  The coalescence of two black holes generates gravitational waves that carry
detailed information about the properties of those black holes and their binary
configuration. The final coalescence cycles are in the form of a {\it
ringdown}: a superposition of quasi-normal modes of the merged remnant black
hole. Each mode has an oscillation frequency and decay time that in general
relativity is determined by the remnant's mass and spin. Measuring the
frequency and decay time of multiple modes makes it possible to measure the
remnant's mass and spin, and to test the waves against the predictions of
gravity theories. In this {\it Letter}, we show that the relative amplitudes of
these modes encodes information about a binary's {\it geometry}. Focusing on
the large mass-ratio limit, which provides a simple-to-use tool for effectively
exploring parameter space, we demonstrate how a binary's geometry is encoded in
the relative amplitudes of these modes, and how to parameterize the modes in
this limit. Although more work is needed to assess how well this carries over
to less extreme mass ratios, our results indicate that measuring multiple
ringdown modes from coalescence may aid in measuring important source
properties, such as the misalignment of its members' spins and orbit.
