+50*b+32*a+73*c]

[1.529904994, [4375/4374, 1728/1715, 126/125], 43*b+27*a+63*c+76*d]

[1.539376974, [4375/4374, 1728/1715, 245/243], 43*b+27*a+63*c+76*d]

[1.690915994, [4375/4374, 1728/1715, 81/80], 60*c+73*d+41*b+26*a]

[1.887533019, [4375/4374, 1728/1715, 64/63], 43*b+27*a+63*c+76*d]

[2.070013772, [4375/4374, 1728/1715, 50/49], 60*c+73*d+41*b+26*a]

[1.028806584, [4375/4374, 1728/1715, 49/48], a+2*b+3*c+3*d]

[1.036605118, [4375/4374, 1728/1715, 36/35], a+2*b+3*c+3*d]

[3.442298035, [4375/4374, 1728/1715, 28/27], 45*b+28*a+66*c+79*d]

[3.370477905, [4375/4374, 1728/1715, 25/24], 57*c+70*d+39*b+25*a]

[1.479592122, [4375/4374, 126/125, 81/80], 19*a+30*b+44*c+53*d]

[1.908883188, [4375/4374, 126/125, 64/63], 43*b+27*a+63*c+76*d]

[1.255075583, [4375/4374, 126/125, 50/49], 13*b+8*a+19*c+23*d]

[1.728934000, [4375/4374, 126/125, 49/48], 19*a+30*b+44*c+53*d]

[1.337685863, [4375/4374, 126/125, 36/35], 13*b+8*a+19*c+23*d]

[1.428343371, [4375/4374, 126/125, 28/27], 13*b+8*a+19*c+23*d]

[1.714645543, [4375/4374, 126/125, 25/24], 30*d+25*c+17*b+11*a]

[1.486032490, [4375/4374, 245/243, 81/80], 19*a+30*b+44*c+53*d]

[1.920701505, [4375/4374, 245/243, 64/63], 43*b+27*a+63*c+76*d]

[1.257372941, [4375/4374, 245/243, 50/49], 13*b+8*a+19*c+23*d]

[1.736459703, [4375/4374, 245/243, 49/48], 19*a+30*b+44*c+53*d]

[1.340134435, [4375/4374, 245/243, 36/35], 13*b+8*a+19*c+23*d]

[1.430957888, [4375/4374, 245/243, 28/27], 13*b+8*a+19*c+23*d]

[1.718962570, [4375/4374, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[1.219933467, [4375/4374, 81/80, 64/63], 11*b+7*a+16*c+20*d]

[2.349492733, [4375/4374, 81/80, 50/49], 60*c+73*d+41*b+26*a]

[1.881628622, [4375/4374, 81/80, 49/48], 19*a+30*b+44*c+53*d]

[1.330764996, [4375/4374, 81/80, 36/35], 11*b+7*a+16*c+20*d]

[1.800789429, [4375/4374, 81/80, 28/27], 28*c+33*d+19*b+12*a]

[1.453990411, [4375/4374, 81/80, 25/24], 11*b+7*a+16*c+20*d]

[3.421600769, [4375/4374, 64/63, 50/49], 79*c+96*d+54*b+34*a]

[2.079078641, [4375/4374, 64/63, 49/48], 47*c+56*d+32*b+20*a]

[1.362109799, [4375/4374, 64/63, 36/35], 11*b+7*a+16*c+20*d]

[2.848783252, [4375/4374, 64/63, 28/27], 47*c+56*d+32*b+20*a]

[2.093653566, [4375/4374, 50/49, 49/48], 28*b+18*a+41*c+50*d]

[1.475235230, [4375/4374, 50/49, 36/35], 13*b+8*a+19*c+23*d]

[3.011864701, [4375/4374, 50/49, 25/24], 28*b+18*a+41*c+50*d]

[1.050240055, [4375/4374, 49/48, 36/35], a+2*b+3*c+3*d]

[3.140275637, [4375/4374, 49/48, 28/27], 47*c+56*d+32*b+20*a]

[3.034533867, [4375/4374, 49/48, 25/24], 28*b+18*a+41*c+50*d]

[1.678896865, [4375/4374, 36/35, 28/27], 13*b+8*a+19*c+23*d]

[1.623444754, [4375/4374, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[1.362979272, [4375/4374, 28/27, 25/24], 6*b+4*a+9*c+10*d]

[1.296835720, [2401/2400, 6144/6125, 1029/1024], 49*b+31*a+87*d+72*c]

[1.409207770, [2401/2400, 6144/6125, 1728/1715], 49*b+31*a+87*d+72*c]

[1.427524260, [2401/2400, 6144/6125, 126/125], 49*b+31*a+87*d+72*c]

[2.217365728, [2401/2400, 6144/6125, 245/243], 158*c+191*d+108*b+68*a]

[2.039491520, [2401/2400, 6144/6125, 64/63], 104*d+59*b+37*a+86*c]

[1.152921505, [2401/2400, 6144/6125, 50/49], 17*d+14*c+10*b+6*a]

[1.155806812, [2401/2400, 6144/6125, 49/48], 17*d+14*c+10*b+6*a]

[2.208066855, [2401/2400, 6144/6125, 36/35], 39*b+25*a+58*c+70*d]

[5.556136148, [2401/2400, 6144/6125, 28/27], 69*b+43*a+100*c+121*d]

[1.313691630, [2401/2400, 3136/3125, 1029/1024], 49*b+31*a+87*d+72*c]

[1.427524260, [2401/2400, 3136/3125, 1728/1715], 49*b+31*a+87*d+72*c]

