When two notes are an octave apart the higher note is vibrating at twice the speed of the lower note. For example: if a string is vibrating at 440 Hz (440 times a second) the note we hear is an A (the A above Middle C on the piano). If the string is stopped halfway up (e.g. because the player presses it down with a finger) it will vibrate at 880 Hz and we will hear the note one octave higher.