  We show that by averaging over transitions to multiple hyperfine levels,
quadrupole shifts and dominant Zeeman effects exactly cancel whenever the
nuclear spin, I, is at least as large as the total electronic angular momentum,
J. The average frequency thus defines a frequency reference which is inherently
independent of external magnetic fields and electric field gradients. We use
Lu+ to illustrate the method although the approach could be readily adapted to
other atomic species. This approach practically eliminates the quadrupole and
Zeeman shift considerations for many clock candidates.
