pythagoras, that magical genius of greece, figured out that if you take a lute's string, for instance, and pluck it, and it resonates at a C for instance, if you cut the string in half, it also resonates at a C but at a higher pitch. if you double the length of that string, it is a C but at a lower pitch.
the "octave" is the space betwee those pitches, and is composed of 8 (octo) notes.
do -re-mi-fah-so-lah-ti- and back to DOH
it is a physical result of how nature is constructed, and when pythagoras found this out he went apeshit thinking the entire universe was composed of music, the music of the spheres. it appears to be an underlying order of nature.
a note at 400 hertx doubled, is 800 hertz, and an octave higher. One at 200 hertz is an octave lower.