> There are also inharmonics—frequencies that aren't multiples of the fundamental. Those tend to die out quickly because they don't form standing waves in the resonating body.
This is only true for one-dimensional vibrations on strings and the air inside a long tube. Two and three-dimensional bodies have overtones with arbitrary ratios to the fundamental, depending on the shape of the object. Bells need to be carefully tuned to have harmonic spectrum, but this is an artificial construction based on western music tastes (who wants octaves to not be dissonant). Other percussion instruments (e.g., the indonesian gamelan) are deliberately tuned to a non-harmonic overtone sequence adapted to the local music tastes. You can certainly have inharmonic overtones that don't die out quickly! And they can form standing waves in the resonating body, just like the fundamental.
This is only true for one-dimensional vibrations on strings and the air inside a long tube. Two and three-dimensional bodies have overtones with arbitrary ratios to the fundamental, depending on the shape of the object. Bells need to be carefully tuned to have harmonic spectrum, but this is an artificial construction based on western music tastes (who wants octaves to not be dissonant). Other percussion instruments (e.g., the indonesian gamelan) are deliberately tuned to a non-harmonic overtone sequence adapted to the local music tastes. You can certainly have inharmonic overtones that don't die out quickly! And they can form standing waves in the resonating body, just like the fundamental.