A wide variety of minerals exhibit ordered crystalline phases in which an underlying periodic structure is modulated by small atomic displacements with a wavelength incommensurate with this periodicity. We describe our recent efforts to use state-of-the-art first-principle calculations for the study of the microscopic mechanisms governing the existence of such phases and their transitions. A database containing more than one hundred incommensurate phases, available on the Web, has been constructed, and representative materials (PbO, AuTe2, K2SO4 ···) have been selected for our purpose. The ABINIT software project, thanks to which we can compute and analyze interatomic force constants and lattice instabilities, is described. We present preliminary results for the representative materials, discuss the difficulties that we face, and contrast them with those encountered in the study of ferroelectric materials.