[Insert picture of vibrating strings with explanation]
Applying that all to, for instance, a guitar string, you can see that there is much going on with every pluck of a guitar string, and this theory allows the guitarist, through general feel and practice of how a string responds to a guitar pluck, to shape the sound.
Click on the image to see the full size. The quick Legend in the top right is lists the harmonics of interest. The Legend in the top right corner includes all the practically playable harmonics, and where to play them on the fretboard (note that the numbering corresponds to the variable “n” from above):
1 (black) – Open
2 (dark grey) – Octave
3 (blue) – Octave + Fifth
4 (light grey) – Octave + Octave
5 (green) – Octave + Octave + Major 3rd
6 (light blue) – Octave + Octave + Fifth
7 (red) – Octave + Octave + Minor 7th
8 (thin light grey) – Octave + Octave + Octave
9 (yellow) – Octave + Octave + Octave + Major 2nd
10 (light green) – Octave + Octave + Octave + Major 3rd
11 (pink) – Octave + Octave + Octave + Tritone
12 (thin light blue) – Octave + Octave + Octave + Fifth
Any harmonics higher than this generally fill up a theoretically infinitely increasing spectrum, and are not of practical interest. Anything over about 20 000 [Hz], musicians don’t need to worry about in general.