= Solution
For <drum-frequency dimensional scaling>, let $f\propto r^p\sigma^qT^s$, where $\sigma$ is <surface mass density>. The dimensions are $[\sigma]=ML^{-2}$ and $[T]=MT^{-2}$, so matching <powers> gives $s=1/2$, $q=-1/2$ and $p=-1$:
$$
f=C\frac1r\sqrt{\frac T\sigma},\qquad
\boxed{\frac{f_{\rm new}}f=\frac{\sqrt2}{2}=\frac1{\sqrt2}.}
$$
The dimensionless <normal mode> constant $C$ is unchanged for the same circular shape and <boundary condition>. The doubled <tension> does not offset the doubled <radius>.
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