My guitar has a odd (to me) buzzing problem. Almost all the strings from frets 12-14 buzz when tapped, but not when fretted and plucked. It is not due to lack of neck relief. And like, for example, when I capo the strings on the first fret, the "new" 12th fret, or the 13th fret of the high E string no longer buzzes, but the low E string still does.

Does anyone have any clue what's going on here?
you're guitar may dodgy frets on the 12-14 area. Is the guitar new? If so, take it back. If not, get it looked at by a tech.
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yes, one fret will be higher than the others somewhere.
my 1988 strat does it, but its only a little, so I dont mind.
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Well, technically it could be done, but only in the same way that you could change a cat into a hamburger. It's an unpleasant process, and nobody is happy with the result.
I can't take it back because I ordered online. But I that was my guess....the frets in that area are too high. Maybe some soft tapping with a rubber hammer will work.