An effective Hamiltonian is developed to investigate the magnetic cycloid of the BiFeO3 (BFO) multiferroic. This approach reproduces many complex features of this cycloid, such as its plane of rotation containing the polarization and the newly discovered spin density waves resulting from the canting of magnetic dipoles out of this cycloidal plane. It also suggests that the energetic origin of the cycloid can be thought of in terms of the converse spin-current model, and reveals the mechanisms responsible for the spin density waves. Finally, this atomistic scheme resolves an ongoing controversy about the cycloid anharmonicity, and revisits a recent misconception about the relationship between out-of-plane spin-density waves and the weak magnetization associated with the spin-canted structure of BFO.