What do I have against the 12th root of 2?

Nothing at all, as a number. Unfortunately, it is the sole generator of our musical resources in the modern world. Almost every bit of music you hear has 12 equally spaced pitches in each octave. Therefore, starting at, say, 400 Hz we build a chromatic scale by successive multiplication of the current pitch P_n by 2^1/12. Recursively, we have P_n+1=  2^1/12 P_n. Thus, after 12 such multiplications we have a pitch of 800 Hz. Nice and simple isn't it? But why do we use this method exclusively? Read about the reason here, and about some alternatives here.