The Debye model predicts a T3 dependence of the specific heat Cp at sufficiently low tem- peratures in insulators and it reproduces a wealth of data in isotropic crystals. However, this simple and general law fails to account for the low temperature specific heat of amorphous compounds and this matter is currently the focus of many scientific activities, yet unsettled. At low temperatures there are two anomalies that show up as an excess to the constant Cp(T)/T3 in glasses and amorphous systems: the upturn below 1 K and a broad bump at T≈10 K named Boson peak (BP). Whereas insulating crystals with low-lying optical modes and/or acoustic modes with non-linear dispersions can exhibit a similar bump in the specific heat, the Debye law is expected to re-emerge in these systems at sufficiently low temperatures. (Andres) One question that we want to address is to what extent this is the case. We show that, contrary to the expected, the Debye law is not fully obeyed as soon as translational periodicity is slightly broken like in insulating crystals with incommensurate superstructures. (Our suggestion) We show that, however, as soon as translational periodicity is slightly broken, like in insulating crystals with incommensurate superstructures, the Debye law is not the most important contribution to the low temperature specific heat anymore. On general grounds we show the specific heat of incommensurate modulated systems bears many similarities with that of amorphous systems. Moreover we account for the excess of heat capacity within a simple model in which the energy dispersion and the damping of the lowest energy, non-acoustic, phonon branch are the main parameters.