[jacquesm]

> Most physical objects do not have naturally harmonic vibration spectra.

What is your basis for saying this?

A large number of physical objects, solids as well as hollow can be approximated as systems of springs, surfaces and tensile elements, which all have some frequency response. It isn't rare at all for a physical system to have a very sharp resonant peak in its frequency response, to the point that you'll often find mechanisms to dampen that response so the structure will survive certain inputs.

[HelloNurse]

Many objects don't have audible vibration modes (infrasonic, ultrasonic, so damped that sounds are too brief and too quiet) but it doesn't mean that they don't vibrate.

[enriquto]

what you are describing is a graph. The laplacian spectrum of a graph is arbitrary.

[PEJOE]

Every rigid object has a fundamental frequency, regardless of whether you put it on a graph.

[enriquto]

Sure. But the other frequencies need not be integer multiples of the fundamental.

[jacquesm]

They don't have to, but usually those integer multiples will be present as well. Whether they are dominant or not is another matter but it is quite hard to design something in such a way that if it has a natural resonance at a certain frequency that integer multiples will not be present in the response spectrum.

A typical object will have multiple modes of resonance as well.

[enriquto]

> usually those integer multiples will be present as well

"usually", under what probability model? A random 3d or 2d shape will have zero harmonic partials with probability 1. What is hard to achieve is having even a few harmonic partials. A rectangular wooden piece is painstakingly carved to have a couple of harmonic partials, in order to become a xylophone or marimba bar.

[jacquesm]

Yes, but shapes are not usually random. Bars, cylinders, cubes, rectangles, squares and circles are everywhere. That does not mean that they will have a string like attenuation curve for those higher harmonics, but they'll be there.