how do all the notes of the same name relate?

Question

I suppose that frequencies were randomly assigned letter names here we have A4 = 440Hz http://www.phy.mtu.edu/~suits/notefreqs.html C3 here = 130.81Hz C4 here = 261.63Hz C5 here = 523.25Hz How is C3, C4 and C5 related? How are they all "C"?

Answer 1

You may have noticed that the ratio from one C to the next is 2:1. If you multiply 130.81 by 2, you get 261.63. If you multiply 261.63 by 2, you get 523.25. The ratio of any octave is 2:1. C5 is vibrating twice as fast as C4, which is vibrating twice as fast as C3.

Answer 2

Frequencies were randomly assigned and then chosen as a common note so that music could remain in tune.

The relationship between the notes of the same letter name are double the frequency of the previous one. This is because they are all an octave apart. So if you start with C3 130.81x2 you get 261.63, or C4. Likewise, if you were to divide C5, 523.25, by 2 you would get the same answer as before or the frequency of C4.