Aperiodic structures which are perfectly ordered, but lack translational symmetry, can be considered as intermediate between crystalline and amorphous systems. Their wave-functions and spectra exhibit unusual properties. A one-dimensional model, known as the Thue-Morse lattice, is here investigated; it is particularly interesting because ft is deterministic non-periodic, but not quasi-periodic. The propagation of both electrons and photons is considered and the corresponding spectra are characterized with a multifractal analysis. The multifractal scaling function tau(q) tends in both cases to a non-analytic limit that implies the presence of both extended and exponentially localized states, as confirmed by the behaviour of the wave functions.