In this work we study the dynamic behavior of a class of one-dimensional quasi-periodic media by means of an analytical asymptotic approach. With the devised method, introducing a small parameter by slightly perturbing the periodicity of the domain, we are able to obtain some general properties of the spectrum of the problem, reconstructing asymptotically its band-gaps structure. In particular, we demonstrate that non trivial band-gaps can nucleate and grow with the small parameter. Moreover, we explicitly determine the conditions under which nucleation can happen, showing that they are hierarchical in terms of order of magnitude of the perturbating parameter. Our findings allow to have a new qualitative and quantitative interpretation of the band-gaps structure.