Identification of the techniques employed by artists, e.g. mixing and layering of paints, if used together with information about their colour palette and style, can help to attribute works of art with more confidence. In this study, we show how the pigment composition in binary paint mixtures can be quantified using optical-reflectance spectroscopy, by analysis of the peak features corresponding to colour-transition edges in the first-derivative spectra. This technique is found to be more robust than a number of other spectral-analysis methods, which can suffer due to shifts in the transition edges in mixed paints compared to those observed in spectra of pure ones. Our method also provides a means of distinguishing paint mixtures from layering in some cases. The spectroscopy also shows the presence of multiple electronic transitions, accessible within a narrow energy range, to be a common feature of many coloured pigments, which electronic-structure calculations attribute to shallow band edges. We also demonstrate the successful application of the reflectance-analysis technique to painted areas on a selection of medieval illuminated manuscripts.