I usually think of it as someone hundreds of years ago playing with a tighly stretched string and noticing stuff:

- Half a string's length is the same thing, just higher

- The harmonic at half a string is the same thing, just higher

- The harmonics at 3:2 and 4:3 are really loud and distinct

- If you chop up the rest of the string at the same interval as the 3:2 and 4:3, some have strong harmonics that match tones on the string, some don't

Then I figure with the Octave, 5th, 4th, and M3 as the strongest harmonics that matches other lengths (or octaves thereof), they went from there...