Overtones and harmonics are one thing, but the 12-tone scale is only an approximation of those overtones. The frequency ratio between a note and its overtone is 1.5, while the frequency ratio between a note and its 5th note (in the chromatic 12-tone scale) is 1.498 (which makes a big difference). The overtone's overtone has a ratio of 2.25 to the root note, while in the chromatic scale it's 2.245. If you were to keep on going up a fifth based purely on overtones you'd never actually get back to the root note (even if you do go down an octave to keep yourself within the original octave range).Leaf wrote:Furthermore, the overtone series works in a way where you have your fundemental pitch (let's use A again) and in the series, the next strongest harmonic is the dominant (E, the fifth, in this case) followed by the subdominant (D, the fourth) the mediant (c#, the major third) the submediant,( F#, the major 6th) , the ... damn... I'm doing this from memory.. and I'm at work, so I don't have my Mergel text book handy... anyway the seventh is called the leading tone..
Here's a table of the frequency ratio between the chromatic-scale circle of fifths and the frequencies you get by finding subsequent overtones and bringing them back into the original octave:
Code: Select all
fifth overtones chromatic ratio
0 1.000000 1.000000 1.000000
1 1.500000 1.498307 1.001130
2 1.125000 1.122462 1.002261
3 1.687500 1.681793 1.003393
4 1.265625 1.259921 1.004527
5 1.898438 1.887749 1.005662
6 1.423828 1.414214 1.006799
7 1.067871 1.059463 1.007936
8 1.601807 1.587401 1.009075
9 1.201355 1.189207 1.010215
10 1.802032 1.781797 1.011357
11 1.351524 1.334840 1.012499
12 1.013643 1.000000 1.013643
13 1.520465 1.498307 1.014789
14 1.140349 1.122462 1.015935
15 1.710523 1.681793 1.017083
16 1.282892 1.259921 1.018232
17 1.924338 1.887749 1.019383
18 1.443254 1.414214 1.020535
19 1.082440 1.059463 1.021688
20 1.623661 1.587401 1.022842
21 1.217745 1.189207 1.023998
22 1.826618 1.781798 1.025155
23 1.369964 1.334840 1.026313
24 1.027473 1.000000 1.027473
You can actually generate any tempered scale with any arbitrary number of divisions by using the frequency ratios in the 'overtone' column above and you'll still get a good-sounding scale out of it (though it won't be readily transposable).