  We demonstrate that quasiperiodicity can radically change the ground state
properties of 1D moir\'e systems with respect to their periodic counterparts.
By studying an illustrative example we show that while narrow bands play a
significant role in enhancing interactions both for commensurate and
incommensurate structures, only quasiperiodicity is able to extend the ordered
phase down to an infinitesimal interaction strength. In this regime, the
quasiperiodic-enabled state has contributions from infinitely many wave
vectors. This quasi-factal regime cannot be stabilized in the commensurate case
even in the presence of a narrow band. These findings suggest that
quasiperiodicity may be a critical factor in stabilizing non-trivial ordered
phases in interacting moir\'e structures and signal out multifractal
non-interacting phases, recently found in 2D incommensurate moir\'e systems, as
particularly promising parent states.
