By With an introd. by Martin Gardner
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Those are the multiples 2x, 3x, 4x, 5x, 6x, and 7x. In general, the end-harmonic above x will be nx. Note that this works only for positive n. There are no harmonics below the fundamental, at least in natural sounds. To go from one to the next, we are just adding x again. The general formula here is n × x, and it’s multiplication because multiplication is just repeated addition. Comparing Overtones and Octaves • Let’s compare overtones and octaves played at frequency x. We hear the overtones x, 2x, 3x, 4x, 5x.
C-sharp is the fifth harmonic, so it’s five times the frequency of A, but it’s two octaves too high. We divide by 2 twice, and we get 5/4 for our C-sharp. o We can take as many of the overtones of A as we want and bring them down an octave; the result is a just scale. A Pentatonic Scale • To get a just-tuned pentatonic scale, we have to do only the notes we just discussed. 37 o We have an A (the fundamental), and the second one is also an A. The third one is an E; that’s a new note. The fourth is an A again.
One of the key differences between Western music and music from other cultures is that they use different musical scales—they make different choices about notes. Let’s review what we need to construct scales. o We know that objects vibrate in different modes (overtones), and we know that the wavelengths of the most common overtones are in a ratio of 1/1 to 1/2, 1/3, 1/4. The harmonic series and the frequencies are just multiples, 1, 2, 3, 4; those are arithmetic series. o What we learn from the overtone series is that to go up an octave, we multiply frequencies by 2, and to go down an 35 octave, we divide frequencies by 2.