The earlier suggested energy-level model based on an orbital singlet ground state for 3d 6 and 3d 4 ions at trigonal sites with large triclinic distortion is adopted to develop the single-ion theory of magnetic anisotropy. The Hamiltonian consisting of eight spin Hamiltonian terms and the molecular field is solved by perturbation theory. The resulting energies E Ms with M S = 0, ± 1, ± 2 are applied to calculate the free energy for Fe 2+ ions in Si- or Ge-substituted yttrium iron garnets where a uniform distribution of Fe 2+ ions over the 12 inequivalent sites is assumed. It turns out that the first two cubic anisotropy constants K 1 , and K 2 are insufficient to describe the anisotropy at high temperatures in the present model. By a least-squares method it is established that the anisotropy expansion series can be terminated at the fourth-order term for the present model. Thus K 1 , K 2, K 3 and K 4 are derived analytically in terms of the free energy for some choosen directions of magnetization. The analytical results agree very well with the corresponding ones obtained by the least-squares method. The temperature dependence of K i, i = 1. 2, 3, and 4, is calculated for a wide range of the spin Hamiltonian parameters (B q (k) ) and the molecular field (h). The theoretical K 1 , and K 2 are fitted to the experimental values of K 1 , and K 2 at low temperatures obtained by neglecting the higher-order anisotropy terms, to get the values of B q (h) and for YIG:Si and YIG:Ge. A comparison of the present results with The corresponding ones of the previous doublet model, is also discussed. The theoretical account of the experimentally observed temperature dependence of the ratio K 2 /K 1 for Fe 2+ in YIG:Ge speak in favour of the present model rather than the doublet model. The change in sign of K 1 observed for Fe 2+ in YIG:Si, which could not be explained by the doublet model with uniform distribution of Fe 2+ ions, is well accounted for by the present model. The recently observed spin reorientation in YIG:Si can also be explained by the present model without resorting to a nonuniform distribution of Fe 2+ ions required by the previous model. This study indicates that the higher-order constants K 3 and K 4 are significant at high temperatures according to the present model, whereas at low temperatures according to the doublet model. Hence an experimental determination of K 3 and K 4 over a wide temperature range may provide a test of the applicability of the two models.