frequency = (1 / (2 * L)) * root (T / u)
f = k / L where k= constant
the factor of the twelfth root of 2=1.059463=b
shifting up in frequency by 1 semitone when it is fingered to touch the first fret, its length changes to
length=0.628 / 1.059463 = 0.628 / b
j th fret the length will given by 0.628 / b^j
for the j+1 fret,length=0.628 / b^(j + 1)
distance between the frets is:
(0.628 / b^j) - (0.628 / b^(j+1)) = 0.628 * (1/b^j - 1/b^(j+1)) = 0.628 * (b^(j+1) - b^j) / (b^j * (b^(j+1)) = 0.628 * b^j * (b - 1) / (b^j * (b^(j+1)) = 0.628 * (b - 1) / b^(j+1) = 0.037343 / 1.059463^(j+1) m = 3.7343 / 1.059463^(j+1) cm
a) Between fret 4 and 5, j = 4
spacing =3.7343 / 1.059463 ^ 5 = 2.7976 cm
b) Between fret 9 and 10, j = 9
spacing =3.7343 / 1.059463^10 = 2.0958 cm
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