[1.446078824, [2401/2400, 3136/3125, 126/125], 49*b+31*a+87*d+72*c]

[2.281076121, [2401/2400, 3136/3125, 245/243], 158*c+191*d+108*b+68*a]

[2.071170648, [2401/2400, 3136/3125, 64/63], 104*d+59*b+37*a+86*c]

[1.155806812, [2401/2400, 3136/3125, 50/49], 17*d+14*c+10*b+6*a]

[1.158699341, [2401/2400, 3136/3125, 49/48], 17*d+14*c+10*b+6*a]

[2.231182923, [2401/2400, 3136/3125, 36/35], 39*b+25*a+58*c+70*d]

[5.656559608, [2401/2400, 3136/3125, 28/27], 69*b+43*a+100*c+121*d]

[1.488862040, [2401/2400, 1029/1024, 1728/1715], 49*b+31*a+87*d+72*c]

[1.508213854, [2401/2400, 1029/1024, 126/125], 49*b+31*a+87*d+72*c]

[1.738209169, [2401/2400, 1029/1024, 245/243], 65*b+41*a+115*d+95*c]

[1.731539563, [2401/2400, 1029/1024, 81/80], 49*b+31*a+87*d+72*c]

[1.234120624, [2401/2400, 1029/1024, 64/63], 16*b+10*a+23*c+28*d]

[1.290335387, [2401/2400, 1029/1024, 50/49], 16*b+10*a+23*c+28*d]

[1.295721876, [2401/2400, 1029/1024, 49/48], 16*b+10*a+23*c+28*d]

[2.019048032, [2401/2400, 1029/1024, 36/35], 33*b+21*a+49*c+59*d]

[1.547209978, [2401/2400, 1728/1715, 245/243], 43*b+27*a+63*c+76*d]

[1.881579115, [2401/2400, 1728/1715, 81/80], 49*b+31*a+87*d+72*c]

[1.897137590, [2401/2400, 1728/1715, 64/63], 43*b+27*a+63*c+76*d]

[1.119277634, [2401/2400, 1728/1715, 50/49], 9*c+6*b+4*a+11*d]

[1.121144263, [2401/2400, 1728/1715, 49/48], 9*c+6*b+4*a+11*d]

[1.155526618, [2401/2400, 1728/1715, 36/35], 9*c+6*b+4*a+11*d]

[2.772428686, [2401/2400, 1728/1715, 28/27], 37*b+23*a+65*d+54*c]

[1.564710702, [2401/2400, 126/125, 245/243], 43*b+27*a+63*c+76*d]

[1.906035357, [2401/2400, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[1.918596397, [2401/2400, 126/125, 64/63], 43*b+27*a+63*c+76*d]

[1.121144263, [2401/2400, 126/125, 50/49], 9*c+6*b+4*a+11*d]

[1.123014005, [2401/2400, 126/125, 49/48], 9*c+6*b+4*a+11*d]

[1.157453700, [2401/2400, 126/125, 36/35], 9*c+6*b+4*a+11*d]

[2.799119925, [2401/2400, 126/125, 28/27], 37*b+23*a+65*d+54*c]

[1.342457835, [2401/2400, 245/243, 81/80], 22*b+14*a+32*c+39*d]

[1.930474851, [2401/2400, 245/243, 64/63], 43*b+27*a+63*c+76*d]

[1.496942601, [2401/2400, 245/243, 50/49], 22*b+14*a+32*c+39*d]

[1.505698455, [2401/2400, 245/243, 49/48], 22*b+14*a+32*c+39*d]

[1.613159878, [2401/2400, 245/243, 36/35], 21*b+13*a+31*c+37*d]

[1.794547474, [2401/2400, 245/243, 28/27], 21*b+13*a+31*c+37*d]

[1.625778481, [2401/2400, 81/80, 64/63], 27*b+17*a+40*c+48*d]

[1.588174591, [2401/2400, 81/80, 50/49], 22*b+14*a+32*c+39*d]

[1.597464076, [2401/2400, 81/80, 49/48], 22*b+14*a+32*c+39*d]

[2.008061053, [2401/2400, 81/80, 36/35], 27*b+17*a+40*c+48*d]

[1.159072634, [2401/2400, 81/80, 28/27], 8*c+9*d+5*b+3*a]

[1.438608665, [2401/2400, 64/63, 50/49], 16*b+10*a+23*c+28*d]

[1.444614119, [2401/2400, 64/63, 49/48], 16*b+10*a+23*c+28*d]

[2.124866198, [2401/2400, 64/63, 36/35], 27*b+17*a+40*c+48*d]

[1.691009707, [2401/2400, 64/63, 28/27], 16*b+10*a+23*c+28*d]

[1.215506250, [2401/2400, 50/49, 36/35], 9*c+6*b+4*a+11*d]

[1.768035978, [2401/2400, 50/49, 28/27], 16*b+10*a+23*c+28*d]

[1.217533360, [2401/2400, 49/48, 36/35], 9*c+6*b+4*a+11*d]

[1.775416623, [2401/2400, 49/48, 28/27], 16*b+10*a+23*c+28*d]

[2.326619401, [2401/2400, 36/35, 28/27], 21*b+13*a+31*c+37*d]

[1.427524260, [6144/6125, 3136/3125, 1029/1024], 49*b+31*a+87*d+72*c]

[1.551220597, [6144/6125, 3136/3125, 1728/1715], 49*b+31*a+87*d+72*c]

[1.571382931, [6144/6125, 3136/3125, 126/125], 49*b+31*a+87*d+72*c]

