As the drawing shows, the length of a guitar string is L = 0.637 m. The frets ar

Question

As the drawing shows, the length of a guitar string is L = 0.637 m. The frets are numbered for convenience. A performer can play a musical scale on a single string because the spacing between the frets is designed according to the following rule: When the string is pushed against any fret j, the fundamental frequency of the shortened string is larger by a factor of the twelfth root of two than it is when the string is pushed against the fret j - 1. Assuming that the tension in the string is the same for any note, find the spacing (a) between fret 1 and fret 0 and (b) between fret 7 and fret 6.

As the drawing shows, the length of a guitar string is L = 0.637 m. The frets are numbered for convenience. A performer can play a musical scale on a single string because the spacing between the frets is designed according to the following rule: When the string is pushed against any fret j, the fundamental frequency of the shortened string is larger by a factor of the twelfth root of two than it is when the string is pushed against the fret j - 1. Assuming that the tension in the string is the same for any note, find the spacing (a) between fret 1 and fret 0 and (b) between fret 7 and fret 6.

Explanation / Answer

The frequency ratio corresponding to the interval between each consecutive semitone is 2^(1/12).

Since the tension and density of the string is constant, then the wave speed in the
string is constant and the wavelengths for a stretched string is given by: lambda = 2L,
and the frequency ratio can be written as:

f'/f = 2L/(2L-d) = 2^(1/12) = 1.059

2L = 2*0.637 = 1.274 m

1.274/(1.274-d) = 1.059

d = 1.274 - (1.274/1.059) = 0.071 m

b) fret 7 corresponds to (1/3) L = (1/3) 0.637 = 0.2123 m

lambda = 2L' = 2*0.2123 = 0.4246 m

2^(1/12) = 0.4246/(0.4246-d)

d = 0.4246 - 0.4246/1.059 = 0.02365 m

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