Quantitative Color Subtraction
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
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.
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
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.
Download in Excel format
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.
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.
1The introduction section was adapted from my earlier essay
for the Science of Color class:
Sprehn, David. "Additive and Subtractive Color" 12/4/2004.
2Williamson, Samuel J. and Herman Z. Cummins. Light and color in nature and art. New York: Wiley, 1983. pp. 22-29, 36-39.