[2.737190864, [6144/6125, 3136/3125, 245/243], 158*c+191*d+108*b+68*a]

[2.287131427, [6144/6125, 3136/3125, 64/63], 104*d+59*b+37*a+86*c]

[1.174547064, [6144/6125, 3136/3125, 50/49], 17*d+14*c+10*b+6*a]

[1.177486492, [6144/6125, 3136/3125, 49/48], 17*d+14*c+10*b+6*a]

[2.385833149, [6144/6125, 3136/3125, 36/35], 39*b+25*a+58*c+70*d]

[6.347647197, [6144/6125, 3136/3125, 28/27], 69*b+43*a+100*c+121*d]

[1.617873353, [6144/6125, 1029/1024, 1728/1715], 49*b+31*a+87*d+72*c]

[2.103467335, [6144/6125, 1029/1024, 245/243], 73*b+46*a+129*d+107*c]

[1.881579115, [6144/6125, 1029/1024, 81/80], 49*b+31*a+87*d+72*c]

[1.427247693, [6144/6125, 1029/1024, 64/63], 15*a+24*b+42*d+35*c]

[1.569463799, [6144/6125, 1029/1024, 50/49], 25*b+16*a+37*c+45*d]

[1.535432120, [6144/6125, 1029/1024, 49/48], 15*a+24*b+42*d+35*c]

[1.782870571, [6144/6125, 1029/1024, 36/35], 25*b+16*a+37*c+45*d]

[1.050000000, [6144/6125, 1029/1024, 25/24], b+a+3*d+2*c]

[1.780914442, [6144/6125, 1728/1715, 126/125], 49*b+31*a+87*d+72*c]

[1.513769837, [6144/6125, 1728/1715, 245/243], 35*b+22*a+51*c+62*d]

[2.044619737, [6144/6125, 1728/1715, 81/80], 49*b+31*a+87*d+72*c]

[1.787357184, [6144/6125, 1728/1715, 64/63], 35*b+22*a+51*c+62*d]

[1.971379996, [6144/6125, 1728/1715, 50/49], 35*b+22*a+51*c+62*d]

[1.325000779, [6144/6125, 1728/1715, 49/48], 25*d+21*c+9*a+14*b]

[1.418184249, [6144/6125, 1728/1715, 36/35], 25*d+21*c+9*a+14*b]

[1.842667322, [6144/6125, 1728/1715, 28/27], 37*d+30*c+21*b+13*a]

[1.952519731, [6144/6125, 1728/1715, 25/24], 37*d+30*c+21*b+13*a]

[2.425554379, [6144/6125, 126/125, 245/243], 73*b+46*a+129*d+107*c]

[2.071195136, [6144/6125, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[1.495120029, [6144/6125, 126/125, 64/63], 15*a+24*b+42*d+35*c]

[1.649199242, [6144/6125, 126/125, 50/49], 25*b+16*a+37*c+45*d]

[1.608449134, [6144/6125, 126/125, 49/48], 15*a+24*b+42*d+35*c]

[1.873447987, [6144/6125, 126/125, 36/35], 25*b+16*a+37*c+45*d]

[1.053257143, [6144/6125, 126/125, 25/24], b+a+3*d+2*c]

[1.766847065, [6144/6125, 245/243, 81/80], 56*c+38*b+24*a+67*d]

[1.812907661, [6144/6125, 245/243, 64/63], 35*b+22*a+51*c+62*d]

[2.150970319, [6144/6125, 245/243, 49/48], 56*c+38*b+24*a+67*d]

[1.082128318, [6144/6125, 245/243, 36/35], 3*b+2*a+5*c+5*d]

[1.100014434, [6144/6125, 245/243, 28/27], 3*b+2*a+5*c+5*d]

[2.835788482, [6144/6125, 245/243, 25/24], 46*c+57*d+32*b+20*a]

[1.244677941, [6144/6125, 81/80, 64/63], 11*b+7*a+16*c+20*d]

[3.886285184, [6144/6125, 81/80, 50/49], 107*d+88*c+60*b+38*a]

[2.380563249, [6144/6125, 81/80, 49/48], 56*c+38*b+24*a+67*d]

[1.357757518, [6144/6125, 81/80, 36/35], 11*b+7*a+16*c+20*d]

[2.415933756, [6144/6125, 81/80, 28/27], 47*d+40*c+27*b+17*a]

[2.360946697, [6144/6125, 64/63, 50/49], 35*b+22*a+51*c+62*d]

[1.807551657, [6144/6125, 64/63, 49/48], 15*a+24*b+42*d+35*c]

[1.389738102, [6144/6125, 64/63, 36/35], 11*b+7*a+16*c+20*d]

[2.289191294, [6144/6125, 64/63, 28/27], 15*a+24*b+42*d+35*c]

[1.518424276, [6144/6125, 64/63, 25/24], 11*b+7*a+16*c+20*d]

[1.301497343, [6144/6125, 50/49, 49/48], 17*d+14*c+10*b+6*a]

[2.278536109, [6144/6125, 50/49, 36/35], 25*b+16*a+37*c+45*d]

[5.315847988, [6144/6125, 50/49, 28/27], 45*b+28*a+65*c+79*d]

[1.469220000, [6144/6125, 50/49, 25/24], 17*d+14*c+10*b+6*a]

[1.595180079, [6144/6125, 49/48, 36/35], 25*d+21*c+9*a+14*b]

[2.462710473, [6144/6125, 49/48, 28/27], 15*a+24*b+42*d+35*c]

