  Accretion flows with pressure gradients permit the existence of standing
waves which may be responsible for observed quasi-periodic oscillations (QPO's)
in X-ray binaries. We present a comprehensive treatment of the linear modes of
a hydrodynamic, non-self-gravitating, polytropic slender torus, with arbitrary
specific angular momentum distribution, orbiting in an arbitrary axisymmetric
spacetime with reflection symmetry. We discuss the physical nature of the
modes, present general analytic expressions and illustrations for those which
are low order, and show that they can be excited in numerical simulations of
relativistic tori. The mode oscillation spectrum simplifies dramatically for
near Keplerian angular momentum distributions, which appear to be generic in
global simulations of the magnetorotational instability. We discuss our results
in light of observations of high frequency QPO's, and point out the existence
of a new pair of modes which can be in an approximate 3:2 ratio for arbitrary
black hole spins and angular momentum distributions, provided the torus is
radiation pressure dominated. This mode pair consists of the axisymmetric
vertical epicyclic mode and the lowest order axisymmetric breathing mode.
