We report measurements of the diffraction pattern of atwo-dimensional photonic quasicrystal structure and use the setof plane waves defined by the diffraction pattern as the basisof a theoretical approach to calculate the photonicband structure of the system. An important feature of the modelis that it retains the essence of the rotational and inflationalproperties of the quasicrystal at all levels of approximation:properties lost in approximate models which artificiallyintroduce elements of periodicity.The calculated density of modes of the quasicrystals is found todisplay a weakly depleted region analogous to the bandgap thatoccurs in a periodic system. The calculated transmissionspectra for different polarizations and directions ofpropagation show features that correlate with the behaviour ofthe density of modes.