[1.472896878, [6144/6125, 49/48, 25/24], 17*d+14*c+10*b+6*a]

[1.144847592, [6144/6125, 36/35, 28/27], 3*b+2*a+5*c+5*d]

[1.656373835, [6144/6125, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[2.839788039, [6144/6125, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[1.660204014, [3136/3125, 1029/1024, 126/125], 49*b+31*a+87*d+72*c]

[4.231649866, [3136/3125, 1029/1024, 245/243], 87*a+138*b+202*c+244*d]

[1.828192297, [3136/3125, 1029/1024, 64/63], 70*d+40*b+25*a+58*c]

[1.187113513, [3136/3125, 1029/1024, 50/49], 14*c+17*d+9*b+6*a]

[2.064939480, [3136/3125, 1029/1024, 49/48], 70*d+40*b+25*a+58*c]

[3.867306154, [3136/3125, 1029/1024, 36/35], 58*b+37*a+86*c+104*d]

[2.547030414, [3136/3125, 1029/1024, 25/24], 53*d+44*c+31*b+19*a]

[1.804062272, [3136/3125, 1728/1715, 126/125], 49*b+31*a+87*d+72*c]

[2.569847702, [3136/3125, 1728/1715, 245/243], 78*b+49*a+138*d+114*c]

[3.720561269, [3136/3125, 1728/1715, 64/63], 78*b+49*a+138*d+114*c]

[1.755627065, [3136/3125, 1728/1715, 50/49], 29*b+18*a+42*c+51*d]

[1.509685048, [3136/3125, 1728/1715, 49/48], 20*b+13*a+30*c+36*d]

[1.665410631, [3136/3125, 1728/1715, 36/35], 20*b+13*a+30*c+36*d]

[1.296200144, [3136/3125, 1728/1715, 25/24], 9*b+5*a+12*c+15*d]

[1.453386524, [3136/3125, 126/125, 245/243], 19*a+30*b+44*c+53*d]

[1.386474440, [3136/3125, 126/125, 64/63], 34*d+28*c+19*b+12*a]

[1.462600997, [3136/3125, 126/125, 50/49], 34*d+28*c+19*b+12*a]

[1.840291986, [3136/3125, 126/125, 49/48], 19*a+30*b+44*c+53*d]

[1.609356470, [3136/3125, 126/125, 36/35], 34*d+28*c+19*b+12*a]

[1.397866900, [3136/3125, 126/125, 28/27], 7*a+11*b+16*c+19*d]

[1.442139658, [3136/3125, 126/125, 25/24], 7*a+11*b+16*c+19*d]

[3.840060916, [3136/3125, 245/243, 64/63], 78*b+49*a+138*d+114*c]

[2.604911795, [3136/3125, 245/243, 50/49], 70*c+85*d+48*b+30*a]

[1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d]

[1.696997541, [3136/3125, 245/243, 28/27], 18*b+11*a+26*c+32*d]

[1.522358304, [3136/3125, 245/243, 25/24], 12*b+8*a+18*c+21*d]

[1.605730195, [3136/3125, 64/63, 50/49], 34*d+28*c+19*b+12*a]

[2.710238360, [3136/3125, 64/63, 49/48], 70*d+40*b+25*a+58*c]

[4.017844284, [3136/3125, 64/63, 28/27], 70*d+40*b+25*a+58*c]

[2.183860583, [3136/3125, 64/63, 25/24], 36*d+13*a+21*b+30*c]

[1.304754477, [3136/3125, 50/49, 49/48], 17*d+14*c+10*b+6*a]

[1.863858578, [3136/3125, 50/49, 36/35], 34*d+28*c+19*b+12*a]

[2.949144340, [3136/3125, 50/49, 28/27], 29*b+18*a+42*c+51*d]

[1.472896878, [3136/3125, 50/49, 25/24], 17*d+14*c+10*b+6*a]

[1.973782564, [3136/3125, 49/48, 36/35], 20*b+13*a+30*c+36*d]

[4.538146945, [3136/3125, 49/48, 28/27], 70*d+40*b+25*a+58*c]

[1.476582958, [3136/3125, 49/48, 25/24], 17*d+14*c+10*b+6*a]

[2.113987015, [3136/3125, 36/35, 28/27], 18*b+11*a+26*c+32*d]

[1.075200000, [3136/3125, 36/35, 25/24], 2*c+2*d+b+a]

[1.759314432, [3136/3125, 28/27, 25/24], 7*a+11*b+16*c+19*d]

[1.881579115, [1029/1024, 1728/1715, 126/125], 49*b+31*a+87*d+72*c]

[1.108598313, [1029/1024, 1728/1715, 245/243], 8*b+5*a+12*c+14*d]

[1.151256995, [1029/1024, 1728/1715, 64/63], 8*b+5*a+12*c+14*d]

[2.335569854, [1029/1024, 1728/1715, 50/49], 60*c+73*d+41*b+26*a]

[1.179639680, [1029/1024, 1728/1715, 49/48], 8*b+5*a+12*c+14*d]

[1.225032023, [1029/1024, 1728/1715, 36/35], 8*b+5*a+12*c+14*d]

[3.059142980, [1029/1024, 1728/1715, 25/24], 48*c+33*b+21*a+59*d]

[2.631752268, [1029/1024, 126/125, 245/243], 73*b+46*a+129*d+107*c]

[2.188267678, [1029/1024, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[1.535432120, [1029/1024, 126/125, 64/63], 15*a+24*b+42*d+35*c]

