The ratio (tempered) between any fret space and
the next fret space is defined to be the 12th root of
2.<br><br>Fn/Fn+1 = 2^(1/12) where F is the fretspace at the nth
fret.<br><br>Fn+1/Fn = 1 / (2^(1/12)) is the equivalent
form<br><br>Since each successive fretspace is smaller than the
previous, the sum of an infinite number of fret spaces will
converge to a real number, and that number is the scale
length used to determine the fret spaces in the first
place.<br><br>We can take the infinite geometric series with the
ratio as the ratio described above and Fn as the first
term.<br><br>Scale length = Sigma( Fn / 2^(i/12) ) from i = 0 to
infinity<br><br>SL = Fn / (1 - 2^(-1/12) )<br><br>Simplifying, we
have SL/Fn = 2^(1/12) / (2^(1/12) - 1)<br><br>Which is
approx. 17.817<br><br>Once you figured out the fret space
for the first fret, don't use the scale length
relationship to figure out the rest because you need to
multiply and subtract and memorize! This is redundant!
Simply use the definition of the ratio to do figure the
rest out.<br><br>Fn+1 = Fn / 2^(1/12)<br>(The next
fret space is the current fret space divided by the
ratio).<br><br>Now you can keep on dividing with your calculator to
get the next fret space instead of memorizing the
fret space, subtracting that from the remaining
length, and perform the 18 rule calculation again and
again.<br><br>If you want to extend frets beyond the nut to maybe
simulate a Baritone neck, then just use this:<br><br>Fn =
2^(1/12) * Fn+1<br>(The previous fret space is the current
fret space multiplied by the ratio).<br><br>I posted
this because I'm tired of seeing the approximated
17.817 all the time instead of the perfect value. The
users should be the one deciding how accurate they want
their fret spaces to be.<br><br>-GM4T