In physics textbook, we can calculate the nth harmonic of a vibrating string of a fixed length. How can we do this in a real guitar?
For example, if I just pick a single open string, how can I get any arbitrary harmonic for this?
More precisely,
There is a technique called flageolet where you damp the string with a finger laid lightly onto the site at the node of a higher harmonic. You do not press the string to the fretboard but just damp the string at a position, where there is a node of the specific harmonic.
When you now pluck the string all harmonics, which do not have a node at the specified position, are damped strongly, the ones which do have nodes there are not, so if you let the finger rest shortly after plucking and then release the string, you will have only specific higher harmonics (and the pattern of nodes can clearly be seen).
This technique is easy for the first three or four harmonics, for the first three ones the nodes coincide with frets (the twelveth, the seventh and the fifth). The fourth one is very close to the fourth fret. Higher harmonics are technically quite difficult to produce cleanly, as the finger placement must be very precise and is not indicated by a fret.
Another technique to excite the higer harmonics is to use a loudspeaker, a frequency generator and let resonance do the job for you.
On a side note, flageolet is commonly used to tune a guitar, as the fifths between adjacent strings assures, that the second harmonic of the higher pitched string matches the third harmonic of the lower pitched string, and the beat frequency of high notes is faster (and therfore easier to hear precisely). Also the second and third harmonic of the deep E string are at the frequencies of the high e and the b string.
