  We study the phonon modes of interacting particles on the surface of a
truncated cone resting on a plane subject to gravity, inspired by recent
colloidal experiments. We derive the ground state configuration of the
particles under gravitational pressure in the small cone angle limit, and find
an inhomogeneous triangular lattice with spatially varying density but robust
local order. The inhomogeneity has striking effects on the normal modes such
that an important feature of the cone geometry, namely its apex angle, can be
extracted from the lattice excitations. The shape of the cone leads to energy
crossings at long wavelengths and frequency-dependent quasi-localization at
short wavelengths. We analytically derive the localization domain boundaries of
the phonons in the limit of small cone angle and check our results with
numerical results for eigenfunctions.
