We report on the tuning of magnetic interactions in superlattices composed of single and bilayer SrIrO$_3$ inter-spaced with SrTiO$_3$. Magnetic scattering shows predominately $c$-axis antiferromagnetic orientation of the magnetic moments for the bilayer justifying these systems as viable artificial analogues of the bulk Ruddlesden-Popper series iridates. Magnon gaps are observed in both superlattices, with the magnitude of the gap in the bilayer being reduced to nearly half that in its bulk structural analogue, Sr$_3$Ir$_2$O$_7$. We assign this to modifications in the anisotropic exchange driven by bending of the $c$-axis Ir-O-Ir bond and subsequent local environment changes, as detected by x-ray diffraction and modeled using spin wave theory. These findings explain how even subtle structural modulations driven by heterostructuring in iridates are leveraged by spin orbit coupling to drive large changes in the magnetic interactions.
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