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Topic: The Best Music Scale: 12 notes/octave?
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Rock Brentwood

Posts: 122
Registered: 6/18/10
The Best Music Scale: 12 notes/octave?
Posted: Mar 27, 2014 8:06 PM
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This is a followup I did on an earlier article on this topic. The situation is this: when pitches harmonize, their frequencies are in small integer ratios. But, pitches are heard on a logarithmic scale, which means the natural configuration of notes, going up a scale, are as powers of 2.

As the Pythagoreans discovered early on, these two facts cannot be reconciled, not even to get a half-octave ratio (i.e. the square root of 2). For simple reasons, no fractional power of any non-zero integer is a rational unless it is also an integer. (Proof outline: decompose the supposed rational N^{q/r}, with q and r mutually prime, uniquely into (-1)^a 2^b 3^c 5^d 7^e ... products taken over the primes, a ranging over {0,1}, b,c,d,e,... over the integers. Take the r power and note that the integer N, itself, also has a unique prime decomposition as N = (-1)^A 2^B 3^C 5^D 7^E ... with B,C,D,E non-negative integers, so that one must have AQ = ar (mod 2), Bq = br, Cq = cr, Dq = dr, Eq = er, ... all multiples of q. Hence b, c, d, e, ... are all multiples of q and B, C, D, E ... are multiples of r.)

So, we have to compromise and ESTIMATE the powers of 2 by suitable small-ratio fractions relative to some fixed pitch (hence the notion of "tuning" on a "key").

Let N be the number of steps per octave. Then the following table gives the ratios of log(k)/log(r), where the step ratio is r = 2^{1/N}. Sorry if line breaks break up the rows.

N = number of notes per octave.
The table shows the ratios of the logarithms to that of the note-to-note ratio 2^{1/N}.

N 2^{1/N} 3 5 7 9 11 13 15 17 19 21 23 25
1 2.000 1.585 2.322 2.807 3.170 3.459 3.700 3.907 4.087 4.248 4.392 4.524 4.644
2 1.414 3.170 4.644 5.615 6.340 6.919 7.401 7.814 8.175 8.496 8.785 9.047 9.288
3 1.260 4.755 6.966 8.422 9.510 10.378 11.101 11.721 12.262 12.744 13.177 13.571 13.932
4 1.189 6.340 9.288 11.229 12.680 13.838 14.802 15.628 16.350 16.992 17.569 18.094 18.575
5 1.149 7.925 11.610 14.037 15.850 17.297 18.502 19.534 20.437 21.240 21.962 22.618 23.219
6 1.122 9.510 13.932 16.844 19.020 20.757 22.203 23.441 24.525 25.488 26.354 27.141 27.863
7 1.104 11.095 16.253 19.651 22.189 24.216 25.903 27.348 28.612 29.735 30.746 31.665 32.507
8 1.091 12.680 18.575 22.459 25.359 27.675 29.604 31.255 32.700 33.983 35.139 36.188 37.151
9 1.080 14.265 20.897 25.266 28.529 31.135 33.304 35.162 36.787 38.231 39.531 40.712 41.795
10 1.072 15.850 23.219 28.074 31.699 34.594 37.004 39.069 40.875 42.479 43.923 45.236 46.439
11 1.065 17.435 25.541 30.881 34.869 38.054 40.705 42.976 44.962 46.727 48.315 49.759 51.082
12 1.059 19.020 27.863 33.688 38.039 41.513 44.405 46.883 49.050 50.975 52.708 54.283 55.726
13 1.055 20.605 30.185 36.496 41.209 44.973 48.106 50.790 53.137 55.223 57.100 58.806 60.370
14 1.051 22.189 32.507 39.303 44.379 48.432 51.806 54.696 57.224 59.471 61.492 63.330 65.014
15 1.047 23.774 34.829 42.110 47.549 51.891 55.507 58.603 61.312 63.719 65.885 67.853 69.658
16 1.044 25.359 37.151 44.918 50.719 55.351 59.207 62.510 65.399 67.967 70.277 72.377 74.302
17 1.042 26.944 39.473 47.725 53.889 58.810 62.907 66.417 69.487 72.215 74.669 76.901 78.946
18 1.039 28.529 41.795 50.532 57.059 62.270 66.608 70.324 73.574 76.463 79.062 81.424 83.589
19 1.037 30.114 44.117 53.340 60.229 65.729 70.308 74.231 77.662 80.711 83.454 85.948 88.233
20 1.035 31.699 46.439 56.147 63.399 69.189 74.009 78.138 81.749 84.959 87.846 90.471 92.877
21 1.034 33.284 48.760 58.954 66.568 72.648 77.709 82.045 85.837 89.206 92.239 94.995 97.521

So, this accounts for small ratios involving odd integers k up to 25. (Even integers do not affect the final assessment.)

The total sum square deviations from integers for the logarithmic ratios is as follows:

With k ranging over { 3, 5, 7, 9 }:
12: 0.344, 10: 0.408, 5: 0.426, 19: 0.441, 7: 0.480, 16: 0.487, 6: 0.519,
15: 0.544, 17: 0.561, 21: 0.572, 1: 0.585, 4: 0.594, 9: 0.611, 2: 0.648,
11: 0.657, 20: 0.681, 3: 0.692, 13: 0.693, 18: 0.697, 14: 0.717, 8: 0.789

With k ranging over { 3, 5, 7, 9, 11, 13, 15, 17, 19 }:
12: 0.732, 10: 0.762, 20: 0.764, 11: 0.773, 13: 0.778, 21: 0.779, 9: 0.780,
19: 0.784, 7: 0.793, 4: 0.824, 1: 0.849, 16: 0.897, 17: 0.904, 3: 0.919,
15: 0.941, 2: 0.948, 5: 0.992, 6: 1.014, 8: 1.020, 14: 1.050, 18: 1.102

With k ranging over { 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 }:
11: 0.814, 19: 0.819, 12: 0.831, 9: 0.857, 13: 0.883, 1: 1.036, 20: 0.906,
17: 0.911, 10: 0.911, 21: 0.914, 4: 0.932, 2: 0.992, 7: 0.992, 15: 1.012,
3: 1.016, 16: 1.018, 6: 1.033, 8: 1.048, 5: 1.085, 14: 1.101, 18: 1.250

In order to best capture small integer ratios -- by this account -- the best number of steps per octave is 12.

(If I recall, my original intent in the previous article was to prove that one of the other candidates for N was the best, I think it was N = 5, 8 or 16.)




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