Classically, in web geometry, we consider families of foliations modulo local diffeomorphisms and we study them by differential invariants of the pseudo-group of diffeomorphisms. In this work we introduce the notion of harmonic foliations and hexagonal harmonic 3-webs, we develop basic properties, provide examples and we devote ourselves to the local study of hexagonal harmonic webs by changing the structure of the pseudo-group to that of the pseudo-group of conformal transformations. In this framework, we highlight a family of infinite dimension (called generic) and that we describe completely. Then, we obtain a finitude result for non-generic hexagonal harmonic 3-webs via abelian relations and using symbolic calculus software. Finally, we construct a complete classification of these 3-webs when 2 foliations among 3 form a constant angle.