Charge Density Waves (CDW) are commonly associated with the presence of near-Fermi level states which are separated from others, or “nested", by a wavector ofq. Here we use Angle-Resolved Photo Emission Spectroscopy (ARPES) on the CDW material Ta2NiSe7and identify a total absence of any plausible nesting of states at the primary CDW wavevectorq. Nevertheless we observe spectral intensity on replicas of the hole-like valence bands, shifted by a wavevector ofq, which appears with the CDW transition. In contrast, we find that there is a possible nesting at2q, and associate the characters of these bands with the reported atomic modulations at2q. Our comprehensive electronic structure perspective shows that the CDW-like transition of Ta2NiSe7is unique, with the primary wavevectorqbeing unrelated to any low-energy states, but suggests that the reported modulation at2q, which would plausibly connect low-energy states, might be more important for the overall energetics of the problem.