  We explain in detail how to calculate the gravitational mass and angular
momentum multipoles of the most general non-extremal four-dimensional black
hole with four magnetic and four electric charges. We also calculate these
multipoles for generic supersymmetric four-dimensional microstate geometries
and multi-center solutions. Both for Kerr black holes and BPS black holes many
of these multipoles vanish. However, if one embeds these black holes in String
Theory and slightly deforms them, one can calculate an infinite set of ratios
of vanishing multipoles which remain finite as the deformation is taken away,
and whose values are independent of the direction of deformation. For
supersymmetric black holes, we can also compute these ratios by taking the
scaling limit of multi-center solutions, and for certain black holes the ratios
computed using the two methods agree spectacularly. For the Kerr black hole,
these ratios pose strong constraints on the parameterization of possible
deviations away from the Kerr geometry that should be tested by future
gravitational wave interferometers.
