That High Pitched Sound - High Harmonic
Often when you tap an unweighted tuning fork you will hear a high pitched tone along with the actual frequency of the tuning fork. That high pitch tone is called a high harmonic and is normal, it is just not comfortable to many. Without going into too much detail, a harmonic is related to the frequency of the tuning fork, only many octaves higher. It can be seen as a good thing however many people find that high pitch tone to be annoying at best, unnerving at worst.
To avoid this, try tapping 1/3 from the top of the fork.
The high harmonic also occurs when you tap the fork too hard or against a surface that is too hard. It is avoidable by tapping against a softer surface such as a hockey puck or something similar in feel. If I am standing then I'll put the puck in my pocket and tap the fork against it. If I am sitting then I'll tap just below the knee on the top of the tibia (shin bone). If I am sitting cross legged I'll tap just above my knee. Experiment to find what works best for you.