Compression behaviors of two Al-rich phases in the lower mantle, hexagonal new aluminum-rich (NAL) phase and its high-pressure polymorph Ca-ferrite-type (CF) phase, were examined for identical Na0.4Mg0.6Al1.6Si0.4O4 (40 % NaAlSiO4-60 % MgAl2O4) composition. The volumes of the NAL and CF phases were obtained at room temperature up to 31 and 134 GPa, respectively, by a combination of laser-annealed diamond-anvil cell techniques and synchrotron X-ray diffraction measurements. Fitting of the third-order Birch-Murnaghan equation of state to such pressure-volume data yields bulk modulus K (0) = 199(6) GPa at 1 bar and its pressure derivative K (0)' = 5.0(6) for the NAL phase and K (0) = 169(5) GPa and K (0)' = 6.3(3) for the CF phase. These results indicate that the bulk modulus increases from 397 to 407 GPa across the phase transition from the NAL to CF phase at 43 GPa, where the NAL phase completely transforms into the CF phase on Na0.4Mg0.6Al1.6Si0.4O4. Density also increases by 2.1 % across the phase transition.