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Here's the A string on a classical guitar (played without the nail):

The harmonic series, up to the 7th harmonic, contained as the sound ingredients within the sound of the string, is as follows. These are not plucked harmonics, but are extracted from the sound of the open string, electronically.

Here they are, "strummed" together (which of course you couldn't do in practice, but you can with sound engineering) which illustrates how they combine to produce the tone of the string:

And here's the plucked string again, for comparison:

You can hear the harmonics by touching the string lightly, directly over the corresponding fret position, and plucking the string fairly close (but not too close) to the bridge. If you can't make one sound, experiment with moving your plucking point!

The frets are:

1st harmonic - just on the fingerboard side of the nut.

2nd harmonic - 12th fret

3rd harmonic - 7th fret

4th harmonic - 5th fret

5th harmonic - 4th fret

6th harmonic - 3rd fret

7th harmonic - halfway between 2nd and 3rd frets.