With reference to
sound, the
harmonic series is a series of frequencies which can be
derived back to a
fundamental frequency. For example you could have a
piano string vibrating at 256Hz or you could double that and the string's vibration would have a frequency of 512Hz. Both situations are possible on that string:
256Hz:
_
/ \
/ \
\ /
\ _ /
512Hz:
_ _
/ \ / \
/ \ / \
\ / \ /
\_/ \_/
This occurs mainly in
standing wave setups such as
brass instruments, some
wind instruments, the piano (to a small degree), etc.
Tonality of music is based upon the harmonic series because when you hear a note the hairs in your ear vibrate at that frequency whilst some hairs vibrate at a frequency in the harmonic series of that note. That is why some chords make you cringe and others sound nice - because the former is messing around your sensitive hairs in your ear and the latter is making them all vibrate together harmoniously.
For a good example of harmonics in action go to a real piano and press down and hold a low C slowly so as to not make any noise. Then play a C one octave higher quickly. Notice how you can stil hear it playing even after you have let go of that top c?
That is because the bottom C is now vibrating at the top C's frequency.