  A covariant, scalar-tensor gravity is constructed such that the static,
spherically symmetric Rezzolla-Zhidenko metric is an exact solution to the
theory. The equations describing gravitational perturbations of this spacetime,
which represents a generic black hole possessing an arbitrary number of hairs,
can then be derived. This allows for a self-consistent study of the associated
quasinormal modes. It is shown that mode spectra are tied to not only the
non-Einstein parameters in the metric but also to those that appear at the
level of the action, and that different branches of the exact theory can, in
some cases, predict significantly different oscillation frequencies and damping
times. For choices which make the theory appear more like general relativity in
some precise sense, we find that a nontrivial Rezzolla-Zhidenko parameter space
is permissible under current constraints on fundamental ringdown modes observed
by Advanced LIGO.
