Question

What is the appropriate unit of measurement for pitch?

Answer 1

The appropriate unit of measurement for pitch is mel.

Answer 2

There are several definitions of 'pitch', but I think you mean how 'high' or 'low' a sound sounds to our ear.  

If that's it, then you're talking about the frequency of the sound, and the unit is 'vibrations per second' or 'Hertz' (Hz) .