Read any popular book on guitar repair and maintenance and it will tell you that the intonation is adjusted by lengthening the sounding length of the string until the note at the 12th fret is the same as the harmonic sounded at the 12th fret.
This technique is OK as far as it goes but you don't get the whole picture from this.
One of the best kept secrets of the trade, known to a few skilled practitioners and rarely divulged is the process of intonation tempering.
Early stringed instruments were really quite unsophisticated. Before the invention of the adjustable truss rod necks were generally designed to be stiff and straight and the low, fast actions demanded by today's players were not thought to be an option.
The adjustable truss rod as we know it today would appear to have made its appearance sometime in the 1920s, probably at the Gibson factory (US Patent 1446758). An earlier incarnation (1910) in the form of a wholly external truss rod - possibly inspired by the mechanism used on the American railroads to support the frames of the largely wooden stock cars - appears on a harp guitar by George Laurian.
The intention of the truss rod was to balance the pull of the strings on the relatively thin wooden necks of early steel string guitars but it soon became apparent that it offered a way of controlling the exact contour of the neck, allowing a skilled maker to curve the fingerboard plane around the string's vibration arc. This introduces variable clearance for the vibrating string, critically around the first quarter of the neck. A major advantage of this was to allow the action to be much lower in the higher reaches of the fingerboard where previously it may have been harder to depress the strings.
The effect of this on intonation is quite dramatic.
I'm sure that most readers of this know by now that correcting the intonation of a fretted instrument involves selectively extending the sounding length of a string by moving the saddle away from the 12th fret by a small amount. This fixed displacement represents a tiny, but changing, percentage of the string's sounding length. To elaborate, a 2mm displacement might represent a 0.003% lengthening of the string at the first fret, but the same displacement is 0.006% of the length at the 12th fret. This increasing percentage lengthening effect is fine when we have a guitar with a dead straight neck and a string whose distance from the fret crown increases with each step but when we introduce arc relief into the equation the picture changes. In a typical guitar the fret plane begins to level out above the 12th fret (in fact this is a continuing process between the nut and the 12th) so that from the 12th fret on there is little change in the distance from string to fret and, by inference, little change in the strain applied to the string when it is depressed. The fixed nature of the intonation compensation means that the increasing percentage extension of the string is actually overcompensating at the higher frets. When this happens the intonation becomes noticeably flatter as you proceed up to the higher registers.
The trick with intonation tempering is to balance the intonation at the higher frets with the intonation at the lower ones. This can be done quite easily on a conventional guitar without any modification (provided it has been built properly in the first place). The success of the technique depends on the fact that displacements of the saddle have more effect on the higher frets than on the lower ones so that a series of increasing approximations are employed. A sensitive ear and a degree of patience are also required.
