  We introduce a class of metric spaces called $p$-additive combinations and
show that for such spaces we may deduce information about their $p$-negative
type behaviour by focusing on a relatively small collection of almost disjoint
metric subspaces, which we call the components. In particular we deduce a
formula for the $p$-negative type gap of the space in terms of the $p$-negative
type gaps of the components, independent of how the components are arranged in
the ambient space. This generalizes earlier work on metric trees by Doust and
Weston. The results hold for semi-metric spaces as well, as the triangle
inequality is not used.
