The full elastic constants tensors of diopside (CaMgSi2O6) and jadeite (NaAlSi2O6) have been calculated using a planewave and pseudopotentials based implementation of density functional theory within the generalised gradient approximation at pressures between 0 and 20 GPa. Both minerals stiffen over this pressure range with the isotropic average bulk moduli increasing by ~50% and the shear moduli by ~20%. However, in detail the behaviour of the individual elastic constants varies and this drives changes in the anisotropy.Overall, and in contrast to predictions based on the extrapolation of calculations based on inter-atomic potential models, the elastic anisotropy of diopside decreases with increasing pressure. The elastic anisotropy of jadeite increases slightly at low pressure, exhibits a maximum at around 10 GPa and then begins to slowly decrease. Despite the small changes in the total and maximum anisotropy, the shear-wave anisotropy for certain propagation directions vary dramatically with pressure. For example, the anisotropy experienced by a shear-wave propagating in the  direction in diopside doubles between 5 and 15 GPa.