Gamut mapping transformations are used to convert an image pixel by pixel so that its gamut fits within the gamut of the desired output device. These transformations can be defined independently of the source of illumination by specifying colours in reflectance coordinates, the most economical of which are those based on linear combinations of basis functions. Of course, basis functions differ from one output device to another, making it necessary to map reflectances defined in terms of one basis into another basis. Projective transformations are the most natural way of doing so but are not satisfactory. This paper develops the formal properties of reflective gamut mappings, defines a mapping algorithm based on the fundamental component of the reflectance, and shows that this method consistently conserves colour sensation better than simple projective mappings.