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