  Complex systems often comprise many kinds of components which vary over many
orders of magnitude in size: Populations of cities in countries, individual and
corporate wealth in economies, species abundance in ecologies, word frequency
in natural language, and node degree in complex networks. Here, we introduce
`allotaxonometry' along with `rank-turbulence divergence' (RTD), a tunable
instrument for comparing any two ranked lists of components. We analytically
develop our rank-based divergence in a series of steps, and then establish a
rank-based allotaxonograph which pairs a map-like histogram for rank-rank pairs
with an ordered list of components according to divergence contribution. We
explore the performance of rank-turbulence divergence, which we view as an
instrument of `type calculus', for a series of distinct settings including:
Language use on Twitter and in books, species abundance, baby name popularity,
market capitalization, performance in sports, mortality causes, and job titles.
We provide a series of supplementary flipbooks which demonstrate the tunability
and storytelling power of rank-based allotaxonometry.
