Harmonics & pitch

I type out a long reply, but when I was looking for an illustration of the effect I found this that says it better :)

http://www.harmony-central.com/Guitar/harmonics.html

Just shout if anything there is not clear :)

Because harmonics are caused by sections of the string, and the sections are getting smaller. Now to get into the technical details :)

When you pluck a string, the pick deflects the string, and then it snaps back. When it does, the energy moves along the length of the string...

Picture a rope lying on a lawn. You grab one end, lift it, and snap it down. This creates a 'wave' shape that moves along the length of the rope, and a guitar string does the same thing - that's called a 'transverse wave'

But when it gets to the end of a guitar string, it hits the bridge saddle or nut, and it's got nowhere to go... it gets 'squeezed' against the nut, and then moves back in the opposite direction, but this time on the bottom half of the string. That's called a 'reflected wave'. You can see an animated illustration here.

So now we've got a wave moving from your deflected point to the bridge, and another moving from your deflected point to the nut. And they're each bouncing off the end and headed back in the opposite direction - so at some points the waves cross each other. When that happens, they interfere - they're either both at a high point (doubling the wave height), one at a high and one at a low (cancelling the wave action), or somewhere in between. After a few iterations, they settle into specific points where the waves are always cancelled, called 'nodes', and places where they are always reinforced. This means only parts of the string are actually vibrating, an effect called 'standing waves'.

These happen in multiples of the fundamental frequency... meaning if your whole string vibrates 100 times per second, each half of the string has a standing wave going 200 times per second, each third moves 300 times per second, etc.

Now for harmonics...

For each standing wave pattern, you've got these nodes - points that aren't moving. But for a different standing wave pattern, they are moving - your whole string is bouncing 100 times per second, but at the halfway point there's a node for the 200 times per second vibration. So if you touch this point (the 12th fret) you cancel the 100 times per second vibration - but you don't affect the faster one. So now none of the string is vibrating 100 times per second, but parts are still going at 200, 400, etc. times - and since 200 times (the lowest, strongest remaining vibration) is one octave higher than 100 times per second, you have an octave harmonic.

Moving to the 7th fret, you're cutting the string in thirds - that's 1/3 of the way from nut to saddle. Since that cancels the fundamental AND the first harmonic, your lowest tone is now 300 times per second - one octave and a perfect fifth above the original. But there's another place to hit this harmonic - it'll be 1/3 of the way from the bridge saddle to the nut, at about your 19th fret.

The 5th fret cuts the string into quarters, and so on. The fifth fret is 20% of the way from nut to bridge saddle, so you'll have duplicates at 40%, 60%, and 80% of the distance - you can also find this one at about the 17th fret.

As you go into smaller slices of string length, you find more places to play the same harmonic. But we learn by starting out from the nut - because that puts the nodes over the fingerboard. The frets help you find your place - it's harder to accurately locate nodes over your pickups than it is over your fretboard.

If you play a harmonic at any position lower than the 12th fret, the section of string that determines the pitch of the harmonic lies between that point and the nut, not the bridge (unlike fretted notes). As you play closer to the nut the string length is becoming shorter so produces a higher pitched harmonic.
For example, if you play a harmonic at the seventh fret, you have divided the string into sections. One is a third of the length of the string (7th fret to the nut). The other (7th fret to to the bridge) is two thirds. Only the shorter section's vibration will determine the pitch of the harmonic.

Thanks guys. Very helpful.

So, if I understand this correctly, a harmonic is played over the whole string, not just between your finger and bridge. So it follows that...

1. A 12th fret harmonic will give the lowest possible tone since it is half-way between the nut and saddle giving the longest possible length on both sides.
2. A 19th fret harmonic will give the same note as a 7th fret one because the ratio of the two sides is the same. (1/3-2/3 vs. 2/3-1/3)

Correct?

You got it :)

Thanks guys. Very helpful.

So, if I understand this correctly, a harmonic is played over the whole string, not just between your finger and bridge. So it follows that...

1. A 12th fret harmonic will give the lowest possible tone since it is half-way between the nut and saddle giving the longest possible length on both sides.
2. A 19th fret harmonic will give the same note as a 7th fret one because the ratio of the two sides is the same. (1/3-2/3 vs. 2/3-1/3)

Correct?
Yep. Twelth fret is the lowest harmonic you can get, and all the others are mirrored around the twelth fret. Remember that because it's based on string length, you don't have to be directly over a fret either - there's one I like that's just a little past the third fret, it's about 3 and 1/3 or so.