  Given a finite metric, one can construct its tight span, a geometric object
representing the metric. The dimension of a tight span encodes, among other
things, the size of the space of explanatory trees for that metric; for
instance, if the metric is a tree metric, the dimension of the tight span is
one. We show that the dimension of the tight span of a generic metric is
between the ceiling of n/3 and the floor of n/2, and that both bounds are
tight.
