What determines the pitch of a vibrating string?

The fundamental frequency is f = sqrt(T/rho) / (2L) where T is tension, rho is linear density, and L is string length. Higher tension or shorter string means higher pitch. Thicker heavier strings vibrate slower giving lower pitch. This is why bass guitar strings are thick and tuning pegs adjust tension.

What are standing waves and harmonics?

A string fixed at both ends supports standing waves at frequencies f_n = n times c divided by 2L. The fundamental n=1 is the lowest pitch. Higher harmonics give richer sound. A plucked string excites all harmonics simultaneously which is why it sounds rich compared to a pure sine tone.

Vibrating String — Wave Equation Simulator

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Vibrating String