additive & subtractive color mixing
The previous pages have described the fundamentals of color vision. Now the focus narrows to a single issue: how color mixtures can be explained.
Painters mix their paints to shape the light reflected from a painting, and the viewer's eye interprets this reflected light as color in space. These two extremes of color experience the mixed paints, and the interpreting eye are described by two separate and unequal color mixing theories.
Isaac Newton's hue circle, a geometrical arrangement of the different colors seen in a solar spectrum, is the original statement of additive color mixing. Newton explained that the chromaticity (combined hue and saturation) of a light mixture could be predicted as the weighted average of the ingredient hues around the hue circle. However, the most important feature of additive mixing, as specified by Newton's averaging method, is that the chromaticity of ingredient lights always determines the chromaticity of their mixture.
Newton explicitly stated that color is a perceptual property, not a physical attribute, which meant that the light mixtures occurred in the eye, not in the light. Today we define the chromaticity of light mixtures as the proportional stimulation induced in the separate L, M and S cones relative to the stimulation across them all (the color's brightness). Three red orange, green and blue violet (RGB) lights are used to demonstrate additive color mixing, because they are the most direct way to stimulate the separate L, M and S cones.
However, painters and dyers had long experience with paints and dyes, and they affirmed that material color mixtures behaved very differently from light mixtures: they get darker rather than brighter, and they seem defined by red, yellow and blue primary paints (now replaced by the modern choice of cyan, yellow and magenta or CYM). In subtractive color mixing, colorants absorb or subtract wavelengths from filtered or reflected light. This was pointed out by several 18th century artists and naturalists, including Jakob Le Blon in 1725, Brook Taylor, Moses Harris, Philipp Otto Runge and Thomas Young. But it was only in the late 19th century that material color mixtures were understood as an indirect form of additive color mixing. Until then, artists were taught that light mixtures and paint mixtures were both produced by the same red, yellow and blue "primary" colors.
Unfortunately, the color mixture "predictions" made by subtractive color theory are often inaccurate, because the light absorbing properties of a colorant are affected by its physical state its particle size, transparency, density, dispersion or medium, the color of the substrate, the other colorants it is mixed with, the thickness of the color layer, and so on. I call these problems substance uncertainty: because of them, the color of ingredient substances does not determine the color of their mixtures. Often, colorants must be physically mixed in order to find out what their mixture color will be.
Even among artists today, misconceptions about additive or subtractive color mixing are the cause of many misleading ideas about color. Most of these misconceptions are taught to artists as "color theory." I explain why artists should rely on mixing experience instead.
additive color mixing
Additive color mixing explains how the eye interprets light wavelengths in the perception of color. It describes the color structure of light perception from four cardinal lights: red orange, middle green and blue violet, plus the white light (or white point) defined by mixing the three colored lights together. This trichromatic foundation is in turn the basis for all modern chromaticity diagrams, the identification of visual complementary colors, and the definition of modern trichromatic color models.
Additive Mixtures Occur In The Eye. The beauty of additive color mixing principles is in their narrow scope. They are limited to a single sensory process for the explanation of color mixtures: the average or typical responses of the L, M and S photoreceptors to light.
These LMS cone outputs can be predicted fairly accurately from the light's spectral emittance curve and the cone sensitivity curves. In fact, the LMS cone sensitivity curves are actually only a mathematical restatement of the quantities of three "primary" lights necessary to mix a specific wavelength of spectral light the RGB color matching functions. This close link between light energy and cone outputs allows us to describe accurately the resulting color perception for those with "normal" color vision.
But wait ... isn't additive color mixture really a theory of how light mixtures behave? No, it is not. This misconception arises because light is obviously the only stimulus that the eye normally responds to, and because lights of various colors are explicitly manipulated in color matching experiments used to measure additive color mixtures. But light is the stimulus, and additive color mixing describes the response of the eye to a light stimulus.
This description only applies to unrelated colors that is, a light stimulus perceived without any surrounding physical context. Unrelated colors can be created by shining a diffuse light directly into the eye, or by reflecting the light into the eye from a colorless (white or gray) surface; the source of the light in an unrelated color does not matter, because additive color mixing happens in the retina, not in the light. Special steps must be taken to reduce the effects of a visible context. When this is done the connection between light stimulus and color response is predictable.
