Psychofish
Harmonics? As in how they carry a vibration?
I'm sure it's not that simple, but I'm not sciencey enough to make any more specific guesses.
Essentially.
In a nut shell its more about oscillation than true vibration.
Harmonics is the frequency at which the barrel oscillates. The barrel oscillates (rotates in a circular pattern) during firing due to the linear and rotational stresses placed on it due to a bullet acting on it at 2500 fps and something like 40,000 CUP.
Good harmonics is where the barrel behaves in a uniform way every shot, so that any distortion caused is consistent and can therefore be corrected with proper sight and shot placement. The problem comes in with how the barrel (which has its own harmonics) interacts with the frame (or in rifles, the bed) and effects the harmonics of the barrel.
This is why you hear the term 'free floating barrel' a lot, it means that barrel is not restricted or effected by the frame by literally not letting it touch the frame.