  Causal processes can give rise to distinctive distributions in the linguistic
variables that they affect. Consequently, a secure understanding of a
variable's distribution can hold a key to understanding the forces that have
causally shaped it. A storied distribution in linguistics has been Zipf's law,
a kind of power law. In the wake of a major debate in the sciences around
power-law hypotheses and the unreliability of earlier methods of evaluating
them, here we re-evaluate the distributions claimed to characterize phoneme
frequencies. We infer the fit of power laws and three alternative distributions
to 166 Australian languages, using a maximum likelihood framework. We find
evidence supporting earlier results, but also nuancing them and increasing our
understanding of them. Most notably, phonemic inventories appear to have a
Zipfian-like frequency structure among their most-frequent members (though
perhaps also a lognormal structure) but a geometric (or exponential) structure
among the least-frequent. We compare these new insights the kinds of causal
processes that affect the evolution of phonemic inventories over time, and
identify a potential account for why, despite there being an important role for
phonetic substance in phonemic change, we could still expect inventories with
highly diverse phonetic content to share similar distributions of phoneme
frequencies. We conclude with priorities for future work in this promising
program of research.
