Intonation error from pulling a string to the top of a fret
When we fret a string, we add tension to the string, raising the pitch. Lengthening the string at the saddle compensates for that added tension by resulting in lower pitch for any fretted note, compared to the open string. We increase the string length by changing the point of suspension on the bridge saddle.
Generally, as we move up the fretboard, the tension becomes greater, because the strings are higher off the fretboard. Compensation at the bridge takes care of this nicely because the added distance to the saddle becomes a greater proportion of the vibrating length as you move up the fretboard.
Intonation error from the stiffness of a string
Because of the stiffness of a string, the string does not bend perfectly at the edge of the saddle. The effective point of suspension, as far as vibration is concerned, is beyond the actual point of suspension. The effective point of suspension at the bridge is just a bit in front of the saddle. Also, the effective point of suspension of any fretted note will be ahead of the top of the fret. Both of these contribute to the need of saddle compensation.
We may not have thought of the stiffness effect before, because we compensate by trial and error, and make the required adjustments irrespective of the specific causes of intonation error.
We can easily see now, why the smallest wound string needs less compensation than the biggest unwound string: the stiffness of a string is mostly determined by the string’s core size, not its overall size, and the core size of the smallest wound string is small compared to its overall size, and it is also smaller than the biggest unwound string.
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