  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.
