We show that the specific heat of incommensurately modulated crystals with broken translational periodicity presents similar features at low temperatures to those of amorphous and glass materials. Here we demonstrate that the excess to the constant C_{p}(T)/T^{3} law (or Debye limit) is made up of an upturn below 1 K and of a broad bump at T≈10 K that directly originates from the gapped phase and amplitude modes of the incommensurate structure. We argue that the low-energy dynamics of incommensurate systems constitute a plausible simplification of the landscape of interactions present in glasses, giving rise to their low-temperature anomalies.