  A converse is given to the well-known fact that a hyperplane localised zero
mode of a crystallographic bar-joint framework gives rise to a line or lines in
the zero mode (RUM) spectrum. These connections motivate definitions of linear
zero mode spectra, for an aperiodic bar-joint framework $G$, that are based on
relatively dense sets of linearly localised flexes. For a Delone framework in
the plane the limit spectrum ${\bf L}_{lim}(G, a)$ is defined in this way, as a
subset of the reciprocal space for a reference basis $a$ of the ambient space.
A smaller spectrum, the slippage spectrum ${\bf L}_{slip}(G, a)$, is also
defined. In the case of the quasicrystal parallelogram frameworks associated
with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra
coincide and are determined in terms of the geometry of $G$.
