[dfan]

With an equal-tempered n-tone scale, two pitches that are x units apart have a frequency ratio of 2^(x/n).

"Nice-sounding" intervals have small integer frequency ratios.

It so happens that 3/2 = 1.5 is very close to 2^(7/12) = 1.498, 4/3 = 1.333 is very close to 2^(5/12) = 1.335, and 5/4 = 1.25 is very close to 2^(4/12) = 1.260. So 12 works very well as the number of pitches to divide an octave into.

The fact that 12 is highly divisible is handy for certain aspects of music composition. For example, if you have a diminished seventh chord, which consists of every third pitch, due to its symmetry you can think of it as being in any of four different keys. So it is easy to use it to pivot from one key to another. But none of that has to do with the reasons that it sounds good.