  Complex periodic structures inherit spectral properties from the constituent
parts of their unit cells, chiefly their spectral band gaps. Exploiting this
intuitive principle, which is made precise in this work, means spectral
features of periodic systems with very large unit cells can be predicted
without numerical simulation. We study a class of difference equations with
periodic coefficients and show that they inherit spectral gaps from their
constituent elements. This result shows that if a frequency falls in a band gap
for every constituent element then it must be in a band gap for the combined
complex periodic structure. This theory and its instantaneous utility is
demonstrated in a series of vibro-acoustic and mechanical examples.
