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Which music scale produces the best results? 12 or 15 or ...?
Posted:
Jan 11, 2014 2:43 PM


We hear waveforms on a logarithmic scale. The reason for this is that the optimal windowing of the timefrequency plane is to have the time window proportional to the number of cycles that happen (i.e. inversely proportional to the frequency). By the Heisenberg limit, this implies that the window for the frequency should be proportional to the frequency  i.e. that the frequencies should be on a logarithmic scale.
But this creates a fundamental conflict: the harmonics are on a linear scale, while  as the Pythagoreans were one of the first to note (in the process of trying to construct an ideal theory of music)  none of the Nth roots of any whole number is a whole number ratio. Roots of whole numbers are either irrational or integers. Hence, no logarithmic scale can be consistent with the requirement of producing harmonics at regular spacings of the notes.
The compromise, therefore, is to find a logarithmic scale the approximates the logarithms of the lowest whole numbers. The Western music scale uses 12 notes per octave. But is that the best?
Here is a table showing the results for the N note scale (N = 1 to 20, for odd whole numbers up to 19):
N 2^{1/N} 3 5 6 7 9 11 13 15 17 19 1 2.000 1.585 2.322 2.585 2.807 3.170 3.459 3.700 3.907 4.087 4.248 2 1.414 3.170 4.644 5.170 5.615 6.340 6.919 7.401 7.814 8.175 8.496 3 1.260 4.755 6.966 7.755 8.422 9.510 10.378 11.101 11.721 12.262 12.744 4 1.189 6.340 9.288 10.340 11.229 12.680 13.838 14.802 15.628 16.350 16.992 5 1.149 7.925 11.610 12.925 14.037 15.850 17.297 18.502 19.534 20.437 21.240 6 1.122 9.510 13.932 15.510 16.844 19.020 20.757 22.203 23.441 24.525 25.488 7 1.104 11.095 16.253 18.095 19.651 22.189 24.216 25.903 27.348 28.612 29.735 8 1.091 12.680 18.575 20.680 22.459 25.359 27.675 29.604 31.255 32.700 33.983 9 1.080 14.265 20.897 23.265 25.266 28.529 31.135 33.304 35.162 36.787 38.231 10 1.072 15.850 23.219 25.850 28.074 31.699 34.594 37.004 39.069 40.875 42.479 11 1.065 17.435 25.541 28.435 30.881 34.869 38.054 40.705 42.976 44.962 46.727 12 1.059 19.020 27.863 31.020 33.688 38.039 41.513 44.405 46.883 49.050 50.975 13 1.055 20.605 30.185 33.605 36.496 41.209 44.973 48.106 50.790 53.137 55.223 14 1.051 22.189 32.507 36.189 39.303 44.379 48.432 51.806 54.696 57.224 59.471 15 1.047 23.774 34.829 38.774 42.110 47.549 51.891 55.507 58.603 61.312 63.719 16 1.044 25.359 37.151 41.359 44.918 50.719 55.351 59.207 62.510 65.399 67.967 17 1.042 26.944 39.473 43.944 47.725 53.889 58.810 62.907 66.417 69.487 72.215 18 1.039 28.529 41.795 46.529 50.532 57.059 62.270 66.608 70.324 73.574 76.463 19 1.037 30.114 44.117 49.114 53.340 60.229 65.729 70.308 74.231 77.662 80.711 20 1.035 31.699 46.439 51.699 56.147 63.399 69.189 74.009 78.138 81.749 84.959
Just for yourself, which works the best. At least for the smallest whole numbers (those under 10), 15 works slightly better than 12.
(One way to estimate accuracy is to take the total square deviation of these values from integers. I think 15 will come out better than 12).



