  We calculate the low-lying part of the spectrum of the $Z_3$-symmetrical
chiral Potts quantum chain in its self-dual and integrable versions, using
numerical diagonalisation of the hamiltonian for $N \leq 12$ sites and
extrapolation $N \ra \infty$. From the sequences of levels crossing we show
that the massive phases have oscillatory correlation functions. We calculate
the wave vector scaling exponent. In the high-temperature massive phase the
pattern of the low-lying levels can be explained assuming the existence of two
particles, with $Z_3$-charge $Q\!=\!1$ and $Q\!=\!2$, and their scattering
states. In the superintegrable case the $Q\!=\!2$-particle has twice the mass
of the $Q\!=\!1$-particle. Exponential convergence in $N$ is observed for the
single particle gaps, while power convergence is seen for the scattering
levels. In the high temperature limit of the self-dual model the parity
violation in the particle dispersion relation is equivalent to the presence of
a macroscopic momentum $P_m = \pm \vph/3$, where $\vph$ is the chiral angle.
