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 in and the scale length between the bridge and nut (the effective length of the string) is 25.5 in. 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 A# ( 58.3 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. diameter of string: inches Unfortunately, none of the strings in your collection have such a large diameter. In fact, the largest diameter you possess is 0.06033 in. If the tension on your existing string is denoted ?before , by what fraction will you need to detune (that is, lower the tension) of this string to achieve the desired A# note? ?after?before=

Answer 1

Assuming that the wavelength of the sound in guitar remains the same (different overtones are possible in a vibrating string and frequency changes with wavelength as well) , the frequency of vibration is proportional to

Where T is the tension on the string and is the linear mass density of the string given by

= density of string * area of cross section =

where d is the diameter of the string.

For a constant tension and same material(same density)

So,

Given that f1 = 82.4 Hz

d1 = 0.05 in

and f2 = 58.3 Hz,

d2 = 82.4*0.05/58.3 = 0.07067 in

b) For the present tension, the frequency of the largest string (d2 = 0.6033) is

f2 = 82.4*0.05/0.06033 = 68.29 Hz.

Now, the relation between tension and frequency is given by

The desired frequency is 58.3 Hz.

So,