Color scientists diagram this connection between cone responses and perceived color using the trilinear mixing triangle devised by James Clerk Maxwell. This triangle defines the chromaticity (hue and saturation) of any unrelated color as a proportional mixture of the three cone outputs; the "white" brightness is approximately equal to their sum. These outputs, in turn, can be exactly reproduced by a specific mixture of three actual (visible) "primary" lights typically red, green and blue violet. All modern color models are based on additive trilinear values that specify both the chromaticity and luminance of a color. In fact, many late 19th and early 20th century artists learned the basics of color theory in terms of a mixing triangle not a color wheel.
The Additive "Primary" Lights. Now, how do we illustrate, verify or measure the rules of additive color mixing? Obviously, by manipulating the outputs of the separate L, M and S cones. How do we manipulate these outputs? By stimulating them with three colored lights red, green, and blue violet (RGB). Necessarily, these lights create a fourth "primary": the "white" light mixture of them all. So "white" light can be substituted for any of the colored lights in a color mixing demonstration.
The basis of additive color mixing is trichromatic metamerism: the color produced by any spectral emittance curve, no matter how complex the curve may be, can be exactly matched by the visual mixture of no more than three lights: either three strongly saturated (single wavelength or monochromatic) lights, or at most two monochromatic lights mixed with a "white" light. All physically possible light colors can be reduced to the mixture of just three simple lights.
Here, for example, are the "primary" RGB colors of your computer monitor. Note that the green primary contains too much yellow, and the blue primary not enough violet, dulling all purple and blue green mixtures.
There is an important limitation to the use of trichromatic primaries: we can't mix all possible colors with the same three lights, as explained below. Even so, the incredible complexity of light in the physical world is reduced by the eye to just four primaries three independent cone outputs, and the "white" light that defines their achromatic mixture.
How do color scientists create the strongly saturated RGB lights used in color matching experiments? For precise color measurement, the primary lights are single wavelength or monochromatic lights, isolated from the visible spectrum by a system of prisms, that are mixed by shining them onto a diffusion glass or white surface visible through an eyepiece. Less saturated but higher luminance lights have been created by passing three separate beams of "white" light through separate broadband red, green or blue transmission filters, and mixing the colored beams as before. A third (and relatively weak) method uses partly overlapping disks of colored or painted paper that are visually mixed by spinning them rapidly on a color top.
additive color mixtures
as demonstrated with filtered lights; note that each pair of RGB primaries mixes one of the CMY primaries
The illustration (above) shows the typical demonstration of additive light mixtures, made by shining three overlapping circles of filtered light onto an achromatic (gray or white) surface. If the surface is illuminated by both the red and green lights, but not by the blue light, then the eye responds with the color sensation of yellow. A magenta color results from the mixture of red and blue violet light, and cyan from the mixture of blue violet and green. In additive color mixing, yellow and blue don't make green they make white!
The "White" Color Theory. It's handy to think of additive mixing as the "white" color theory. Mixing light wavelengths from the "red," "green" and "blue violet" parts of the spectrum adds luminosity and negates hue to shift the mixture color of lights from dim pure hues toward bright whites. The key principle is that the eye always adds together all the wavelengths of light incident on the retina nothing is lost and it is this total light sensation that the eye interprets as color.
This additive behavior leads to an important constant in color vision: the chromaticity and brightness of lights always predicts the chromaticity and brightness of their mixture, for lights from moderately dim to bright but not dazzling. This is true regardless of whether the lights are monochromatic (a very pure hue, as we see in homogenous or single wavelength light) or complex (as we see in a mixture of many different spectral wavelengths, for example a "white" light passed through a colored filter). In additive color mixing, for both normal and colorblind vision:
the brightness, saturation and hue of any two or more lights predicts the brightness, saturation and hue of their mixture
any two lights that appear to be the same color will mix identical colors with any third light even if the spectral emittance profiles of the lights are different (that is, they contain different wavelengths in different proportions)
if two separate light mixtures have an identical color, then adding a third light in the same quantity to both of them will result in identical color mixtures
these points are true, even though is not possible to deduce the spectral profile of a light from its color alone; for example, the chromaticity (hue and saturation) of any spectrally complex light can always be exactly matched by one or two monochromatic wavelengths mixed with some quantity of "white" (achromatic) light.
These principles summarize the metameric mixture rules of additive color mixture. They were first stated by Hermann Grassmann in 1853 and are known today as Grassmann's laws, though in fact they are not laws but generally accurate descriptions of color mixing in mesopic and moderate photopic light sources.
We will discover that equivalent subtractive metameric rules do not exist in the many examples of material color mixing, and that lack of predictable consistency in substance mixtures is the most important difference between the additive and subtractive color mixing frameworks.
