It has been demonstrated by many theoretical and experimentals works that Mie resonances are at the heart of the effective properties of dielectric metamaterials. These resonances indeed allow for the onset of tailorable macroscopic magnetic properties. They were shown to provide a convenient way to study the transition between photonic crystals and metamaterials. In the present work, we show that the band structure linked to theses resonances is largely scale invariant and also, to some extend, robust with regard to disorder. These results do not rely heavily on a specific type of wave, suggesting that the same kind of results can be obtained for acoustic or gravity waves.