A series of numerical electronic density-of-states calculations is performed for rational approximants to a model one-electron potential based on icosahedrally arranged plane-wave components. It is found that high-order approximants can have band gaps even if the low-order approximants do not; furthermore, the magnitude of the gap increases with the order of the approximant. The results are interpreted via a two- and three-wave analysis of the energy eigenvalues at the pseudo-Jones-zone faces and edges. It is also found that the mechanism of band-gap reduction in the rational approximants is the presence of a small density of gap states. An analytic calculation shows that these gap states result from a splitting of threefold and pseudothreefold states at the valence-band edge when the icosahedral symmetry is broken. The splitting is proportional to the error with which the ratio between the approximant indices approximates τ, the golden mean. Finally, an application to the AlCuLi system is presented.