The frequency of a vibrating string is given by $f = (1/2L)\sqrt{T/\mu}$. If both tension (T) and mass per unit length (μ) are doubled, the new frequency $f' = (1/2L)\sqrt{2T/2\mu} = (1/2L)\sqrt{T/\mu} = f$. The frequency remains the same. [cite: 1, 1423, 1426]