testing the color wheel

The previous pages explained the traditional color wheel and how to go beyond it to a practical method of color mixing. We had to keep in mind that the color wheel depends on a basic fallacy and contains mixing biases that appear in mixing step scales among "primary" colors.

This page presents my research to measure the biases in the color wheel more precisely, using color mixture measurements made with a GretagMacbeth spectrophotometer. These are measurements that paint manufacturing color technicians must have already done many times, but I have never seen similar results published before.

My test results clarify the significant biases in color wheel geometry, and also explain our color experience as we mix paints. We will get a clearer picture of the biases in the color wheel, but will also find it is not so easy to "fix" these biases in a systematic way.

 testing the mixing method
To begin, let's translate the basic assumptions of the geometrical mixing method into something we can measure precisely.

If we make allowances for differences in the relative tinting strength among paints, then the color wheel assumes that the hue and saturation of color mixtures can be described by a straight line between the two mixed colors.

The figure below shows a straight mixing line between the "primary" colors magenta (M) and light yellow (Y). Five mixing points are shown along the line. Each point marks the hue angle of a magenta+yellow mixture, as these pass from magenta through carmine, middle red, red orange, deep yellow and middle yellow to light yellow.

changes in chroma along a mixing line

The seven gray lines drawn from the center of the wheel to the mixing line indicate the chroma of the mixtures at each mixing point, and the chroma of the "primary" colors yellow and magenta. We observe two things as colors (paint mixtures) shift along the mixing line:

• As a small amount of one paint is mixed with the other, there is a sharp drop in the chroma of the mixture, because the mixing line is at an angle close to the center of the wheel.

• The chroma reaches its minimum at the midpoint of the line, and here changes in chroma are quite small across large changes in hue, since the mixing line is at a tangent to the center of the wheel but roughly parallel to the hue gradations along the circumference.

So the basic rule is roughly:

drop in chroma —> change in hue —> rise in chroma.

If we stand these seven chroma lines in a row, spaced horizontally by hue angle, the tops of the lines describe a curve like a chain hanging between the two "primary" color poles (chroma is represented by the height of the chain).

changes in chroma across a mixing line

And this hanging chain is what we would expect to appear between any two widely spaced, saturated paints, if we measure both the hue angle and chroma of their mixtures. We will test to see if this is true.

I should clarify an apparent contradiction in color mixing that seems hidden in this "hanging chain." The subjective experience of mixing one intensely colored (saturated) paint with another seems to the eye to produce strong changes in color in mixtures near the original paints, but rather small changes in color across the middle hues between them. Does this contradict what I just said about the sharp drop in chroma near the original colors and large changes in hue across the middle mixtures? No: chroma or saturation is basically our ability to discriminate hues, and hue differences near the intense colors are brighter and so easier to see. The hanging chain really tells us that accurate color mixing is difficult for two completely different reasons — near the "primary" colors, because small quantities of paint will have large effects on the apparent color; near the middle mixtures, because they are significantly duller and therefore harder to identify visually.

 mixtures between two "primary" colors
Now the test is easy to design. We simply pick three "primary" color paints (a light yellow, a magenta, and a greenish blue), and mix pairs of them in various proportions. A spectrophotometer measures the chroma and hue angle of these mixtures, and the chroma can be plotted against the hue angle to see how well the "hanging chain" describes changes in chroma across the hues from one "primary" color to the next.

I actually chose two different paints for each of the three "primary" colors, to minimize effects peculiar to any single pigment. The paints were: Winsor & Newton cadmium lemon (PY37) and benzimidazolone yellow (winsor yellow, PY154) as the "primary" yellows, Winsor & Newton permanent [quinacridone] rose (PV19) and quinacridone magenta (PR122) as the "primary" magentas, and Winsor & Newton phthalocyanine blue (winsor blue GS, PB15:3) and Holbein phthalocyanine cyan (peacock blue, PB17) as the "primary" blues.

Each "primary" paint was mixed with the four paints in the other two "primary" colors, creating 12 unique mixing combinations. For each combination, seven test swatches were prepared from mixtures in the approximate proportions (paint 1/paint 2) of 1/7, 2/6, 3/5, 4/4, 5/3, 6/2 and 7/1, or 13%, 25%, 38%, 50%, 63%, 75% and 88%. (Actual paint proportions varied slightly depending on the tinting strength and darkness of the paints, and my errors in making the mixtures.) Then the actual chroma and hue angle of these 84 mixtures, and the 6 pure "primary" paints, was measured with a GretagMacbeth Spectrolino™ spectrophotometer. The results were plotted using Microsoft Excel spreadsheet software. The following graph shows the results.

saturation costs between "primary" color mixtures

The vertical scale shows the chroma of the six "primary" paints and their mixtures; the horizontal scale shows the hue angle in the CIELAB space where 0 (360) degrees is approximately the position of a "primary" magenta color. Each color measurement is plotted as a blue point, with the average of the measurements shown as a red line.

