An analytical perturbative approach is presented to calculate the out-of-plane precessional motion of the magnetization of an in-plane free layer submitted to the spin torque of a perpendicular polarizer in the macrospin model. We consider the effects of the uniaxial anisotropy field, of an in-plane applied field, and of the perturbation due to the in-plane reference layer on the otherwise circular and monochromatic oscillations of the free layer. We calculate the frequency change due to these perturbations, the amplitude of the second harmonics, and the critical current for the existence of oscillations. This approach is rather general in the treatment of harmonics in spin torque oscillators