Question

The two highest-pitch strings on a violin are tuned to 440 Hz (the A string) and 659 Hz (the E string). What is the ratio of the mass of the A string to that of the E string? Violin strings are all the same length and under essentially the same tension.

Short answer

1 : 2.243

Step-by-step solution

Given parameters

Two strings of same length and tension but different masses and different fundamental frequencies.

Frequencies are: 659Hz and 440Hz.

Applying the formula

$f=\frac{1}{2L}\sqrt{\frac{T_s}{\mu }}$

Taking ratio

$$\begin{gathered} \frac{f_1=\frac{1}{2L}\sqrt{\frac{T_s}{{\mu }_1}}}{f_2=\frac{1}{2L}\sqrt{\frac{T_s}{{\mu }_2}}} \\ \frac{f_1}{f_2}=\sqrt{\frac{{\mu }_2}{{\mu }_1}} \\ \frac{f_1}{f_2}=\sqrt{\frac{m_2}{m_1}} \\ {\left(\frac{440}{659}\right)}^2=\frac{m_2}{m_1} \\ 0.445=\frac{m_2}{m_1} \\ \frac{m_1}{m_2}=2.243 \\ {m}_1:m_2=1:2.243 \end{gathered}$$