  In a language corpus, the probability that a word occurs $n$ times is often
proportional to $1/n^2$. Assigning rank, $s$, to words according to their
abundance, $\log s$ vs $\log n$ typically has a slope of minus one. That simple
Zipf's law pattern also arises in the population sizes of cities, the sizes of
corporations, and other patterns of abundance. By contrast, for the abundances
of different biological species, the probability of a population of size $n$ is
typically proportional to $1/n$, declining exponentially for larger $n$, the
log series pattern. This article shows that the differing patterns of Zipf's
law and the log series arise as the opposing endpoints of a more general
theory. The general theory follows from the generic form of all probability
patterns as a consequence of conserved average values and the associated
invariances of scale. To understand the common patterns of abundance, the
generic form of probability distributions plus the conserved average abundance
is sufficient. The general theory includes cases that are between the Zipf and
log series endpoints, providing a broad framework for analyzing widely observed
abundance patterns.
