The dynamic properties of periodic generalised two-phase quasicrystalline phononic rods [1] are studied through the method of the toroidal manifold. The method allows all stop and pass bands featuring the frequency spectrum to be represented in a compact form with a frequency dependent flow line on the surface describing their ordered sequence. The flow lines on the torus can be either closed or open: in the former case, (i) the frequency spectrum is periodic and the el-ementary cell corresponds to a canonical configuration [2], (ii) the stop-band density depends on the lengths of the two phases; in the latter, the flow lines cover the torus and the stop-band density is independent of those lengths. It is then shown how the proposed compact description of the spectrum can be exploited to optimise the layout of the elementary cell in order to maximise the low-frequency stop band. The scaling property of the frequency spectrum, that is a distinctive feature of quasicrystalline generated phononic media, is rigorously introduced. The obtained re-sults represent both a key to a better understanding of the properties of classical two-phase com-posite waveguides and an important advancement towards the realisation of composite quasi-crystalline metamaterials.
