Yahoo Tuning Groups Ultimate Backup tuning-math "Good sounding" pairs of homometric frequency-sets [corrected]

There was a bug in my computer code.
Here is a re-run with bug repaired. (Still might be bugs, but that one repaired!)

Here are all homometric pairs of N-note chords, N<=7, omitting "sporadic miracles," with
1.105<MaxRatio<24.533 and FigureOfMerit<0.160 for 7-sets and FigureOfMerit<0.128 for 6-sets.
Warren D. Smith Oct. 2011:

FAM(PARAM1, PARAM2)=FoM...............FreqSet1.....................................FreqSet2
7I(-2.90,0.52)=0.154: {1.000,4.745,13.397,22.511,1.234,9.836,16.527} {1.000,7.973,13.397,0.734,3.483,9.836,16.527}
7I(-2.74,0.52)=0.137: {1.000,4.745,13.397,22.511,1.456,11.612,19.511} {1.000,7.973,13.397,0.867,4.112,11.612,19.511}
7I(-2.22,0.42)=0.148: {1.000,3.536,8.207,12.503,1.354,7.294,11.112} {1.000,5.387,8.207,0.889,3.142,7.294,11.112}
7VI(-2.03,2.08)=0.159: {1.000,1.202,9.602,1.259,11.023,12.085,12.654} {1.000,1.148,1.259,10.054,1.318,12.085,12.654}
7I(-1.70,0.42)=0.148: {1.000,3.536,8.207,12.503,2.277,12.268,18.690} {1.000,5.387,8.207,1.495,5.286,12.268,18.690}
7VI(-1.28,1.38)=0.147: {1.000,1.486,5.894,1.640,7.933,9.670,10.676} {1.000,1.346,1.640,6.508,1.811,9.670,10.676}
7VI(-1.02,1.21)=0.158: {1.000,2.079,6.952,2.497,12.037,17.357,20.843} {1.000,1.732,2.497,8.348,2.998,17.357,20.843}
6II(-0.49,0.68)=0.123: {1.000,1.212,6.959,4.263,10.216,20.206} {1.000,2.396,3.518,16.676,10.216,20.206}
7V(-0.18,0.51)=0.143: {1.000,1.158,3.714,3.099,5.155,5.972,8.290} {1.000,1.608,2.675,5.155,7.156,5.972,8.290}
6II(-0.18,0.40)=0.123: {1.000,1.241,4.212,3.522,6.488,9.631} {1.000,1.842,2.838,7.760,6.488,9.631}
6II(-0.14,0.42)=0.120: {1.000,1.320,5.280,4.604,9.207,13.943} {1.000,2.000,3.487,10.559,9.207,13.943}
7VI(-0.12,0.27)=0.152: {1.000,1.829,2.406,2.128,3.785,5.119,5.954} {1.000,1.573,2.128,2.798,2.474,5.119,5.954}
7VI(-0.12,0.22)=0.153: {1.000,1.462,1.813,1.608,2.411,2.915,3.206} {1.000,1.330,1.608,1.994,1.768,2.915,3.206}
7VI(-0.01,0.28)=0.158: {1.000,2.921,3.854,3.819,8.611,14.717,19.240} {1.000,2.234,3.819,5.038,4.993,14.717,19.240}
7V(0.00,0.17)=0.151: {1.000,1.188,1.982,1.984,2.351,2.793,3.313} {1.000,1.409,1.670,2.351,2.790,2.793,3.313}
6II(0.04,0.19)=0.123: {1.000,1.259,2.915,3.034,4.618,5.585} {1.000,1.522,2.411,4.437,4.618,5.585}
7V(0.05,0.09)=0.153: {1.000,1.215,1.779,1.872,2.054,2.497,2.883} {1.000,1.404,1.540,2.054,2.373,2.497,2.883}
7IV(0.06,0.86)=0.142: {1.000,1.201,2.826,1.357,1.442,3.607,3.834} {1.000,2.659,1.276,1.357,3.193,3.607,3.834}
