An energetic expression containing four different macroscopic terms is proposed to explain and understand coupled magnetic orders (and the directions of the simultaneously occurring ferromagnetic and/or antiferromagnetic vectors) in terms of anti-phase and/or in-phase tilting of oxygen octahedra in magnetic and multiferroic perovskites. This expression is derived from a suggested simple microscopic formula, and has its roots in the Dzyaloshinsky-Moriya interaction. Comparison with data available in the literature and with first-principles calculations we conduct here confirms the validity of such a simple and general law for any tested structural paraelectric and even ferroelectric phase, and for any chosen direction of any selected primary magnetic vector.
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