We present a thorough analysis of the effects of the tensor interaction on the multipole response of magic nuclei, using the fully self-consistent Random Phase Approximation (RPA) model with Skyrme interactions. We disentangle the modifications to the static mean field induced by the tensor terms, and the specific features of the residual particle-hole (p-h) tensor interaction, for quadrupole (2+), octupole (3-), and also magnetic dipole (1+) responses. It is pointed out that the tensor force has a larger effect on the magnetic dipole states than on the natural parity states 2+ and 3-, especially at the mean field level. Perspectives for a better assessment of the tensor force parameters are eventually discussed.
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