An investigation has been undertaken into acoustic iridescence, exploring how a device can be constructed which alter sound waves, in a similar way to structures in nature that act on light to produce optical iridescence. The main construction had many thin perforated sheets spaced half a wavelength apart for a specified design frequency. The sheets create the necessary impedance discontinuities to create backscattered waves, which then interfere to create strongly reflected sound at certain frequencies. Predictions and measurements show a set of harmonics, evenly spaced in frequency, for which sound is reflected strongly. And the frequency of these harmonics increases as the angle of observation gets larger, mimicking the iridescence seen in natural optical systems. Similar to optical systems, the reflections become weaker for oblique angles of reflection. A second construction was briefly examined which exploited a metamaterial made from elements and inclusions which were much smaller than the wavelength. Boundary element method predictions confirmed the potential for creating acoustic iridescence from layers of such a material.