Introduction¹

There are many cases in nature, technology, and everyday life in which two visually observable colors are combined into a new color.  It would of course be useful to be able to predict quantitatively what color would be produced by the combination of two given colors.  Unfortunately, the appearance of this third color depends not only on the first two colors, but also on the way that they were produced.

Most encountered color combinations of this sort fall into two categories: additive color and subtractive color.  The additive nature of color is encountered whenever two light sources are visually merged into one, giving a resulting color which is less saturated than either of its components, greater in lightness, and intermediate in hue.  The subtractive nature is seen in combinations of colored pigments, inks, or other objects that do not give off light, but absorb or transmit it in various amounts depending on its wavelength.  A subtractive combination of two colors gives a result that is lower in lightness and intermediate in hue.  In additive color combination, two visible light spectra are successively added to black (no light).  In subtractive color, two visible absorption spectra are successively subtracted from the visible spectrum of the light source (usually white, containing all wavelengths in nearly equal amounts).  These two processes often result in quite different colors.

The additive result can easily be predicted in terms of the visible spectra of the light sources being added: just add together the spectra!  There is also a simple rule in terms of the three-dimensional "Yxy" CIE color coordinates of the sources: the combination has chromaticity coordinates (x, y) that are the average of the two light sources, weighted by their relative lightnesses (Y).  The goal of this project is to determine experimentally whether or not a subtractive color combination can be predicted in either of these two manners: with the visible transmittance spectra, and/or with the three-dimensional color coordinates.

Background

My hypothesis was as follows: a subtractive color combination can be predicted from the percent transmittance spectra of two color filters by multiplying them together, giving the percent transmittance spectrum of the combination; the three-dimensional color coordinates of a combination, on the other hand, will not be predicable in terms of the color coordinates of the two filters.

The main source of deviation from this ideal of "spectral multiplication" will be that a substantial amount of light is reflect off of the filter's surface, rather than transmitted or absorbed.  This light is not taken into account in the hypothesis.  Furthermore, this amount increases significantly when two or three filters are present rather than just one, as more surfaces are available for reflection.

Procedure

My experiment was run in two parts.  The first part aimed to establish whether my hypothesis was correct about the transmittion spectra, while the second part was designed to determine whether any connection exists between the color coordinates of two color filters and the coordinates of their combination.  The colors used in the experiment were three color filters kindly supplied by Dr. Bordley: colors #2, "Bastard Amber", #53, "Pale Lavendar", and #87, "Pale Yellow-Green".  These filters were chosen for their low absorbance (high transmittance) throughout the visible spectrum, in order to minimize the uncertainty in the measurements.

For the first part of the experiment, a spectrophotometer was used.  This device emits a beam of light, all of which is at a particular wavelength which is specified. This light then passes through any filters inserted, and is detected in order to determine what percent of the emmitted light was transmitted through the filter(s).  In particular, I took percent transmittance data in this manner for each of the three color filters individually, for each of the three possible combinations of two filters (one on top of the other), and finally for all three filters together.  Each of these measurements was taken at six equally spaced wavelengths of light, so that the shape of the visible transmittance spectrum could be roughly determined.

In the second part of the experiment, I used a colorimeter -- a device that contains its own calibrated light source and uses it to determine three-dimensional color coordinates for a surface -- and recorded the resulting color data for each filter/filter combination in CIE Yxy, Lab, Munsell, and LCH color coordinate systems.

Observations and Data

These are the data that was collected, together with calculations of what the filter combination values were expected to be -- calculated according to the hypothesis by taking the product of the values for the combined filters.

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In the above graph, the three lines on the top (with diamond markers) represent the percent transmittance data collected using the spectrophotometer for each of the three filters.  The four lower lines (with triangle markers) represent these data collected for the 2- and 3-filter combinations.  The dotted blue lines represent the predicted values based on the hypothesis.  The fact that the dotted lines follow the triangle lines quite closely substantiates the hypothesis.

This figure shows the approximate colors of each filter and filter combination, as determined in the second part of the experiment, using the colorimeter.  The colors are approximate only because not all web browsers render color similarly, or accurately, which makes all web page color of limited accuracy.

Conclusions

The first part of the experiment gives very good evidence for the correctness of the "spectral multiplication" hypothesis.  As for the second part of the question, the data were graphed together in the Yxy and Lab systems, and no consistent pattern is obvious.  It is clear that each color combination results in a color that is lower in lightness than either of its components, and intermediate in hue.  However, the saturation of the resulting color varies unpredictably.  

References

¹The introduction section was adapted from my earlier essay for the Science of Color class:
Sprehn, David. "Additive and Subtractive Color" 12/4/2004.

Background Source:
²Williamson, Samuel J. and Herman Z. Cummins. Light and color in nature and art.  New York: Wiley, 1983. pp. 22-29, 36-39.

Acknowledgements