7IV(0.06,1.83)=0.157: {1.000,1.201,7.516,1.357,1.442,9.593,10.196} {1.000,7.071,1.276,1.357,8.491,9.593,10.196}
6II(0.08,0.34)=0.124: {1.000,1.531,7.128,7.745,16.710,23.547} {1.000,2.158,5.058,15.379,16.710,23.547}
7IV(0.11,0.60)=0.147: {1.000,1.391,2.535,1.733,1.935,3.935,4.393} {1.000,2.270,1.553,1.733,3.158,3.935,4.393}
7IV(0.11,0.84)=0.156: {1.000,1.391,3.225,1.733,1.935,5.008,5.590} {1.000,2.889,1.553,1.733,4.019,5.008,5.590}
7IV(0.14,1.51)=0.145: {1.000,1.508,6.807,1.984,2.275,11.775,13.504} {1.000,5.936,1.730,1.984,8.953,11.775,13.504}
7V(0.14,0.19)=0.160: {1.000,1.602,3.758,4.323,5.233,8.381,11.670} {1.000,2.230,2.699,5.233,7.286,8.381,11.670}
7IV(0.15,0.31)=0.159: {1.000,1.587,2.155,2.160,2.519,3.991,4.655} {1.000,1.848,1.852,2.160,2.933,3.991,4.655}
7IV(0.15,1.76)=0.149: {1.000,1.587,9.253,2.160,2.519,17.133,19.985} {1.000,7.933,1.852,2.160,12.591,17.133,19.985}
7IV(0.17,0.67)=0.153: {1.000,1.645,3.228,2.293,2.707,6.271,7.404} {1.000,2.735,1.943,2.293,4.500,6.271,7.404}
7IV(0.17,1.60)=0.155: {1.000,1.650,8.158,2.305,2.724,15.911,18.803} {1.000,6.903,1.950,2.305,11.393,15.911,18.803}
7IV(0.17,1.50)=0.155: {1.000,1.675,7.516,2.363,2.807,14.954,17.761} {1.000,6.328,1.990,2.363,10.602,14.954,17.761}
7V(0.18,0.15)=0.153: {1.000,1.669,3.743,4.491,5.207,8.688,12.085} {1.000,2.321,2.691,5.207,7.243,8.688,12.085}
7IV(0.24,0.41)=0.159: {1.000,2.054,3.108,3.320,4.221,8.117,10.319} {1.000,2.445,2.612,3.320,5.023,8.117,10.319}
7V(0.24,-0.07)=0.156: {1.000,1.510,1.974,2.514,2.340,3.532,4.187} {1.000,1.790,1.665,2.340,2.773,3.532,4.187}
6II(0.24,0.22)=0.114: {1.000,1.592,6.278,8.012,15.911,19.846} {1.000,1.986,5.033,12.466,15.911,19.846}
7IV(0.26,1.05)=0.155: {1.000,2.175,6.190,3.651,4.730,17.444,22.601} {1.000,4.778,2.818,3.651,10.392,17.444,22.601}
7V(0.27,0.08)=0.152: {1.000,1.846,4.007,5.228,5.669,10.465,14.806} {1.000,2.612,2.832,5.669,8.020,10.465,14.806}
7IV(0.29,-0.50)=0.157: {1.000,2.394,1.445,4.284,5.732,4.627,6.190} {1.000,1.080,3.203,4.284,2.586,4.627,6.190}
7V(0.32,-0.13)=0.154: {1.000,1.657,2.096,2.886,2.522,4.179,5.028} {1.000,1.994,1.742,2.522,3.034,4.179,5.028}
7IV(0.33,0.40)=0.157: {1.000,2.667,3.975,5.129,7.114,14.702,20.389} {1.000,2.866,3.699,5.129,7.645,14.702,20.389}
7VI(0.36,-0.14)=0.157: {1.000,2.411,2.088,3.004,4.039,6.271,7.815} {1.000,1.935,3.004,2.601,3.743,6.271,7.815}
7VI(0.62,-0.40)=0.153: {1.000,2.354,1.570,2.915,2.983,4.577,5.669} {1.000,1.900,2.915,1.944,3.611,4.577,5.669}
