Every kind of Asian music west of India does too. Counting is used to represent how high the pitch of a sound is. It becomes tied up with counting. So for example, if we're talking about the relation between 1and 2, then that's an octave. That kind of situation.
Indian music is the same, so is Greek, so is Arab, and so is Persian.
By the time pitch entered western music, it was really rather late. By then it had become typical for western music to have pitch too. But before I explain any more about what tone-color is, maybe I should say a little bit more about what pitch is. Pitch is connected to counting. Then if you have things with a different quality, saying 1 or 2 doesn't have any meaning, so there is a tendency to make everything equal.