Having finished the incomparable Robert Greenberg's Teaching Company courses The Symphony and How to Listen to and Understand Music, I have moved on to Understanding the Fundamentals of Music (a.k.a music theory.)
One of the cool things I learned was that Pythagoras discovered the ratios between different harmonic intervals. If you take a string, and another string that is twice the length of the first string, and pluck both of them, you've got an octave. In fact, that shorter string is vibrating twice as fast as the longer string. If you take a string that is 1.5 times (or 0.75 times, either way) as long as that first string, when you pluck those you've got a perfect fifth - for example, a C and a G (which sound really good together). To read more about this, check out this site.
It gets cooler. Play a note, starting with an F, then a note a perfect fifth above that, and so on until 7 notes have been played. Then put all of of those notes within the same octave: you've got the white keys on a piano. During the Renaissance, people tried stacking even more perfect fifths on, and found that the 13th pitch was the same as the first. That gave us the white AND black keys on a piano. For a better explanation of this than I can offer, see Wikipedia's article on Pythagorean Tuning.