[1.696672390, [1029/1024, 126/125, 50/49], 25*b+16*a+37*c+45*d]

[1.651816855, [1029/1024, 126/125, 49/48], 15*a+24*b+42*d+35*c]

[1.927376265, [1029/1024, 126/125, 36/35], 25*b+16*a+37*c+45*d]

[1.055126953, [1029/1024, 126/125, 25/24], b+a+3*d+2*c]

[1.135929305, [1029/1024, 245/243, 81/80], 8*b+5*a+12*c+14*d]

[1.154976817, [1029/1024, 245/243, 64/63], 8*b+5*a+12*c+14*d]

[3.312603948, [1029/1024, 245/243, 50/49], 57*b+36*a+83*c+101*d]

[1.183451209, [1029/1024, 245/243, 49/48], 8*b+5*a+12*c+14*d]

[5.315252144, [1029/1024, 245/243, 25/24], 31*a+87*d+71*c+49*b]

[1.179639680, [1029/1024, 81/80, 64/63], 8*b+5*a+12*c+14*d]

[2.650902363, [1029/1024, 81/80, 50/49], 60*c+73*d+41*b+26*a]

[1.208722102, [1029/1024, 81/80, 49/48], 8*b+5*a+12*c+14*d]

[1.255233531, [1029/1024, 81/80, 36/35], 8*b+5*a+12*c+14*d]

[3.388625357, [1029/1024, 81/80, 25/24], 48*c+33*b+21*a+59*d]

[1.504137953, [1029/1024, 64/63, 50/49], 16*b+10*a+23*c+28*d]

[1.276281563, [1029/1024, 64/63, 36/35], 8*b+5*a+12*c+14*d]

[1.848570832, [1029/1024, 64/63, 25/24], 16*b+10*a+23*c+28*d]

[1.579217147, [1029/1024, 50/49, 49/48], 16*b+10*a+23*c+28*d]

[2.344125081, [1029/1024, 50/49, 36/35], 25*b+16*a+37*c+45*d]

[1.932774085, [1029/1024, 50/49, 25/24], 16*b+10*a+23*c+28*d]

[1.307746559, [1029/1024, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[1.940842427, [1029/1024, 49/48, 25/24], 16*b+10*a+23*c+28*d]

[2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d]

[1.897137590, [1728/1715, 126/125, 245/243], 43*b+27*a+63*c+76*d]

[2.377883156, [1728/1715, 126/125, 81/80], 49*b+31*a+87*d+72*c]

[2.326207228, [1728/1715, 126/125, 64/63], 43*b+27*a+63*c+76*d]

[1.153602745, [1728/1715, 126/125, 50/49], 9*c+6*b+4*a+11*d]

[1.155526618, [1728/1715, 126/125, 49/48], 9*c+6*b+4*a+11*d]

[1.190963383, [1728/1715, 126/125, 36/35], 9*c+6*b+4*a+11*d]

[3.298310219, [1728/1715, 126/125, 28/27], 37*b+23*a+65*d+54*c]

[1.151256995, [1728/1715, 245/243, 81/80], 8*b+5*a+12*c+14*d]

[2.205432014, [1728/1715, 245/243, 50/49], 35*b+22*a+51*c+62*d]

[1.199420137, [1728/1715, 245/243, 49/48], 8*b+5*a+12*c+14*d]

[1.245573629, [1728/1715, 245/243, 36/35], 8*b+5*a+12*c+14*d]

[1.297682532, [1728/1715, 245/243, 28/27], 8*b+5*a+12*c+14*d]

[2.616129604, [1728/1715, 245/243, 25/24], 27*b+17*a+39*c+48*d]

[1.195557177, [1728/1715, 81/80, 64/63], 8*b+5*a+12*c+14*d]

[2.842253559, [1728/1715, 81/80, 50/49], 60*c+73*d+41*b+26*a]

[1.225032023, [1728/1715, 81/80, 49/48], 8*b+5*a+12*c+14*d]

[1.272171056, [1728/1715, 81/80, 36/35], 8*b+5*a+12*c+14*d]

[3.584855584, [1728/1715, 81/80, 25/24], 48*c+33*b+21*a+59*d]

[2.604025167, [1728/1715, 64/63, 50/49], 35*b+22*a+51*c+62*d]

[1.245573629, [1728/1715, 64/63, 49/48], 8*b+5*a+12*c+14*d]

[1.293503099, [1728/1715, 64/63, 36/35], 8*b+5*a+12*c+14*d]

[1.347617144, [1728/1715, 64/63, 28/27], 8*b+5*a+12*c+14*d]

[2.974491206, [1728/1715, 64/63, 25/24], 27*b+17*a+39*c+48*d]

[1.213482515, [1728/1715, 50/49, 49/48], 9*c+6*b+4*a+11*d]

[1.250696625, [1728/1715, 50/49, 36/35], 9*c+6*b+4*a+11*d]

[3.171469175, [1728/1715, 50/49, 28/27], 29*b+18*a+42*c+51*d]

[1.315616223, [1728/1715, 50/49, 25/24], 9*c+6*b+4*a+11*d]

[1.380840823, [1728/1715, 49/48, 28/27], 8*b+5*a+12*c+14*d]

[1.317810287, [1728/1715, 49/48, 25/24], 9*c+6*b+4*a+11*d]

[1.433975353, [1728/1715, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.358223838, [1728/1715, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[1.721440913, [126/125, 245/243, 81/80], 19*a+30*b+44*c+53*d]

[2.367084273, [126/125, 245/243, 64/63], 43*b+27*a+63*c+76*d]

