Vibrations of Strings

Fundamental frequency of a stretched wire: $n = \dfrac{1}{2l}\sqrt{\dfrac{T}{\mu}}$, where the mass per unit length $\mu = \pi r^2\rho$.
$n = \dfrac{1}{2l}\sqrt{\dfrac{T}{\pi r^2\rho}}$
New values: $T_1 = 2T$, diameter doubled so $r_1 = 2r$, $\rho_1 = \dfrac{\rho}{2}$
$n_1 = \dfrac{1}{2l}\sqrt{\dfrac{2T}{\pi(2r)^2\frac{\rho}{2}}} = \dfrac{1}{2l}\sqrt{\dfrac{2T}{2\pi r^2\rho}} = \dfrac{1}{2l}\sqrt{\dfrac{T}{\pi r^2\rho}} = n$
So there is no change: the frequency remains n.

For the same tension and mass per unit length, the fundamental frequency $n = \dfrac{1}{2l}\sqrt{\dfrac{T}{\mu}}$, so $l = \dfrac{k}{n}$ with k constant.
The lengths of the pieces add up to the original length: $l = l_1 + l_2 + l_3$
$\dfrac{k}{n} = \dfrac{k}{n_1} + \dfrac{k}{n_2} + \dfrac{k}{n_3}$
$\Rightarrow \dfrac{1}{n} = \dfrac{1}{n_1} + \dfrac{1}{n_2} + \dfrac{1}{n_3}$