As the drawing shows, the length of a guitar string is L = 0.783 m. The frets ar
ID: 1298766 • Letter: A
Question
As the drawing shows, the length of a guitar string is L = 0.783 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 = 0.783 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: ? = 2L,
and the frequency ratio can be written as:
f'/f = 2L/(2L-d) = 2^(1/12) = 1.059
2L = 2*0.783 = 1.566 m
1.566/(1.566-d) = 1.059
d = 1.566 - (1.566/1.059) = 0.0872 m -----------------------------Ans.
b) fret 7 corresponds to ? L = ? 0.783 = 0.261 m
? = 2L' = 2*0.261 = 0.522 m
2^(1/12) = 0.522/(0.522-d)
d = 0.522 - 0.522/1.059 = 0.029 m ------------------------------Ans.
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