 Solution

For drum-frequency dimensional scaling, let $f\propto r^p{\sigma }^q{T}^s$, where $\sigma$ is surface mass density. The dimensions are $[\sigma ]=M{L}^{-2}$ and $[T]=M{T}^{-2}$, so matching powers gives $s=1/2$, $q=-1/2$ and $p=-1$: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.