  Computable reducibility is a well-established notion that allows to compare
the complexity of various equivalence relations over the natural numbers. We
generalize computable reducibility by introducing degree spectra of
reducibility and bi-reducibility. These spectra provide a natural way of
measuring the complexity of reductions between equivalence relations. We prove
that any upward closed collection of Turing degrees with a countable basis can
be realised as a reducibility spectrum or as a bi-reducibility spectrum. We
show also that there is a reducibility spectrum of computably enumerable
equivalence relations with no countable basis and a reducibility spectrum of
computably enumerable equivalence relations which is downward dense, thus has
no basis.
