  We investigate spectral properties of the Neumann Laplacian
$\mathscr{A}_\varepsilon$ on a periodic unbounded domain $\Omega_\varepsilon$
depending on a small parameter $\varepsilon>0$. The domain $\Omega_\varepsilon$
is obtained by removing from $\mathbb{R}^n$ $m\in\mathbb{N}$ families of
$\varepsilon$-periodically distributed small resonators. We prove that the
spectrum of $\mathscr{A}_\varepsilon$ has at least $m$ gaps. The first $m$ gaps
converge as $\varepsilon\to 0$ to some intervals whose location and lengths can
be controlled by a suitable choice the resonators; other gaps (if any) go to
infinity. An application to the theory of photonic crystals is discussed.
