First-principles calculations are carried out to estimate the spontaneous polarization and the band gap bowing in the Sc xAl yGa 1 x yN alloys lattice-matched to GaN based on the density functional theory together with a pseudopotential plane-wave method. The results indicate nonlinear behavior of the spontaneous polarization and the band gap energies in the Sc xAl yGa 1 x yN alloys similar to that in Y xAl yGa 1 x yN (Shimada et al 2011 J. Appl. Phys. 110 074114). The band gap energies and the magnitude of the spontaneous polarization of Sc xAl yGa 1 x yN is larger than that of Y xAl yGa 1 x yN in a wide range of the mole fraction of Ga. We find that the atomic configuration of Sc affects the magnitude of the polarization in Sc xAl yGa 1 x yN. Another finding is the apparent deviation from Vegard's law of the lattice constants of the Sc xAl yGa 1 x yN alloys lattice-matched to GaN.