[1.337685863, [126/125, 245/243, 50/49], 13*b+8*a+19*c+23*d]

[2.011539314, [126/125, 245/243, 49/48], 19*a+30*b+44*c+53*d]

[1.425733632, [126/125, 245/243, 36/35], 13*b+8*a+19*c+23*d]

[1.522358304, [126/125, 245/243, 28/27], 13*b+8*a+19*c+23*d]

[1.871717101, [126/125, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[1.542907401, [126/125, 81/80, 64/63], 34*d+28*c+19*b+12*a]

[1.627623155, [126/125, 81/80, 50/49], 34*d+28*c+19*b+12*a]

[2.179705030, [126/125, 81/80, 49/48], 19*a+30*b+44*c+53*d]

[1.790936736, [126/125, 81/80, 36/35], 34*d+28*c+19*b+12*a]

[1.487814607, [126/125, 81/80, 28/27], 7*a+11*b+16*c+19*d]

[1.534936158, [126/125, 81/80, 25/24], 7*a+11*b+16*c+19*d]

[1.693895333, [126/125, 64/63, 50/49], 34*d+28*c+19*b+12*a]

[1.944562872, [126/125, 64/63, 49/48], 15*a+24*b+42*d+35*c]

[1.863858578, [126/125, 64/63, 36/35], 34*d+28*c+19*b+12*a]

[2.462710473, [126/125, 64/63, 28/27], 15*a+24*b+42*d+35*c]

[1.213629630, [126/125, 64/63, 25/24], 3*a+5*b+8*d+7*c]

[1.215506250, [126/125, 50/49, 49/48], 9*c+6*b+4*a+11*d]

[1.675829337, [126/125, 50/49, 28/27], 13*b+8*a+19*c+23*d]

[1.317810287, [126/125, 50/49, 25/24], 9*c+6*b+4*a+11*d]

[1.254871699, [126/125, 49/48, 36/35], 9*c+6*b+4*a+11*d]

[2.649382292, [126/125, 49/48, 28/27], 15*a+24*b+42*d+35*c]

[1.320008011, [126/125, 49/48, 25/24], 9*c+6*b+4*a+11*d]

[1.786134033, [126/125, 36/35, 28/27], 13*b+8*a+19*c+23*d]

[1.360488960, [126/125, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[1.815034831, [126/125, 28/27, 25/24], 7*a+11*b+16*c+19*d]

[1.199420137, [245/243, 81/80, 64/63], 8*b+5*a+12*c+14*d]

[1.770935474, [245/243, 81/80, 50/49], 22*b+14*a+32*c+39*d]

[1.276281563, [245/243, 81/80, 36/35], 8*b+5*a+12*c+14*d]

[1.329675140, [245/243, 81/80, 28/27], 8*b+5*a+12*c+14*d]

[2.363581242, [245/243, 81/80, 25/24], 22*b+14*a+32*c+39*d]

[2.641250007, [245/243, 64/63, 50/49], 35*b+22*a+51*c+62*d]

[1.249598197, [245/243, 64/63, 49/48], 8*b+5*a+12*c+14*d]

[1.297682532, [245/243, 64/63, 36/35], 8*b+5*a+12*c+14*d]

[1.351971425, [245/243, 64/63, 28/27], 8*b+5*a+12*c+14*d]

[3.007295043, [245/243, 64/63, 25/24], 27*b+17*a+39*c+48*d]

[1.986278255, [245/243, 50/49, 49/48], 22*b+14*a+32*c+39*d]

[1.572336629, [245/243, 50/49, 36/35], 13*b+8*a+19*c+23*d]

[1.678896865, [245/243, 50/49, 28/27], 13*b+8*a+19*c+23*d]

[2.635572871, [245/243, 50/49, 25/24], 22*b+14*a+32*c+39*d]

[1.329675140, [245/243, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[1.385302452, [245/243, 49/48, 28/27], 8*b+5*a+12*c+14*d]

[2.650988754, [245/243, 49/48, 25/24], 22*b+14*a+32*c+39*d]

[1.260576198, [245/243, 36/35, 25/24], 7*c+9*d+5*b+3*a]

[1.291958414, [245/243, 28/27, 25/24], 7*c+9*d+5*b+3*a]

[1.786901316, [81/80, 64/63, 50/49], 34*d+28*c+19*b+12*a]

[1.276281563, [81/80, 64/63, 49/48], 8*b+5*a+12*c+14*d]

[1.380840823, [81/80, 64/63, 28/27], 8*b+5*a+12*c+14*d]

[1.620849023, [81/80, 64/63, 25/24], 11*b+7*a+16*c+20*d]

[2.107333077, [81/80, 50/49, 49/48], 22*b+14*a+32*c+39*d]

[2.074153775, [81/80, 50/49, 36/35], 34*d+28*c+19*b+12*a]

[1.147959184, [81/80, 50/49, 28/27], 3*b+2*a+4*c+5*d]

[2.796199310, [81/80, 50/49, 25/24], 22*b+14*a+32*c+39*d]

[1.358068434, [81/80, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[1.414883587, [81/80, 49/48, 28/27], 8*b+5*a+12*c+14*d]

[2.812554722, [81/80, 49/48, 25/24], 22*b+14*a+32*c+39*d]

[1.469328077, [81/80, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.768103919, [81/80, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[1.872519988, [81/80, 28/27, 25/24], 7*a+11*b+16*c+19*d]

[1.760686016, [64/63, 50/49, 49/48], 16*b+10*a+23*c+28*d]

