  We demonstrate that all of the salient features of the Harper-Hofstadter
model can be implemented with ultracold atoms trapped in a bichromatic
ring-shaped lattice. Using realistic sinusoidal lattice potentials rather than
assume the idealized tight-binding picture, we determine the optimal conditions
necessary to realize the critical point where the spectrum becomes fractal, and
identify the nature and cause of the departures from the discrete model
predictions. We also show that even with a commensurate ring with a few lattice
sites, the Aubry-Andr\'e localization transition can be realized. Localized
states that behave like edge states with energies that reside in the band gaps
can be generated by introducing a surprisingly small local perturbation within
the ring. Spectrum oscillation arising from complex coupling can be implemented
by uniform rotation of the ring, but with certain significant differences that
are explained
