  The cuprate high temperature superconductors exhibit a pronounced trend in
which the superconducting transition temperature, $T_{\rm c}$, increases with
the number of CuO$_2$ planes, $n$, in the crystal structure. We compare the
magnetic excitation spectrum of Bi$_{2+x}$Sr$_{2-x}$CuO$_{6+\delta}$ (Bi-2201)
and Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10 + \delta}$ (Bi-2223), with $n=1$ and $n=3$
respectively, using Cu $L_3$-edge resonant inelastic x-ray scattering (RIXS).
Near the anti-nodal zone boundary we find the paramagnon energy in Bi-2223 is
substantially higher than that in Bi-2201, indicating that multilayer cuprates
host stronger effective magnetic exchange interactions, providing a possible
explanation for the $T_{\rm c}$ vs.\ $n$ scaling. In contrast, the nodal
direction exhibits very strongly damped, almost non-dispersive excitations. We
argue that this implies that the magnetism in the doped cuprates is partially
itinerant in nature.
