  While recent work has established divergence as a key framework for
understanding evenness, there is currently no research exploring how the
families of measures within the divergence-based framework relate to each
other. This paper uses geometry to show that, holding order and richness
constant, the families of divergence-based evenness measures nest. This
property allows them to be ranked based on their reactivity to changes in
relatively even assemblages or changes in relatively uneven ones. We establish
this ranking and explore how the distance-based measures relate to it for both
order q=2 and q=1. We also derive a new family of distance-based measures that
captures the angular distance between the vector of relative abundances and a
perfectly even vector and is highly reactive to changes in even assemblages.
Finally, we show that if we only require evenness to be a divergence, then any
smooth, monotonically increasing function of diversity can be made into an
evenness measure. A deeper understanding of how to measure evenness will
require empirical or theoretical research that uncovers which kind of
divergence best reflects the underlying concept.
