  We study the connection between triangulations of a type $A$ root polytope
and the resonance arrangement, a hyperplane arrangement that shows up in a
surprising number of contexts. Despite an elementary definition for the
resonance arrangement, the number of resonance chambers has only been computed
up to the $n=8$ dimensional case. We focus on data structures for labeling
chambers, such as sign vectors and sets of alternating trees, with an aim at
better understanding the structure of the resonance arrangement, and, in
particular, enumerating its chambers. Along the way, we make connections with
similar (and similarly difficult) enumeration questions. With the root polytope
viewpoint, we relate resonance chambers to the chambers of polynomiality of the
Kostant partition function. With the hyperplane viewpoint, we clarify the
connections between resonance chambers and threshold functions. In particular,
we show that the base-2 logarithm of the number of resonance chambers is
asymptotically $n^2$.
