  We consider hadronic axion models that extend the Standard Model by one
complex scalar field and one or more new heavy quarks, i.e. $N_\mathcal{Q} \geq
1$. We review previously suggested selection criteria as well as categorize and
catalog all possible models for $N_\mathcal{Q} \leq 9$. In particular, allowing
for $N_\mathcal{Q} > 1$ can introduce models that spoil the axion solution of
the strong CP problem. Demanding that Landau poles do not appear below some
energy scale limits the number of preferred models to a finite number. For our
choice of criteria, we find that $N_\mathcal{Q} \leq 28$ and only 820 different
anomaly ratios $E/N$ exist (443 when considering additive representations, 12
when all new quarks transform under the same representation). We analyze the
ensuing $E/N$ distributions, which can be used to construct informative priors
on the axion-photon coupling. The hadronic axion model band may be defined as
the central region of one of these distributions, and we show how the band for
equally probable, preferred models compares to present and future experimental
constraints.
