How Do You Calculate Guitar String Length and Fret Placement Based on Frequency?

[Beasticly]

Homework Statement

Work out the length of a guitar string, and where the frets will be placed from nut to bridge given the starting frequency.

Frequency = 82.4Hz (E)

Homework Equations

12√2 between each fret

v = fλ

v = √(T/u)

u = m/L

L = (1/2)*λ

The Attempt at a Solution

I'm actually not too sure how to go about this question because I only have the frequency.

I guess I could approximate a tension and density by research but other than that I'm quite stumped.

Thanks in advance.

 

[mjsd]

how many frets in total do you want, it is 12?

 

[m2003]

I understand your confusion and uncertainty with this problem. The first step would be to determine the length of the guitar string, which can be calculated using the formula v = fλ, where v is the speed of sound (which can be approximated based on the material of the string), f is the frequency, and λ is the wavelength. Once the length of the string is determined, you can use the formula u = m/L to calculate the linear density of the string (u), where m is the mass of the string and L is the length of the string. With these values, you can then use the formula v = √(T/u) to calculate the tension (T) of the string.

Once you have the tension and linear density of the string, you can use the equation L = (1/2)*λ to determine the length of each fret, where λ is the wavelength of the string. The distance between each fret can be calculated using the formula 12√2, as stated in the homework statement. By determining the length of the string and the distance between each fret, you can then determine the placement of each fret from the nut to the bridge.

It is important to note that these calculations may provide an approximation, as the actual placement of frets on a guitar can also be affected by factors such as string thickness, fretboard material, and individual playing style. However, using these equations can provide a good starting point for determining the placement of frets on a guitar string.