[2.158607408, [64/63, 50/49, 36/35], 34*d+28*c+19*b+12*a]

[2.060991308, [64/63, 50/49, 28/27], 16*b+10*a+23*c+28*d]

[2.154870412, [64/63, 50/49, 25/24], 16*b+10*a+23*c+28*d]

[1.380840823, [64/63, 49/48, 36/35], 8*b+5*a+12*c+14*d]

[2.163865892, [64/63, 49/48, 25/24], 16*b+10*a+23*c+28*d]

[1.493966092, [64/63, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.809749792, [64/63, 36/35, 25/24], 11*b+7*a+16*c+20*d]

[2.532938159, [64/63, 28/27, 25/24], 16*b+10*a+23*c+28*d]

[1.317810287, [50/49, 49/48, 36/35], 9*c+6*b+4*a+11*d]

[2.163865892, [50/49, 49/48, 28/27], 16*b+10*a+23*c+28*d]

[1.969795692, [50/49, 36/35, 28/27], 13*b+8*a+19*c+23*d]

[1.428724824, [50/49, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[2.648314659, [50/49, 28/27, 25/24], 16*b+10*a+23*c+28*d]

[1.530797807, [49/48, 36/35, 28/27], 8*b+5*a+12*c+14*d]

[1.431107521, [49/48, 36/35, 25/24], 9*c+6*b+4*a+11*d]

[2.659370017, [49/48, 28/27, 25/24], 16*b+10*a+23*c+28*d]

[1.371742112, [36/35, 28/27, 25/24], 7*c+9*d+5*b+3*a]

As you can see if you sort through this, there are quite a few valid

sets with rather exotic numbers of steps, such as 11 or 13. Perhaps

we should learn to celebrate the fact?

--- In tuning-math@y..., genewardsmith@j... wrote:

> I took the same 15 7-limit intervals as before, and this time took

> sets of three. After eliminating the ones with linear dependence or

> torsion, I obtained the following, which lists the validity

measure,

> the intervals, and the dual val:

>

[...]

>

> [1.055126953, [1029/1024, 126/125, 25/24], b+a+3*d+2*c]

[...]

> [2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d]

>

[...]

>

> [3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[...]

> As you can see if you sort through this, there are quite a few

valid

> sets with rather exotic numbers of steps, such as 11 or 13. Perhaps

> we should learn to celebrate the fact?

There are "valid" sets with 1 step (see above). So what? This sort

of "validity" doesn't do much for me.

There _is_ a decent MOS for 7-limit with 11 steps:

http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm

But clearly not all the numbers in your list are to be taken

seriously.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> There are "valid" sets with 1 step (see above). So what? This sort

> of "validity" doesn't do much for me.

The Paul Theorem applies, it just isn't very interesting. However,

it's easy enough to cull out everything which does not have a certain

minimal number of scale steps--what's the big deal?

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>

> > There are "valid" sets with 1 step (see above). So what? This

sort

> > of "validity" doesn't do much for me.

>

> The Paul Theorem applies, it just isn't very interesting. However,

> it's easy enough to cull out everything which does not have a

certain

> minimal number of scale steps--what's the big deal?

It's not about the number of scale steps -- it's about the "skewness"

of the block. Ultimately, this translates into greater errors with

respect to the JI intervals -- which is what I really care about.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> There _is_ a decent MOS for 7-limit with 11 steps:

>

> http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm

>

> But clearly not all the numbers in your list are to be taken

> seriously.

Aside from the ones with a low number of notes or a bad validity,

which ones? It might be interesting to check on these 11 and 13 note

PBs, and see if they make sense.

11 notes:

[1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d]

[1.696997541, [3136/3125, 245/243, 28/27], 18*b+11*a+26*c+32*d]

[1.714645543, [4375/4374, 126/125, 25/24], 30*d+25*c+17*b+11*a]

[1.718962570, [4375/4374, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[1.871717101, [126/125, 245/243, 25/24], 30*d+25*c+17*b+11*a]

[2.113987015, [3136/3125, 36/35, 28/27], 18*b+11*a+26*c+32*d]

[2.253538886, [1029/1024, 36/35, 25/24], 17*b+11*a+25*c+31*d]

We are getting 11*a+17*b+25*c+30*d, 11*a+17*b+25*c+31*d, and

11*a+18*b+26*c+32*d.

13 notes:

[1.509685048, [3136/3125, 1728/1715, 49/48], 20*b+13*a+30*c+36*d]

[1.613159878, [2401/2400, 245/243, 36/35], 21*b+13*a+31*c+37*d]

[1.665410631, [3136/3125, 1728/1715, 36/35], 20*b+13*a+30*c+36*d]

[1.794547474, [2401/2400, 245/243, 28/27], 21*b+13*a+31*c+37*d]

[1.842667322, [6144/6125, 1728/1715, 28/27], 37*d+30*c+21*b+13*a]

[1.952519731, [6144/6125, 1728/1715, 25/24], 37*d+30*c+21*b+13*a]

[1.973782564, [3136/3125, 49/48, 36/35], 20*b+13*a+30*c+36*d]

[2.183860583, [3136/3125, 64/63, 25/24], 36*d+13*a+21*b+30*c]

[2.326619401, [2401/2400, 36/35, 28/27], 21*b+13*a+31*c+37*d]

[2.839788039, [6144/6125, 28/27, 25/24], 37*d+30*c+21*b+13*a]

[3.009084772, [1728/1715, 28/27, 25/24], 37*d+30*c+21*b+13*a]

With 13*a+20*b+30*c+36*d, 13*a+21*b+30*c+37*d, and

13*a+21*b+31*c+37*d.

