  It is well established that the response of a black hole to a generic
perturbation is characterized by a spectrum of damped resonances, called
quasinormal modes; and that, in the limit of large angular momentum ($l \gg
1$), the quasinormal mode frequency spectrum is related to the properties of
unstable null orbits. In this paper we develop an expansion method to explore
the link. We obtain new closed-form approximations for the lightly-damped part
of the spectrum in the large-$l$ regime. We confirm that, at leading order in
$l$, the resonance frequency is linked to the orbital frequency, and the
resonance damping to the Lyapunov exponent, of the relevant null orbit. We go
somewhat further than previous studies to establish (i) a spin-dependent
correction to the frequency at order $1 / l$ for equatorial ($m = \pm l$)
modes, and (ii) a new result for polar modes ($m = 0$). We validate the
approach by testing the closed-form approximations against frequencies obtained
numerically with Leaver's method.
