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.
> 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.
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.
> Yes, but shapes are not usually random. Bars, cylinders, cubes, rectangles, squares and circles are everywhere.
While there are some exceptions, this is skewed in the modern industrial world. If we're making evolutionary scale arguments about sound perception it's a much tougher sell.
Consider one of the only objects that actually matters in this context: vocal cords. Of course it's true that many inert objects don't have audible overtones or resonate at all, but nearly all animal vocalizations do. More complex auditory processing means better ability to distinguish between kin and predators. The fact that non-living objects tend to produce sounds via the same principles is just icing on the cake.