7V(0.67,-0.34)=0.150: {1.000,2.746,3.850,7.546,5.392,14.806,20.739} {1.000,3.846,2.748,5.392,7.553,14.806,20.739}
7VI(0.69,-0.47)=0.146: {1.000,2.411,1.513,3.004,2.927,4.545,5.663} {1.000,1.935,3.004,1.885,3.743,4.545,5.663}
7V(0.69,-0.52)=0.151: {1.000,2.358,1.982,3.939,2.351,5.546,6.580} {1.000,2.798,1.670,2.351,2.790,5.546,6.580}
6I(0.71,1.23)=0.102: {1.000,1.207,3.414,6.931,2.956,4.973} {1.000,0.717,2.030,0.866,2.956,4.973}
6I(0.85,1.18)=0.121: {1.000,1.677,3.264,7.637,6.580,9.180} {1.000,1.202,2.340,2.016,6.580,9.180}
7V(0.86,-0.65)=0.140: {1.000,2.945,2.373,5.629,2.945,8.671,10.762} {1.000,3.655,1.912,2.945,3.655,8.671,10.762}
6I(0.87,0.64)=0.107: {1.000,3.013,1.895,4.527,21.693,17.202} {1.000,3.800,2.389,11.450,21.693,17.202}
7V(0.90,-0.75)=0.156: {1.000,2.829,1.779,4.358,2.054,5.812,6.713} {1.000,3.267,1.540,2.054,2.373,5.812,6.713}
6I(0.95,1.10)=0.128: {1.000,2.241,2.995,7.760,13.014,15.044} {1.000,1.939,2.591,4.345,13.014,15.044}
6I(0.96,1.10)=0.128: {1.000,2.248,3.004,7.807,13.131,15.180} {1.000,1.944,2.599,4.371,13.131,15.180}
6I(0.96,1.51)=0.122: {1.000,1.498,4.545,11.858,5.853,10.196} {1.000,0.860,2.609,1.288,5.853,10.196}
6I(1.01,1.86)=0.116: {1.000,1.166,6.449,17.690,3.732,8.776} {1.000,0.496,2.743,0.579,3.732,8.776}
6I(1.03,1.20)=0.123: {1.000,2.351,3.317,9.263,15.441,18.338} {1.000,1.980,2.793,4.655,15.441,18.338}
6I(1.03,1.84)=0.127: {1.000,1.249,6.297,17.655,4.371,9.816} {1.000,0.556,2.804,0.694,4.371,9.816}
7VI(1.09,-0.77)=0.157: {1.000,3.597,1.665,4.953,4.349,8.248,11.359} {1.000,2.612,4.953,2.293,6.821,8.248,11.359}
6I(1.15,1.92)=0.122: {1.000,1.470,6.855,21.758,6.855,14.806} {1.000,0.680,3.174,1.000,6.855,14.806}
7V(1.40,-1.20)=0.159: {1.000,5.023,2.307,9.403,2.843,14.282,17.602} {1.000,6.190,1.872,2.843,3.504,14.282,17.602}
7V(1.41,-1.25)=0.156: {1.000,4.836,1.912,7.862,2.248,10.870,12.782} {1.000,5.686,1.626,2.248,2.643,10.870,12.782}
6II(1.62,-0.92)=0.126: {1.000,1.998,1.256,6.322,5.013,1.990} {1.000,0.793,3.165,0.996,5.013,1.990}
7V(1.63,-1.47)=0.154: {1.000,5.995,1.889,9.660,2.214,13.277,15.565} {1.000,7.029,1.611,2.214,2.596,13.277,15.565}
7V(2.04,-1.88)=0.118: {1.000,8.998,1.874,14.411,2.192,19.727,23.081} {1.000,10.528,1.602,2.192,2.565,19.727,23.081}
6II(2.23,-1.32)=0.127: {1.000,2.497,1.115,10.402,6.952,1.861} {1.000,0.668,4.166,0.745,6.952,1.861}
6II(2.35,-1.32)=0.123: {1.000,2.812,1.594,16.743,12.604,3.374} {1.000,0.753,5.954,1.200,12.604,3.374}
6II(2.47,-1.38)=0.128: {1.000,2.977,1.657,19.688,14.688,3.680} {1.000,0.746,6.613,1.236,14.688,3.680}
all done.

