For any finite path v on the square lattice consisting of north and east unit steps, we construct a poset Tam(v) that consists of all the paths lying weakly above v with the same endpoints as v. For particular choices of v, we recover the traditional Tamari lattice and the m-Tamari lattice. In particular this solves the problem of extending the m-Tamari lattice to any pair (a, b) of relatively prime numbers in the context of the so-called rational Catalan combinatorics. For that purpose we introduce the notion of canopy of a binary tree and explicit a bijection between pairs (u, v) of paths in Tam(v) and binary trees with canopy v. Let ←−v be the path obtained from v by reading the unit steps of v in reverse order and exchanging east and north steps. We show that the poset Tam(v) is isomorphic to the dual of the poset Tam(←−v ) and that Tam(v) is isomorphic to the set of binary trees having the canopy v, which is an interval of the ordinary Tamari lattice. Thus the usual Tamari lattice is partitioned into (smaller) lattices Tam(v), where the v’s are all the paths of length n − 1 on the square lattice. We explain possible connections between the poset Tam(v) and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.