Real Lights and True Primaries. Let's examine further the additive color mixing demonstrations with colored lights, as they are probably the main reason why artists believe that the RGB primary colors can reproduce all color mixtures, or that the additive primaries are "real" colors (that is, visible physical lights), or that the lights used in additive color mixing demonstrations must be RGB lights and no others the choice of lights is fixed rather than arbitrary. All three beliefs are false.
The Additive Primaries Are Invisible. The diagram at right shows the location on the CIELUV chromaticity diagram of three monochromatic lights (at 460nm, 530nm and 650nm) that have frequently been used in color vision research to analyze trichromatic color matches and opponent color mixtures.
The focus here is on the white triangle or gamut that connects the three primary lights. This defines the range of actual additive color mixtures it is possible to make with those three primaries. This gamut encloses most, but not all, of the chromaticity area, which defines the area of all physically possible light colors. A significant portion of the chromaticity diagram is outside the gamut. In other words, the "real" RGB primary lights cannot mix all visible colors.
Thus, the "green primary" gives full mixing coverage along the red to yellow colors, but it cannot mix (with the "blue violet" primary) the most intense greens, blue greens and blues. In addition, the "blue violet" and "red" monochromatic primaries cannot mix the most intense purples and red violets.
The true additive primaries, the only "primaries" that can mix all possible colors, are the outputs from the L, M and S cones. We are never aware of these outputs directly and therefore they are invisible. We only experience them as the tendency toward a red, green or blue color sensation that results from the combination and interpretation of these outputs in the visual cortex.
How Do We Choose the RGB Lights? Some artists think that these primary lights are the same hues that most stimulate the three receptor cones. This also is false. The cones are actually most sensitive to "greenish yellow," "green" and "blue violet" wavelengths, as shown below. Red orange, green and blue violet lights are used by convention and convenience, and it is from these color matching lights that we get the names red, green and blue assigned to the additive primaries.
additive primary colors are illustrative only
the wavelengths of maximum sensitivity for the L, M and S cones (top) are unrelated to the colored lights used to simulate the cones in additive color mixing demonstrations (bottom)
There's a simple logic for choosing these primary lights. Almost any light wavelength that stimulates one cone will also stimulate one or both of the other cones, because the cone sensitivity curves (especially L and M) overlap. To explain color mixing as the result of three independent types of photoreceptor response, we need three light wavelengths that each stimulate one cone much more than the other two. In other words:
An ideal additive primary color must stimulate only one type of receptor cone (L, M or S) as strongly as possible, and stimulate the other two types of cone as little as possible.
So, within each section of the spectrum where the L, M or S cone is the dominant receptor, we pick a wavelength that creates the greatest difference in response between that cone and the other two. This occurs at around 420 nm in "violet" light and above 680 nm in "red" light. However, these monochromatic lights are very close to the spectrum extremes, and are therefore visually quite dim. In practice, the hues of the R and B primary lights are often shifted away from the extreme ends of the spectrum to provide more luminance in the lights, given the method used to generate them. The G light is always bright enough, so it is usually positioned at the point where it achieves both a high relative contribution and the most saturated yellow when mixed with the R light. This is usually a green with a dominant wavelength between 510 nm to 530 nm. Sometimes a very small quantity of "violet" light is mixed with the red primary, to eliminate the yellow tint in spectral "red" light.
Does Additive Mixture Require RGB Lights? Many artists assume that red, green and blue violet lights must be used to explain or demonstrate additive color mixing. Not true. The choice of lights is arbitrary, and one selection of primaries is better than another only if we require the mixture gamut to be as large or comprehensive as possible.
We could just as easily demonstrate additive color mixing with colored lights representing the subtractive primary colors cyan, yellow and magenta, although most of the blues, greens and reds that we could mix with these lights would appear quite whitish or unsaturated.
Again, the somewhat arbitrary procedures for choosing the additive primary lights are acceptable because the real lights are not the actual basis of additive color mixing. The true additive primary colors are the photoreceptor outputs. We use RGB colored lights to symbolize the LMS receptor outputs, because they are also the most effective way to manipulate those outputs.
the gamut of RGB primaries
additive light mixing gamut defined by lights at 460, 530 and 650 nm
Additive Prismatic Mixtures. There is an arcane but influential line of discussion in color theory that is based on mixtures of light and color produced by a prism. This merits a brief discussion, as it contributes to a deeper understanding of additive mixture.