🔗WarrenS <warren.wds@gmail.com>

And here is a smaller list, now only including examples spanning <=3
octaves. Again this is assuming no more bugs in the program.

Here are all homometric pairs of N-note chords, N<=7, omitting "sporadic miracles," with
1.414<MaxRatio<8.000 and FigureOfMerit<0.160. FigureOfMerit is defined
in my debugging paper and is allegedly <0.16 for ok-sounding
chords and <0.08 for good-sounding chords.
Warren D. Smith Oct. 2011:

FAM(PARAM1, PARAM2)=FoM...............FreqSet1.....................................FreqSet2
7VI(-0.12,0.27)=0.152: {1.000,1.829,2.406,2.128,3.785,5.119,5.954} {1.000,1.573,2.128,2.798,2.474,5.119,5.954}
7VI(-0.12,0.22)=0.153: {1.000,1.462,1.813,1.608,2.411,2.915,3.206} {1.000,1.330,1.608,1.994,1.768,2.915,3.206}
7IV(0.15,0.31)=0.159: {1.000,1.587,2.155,2.160,2.519,3.991,4.655} {1.000,1.848,1.852,2.160,2.933,3.991,4.655}
7IV(0.17,0.67)=0.153: {1.000,1.645,3.228,2.293,2.707,6.271,7.404} {1.000,2.735,1.943,2.293,4.500,6.271,7.404}
7V(0.24,-0.07)=0.156: {1.000,1.510,1.974,2.514,2.340,3.532,4.187} {1.000,1.790,1.665,2.340,2.773,3.532,4.187}
7IV(0.29,-0.50)=0.157: {1.000,2.394,1.445,4.284,5.732,4.627,6.190} {1.000,1.080,3.203,4.284,2.586,4.627,6.190}
7V(0.32,-0.13)=0.154: {1.000,1.657,2.096,2.886,2.522,4.179,5.028} {1.000,1.994,1.742,2.522,3.034,4.179,5.028}
7VI(0.36,-0.14)=0.157: {1.000,2.411,2.088,3.004,4.039,6.271,7.815} {1.000,1.935,3.004,2.601,3.743,6.271,7.815}
6I(0.58,0.44)=0.142: {1.000,2.071,1.553,2.784,7.691,6.659} {1.000,2.392,1.793,4.953,7.691,6.659}
7VI(0.62,-0.40)=0.153: {1.000,2.354,1.570,2.915,2.983,4.577,5.669} {1.000,1.900,2.915,1.944,3.611,4.577,5.669}
6I(0.68,0.85)=0.153: {1.000,1.664,2.347,4.637,5.468,6.495} {1.000,1.401,1.976,2.330,5.468,6.495}
6I(0.68,0.85)=0.153: {1.000,1.662,2.349,4.641,5.458,6.488} {1.000,1.398,1.976,2.323,5.458,6.488}
7VI(0.69,-0.47)=0.146: {1.000,2.411,1.513,3.004,2.927,4.545,5.663} {1.000,1.935,3.004,1.885,3.743,4.545,5.663}
7V(0.69,-0.52)=0.151: {1.000,2.358,1.982,3.939,2.351,5.546,6.580} {1.000,2.798,1.670,2.351,2.790,5.546,6.580}
6I(0.73,0.90)=0.157: {1.000,1.744,2.460,5.094,6.297,7.478} {1.000,1.468,2.071,2.560,6.297,7.478}
6I(0.73,0.90)=0.151: {1.000,1.744,2.465,5.109,6.303,7.493} {1.000,1.467,2.073,2.557,6.303,7.493}
6II(0.81,-0.33)=0.155: {1.000,1.608,2.140,4.797,5.534,3.971} {1.000,1.154,2.983,2.469,5.534,3.971}
6II(0.84,-0.33)=0.129: {1.000,1.655,2.335,5.387,6.398,4.591} {1.000,1.188,3.254,2.773,6.398,4.591}
6II(0.86,-0.34)=0.159: {1.000,1.670,2.347,5.523,6.547,4.646} {1.000,1.185,3.307,2.782,6.547,4.646}
7V(0.90,-0.75)=0.156: {1.000,2.829,1.779,4.358,2.054,5.812,6.713} {1.000,3.267,1.540,2.054,2.373,5.812,6.713}
6II(0.96,-0.39)=0.157: {1.000,1.763,2.487,6.508,7.729,5.207} {1.000,1.188,3.691,2.954,7.729,5.207}
6II(1.53,-0.84)=0.157: {1.000,2.000,1.502,6.931,6.007,2.604} {1.000,0.867,3.466,1.302,6.007,2.604}
6II(1.59,-0.88)=0.157: {1.000,2.040,1.467,7.199,6.104,2.537} {1.000,0.848,3.529,1.244,6.104,2.537}
all done.

