  Using single-crystal X-ray diffraction we characterise the 235\,K
incommensurate phase transition in the hybrid molecular framework
tetraethylammonium silver(I) dicyanoargentate, [NEt$_4$]Ag$_3$(CN)$_4$. We
demonstrate the transition to involve spontaneous resolution of chiral
[NEt$_4$]$^+$ conformations, giving rise to a state in which molecular
chirality is incommensurately modulated throughout the crystal lattice. We
refer to this state as an incommensurate chirality density wave (XDW) phase,
which represents a fundamentally new type of chiral symmetry breaking in the
solid state. Drawing on parallels to the incommensurate ferroelectric
transition of NaNO$_2$ we suggest the XDW state arises through coupling between
acoustic (shear) and molecular rotoinversion modes. Such coupling is
symmetry-forbidden at the Brillouin zone centre but symmetry-allowed for small
but finite modulation vectors $\mathbf q=[0,0,q_z]^\ast$. The importance of
long-wavelength chirality modulations in the physics of this hybrid framework
may have implications for the generation of mesoscale chiral textures, as
required for advanced photonic materials.
