The overtones (not so correctly called harmonics) of a single string should not be exactly x2, x3, x4 etc as this sounds very tiring, shrinked, boring to our brain.
They have to be a bit more. How more? Our brain knows and instantly decides the best subjective octave, double octave etc by comparison.
Steinway strings for example are tensioned so much that the intenger numbers 2x, 3x, 4x in frequency increase subtly. This makes a whole world of difference to our brain.
It can be the difference of a cheap toy (with no user programable all no scale) to a beautiful sounding instrument such as a Steinway or a Pleyel piano.
And if overtones were exact multiples of 2, 3, 4 etc the waveform would be periodic, each cycle exactly the same to the next, to infinity. How boring. Real good instruments are not this way.
The waveform changes all the time even on just a single vibrating string. No beats are heard because each overtone is by definition a sinewave and each overtone sinewave is far apart from each other. What we see on the oscilloscope changes and at the same time how sweet the intervals feel changes.
See also nearby euroelectron posts.
Much of this content came from a discussion with the piano tuner Albert Ketenzian.
For references see nearby euroelectron posts.
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