As you can see, the height of the string on the 24th fret is more than on the 12th fret.
When we press a string to a fret, we are acting on the string with some force. This additional force is added to the resting tension force and results in an increase in the vibration frequency of the string. And this force increases as the fret number increases, because the distance from the string to the fret at each successive fret is slightly greater than at the previous one.
In normal guitar fine tuning practice, this extra tension is compensated for once by increasing the length of the string. For electric guitars, it’s tuning with the saddles of each string. For acoustic and classical guitars, it is the displacement of the bridge with the saddle by a certain distance. For guitars with a movable bridge – it is moved all together. Usually this value for the first string is about 2 mm – for the 6th string – about 5 mm.
The thicker and therefore more rigid the string, the greater this additional force.
Consequently, the distance by which you have to move the saddle increases.
Usually for this kind of tuning, you compare the natural harmonic at the 12th fret to the frequency of the string vibrating at the 12th fret. The natural harmonic is free of additional tension. The string pressed at the 12th fret has some additional tension. The saddle is moved to such a length that the oscillation frequency of the natural harmonic and the pressed string are the same.
However, as I said earlier, string height is a value that grows from the first to the last fret.
And we compare it once on the 12th fret.
So the guitar has the exact tuning only on the 12th fret.
It is important to note that this calculation must be done for different types of strings, material and gauge.
However, even with the averaged values, this fret location scheme will be more accurate than the geometrical one that manufacturers use now.
Some tuning error can then be compensated by adjusting the saddle position.