  We make two observations on the motion of coupled particles in a periodic
potential. Coupled pendula, or the space-discretized sine-Gordon equation is an
example of this problem. Linearized spectrum of the synchronous motion turns
out to have a hidden asymptotic periodicity in its dependence on the energy;
this is the gist of the first observation. Our second observation is the
discovery of a special property of the purely sinusoidal potentials: the
linearization around the synchronous solution is equivalent to the classical
Lam\`e equation. As a consequence, {\it all but one instability zones of the
linearized equation collapse to a point for the one-harmonic potentials}. This
provides a new example where Lam\'e's finite zone potential arises in the
simplest possible setting.
