  We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ of
$\varepsilon\mathbb{Z}^n$-periodic Schr\"odinger operators with
$\delta'$-interactions supported on a lattice of closed compact surfaces;
within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in
the limit when $\varepsilon\to 0$ and the interactions strengths are
appropriately scaled, $\mathcal{H}^\varepsilon$ has at most $m$ gaps within
finite intervals, and moreover, the limiting behavior of the first $m$ gaps can
be completely controlled through a suitable choice of those surfaces and of the
interactions strengths.
