We investigate the photonic band structure, using the transfer-matrix method, in one-dimensional structures composed of a dispersive metamaterial A juxtaposed with a non-dispersive dielectric B according to the generalized Fibonacci and Thue–Morse sequences, which are ruled and characterized by two positive integer numbers, p and q. We present the band structures of different generations of these sequences for both TE and TM modes and discuss how they are affected by the p and q parameters and the ratio between the thicknesses of the building blocks. We obtain a new and very important analytical expression for the frequency in which the average refraction index vanishes, i.e. when the gap condition is satisfied, and we investigate, in detail, the behavior of this emergent scale insensitive gap, as a function of the thicknesses ratio, for all structures considered. For the metamaterial considered, we show that the reduced frequency for converges faster or slower to 0.41 depending on the sequence. By adjusting the p and q parameters, we can obtain a higher concentration of the pass bands inside (outside) the nA < 0 region when there are more (less) blocks A than B in the unit cell. Our results also show the presence of omnidirectional and complete band gaps, which have various practical and technological applications.