We show that a natural reinterpretation of the classical Bochner identity for harmonic one-forms reveals an intriguing relationship between the scalar curvature of a manifold M and the geometry of the level sets of harmonic (circle-valued) functions on M. These ideas provide new tools for the study of positive scalar curvature, and lead to several sharp geometric inequalities relating the scalar curvature to other geometric invariants of three-manifolds.