Which is the appropriate unit of measurement for pitch?

Well, the unit of measurement would be the Hertz (after Heinrich H) Hz. But, that doesn't by itself, cover the harmonic content of pitch. Does middle C on a piano sound the same as it does on a trumpet or a violin? Answer is prob not. Reason is that apart from C (about 256Hz) there are harmonics. The harmonics are also in Hz, and spectrum analysis is one way of seeing the "signatures" of different sound sources of the same "pitch" but different sources. Tuning forks sort of work sort of similarly sort of ...

Pitch and frequency both are measured in simple hertz.