🔗Paul <phjelmstad@msn.com>

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
>
> And here is a smaller list, now only including examples spanning <=3
> octaves. Again this is assuming no more bugs in the program.
>
> Here are all homometric pairs of N-note chords, N<=7, omitting "sporadic miracles," with
> 1.414<MaxRatio<8.000 and FigureOfMerit<0.160. FigureOfMerit is defined
> in my debugging paper and is allegedly <0.16 for ok-sounding
> chords and <0.08 for good-sounding chords.
> Warren D. Smith Oct. 2011:
>
> FAM(PARAM1, PARAM2)=FoM...............FreqSet1.....................................FreqSet2
> 7VI(-0.12,0.27)=0.152: {1.000,1.829,2.406,2.128,3.785,5.119,5.954} {1.000,1.573,2.128,2.798,2.474,5.119,5.954}
> 7VI(-0.12,0.22)=0.153: {1.000,1.462,1.813,1.608,2.411,2.915,3.206} {1.000,1.330,1.608,1.994,1.768,2.915,3.206}
> 7IV(0.15,0.31)=0.159: {1.000,1.587,2.155,2.160,2.519,3.991,4.655} {1.000,1.848,1.852,2.160,2.933,3.991,4.655}
> 7IV(0.17,0.67)=0.153: {1.000,1.645,3.228,2.293,2.707,6.271,7.404} {1.000,2.735,1.943,2.293,4.500,6.271,7.404}
> 7V(0.24,-0.07)=0.156: {1.000,1.510,1.974,2.514,2.340,3.532,4.187} {1.000,1.790,1.665,2.340,2.773,3.532,4.187}
> 7IV(0.29,-0.50)=0.157: {1.000,2.394,1.445,4.284,5.732,4.627,6.190} {1.000,1.080,3.203,4.284,2.586,4.627,6.190}
> 7V(0.32,-0.13)=0.154: {1.000,1.657,2.096,2.886,2.522,4.179,5.028} {1.000,1.994,1.742,2.522,3.034,4.179,5.028}
> 7VI(0.36,-0.14)=0.157: {1.000,2.411,2.088,3.004,4.039,6.271,7.815} {1.000,1.935,3.004,2.601,3.743,6.271,7.815}
> 6I(0.58,0.44)=0.142: {1.000,2.071,1.553,2.784,7.691,6.659} {1.000,2.392,1.793,4.953,7.691,6.659}
> 7VI(0.62,-0.40)=0.153: {1.000,2.354,1.570,2.915,2.983,4.577,5.669} {1.000,1.900,2.915,1.944,3.611,4.577,5.669}
> 6I(0.68,0.85)=0.153: {1.000,1.664,2.347,4.637,5.468,6.495} {1.000,1.401,1.976,2.330,5.468,6.495}
> 6I(0.68,0.85)=0.153: {1.000,1.662,2.349,4.641,5.458,6.488} {1.000,1.398,1.976,2.323,5.458,6.488}
> 7VI(0.69,-0.47)=0.146: {1.000,2.411,1.513,3.004,2.927,4.545,5.663} {1.000,1.935,3.004,1.885,3.743,4.545,5.663}
> 7V(0.69,-0.52)=0.151: {1.000,2.358,1.982,3.939,2.351,5.546,6.580} {1.000,2.798,1.670,2.351,2.790,5.546,6.580}
> 6I(0.73,0.90)=0.157: {1.000,1.744,2.460,5.094,6.297,7.478} {1.000,1.468,2.071,2.560,6.297,7.478}
> 6I(0.73,0.90)=0.151: {1.000,1.744,2.465,5.109,6.303,7.493} {1.000,1.467,2.073,2.557,6.303,7.493}
> 6II(0.81,-0.33)=0.155: {1.000,1.608,2.140,4.797,5.534,3.971} {1.000,1.154,2.983,2.469,5.534,3.971}
> 6II(0.84,-0.33)=0.129: {1.000,1.655,2.335,5.387,6.398,4.591} {1.000,1.188,3.254,2.773,6.398,4.591}
> 6II(0.86,-0.34)=0.159: {1.000,1.670,2.347,5.523,6.547,4.646} {1.000,1.185,3.307,2.782,6.547,4.646}
> 7V(0.90,-0.75)=0.156: {1.000,2.829,1.779,4.358,2.054,5.812,6.713} {1.000,3.267,1.540,2.054,2.373,5.812,6.713}
> 6II(0.96,-0.39)=0.157: {1.000,1.763,2.487,6.508,7.729,5.207} {1.000,1.188,3.691,2.954,7.729,5.207}
> 6II(1.53,-0.84)=0.157: {1.000,2.000,1.502,6.931,6.007,2.604} {1.000,0.867,3.466,1.302,6.007,2.604}
> 6II(1.59,-0.88)=0.157: {1.000,2.040,1.467,7.199,6.104,2.537} {1.000,0.848,3.529,1.244,6.104,2.537}
> all done.
>

Nice, could you explain how the sporadic miracles work, especially the polynomial
that is the generating function. Are they a smaller percentage of the "findings" ? pgh