  We propose and experimentally demonstrate a scheme which effects hyperfine
averaging during a Ramsey interrogation of a clock transition. The method
eliminates the need to average over multiple optical transitions, reduces the
sensitivity of the clock to its environment, and reduces inhomogeneous
broadening in a multi-ion clock. The method is compatible with auto-balanced
Ramsey spectroscopy, which facilitates elimination of residual shifts due to
imperfect implementation and ac Stark shifts from the optical probe. We
demonstrate the scheme using correlation spectroscopy of the $^1S_0$-to-$^3D_1$
clock transition in a three-ion Lu+ clock. From the demonstration we are able
to provide a measurement of the $^3D_1$ quadrupole moment,
$\Theta(^3D_1)=0.634(9)ea_0^2$.
