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> Now what happens when I play a chord and get overtones out of the interaction between two strings or a choir? What's happening there?

Nothing. Sound superposition is linear. There's no interference between different frequencies. By playing several strings together you obtain the sum of the sounds played by each of them separately. No new frequencies can appear.

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Overly simplistic. You can get beats when you combined two nearby frequencies. The beats have a frequency and are audible. Similarly, sounds can be reinforced or hidden in the interaction between instruments; it might be linear, but that does not say anything about what you actually hear.

Overly confrontational. You can and do get beats when combining nearby frequencies, and it's not at odds with sound superposition being linear.

The beats have a frequency and are audible, but their frequency is not a pitch.

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