This became too long to include in comments.
If someone wants to construct a better answer using this as a foundation, be my guest; I figure it's better to get the knowledge out there.
F=(nv)/(2L) is for an open tube, like a flute, not for a free reed instrument like a harmonica. Where L is the length of the tube, v is the wave speed (in this case the speed of sound in air, 343) and n is the harmonic. Google "open end air column" if that doesn't make sense to you.
This paper goes into great detail about the physics of sound production within a harmonica. Once you get past the harmonica as a "pitch pipe" and look at bends (sympathetic resonance between reeds) and overblows, it gets pretty interesting.
If you ignore bending and overblows, the frequency produced by a harmonica is the question of the fundamental frequency of a free reed. The physics of free reed instruments is more complicated than flutes. The pitch is affected by the chamber length where the reed sits, but in a harmonica this is negligible (the pitch of the harmonica is roughly the same as the pitch of the reed plate on its own). The question of deriving the fundamental pitch of a free reed is more difficult than it is for a flute, because whereas a flute is only affected by the physical dimensions of the tube (a wood pipe makes the same pitch as a metal one) a free freed is entirely dependent on the properties of the material, the thickness, the way its fastened to the reedplate etc. etc.
There are various journal articles that get into some of the details of how it works, but I'm afraid there is no "short answer" like there is for say, a guitar, a flute, a clarinet, a piano etc.