Imperfect Harmonics of String Instruments

A question appeared on Physics Stack Exchange asking why the harmonics of a piano string aren’t the exact integer multiples of the fundamental as Pythagoras and Fourier had taught.  The question asks about pianos, but this applies as well to guitars, violins, and all other string instruments.

There are several effects to consider.

  • Stiffness of a non-zero thickness wire means higher harmonics are harder to make, due to having more bends, thus a teeny bit higher in frequency, more so the higher the harmonic.  There is a fourth-order differential equation to replace the usual wave equation, from which we obtain the frequency deviations for each harmonic.
  • The ends of the wire are clamped.  Naively, these would be nodes (zero crossings) of a sine wave.  But the clamp prevents the wire from being at an angle.  In effect, the standing wave has less room to exist in that you’d think when measuring the distance between clamps.  This shifts everything up, including the fundamental. Higher harmonics with the same partial amplitude want greater angles at their nodes, thus are more affected.
  • The string is pushing against air as it vibrates.  Resistance tends to slow things down. But does that mean harmonics would be higher or lower relative to their expected naive frequency based on the fundamental?
  • The wire at rest is a straight line between two points. When vibrating, it’s longer. The higher harmonics may be thought of as fast short-scale activity riding on a curved slightly longer string.  This would lower the harmonic frequencies relative to the fundamental.  But this effect is smaller, and not yet mentioned in the answers (as of this writing.)

One fine day at a music store in Fort Collins, years ago, I saw a whimsical bass guitar, much smaller than normal, more like lute sized, and with strings of soft rubbery stuff, some kind of plastic or silicone probably.  It made an interesting bass sound.  Whereas a normal bass guitar string is under great tension, these strings were maybe under just a few pounds.  It made an interesting and different sound.  I suppose that instrument would exhibit interesting nonlinearities and off-integer harmonics in easy to measure ways.  But I had more important things to spend my money on.