  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 of $\mathbf{q}$. Here we use Angle-Resolved Photo Emission
Spectroscopy (ARPES) on the CDW material Ta$_2$NiSe$_7$ and identify a total
absence of any plausible nesting of states at the primary CDW wavevector
$\mathbf{q}$. Nevertheless we observe spectral intensity on replicas of the
hole-like valence bands, shifted by a wavevector of $\mathbf{q}$, which appears
with the CDW transition. In contrast, we find that there is a possible nesting
at $\mathbf{2q}$, and associate the characters of these bands with the reported
atomic modulations at $\mathbf{2q}$. Our comprehensive electronic structure
perspective shows that the CDW-like transition of Ta$_2$NiSe$_7$ is unique,
with the primary wavevector $\mathbf{q}$ being unrelated to any low-energy
states, but suggests that the reported modulation at $\mathbf{2q}$, which would
plausibly connect low-energy states, might be more important for the overall
energetics of the problem.
