Reappraisal of colour matching and Grassmann's laws

CIE 185:2009
Division 1
978 3 901906 78 7

The laws of additivity and proportionality of colour matches, Grassmann’s laws, are the basis of all colour theory, but are not axiomatically true. The extent of departure of human vision from Grassmann’s laws has been periodically examined. One exploration, by W. A. Thornton, found considerable failure of transformability of primaries - a symptom of Grassmann additivity failure. In the 14 years since Thornton’s finding, several groups have formed to replicate and understand Thornton’s results and the limitations of Grassmann’s laws. CIE TC 1-56 is the latest of these. During the ten years of this committee’s existence, statistical simulations indicated that replicate matches by the same observer (not present in Thornton’s data) are required to suppress random errors, and accordingly three laboratories generated intra-observer matching results in three different luminance domains. Two of the studies, respectively conducted at 300 cd•m-2 and 30 cd•m-2, confirm Grassmann additivity, but the third study shows failure of additivity at 3 cd•m-2. In addition, Maxwell and maximum-saturation colour matches have long been known to be inconsistent even at high luminance levels and with intra-observer match replication to suppress noise. A practical consequence of the failure of additivity could be problems observed in cross-media colour matching, although cross-media studies also have other well known sources of imprecision when the colour-matching is asymmetric. Some suggestions are made for a covering theory of Grassmann’s laws that might accommodate both Maxwell and maximum-saturation match data while still maintaining consistency with high-luminance success in experiments such as reported recently. Further investigations are indicated for a successor to TC 1-56.

The publication is written in English, with a short summary in French and German. It consists of 20 pages with 3 figures, and is readily available at the National Committees of the CIE or via the website of the Central Bureau of the CIE (

The following members and advisors of TC 1-56 "Improved Colour Matching Functions" took part in the preparation of this Technical Report:


  • M.H. Brill, USA (Chair)
  • M.D. Fairchild, USA
  • H.S. Fairman, USA
  • K. Houser, USA
  • R.G. Kuehni, USA
  • R. Luo, Great Britain
  • B. Oicherman, Israel
  • C. Oleari, Italy
  • D. Oulton, Great Britain
  • D. Rich, USA
  • A. Robertson, Canada
  • J. Schanda, Hungary
  • A.W.Tarrant, Great Britain
  • W.A. Thornton†, USA
  • K. Wenzel, Hungary
  • J. Wold, Norway


  • A. Stockman, Great Britain
  • F. Vienot, France