We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form φ M ≤ r, and the open diagram, which encapsulates strict inequalities of the form φ M < r. We show that the closed and open Σ N diagrams are Π 0 N+1 and Σ N respectively, and that the closed and open Π N diagrams are Π 0 N and Σ 0 N+1 respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.