An asymptotic scheme is generated that captures the motion of waves within discrete, semi-discrete and continuous periodic media by creating continuum homogenised equations. Conventional homogenisation theory is a well-known classical method valid when the wavelength of any disturbance is long relative to the microstructure. Unfortunately many of the features of interest in real applications involve wave oscillations that are of high frequency and that have wavelength of the same, or similar, order to the microstructure; this requires a new version of homogenisation theory: High frequency homogenisation. This has already been introduced for periodic microstructured continua and extended to discrete systems. Herein we extend high frequency homogenisation further, to deal with localised defect states and non-orthogonal geometries for both discrete and continuous media. We also apply the asymptotic theory to new models, such as in-plane oscillations of the discrete vector system. In each of the studies presented herein, the homogenisation method is verified using numerical and/or analytical solutions.