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When electrons are subject to a potential with two incommensurate periods, translational invariance is lost, and no periodic band structure is expected. However, model calculations based on nearly free one-dimensional electrons and experimental results from high-resolution photoemission spectroscopy on a quasi-one-dimensional material do show dispersing band states with signatures of both periodicities. Apparent band structures are generated by the nonuniform distribution of electronic spectral weight over the complex eigenvalue spectrum.