  Among the quasi one-dimensional transition metal tetrachalcogenides
(MSe$_4$)$_n$I (M=Nb,Ta), the $n=3$ compounds are the only ones not displaying
charge density waves. Instead, they show structural transitions with puzzling
transport behavior. They are semiconductors at the lowest temperatures, but
their transport gaps are significantly smaller than those inferred from ARPES
and optical conductivity. Recently, a metallic polytype of (TaSe$_4$)$_3$I has
been found with ferromagnetism and superconductivity coexisting at low
temperature, in contrast to previous reports. In this work we present detailed
ab-initio and tight binding band structure calculations for the different
(MSe$_4$)$_n$I reported structures. We obtain good agreement with the observed
transport gaps, and explain how ARPES and optics experiments effectively probe
a gap between different bands due to an approximate translation symmetry,
solving the controversy. Finally, we show how small extrinsic hole doping can
tune the Fermi level through a Van Hove singularity in (TaSe$_4$)$_3$I and
discuss the implications for magnetism and superconductivity.
