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Does there exist a periodic sound wave that has partials which are not integer multiples of the fundamental frequency?

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A pure sinusoid does not have any harmonics except the fundamental.

Otherwise, if a frequency is present that is not an integer multiple of the fundamental, then you have not properly identified the fundamental.

For example, say you have a 100 Hz and a 150 Hz signal superposed. This signal will not repeat ever 10 ms, therefore 100 Hz is not its fundamental frequency. It's actual fundamental is 50 Hz, and the signal repeats every 20 ms. However if you measure it with a spectrum analyzer, you'll find no energy present at 50 Hz.

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