Let $mathcal A$ be a mathematical structure with an additional relation $R$. The author is interested in the degree spectrum of $R$, either among computable copies of $mathcal A$ when $(mathcal A,R)$ is a ``natural'' structure, or (to make this rigorous) among copies of $(mathcal A,R)$ computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov--that, assuming an effectiveness condition on $mathcal A$ and $R$, if $R$ is not intrinsically computable, then its degree spectrum contains all c.e. degrees--the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.