On a guitar, the lowest toned string is usually strung to the E note, which prod

Question

On a guitar, the lowest toned string is usually strung to the E note, which produces sound at 82.4 Hz. The diameter of E guitar strings is typically 0.0500 inches and the scale length between the bridge and nut (the effective length of the string) is 25.5 inches. Various musical acts tune their E strings down to produce a "heavier" sound or to better fit the vocal range of the singer. As a guitarist you want to detune the E on your guitar to C# (69.3 Hz). If you were to maintain the same tension in the string as with the E string, what diameter of string would you need to purchase to produce the desired note? Assume all strings available to you are made of the same material. Unfortunately, none of the strings in your collection have such a large diameter. In fact, the largest diameter you possess is 0.05473 inches. If the tension on your existing string is denoted T_before, by what fraction will you need to detune (that is, lower the tension) of this string to achieve the desired C# note?

Explanation / Answer

Part - A

f=(1/2L)(T/)

frequency is proportional to (1/)
is proportional to the square of the radius
82.4Hz --> 69.3Hz ,( 0.05inch -->??)
69.3/82.4 = (1/)
1/= 0.70731
= 1/0.70731= r^2/0.025^2
r^2= 0.025^2/0.70731
r = 0.0297259 inch
diameter= 2r= 2*0.0297259= 0.05945 = 0.0595[inch] approx

Part - B

L = length of string
T = tension
= linear density (mass per unit length)

The frequency is given by f = (1/(2L))(T/)
_____________________________________

For 2 different strings we can write:
f = (1/(2L))(T/)
f = (1/(2L))(T/)

Squaring gives:
f² = (1/(4L²))(T/)
f² = (1/(4L²))(T/)

Since L = L, taking the ratios of the equations gives:
(f/f)² = (T/T)(/)           …………… (equation 1)
______________________________

f/f = 82.4/69.3 = 1.189
is proportional to cross-sectional area squared, hence is proportional to diameter squared:
/ = (0.05473/0.0500)² = 1.198

Plug in these values into equation 1.

(f/f)² = (T/T) (/)

1.189² = (T/T)1.198

1.4138 = (T/T) 1.198

(T/T) = 1.18

Therefore

T/T = 1/1.18 = 0.8475

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