Method

ACCORDING to tradition, a gultar is fretted by sing the geometric progression 
worked out by the German physicist Frederiek Chlandi in 1809. 
He placed the twelfth fret in the middle of the string where the second harmonic is produced. 
The sound made by a free string should, then, have half the vibrations produced
by the second harmonic or pressed fret 12. 
And it would indeed be so, were it not for the fact that the string lncreases
its tension when pressed against the fingerboard. In order to solve this problem, 
Iuthiers have had, so far, to displace the support of the string on the bridge to a convenient 
measure Thus, the subsequent divisions of the fingerboard are placed according to a
geometric progression  whìch bears no connection to the new length of the string.

New geometric progression for fretting stringed musical instruments - (tempered scale)

1) Set the length of the string (diag.1 N22)

2) Place a fret so that by plucking a free string Ifirst harmonic) you can produce half 
the vibratlons of the pressed strtng (second harmonic). 
Obviously, to do so the fret must be placed approximately in the middle of the string.
I say 'approximately'  because the pressure of the strlng being pressed against the fret 
produces a slight increase in its tension (increase of frequency) 
that must be compensated by making part a slightly shorter than part 'b' (see diag. 1).

3) Between the head support and the placed fret we have to fix eleven frets at the distances 
obtained by working out
                                                          

e.g.
The length of a guitar string is 65O.OOO mm. My auditive measurements indicate
that the correct lengths are: for ' A ' 323.750mm,
and for 'b' 326.250mm, which is equal to 650.00 mm. 
These flgures have been obtained by working out 
650.000 divided by 1.05912 = 613.714. subtracting 613.714 from 650.000 = 36.285