  \citet{Shmuel2016JMPS} discovered that all infinite band structures of waves
at normal incidence in two-phase laminates are encapsulated in a compact
universal manifold. We show that manifolds of higher dimensionality encapsulate
the band structures of all multiphase laminates. We use these manifolds to
determine the density of gaps in the spectrum, and prove it is invariant with
respect to certain properties. We further demonstrate that these manifolds are
useful for formulating optimization problems on the gaps width, for which we
develop a simple bound. Using our theory, we numerically study the dependency
of the gaps density and width on the impedance and number of phases. Finally,
we show that in certain settings, our analysis applies to non-linear multiphase
laminates, whose band diagram is tunable by finite pre-deformations. Through
simple examples, we demonstrate how the universality of our representation is
useful for characterizing this tunability in multiphase laminates.
