For understanding harmonics better we need a little music theory.let's get started!
Theory
When a string is plucked it vibrates and creates different sound waves.We hear a single sound because all of those sounds are created at the same moment and the soundwave we hear is a mathematical sum of all the created soundwaves.Each of those waves are called harmonics.See the picture (I hope it's not copyrighted if it is I'm sorry I thought I'm free for using it! )
(The picture shows vibration of a string in different harmonics of a stringed instrument which has two fixed points(nut and bridge)just like guitar. There are other forms of harmonic vibration in other instruments for example brass instruments like trumpet)
Harmonics are created when a string is divided to equal parts.Points of the string which doesn't vibrate are called nodes. Every harmonic has at least two nodes at the nut and bridge.Two noded harmonic is called 1st harmonic or fundamental(which souns just like the open string because it's the loudest harmonic). Three noded one is called 2nd harmonic(an octave higher than fundamential),four noded is3rd harmonic(an octave higher than the note on 7th fret) and it goes like this(there are lots of harmonics the picture just shows first six).
As you see in each of these harmonics the string is divided to equal parts for example 2nd harmonic 2 parts,3rd harmonic 3 parts. The harmonics other than 1st one are called overtones.The frequency of 2nd harmonic is 2 times frequency of 1st one.3rd harmonics frequency is 3 times frequency of 1st one(I think you understood the rest of story!).
Ok this much theory is enough.for more information and physical and mathematical relations you can see different references.
Playing Harmonics
For playing a harmonic you must mute all other sounds(not normal muting with side of picking hand)in a way that just the harmonic you want can be heard. You can do it on a open or fretted string.There are different techniques.
Natural harmonics:
Natural harmonics are played on open strings and also called pitch harmonics(in guitar pro software pitch harmonic is the same with artificial harmonic(+12))are played by touching a string softly(not pressing)on a fret(node) and then picking the string.For louder sound remove your fretting hand right after picking or plucking.Natural harmonics can be played on all of the frets but they sound loyder on 4th, 5th, 7th, 9th, 12th and 19th frets.For playing on other frets it's better to use an amp. Don't touch the frets on the point you regulary push.Touch them right on the metal object which divides two frets very softly(i say again don't press down). Natural harmonics are shown by N.H. and in a tab it's shown by <> with a number in it. Try this.
|-<12>------------------------------------<7>---|
|-----<12>-----------------------------<7>------|
|---------<12>----------------------<7>---------|
|-------------<12>---------------<7>------------|
|-----------------<12>--------<7>---------------|
|---------------------<12>-<7>------------------|
Artificial harmonics:
Artificial harmonics are played on fretted strings. When a string is pushed it gets shorter(the space between finger and bridge)so the location of the nodes changes.Performing an artificial harmonic means using some techniques that allows us playing harmonics(just like natural ones)on the fretted string. So we need to know where the nodes are.We know that on a open string the node for playing 1st natural harmonic is at 12th fret so if we push a string on 3rd fret the node goes to 12+3=15th fret.So we will add the number of fret which is pushed to the number of fret which we know there is a node to find it's new location. This will work the whole neck(fretboard)long where we can find number of the frets but for points between pickups and bridge(points with no frets)you have to practice to find location of the nodes.I mean it's totally experimental.Maybe you can remember places of nodes by relating them to some parts of guitar like pickups.For example on my guitar if I push 11th fret one of the nodes are right on the 2nd pickup.But maybe it's different on your guitar.
Artificial harmonics can be classified by how they sound. If it's one octave(12 frets) higher than the original note it's shown by A.H(12). If it's two octaves(24 frets)higher than the original note it's A.H(5). An octave and fifth(19 frets)higher than the original sound A.H(7), Two octaves and major third(28 frets)higher than original sound A.H(4) or A.H(9).Two octaves and fifth higher than original sound A.H(3).
How to play a artificial harmonic?
