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In this paper we investigate via an asymptotic method the opening of gaps inthe spectrum of a stiff problem for the Laplace operator $-\Delta$ in$\mathbb{R}^2$ perforated by contiguous circular holes. The density and thestiffness constants are of order $\varepsilon^{-2m}$ and $\varepsilon^{-1}$ inthe holes with $m\in (0,1/2)$. We provide an explicit expression of the leadingterms of the eigenvalues and the corresponding eigenfunctions which are relatedto the Bessel functions of the first kind.

Gaps in the spectrum of two-dimensional square packing of stiff disks