that 5th interval you get by plucking the 7th fret harmonic is a "pure" 5th, a perfect mathematical ratio to the open string.
the problem is, pure intervals will sound perfect in certain keys, and totally off in other keys. the math just doesn't quite work out.
what we as modern musicians in the western world use is "equal temperament", where every note is "averaged out" to where they all sound equally decent. no interval (except the octave) is perfect anymore, but they all work OK in every key.
bottom line, the notes you get by plucking harmonics other than octaves will be slightly off from equal temp.; the 7th fret one is about 2 cents off, which is not huge, but it's not right. if you go across all 4 (or 5 or 6) strings trying to "tune" them like this, the error will add up until your top string is really off from your bottom one.
on top of all that, higher harmonics including octaves will tend to be sharp for physical reasons, too; the stiffness of the string makes a bigger and bigger difference the higher (and therefore smaller) the harmonic.
dig out those really high chimes from the 5th fret back to the nut and try to tune them with a tuner, and you'll see that they're all way sharp.
the 5th/7th chime trick will get you in the ballpark, but not over the plate.
here, have some juicy search terms: just intonation, equal temperament, pythagoras, stretch tuning
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