  We consider the quasinormal modes for a class of black hole spacetimes that,
informally speaking, contain a closely ``squeezed'' pair of horizons. (This
scenario, where the relevant observer is presumed to be ``trapped'' between the
horizons, is operationally distinct from near-extremal black holes with an
external observer.) It is shown, by analytical means, that the spacing of the
quasinormal frequencies equals the surface gravity at the squeezed horizons.
Moreover, we can calculate the real part of these frequencies provided that the
horizons are sufficiently close together (but not necessarily degenerate or
even ``nearly degenerate''). The novelty of our analysis (which extends a
model-specific treatment by Cardoso and Lemos) is that we consider ``dirty''
black holes; that is, the observable portion of the (static and spherically
symmetric) spacetime is allowed to contain an arbitrary distribution of matter.