The expected "hanging chain" curve appears between the "primary" color pairs, but we also see several imbalances in the color wheel geometry:

• The "primary" colors should be separated by approximately 120°, or one third of the total circumference of the color circle, but the distance between "primary" yellow and "primary" blue is around 160°, and the distance between yellow and "primary" magenta is less than 90°. The "primary" colors are not equally spaced around the color circle.

• The "primary" paints are not equal in chroma to begin with: magenta is significantly less intense than yellow, and cyan is less intense than magenta. Placing these paints equally on the circumference of a color circle disguises the fact that "primary" colors are not equal in chroma.

• As shown by the vertical dip in the three "hanging chains," the "primary" colors are not equal in saturation costs. This is not simply because they are not equally spaced on the color wheel: the chroma of mixed greens (between yellow and blue) is higher than the chroma across mixed violets (between blue and magenta), even though yellow and blue are slightly farther apart on the color wheel than blue and magenta.

What do these biases mean to practical color mixing? First, saturation costs are attributes of specific paint colors; they depend on which paints we are talking about, as well as how far apart the two mixed colors are on the color circle. Yellow tends to sustain and even raise the chroma of paints mixed with it, while violet has the most intense dulling effect on all other colors.

The change in chroma between magenta and yellow is quite small: the "hanging chain" is almost flat, and never dips much below the chroma of the original "primary" magenta paint. So yellow's "chroma raising" effect is much stronger on the warm than the cool side of the color wheel. Notice also that the scatter of mixture measurements above and below the average line is very large, indicating that there is a lot of variability in the measured chroma. This variability probably comes from very specific interactions between the different pigments — some mixtures are more potent than others.

As reference, I've added markers for the hue and chroma of burnt sienna and burnt umber, a moderately dull and very dull red orange. Notice that "dull" burnt sienna has a higher chroma (is more saturated) than most of the mixed violets and greens! All the mixed warm colors must all have higher chroma than the burnt sienna, or else they would appear brown. (This is the real reason for the old cliche that "warm colors advance and cool colors recede." If you leave browns and tans and pinks out of the picture, then warm colors are all more intense, and often lighter valued, than cool colors.)

Now, if we wanted to straighten out these differences in the saturation costs between "primary" colors, how would we change the color wheel to do that? First, we would have to move magenta closer to yellow; yet most color wheels distort this side of the color space by spacing the warm colors too far apart. Unfortunately, the variability above and below the "hanging chain" shows that a mixing line won't predict the chroma of a warm hued mixture very accurately.

In addition to a closer spacing of the "primary" colors magenta and yellow, the "primary" yellow and blue must be spaced farther apart. So why is the "hanging chain" here higher than the one between magenta and blue, indicating more intense color mixtures? One way to get an answer is to look at the mixing lines plotted on the CIELAB a*b* plane:

mixing lines on the CIELAB a*b* plane

This shows that the lines between magenta and yellow are overall fairly straight: the reason they dip so little in chroma is simply because warm colors are closer together in the color space than blues or greens. (The mixing lines are also highly variable or wavy, effects specific to the different pigment combinations.)

The mixing lines between blue and magenta are straight and consistent, indicating that the color wheel models mixtures in the violet and blue part of the space almost perfectly. This may be due primarily to the fact that these are also the dullest mixtures.

The surprise is that the mixing lines between yellow and blue are strongly curved rather than straight. They bend away from the neutral center of the wheel, making the mixtures of cyan and yellow more intense than we would expect them to be. This effect also happens with other cool mixtures, for example ultramarine blue and lemon yellow (which makes a dull green, even though the mixing line between them suggests it should make a dark maroon) or between phthalo green BS and cadmium orange. These funhouse mirror distortions in the cool mixing lines are one reason that green mixtures are so hard to judge accurately.

You can make fairly accurate color mixing tests of your own, if you have access to a color scanner and Adobe Photoshop or similar image processing software. Photoshop contains a color sampling utility that provides the CIE L*a*b* coordinates for the color of any pixel in an image, and these pixels can be sampled from scanned images of your mixing tests and plotted on graph paper or with a spreadsheet program. Scan the unmixed paint colors, and a Kodak color test card, as reference points for lightness, hue and chroma; compare to the CIE measurements for paints in the guide to watercolor pigments or the CIELAB color chart. Use the "5 by 5" pixel sampling option to average the color over several pixels in the test image. This method will not give highly accurate absolute measurements of color, but is quite serviceable for measuring the relative changes in colors as they are mixed.

 mixtures among tertiary colors
Now let's try to map these color wheel biases in more detail. To do this we will shift our focus to the relationships among tertiary hues as a measurement framework.