When we play natural harmonics our fretting hand is being used for touching strings.But our fretting hand is busy pushing strings on a fret so thats why we can't remove our hand from string to touch a node.There are lots of techniques to solve this problem.
Note:With techniques of playing artificial harmonic(pinch,fingerpick,double attack and tapped harmonics)which are explained below you can also play natural harmonics(using this techniques for playing natural harmonics is not common but it's useful if you can). All you have to do is not to press down any fret and perform the technique to make a natural harmonic.
Pinch harmonics:
Also called pick harmonics or squealie or squealer.To play a pinch harmonic do the following:1.Press down a fret on a string(for example 7th fret on 3rd string).2.Hold your pick(with your thumb and index or thumb and middle or with thumb,index and middle.just feel comfortable).It's better to hold it close to the end but practice with every position because you can't say to people who are listening to your music live at a concert:wait i'm trying to hold my pick close to the end because i want to play a pinch harmonic!3.Pick the string(with a pick or fingernail)right above the node which you want to make a harmonic with.4.Touch the node on the string with the side of your thumb(if picked with a pick)or with your pick(if picked with fingernail)lightly and fast.5.Remove your finger from string immediatly after touching the string.6.If you want you can bend it.It sounds great!(in some references bending a pinch harmonic is also called pitch harmonic bend).It's shown by P.H.Try this.
P.H P.H
|-----------------------------------|
|-----------------------------------|
|-7b9r7--5^^^^^--7b9r7--5^^^^^------|
|-----------------------------------|
|-----------------------------------|
|-----------------------------------|
Fingerpick harmonics:
Another kind of artificial harmonics.Do the following.1.Press down a fret on a string.2.Touch the string softly with tip of your index finger(if you want to use other fingers you are free but index is better)3.Pick the string with your 3rd(ring)finger with the same hand(you can use middle or small finger too)4.For a better sound remove your index finger from string after plucking.
Double attack harmonics:
This technique has been attributed to J. K. Hays of Stidham-Hays.For playing it do the following:1.Push down a fret on a string.2.Pluck the string up with your middle or ring finger while holding the pick firmly between your fingers.3.Let the string touch the pick on the node you want to create a harmonic.Cool isn't it?!
Tapped harmonics:
Another frequently used form of artificial harmonics.Just do as I say:1.Press down a fret on a string if you want to play an artificial harmonic if not don't push.2.Pick the string(it's not necessary)3. Tap on a fret(node)you want very quickly(don't press down).You can use your index or middle finger for tapping.Tapped harmonics are shown by T.H. There is another kind of playing ortificial harmonic which I use but I haven't seen anyone using it! You can't use it everywhere on the fretboard(speciallyA.H(12))but it's easy to do for A.H(5) and A.H(7)specially on higher frets. Push down a fret on a string.touch the fret(node)you want with a free finger of your fretting hand(ring or small)just like playing a natural harmonic.Pick the string then if you want you can remove your touching finger from the string(just like natural harmonics). I think it's nice!learn this technique. It can be useful.
Semi harmonic:
A real cool technique.To perform a semi harmonic you should play in a way that you can hear both harmonic and the original note. To play a semi harmonic for open string touch the string with your fretting hand just like what we do for playing natural harmonic but touch it very softer than that.Then pick the string.the same moment you pick it remove your fretting finger from the string.Do this until you can hear both harmonic and original note.For playing a semi harmonic on a fretted string do the same things you did for playing a artificial harmonics(any technique.pinch,fingerpick or others)But in a wrong way! Try hard. it's a kind of trial error.perform the techniques slower and softer when making a semi harmonic with pinch or double attack technique and do it faster and softer when using tapped or fingerpick techniques for creating semi harmonics.When you heard both harmonic and original note try to do that again and again.
Maybe some techniques are hard to do but if you want to learn sit down and practice. Certainly you will improve and will learn all of them. Never give up. Good luck.
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