--- In tuning-math@y..., genewardsmith@j... wrote:

> Aside from the ones with a low number of notes or a bad validity,

> which ones? It might be interesting to check on these 11 and 13

note

> PBs, and see if they make sense.

> 11 notes:

>

> [1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d]

I decided to check on this one, with the best validity score of any

of these 11-note candidates. If I look at the list I uploaded, I have

two notations on it which complete the above:

[3136/3125, 245/243, 50/49, 36/35] [v, 12, v, v]

[[-8, -13, -19, -23], [12, 19, 28, 34], [-11, -18, -26, -32],

[30, 48, 70, 85]]

[3136/3125, 245/243, 36/35, 25/24] [v, v, v, v]

[[3, 5, 7, 9], [1, 1, 2, 2], [8, 12, 18, 21], [11, 18, 26, 32]]

In the first case, we might look at the corresponding 11-30

temperament, and in the second, at the 11-8 temperament; these

correspond to what look like reasonable steps rather than commas.

Doing a least-squares for the 11-8 gives us generators of 70.06 cents

for the 11 val and 53.67 cents for the 8 val; this gives us a 3 which

is 3.13 cents sharp, a five 1.26 cents sharp and a 7 0.11 cents

sharp, so we are looking at something which might end up sounding

like music. To convert from the 11-8 basis to one of an octave plus a

generator, we find the convergent to 11/8, which is 4/3. The fact

that this is a convergent means that the transformation matrix

[11 8]

[ 4 3]

which transforms from the octave-generator basis to the 11-8 basis is

unimodular; we can therefore invert it, getting

[ 3 -8]

[-4 11]

which we may use to transform from the 11-8 basis to the octave-

generator basis.

Our generator is therefore 4*70.06 + 3*53.67 = 441.24 cents; to a

good approximation this is 4*(4/68) + 3*(3/68) = 25/68 in terms of

octaves. Convergents to 25/68 are 3/8, 4/11 and 7/19, suggesting the

8, 11 and 19 MOS with this generator in the 68-et as scale

possibilities; if we add the semiconvergents to the list we get 30

and 49 as well.

In the 68 division, we have 3, 5, 7, 5/3, 7/3 and 7/5 approximated by

108, 158, 191, 50, 83 and 33 respectively; if we find n/25 mod 68 for

all of these we get -12, -10, -25, 2, -13, -15 for a complexity of

27; however note the many 5/3 we are going to get--the generator is

approximately sqrt(5/3) = 442.18 cents.

All of this tells me that this example at least is far from being

completely looney.

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., genewardsmith@j... wrote:

>

> > Aside from the ones with a low number of notes or a bad validity,

> > which ones? It might be interesting to check on these 11 and 13

> note

> > PBs, and see if they make sense.

>

> > 11 notes:

> >

> > [1.550683960, [3136/3125, 245/243, 36/35], 18*b+11*a+26*c+32*d]

>

> I decided to check on this one, with the best validity score of any

> of these 11-note candidates. If I look at the list I uploaded, I

have

> two notations on it which complete the above:

>

> [3136/3125, 245/243, 50/49, 36/35] [v, 12, v, v]

> [[-8, -13, -19, -23], [12, 19, 28, 34], [-11, -18, -26, -32],

> [30, 48, 70, 85]]

>

> [3136/3125, 245/243, 36/35, 25/24] [v, v, v, v]

> [[3, 5, 7, 9], [1, 1, 2, 2], [8, 12, 18, 21], [11, 18, 26, 32]]

>

> In the first case, we might look at the corresponding 11-30

> temperament, and in the second, at the 11-8 temperament; these

> correspond to what look like reasonable steps rather than commas.

>

> Doing a least-squares for the 11-8 gives us generators of 70.06

cents

> for the 11 val and 53.67 cents for the 8 val; this gives us a 3

which

> is 3.13 cents sharp, a five 1.26 cents sharp and a 7 0.11 cents

> sharp, so we are looking at something which might end up sounding

> like music. To convert from the 11-8 basis to one of an octave plus

a

> generator, we find the convergent to 11/8, which is 4/3. The fact

> that this is a convergent means that the transformation matrix

>

> [11 8]

> [ 4 3]

>

> which transforms from the octave-generator basis to the 11-8 basis

is

> unimodular; we can therefore invert it, getting

>

> [ 3 -8]

> [-4 11]

>

> which we may use to transform from the 11-8 basis to the octave-

> generator basis.

>

> Our generator is therefore 4*70.06 + 3*53.67 = 441.24 cents; to a

> good approximation this is 4*(4/68) + 3*(3/68) = 25/68 in terms of

> octaves. Convergents to 25/68 are 3/8, 4/11 and 7/19, suggesting

the

> 8, 11 and 19 MOS with this generator in the 68-et as scale

> possibilities; if we add the semiconvergents to the list we get 30

> and 49 as well.

>

> In the 68 division, we have 3, 5, 7, 5/3, 7/3 and 7/5 approximated

by

> 108, 158, 191, 50, 83 and 33 respectively; if we find n/25 mod 68

for

> all of these we get -12, -10, -25, 2, -13, -15 for a complexity of

> 27; however note the many 5/3 we are going to get--the generator is

> approximately sqrt(5/3) = 442.18 cents.

>

> All of this tells me that this example at least is far from being

> completely looney.

If the complexity is 27 for an 11-note scale, this tells me the PB is

very "skewed" in the lattice -- there isn't even enough room

for "half a tetrad".