  We study the perturbations to General Relativistic black holes (i.e. those
without scalar hair) in Horndeski scalar-tensor gravity. First, we derive the
equations of odd and even parity perturbations of both the metric and scalar
field in the case of a Schwarzschild black hole, and show that the
gravitational waves emitted from such a system contain a mixture of
quasi-normal mode frequencies from the usual General Relativistic spectrum and
those from the new scalar field spectrum, with the new scalar spectrum
characterised by just two free parameters. We then specialise to the sub-family
of Horndeski theories in which gravitational waves propagate at the speed of
light $c$ on cosmological backgrounds; the scalar quasi-normal mode spectrum of
such theories is characterised by just a single parameter $\mu$ acting as an
effective mass of the scalar field. Analytical expressions for the quasi-normal
mode frequencies of the scalar spectrum in this sub-family of theories are
provided for both static and slowly rotating black holes. In both regimes
comparisons to quasi-normal modes calculated numerically show good agreement
with those calculated analytically in this work.
