  The Harper-Hofstadter model provides a fractal spectrum containing
topological bands of any integer Chern number, $C$. We study the many-body
physics that is realized by interacting particles occupying Harper-Hofstadter
bands with $|C|>1$. We formulate the predictions of Chern-Simons or composite
fermion theory in terms of the filling factor, $\nu$, defined as the ratio of
particle density to the number of single-particle states per unit area. We show
that this theory predicts a series of fractional quantum Hall states with
filling factors $\nu = r/(r|C| +1)$ for bosons, or $\nu = r/(2r|C| +1)$ for
fermions. This series includes a bosonic integer quantum Hall state (bIQHE) in
$|C|=2$ bands. We construct specific cases where a single band of the
Harper-Hofstadter model is occupied. For these cases, we provide numerical
evidence that several states in this series are realized as incompressible
quantum liquids for bosons with contact interactions.
