We study the spectrum of fermions hopping on a chain with a weak incommensuration close to dimerization; both q, the deviation of the wave number from pi, and delta, the strength of the incommensuration, are small. For free fermions, we use a continuum Dirac theory to show that there are an infinite number of bands which meet at zero energy as q approaches zero. In the limit that the ratio q/delta–>0, the number of states lying inside the q = 0 gap is nonzero and equal to 2 delta/pi(2). Thus the limit q–>0 differs from q=0; this can be seen clearly in the behavior of the specific heat at low temperature. For interacting fermions or the XXZ spin-1/2 chain close to dimerization, we use bosonization and a renormalization group analysis to argue that similar results hold; as q–>0, there is a nontrivial density of states near zero energy. However, the limit q–>0 and q=0 give the same results near commensurate wave numbers which are different from pi; for both free and interacting fermions, we find that a nonzero value of q is necessary to close the gap. Our results for free fermions are applied to the Azbel-Hofstadter problem of electrons hopping on a two-dimensional lattice in the presence of a magnetic field. Finally, we discuss the complete energy spectrum of free fermions with incommensurate hopping by going up to higher orders in delta.