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We carry out density functional theory (DFT) calculations to explore the antiferromagnetic (AFM) spin spiral in multiferroic BiFeO₃. We calculate the spin spiral energy dispersion E(q) along the high symmetry directions of the pseudo-cubic unit cell, for four different structural phases: cubic, R-3c, R3m and R3c. In all cases, we find a large exchange frustration. The comparison provides detailed insight into how polarization and octahedral anti-phase tilting affect the different magnetic interactions and the magnetic ground state in BiFeO₃. For the R3c structural ground state, we find an AFM spin spiral ground state with a periodicity of ∽80 nm in good agreement with experiments and previous findings. This spin spiral is driven by a Dzyaloshinskii-Moriya interaction stemming from the Fe–Bi ferroelectric displacement. The spiral appears to be stable because the anisotropy energy in R3c BiFeO₃ is too small to  enforce the collinear order. For all the four phases, we discuss the magnetic ground state and identify its stabilization mechanisms.