Having a harmonic at twice the frequency is not arbitrary. Lots of things that vibrate will have a fundamental frequency and harmonics at integer ratios. The

Having a harmonic at twice the frequency is not arbitrary. Lots of things that vibrate will have a fundamental frequency and harmonics at integer ratios. The name "octave" implying 8 is a bit more complicated, but if you just treat it as a name for a moment, the "octave" is the ratio 2:1. Another common ratio is 3:2, and again, if you ignore the meaning of the name, it's called the "fifth". So why 8 notes per octave? Well, it's really 12 notes per octave. Maybe think of it as the white and black keys on a piano. 8 white keys gets you back to where you were, but you skipped some black keys along the way. So why 12 notes per octave? Well, that's because (3/2)^12 (1.5 to the 12th power) is almost a power of 2. So if you step up by fifths 12 times, you very nearly land 7 octaves up. Each of the notes you stepped on along the way becomes one of the 12. Heh, there are also "fourths" at a ratio of 4:3. So music has a "fourth" plus a "fifth" equals an "octave"! It's kind of silly :-) Anyways, that's the quick version. If you go further down this road, there are "wolf fifths", various tunings with subtle (but perceptible) differences, and you can even find 19 and 31 tone scales. It kind of goes on and on.