As the drawing shows, the length of a guitar string is L = 0.709 m. The frets ar

Question

As the drawing shows, the length of a guitar string is L = 0.709 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

Hello,

This question just needs the use of basic fact that the fundamental frequency is inversely proportional to length of the string (since velocity of sound remains same; Tension and mass per unit length is same).

(a)

Now for fret 0, we are talking about the whole length i.e., L
But for fret 1, we'll be talking about unknown length L' (say).

Then using our above knowledge : f1/f0 = L/L'

=> L' = (f0/f1)L = (f0/{f0 (2)1/12})L

=> L' - L = L (1-(2)-1/12) = spacing between fret 1 and 0.

=> L' - L= 0.0397 m

(b)

Spacing between 6th and 7th can be found by applying above repeatedly.

Or we calculate 6th fret's length as L ((2)-1/12)6 = 0.5013 m

And 7th fret's length similarly = 0.4732 m

Then difference is 0.028 m.

Length of guitar is direct consequence of this calculation.

Refer http://ffden-2.phys.uaf.edu/211.web.stuff/billington/strings.html for more info.

Hope this helps.

Pls rate.

:)

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