The arithmetical ratios associated with musical intervals are numerous. The principle ones known by Pythagoras (it is said) are 2:1 for the octave, 3:2 for the perfect fifth, and 4:3 for the perfect fourth. The ratios (or their inverses) themselves come from things like the lengths of musical pipes that produce the notes, or the lengths of musical strings, or in terms of modern acoustics, frequency ratios.
Even today, these intervals are called "perfect", following from Pythagoreansim. Pythagoras saw the "perfection" in the way the musical intervals fit together, arithmetically, as Divine perfection. That's the hearsay, anyway.
3/2 X 4/3 = 2/1
We see this perfection musically, in the fact that an octave consists of putting the two intervals, the perfect fourth, and the perfect fifth, together.
Other musical interval ratios include 5/4 for a major third, 6/5 for a minor third, 5/3 for a major sixth, and 9/8 for a whole tone. There are many others, including several different whole tone and semitone sizes, and many microtonal intervals.
Our Western music is based on 12 notes to an octave, and for more on the network of their relationships see here.