You have to notice of course, that a music box and a harmonica produce sound from two totally different mechanisms.
However amplified though, what you hear is the signal coming directly from the reed (or tine).
In a harmonica airflow is cut by the swinging reed, and what you hear is the pulsating airflow, NOT the reed and if you look at the waveform of harmonia sound it doesn't look nearly like the sinus waveform produced from a swinging tine in a soundbox.
BTW, a soundbox doesn't have more overtones than a harmonica, but far less.
-- Gruß,Frank
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