In the world of electric guitars you may well hear a good deal about harmonics, but ask anyone exactly what is meant by harmonics, or how harmonics effects are produced, and more often than not you will be met with silence. Harmonics in the world of electric guitars are almost a science, and indeed when it comes to modern digital systems, and computer controlled manipulation of electric guitar signals, it is a science which requires a fair bit of understanding.
The strings of a guitar can be said to be divided into lengths, with the frets underneath showing the dividing sections. Normally when a guitar is played the player frets or plucks the guitar string in the usual place, on the baseboard or above where the hollow part of the body might be on an acoustic guitar. Harmonics, at the simplest level, involves gently touching a string at any of the points along the length of the string at these dividing points. By touching the string lightly at these points, it vibrates, but not in the normal way. Instead of vibrating in the normal way it would do it you plucked a string in the usual area, the string vibrates in a completely different way, and produces a sound which is chime like, and which resonates not at the normal note, but resonating with he note that corresponds to the part of the string touched.
When touched lightly on an open string in this way, the chime like sound produced is referred to as natural harmonics, whereas if the string is fretted then this creates a sound more commonly referred to as artificial harmonics. There are a variety of ways in which this harmonic effect can be created.
One of the many ways to achieve a harmonic effect is by picking the string the usual way, and then very quickly touching the string with the thumb, very gently. This has to be achieved very, very quickly, and often the only way to really achieve it is to use a plectrum, and to hold it between the thumb and first finger so that only a very tiny portion of it is sticking out. Typically no more than about two millimetres should be revealed, and in this way the plectrum can strike the string, and the thumb can then almost immediately touch the string. In this way, the main note is silence immediately, but by touching the string only very lightly, the harmonic sound continues. This method is termed pinch harmonics, and takes a good deal of practice to accomplish correctly and smoothly, but is a good way of understanding how harmonics sound.
However, there is more to this method than simply plucking and tapping a string - the key to harmonics is where to pluck the string. Simply doing this action over the centre of the body of the guitar will not achieve the effect. Along each string are nodes, which are referred to as harmonic nodes. Often these are not indicated, so it is necessary to learn them. This relies on a good deal of mathematics, and in fact much of the world of harmonics is mathematically oriented.
As an example, in order to produce a harmonic note which is one full octave higher than a string played normally when, say, the third fret is pressed down, it is necessary to pluck and then lightly tap the string at a point halfway between the bridge of the guitar and the third fret which is pressed down. Plucking and tapping the string at other places along the string will produce entirely different harmonics, and so depending on the note you wish to play, you will need to pluck the string at different points along its length.
However, for the beginner, there are several points which can be practiced which are all based around the bridge, and so an instructor would be able to teach you these in order to get you started. Those points further up the neck of the guitar become fairly challenging to play, as well as to remember.