In his Farbenlehre (1807), Johann Wolfgang von Goethe places considerable emphasis on the colors produced by either passing a beam of light through a prism or viewing a black and white pattern through a prism. Both forms of refraction produce fringes of "red"+"yellow" or "blue"+"violet" light along the edges between light and dark, and these fringes, at a distance, merge in the middle to create the solar spectrum. That 18th century additive mixture demonstration is explained here.
I encountered a more interesting demonstration at the ColorCube web site, where a prism is used to analyze the mixtures of light in the colors displayed on a computer monitor. This use of a prism was first described in Isaac Newton's Opticks (1704), where colored threads and painted cards are used instead of a computer monitor.
To begin, scroll this web page until the image of the prism color bars (below) is approximately at eye level as you look at your computer monitor.
the prism color bars
Align the prism so that it is level end to end and parallel to your computer monitor, with one corner pointing down and the top side flat. Hold it halfway between your eye and your monitor, below the prism bar image, and close one eye. Move the prism up and down until you see part of the prism image in the lower face, then bring the prism closer to your eye to see the whole image (diagram, right).
Once you have the prism properly aligned, the prism color bars on a computer LCD monitor will look like the image (below). In this image:
The fringes that appear at the top and bottom edge of the black rectangle match the fringes that appear when any black and white pattern is viewed through a prism. (Note that the gradient "red">"yellow" is always in the same direction as the flat>corner orientation of the prism.)
The colored bars on the black background are spread out into their spectral components, which match parts of the complete spectral image created by the white dot at the right end of each bar. These bands are not equally pure for example, the green prism image shows ghostly traces of "red" and "blue" light because the computer LCD filters are not monospectral. All the spectral hues appear in one or more of these prism color bar images.
In contrast, the color bars on the white background are all decomposed into three colors only the subtractive "primaries" yellow, green blue and red violet. These three colors align with the prism images of the black dots at the left end of each bar.
prism image of the color bars
viewing the prism color bars
hold the prism lower than the color bars with the top surface flat; the prism image will appear through the bottom face
Why are the spectral images of color bars, displayed on a black background, separated into their component spectral hues, while the spectral images of the same bars, displayed on a white background, decomposed only into the subtractive primary hues? The diagram (right) illustrates the answer.
Because the prism refracts "violet" wavelengths at a greater angle than "red" wavelengths, the spectral image of every part of the prism diagram is elongated vertically by about 4 times. This appears in the spectral image of the white dots on the black background (diagram, above), which appear as four separate dots, in the vertical order red, green, blue and violet. (Note that this elongation varies with the rotation of the prism in relation to the monitor image.)
The image elongation means that the spectral image of any area of the diagram is overlapped with the spectral images of the areas above and below it. The diagram (right) represents this by copying the four colored dots of the "white" spectral image diagonally side by side; the "red" of each spectral image is aligned by a dotted line to the image area that created it. These overlapping spectral images are added together horizontally by the eye along each dotted line: a white area appears "white" in the spectral image because it consists of its own "red" light, plus the "green" light of the area just above, plus the "blue" light of the area above that, plus the "violet" light of the area above that. We see this by adding the colored dots along the dotted line for any white image area: W = R+G+B+V.
The true spectral image of each colored bar appears when it is presented against a black background, because the spectral image is overlapped with the spectral image of black, which creates no color. So, we copy this spectral image (with its vertical black background) into the location corresponding to the spectral image of each R, Y, G, C, V and M colored bar.
These spectral images contain gaps, because certain wavelengths are missing from a color bar. For example, the spectral image of the red bar (viewed against the black background) contains no "green", "blue" or "violet" light. These missing spectrum sections are indicated by white dots in the diagram (right).
We get the subtractive colors by simple additive mixture: magenta results when "green" is missing from the spectral image of a color bar; cyan when "red" is missing, and "yellow" when either "violet" or "blue" are missing.
As confirmation, the diagram explains the relative vertical displacement of the prism images of the same bar on the white and black backgrounds. For example, the "magenta" and "yellow" spectral image of the red bar on the white background is lower than the "red" spectral image of the red bar on the black background; the "cyan" spectral image of the cyan bar on the white background is higher than the "green", "blue" and "violet" spectral image of the cyan bar on the black background.
To test your understanding, use a similar diagram to explain why Goethe's "blue" and "violet" fringes appear at the top edge of the black background, with the "red" and "yellow" fringes along the bottom.
A Scientific Theory of Color Vision. For many centuries, the behavior of color mixtures was difficult to explain because material colors, which seemed to be anchored in "real" objects of the external world, was not conceptually distinguished from the "illusory" colors in rainbows or prisms. The two types of mixtures behaved differently, but the reason for the difference was unknown.
