What’s the difference in frequency between middle C and the Cs an octave above and below?

Middle C is vibrating at a rate of 261.625 hertz. The C above middle C is vibrating at 523.25.

The C below middle C is vibrating at 130.8125. You may notice that the higher C is vibrating exactly twice as fast as middle C.

The lower C is vibrating at half the speed.

This rule can be applied to any two notes separated by an octave.