There have been a couple threads regarding harmonics lately, and I think a little understanding of what is going on might help a bit. So this is how I think of it, it might not be 100% correct as it is just what I have figured out myself.
(the picture was supposed to be here but I cant get it work right, sometimes its there and others its not, refer to the link posted below it you cant see it)
Imagine, or better yet get a 10 foot long rope and tie it to something solid and about shoulder height. Now swing it like a jump rope (the red line) this is it vibrating at its entire length. Now shake the rope, put a wave through it, and start swinging it again, see if you can get it to form two humps instead of one (the black line) notice in the middle the rope
doesn't move at all. Now see it you can get three humps, and notice there are two places the rope
doesnt move at all (the blue line). And so on.
These places where the rope doesnt move are called nodes and can occur where your rope can be evenly divided (1/2, 1/3, 1/4,...)(
where any of the lines crosses the skinny green line)
I think of a guitar string as vibrating very complexly, almost like all these ways at once. When you pluck an open string you hear it vibrating at its entire length but you also hear overtones (which get progressively quieter) as if the string where 1/2, 1/3, 1/4,... its length. The sound you hear is really a combination of all of these sounds.
When you play a natural harmonic you are
isolating one of these sounds, you do this by touching the string at a node just lightly enough to dampen the other vibrations. So notice at the 12th fret the black line is not moving at all, but the red and blue ones most certainly are. so when you touch the 12th fret you are forcing the sting to vibrate only in this way, and the note you hear is the same one as if the string was played normally at the 12th fret. The next harmonic occurs at the 7th and 19th frets, if you touch the 7th or 19th fret the blue line is not moving but the other lines are. You only hear the blue line and it is the same note as the 19th fret played normally(note: the distance from the 19th fret to the bridge is the same as the nut to to 7th fret it doesnt matter which node you touch).
When you play an artifical harmonic you change the length of the string by freting it, but the harmonics occur at the same intervals. For example if you press a string down at the fifth fret the new halfway point is now the 17th fret and this is where the harmonic lies.
Hope this helps someone, feel free to ask any questions or correct me it im wrong
.