We present a review of the work (Raymond in a constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain (1995), [39], Raymond in Etude algébrique de milieux quasipériodiques (1995), [40]). The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in Band, Beckus and Loewy (Dry Ten Martini Problem for Sturmian Hamiltonians, [3]). A Sturmian Hamiltonian is a one-dimensional Schrödinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, \(V\in \mathbb {R}\) . The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If \(V>4\) , then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if \(V>4\) , then all the gaps, as predicted by the gap labeling theorem, are there.