The figure below shows the chroma of all paint mixtures in one of my preferred paint wheels. Each line is color coded to the pigment hue it represents; these are also noted at the bottom of the graph as equal hue increments around the color circle. Follow any curve horizontally across the graph to track the changes in chroma that occur in mixtures of that paint with all other paints.

chroma curves of mixtures around the color wheel

This graph represents the overall pattern of saturation costs in the color wheel, and makes the following points clear:

• The colors move in two contrasted groups: a "warm" group of colors that ranges from cadmium lemon yellow (PY37) to quinacridone magenta (PR122), and a "cool" group of colors that ranges from ultramarine blue (PB29) to permanent sap green (a dull yellow green, labeled Sap in the figure). These two groups move in opposite changes in chroma around the color wheel; the warm colors show very wide changes in chroma (from 90% to 25% or lower), while the cool colors cycle within a narrower range. These differences affirm the importance the warm/cool color contrast.

• Paints on the warm side of the color wheel, from nickel dioxine yellow (PY153) to pyrrole red (PR254), lose little or no chroma when mixed with other paints in the same group; the chroma of quinacridone magenta (PR122) actually increases when mixed with other warm colors.

• Warm colors begin to lose chroma when they are mixed with either cadmium lemon yellow (PY37), which contains a small amount of "cyan" reflectance, or with quinacridone magenta (PR122), which contains a significant amount of "blue violet" reflectance, so "blue" reflectance is the most dulling addition you can make to any warm paint mixture.

• The warm colors cadmium scarlet (PR108) and pyrrole red (PR254) continue to shift in the same way when mixed with cool colors. However, cadmium lemon yellow (PY37) and nickel dioxine yellow (PY153), which reflect significant amounts of "green" light, create fairly intense mixtures with the "green" reflectance of phthalocyanine blue (PB15), cobalt teal blue (PG50), phthalocyanine green (PG7) and sap green (Sap). As a result, the yellows and yellow oranges diverge from the reds and red oranges in mixtures on the cool side of the color wheel. Mixtures of yellows and blues increase in chroma, while mixtures of reds and blues decrease in chroma. So the "warm" colors seem to consist of two color clusters, split at around red orange, which behave basically as a yellow or a red pigment in mixtures with all other colors.

• In contrast to the warm colors, the blue mixture curves are widely spaced rather than moving as a close group around the wheel. (This spacing is caused by differences in the amount of "green" or "red" reflectance each blue paint contains.) The blues also do not have a large area of high reflectance in common, as the warm colors get with the "warm cliff" reflectance curve. There is a similarly large separation between the greens. In general, the "cool" colors form a natural spacing of chroma curves, each acting as a different hue in relation to all the other colors: there is no clustering.

• The curve for dioxazine violet (PV23) is separate from all other paints, showing very even changes in the chroma of mixtures across different hues in the color wheel. This verifies the dull and dark visual impact of the color, and shows the systematic dulling effect it has on the lightness and chroma of mixtures with every other paint. This is in fact the curve we should see among all the paints, if the geometry of the traditional color wheel were true.

This graph affirms Leonardo's insight five centuries ago that red, yellow, green and blue are the artists' primary colors. The red-orange cluster moves as a group, the yellows as another group, and the blues and greens move independently in a range defined by a red blue on one side, and a yellow green on the other.

The most important discovery is that the color wheel is biased in several ways as a representation of color mixing. The chroma curves of different colors do not behave consistently across different segments of the color spectrum; warm colors cluster into two groups but cool colors do not; warm colors move together and cover a wide range of chroma, while cool colors move independently within a narrower range ... and violet doesn't seem to behave at all as we'd expect.

If we wanted to predict color mixtures with the standard "primary" color wheel, we would have to use an outward bowing arc, and not a line, to estimate the hue and chroma of mixtures on the green side of the space. We'd use an inward bowing arc, and not a line, to estimate mixtures between magenta and blue. We'd need to use another outward bowing arc to estimate the hue and chroma of mixtures between magenta and yellow. (We could straighten out one side of the color wheel by spacing the hues differently, but this only creates worse problems across other hues.) And, because we depend on the complementary relationships in the color wheel to define color design, the color wheel is also flawed as a framework for selecting color harmonies. (Certainly, using the color wheel in circular "color calculators" is highly questionable.)

All this reaffirms the importance of using the color wheel as a compass for color improvisation rather than a geometrically precise framework for color calculations. The many biases within the color wheel are not something we can learn abstractly. They are part of the complex terrain of color mixing that we grasp only after taking many journeys through it with many, many paintings.

N E X T :   an artist's color wheel

Last revised 01.12.2004 • © 2004 Bruce MacEvoy