The trichromatic theory provided the clarifying explanation and prediction of all color sensations as arising in the behavior of the eye. Because the L, M and S receptor responses can be predicted mathematically from the summed intensity of all wavelengths in a light stimulus, the additive primaries empirically connect a measurable light stimulus to a measurable (matchable) color sensation at least, in experimentally restricted viewing conditions. This is what makes additive color mixing, in the scientific sense of the word, a theory of color vision.
subtractive color mixing
Subtractive color mixing is, in comparison to additive color mixing, a flawed attempt to describe the colors that result when light absorbing substances are mixed. It is not a rigorous theory at all, but rather a description of the way colors should mix in the ideal case. Subtractive mixing theory imitates the main features of additive color theory, and to understand why subtractive color mixing does not accurately describe the colors of substance mixtures, we need to unmask these points of imitation one by one.
Subtractive Mixtures Occur in Substances. First, let's get clear on what subtractive mixing rules are trying to explain. All subtractive color mixing occurs in the external world, in a wide variety of material substances.
In principle, subtractive color theory ought to be able to explain the color changes that occur in any kind of material mixture. In fact, it should also be able to explain the color changes that occur when a surface is illuminated by different illuminants (colors of light). And this is the fundamental difficulty with subtractive mixing theory: it must explain the behavior of too many different substances. This problem is minimized, but not at all eliminated, by limiting the application of subtractive mixing principles to manufactured colorants. Even here, the variety of materials would include light reflecting substances (such as powders, paints, dyes or inks) and light transmitting substances (such as photographic filters, stained glass or tinted liquids).
If we are only interested in the color of a pigment or dye in isolation, then we can define its material color attributes by measuring its spectral reflectance curve. This defines the light mixture that is reflected to the eye in fact, all reflectance curves define the subtractive mixture of a pigment or dye with "white" light. The reflectance profile in turn defines the photoreceptor responses under normal viewing conditions, or the material's color. So long as we only consider the spectral profile of the light entering the eye, or the mixture of spectral profiles as light mixtures entering the eye, we are in the domain of additive color theory and predicting the color produced by the reflectance curves and their mixture is a straightforward problem. But when two or more colors are physically mixed, or combined as light filters, all the physical qualities of the substances interact, which can cause their reflectance curves to combine in unexpected ways and produce an unexpected color in the mixture. The most important of these physical mixture issues are:
1. The apprarent color does not define a unique reflectance curve. The same green color of paint can be produced by many different reflectance curves, and these different curves will produce different blue colors when each is mixed with the same purple paint. This is the problem of material metamerism. The apparent color, not the reflectance curve, is all that a painter has to work with. The painters' subtractive mixing rules are not stated in terms of reflectance curves, as "this reflectance curve mixed with that reflectance curve yields this apparent color", but in terms of categorical color labels, as "yellow and blue make green".
subtractive colors from prism color bars
2. The reflectance curve changes with the physical state of the colorant. A pigment such as quinacridone violet (PV19) does not have fixed, unchanging color attributes. The reflectance curve, and hence the apparent color under standard viewing conditions, changes with the physical state of the pigment the pigment may be dry or wet, it may be suspended in water or oil, it may be diluted or concentrated, it may be displayed as a thin or thick layer (diagram, right). In most colorants, each of these physical changes will alter the reflectance curve significantly.
3. Separate colorant reflectance curves do not specify the color of the physical colorant mixture. This problem arises because there are many more physical attributes to a colorant than its reflectance properties. The same reflectance curve can be produced by substances that differ greatly in particle size, refractive index, transparency (hiding power) and tinting strength, and these all can affect how the colorants will appear when dispersed in a vehicle, or which colorant will dominate when used in a mixture with other dyes or pigments.
4. Mixture colors are different in different types of subtractive mixture. Even if we find out the mixture color of two colorants by actually mixing them, that color does not necessarily predict the color that will result if they are mixed in other ways there are different kinds of subtractive mixture. Subtractive mixtures of reflecting paints or dyes obey different mixing rules than subtractive mixtures of transmitting filters; paints applied to highly absorbent white paper appear duller and whiter than paints applied to heavily sized white paper; pigments applied as watercolor (which does not form a paint layer) appear different from paints applied as oils or acrylics (which do form a paint layer).
Of all these issues, material metamerism (1) is probably the most troublesome. In additive color mixing, metamerism does not conflict with our ability to describe the unrelated color perception that results from light mixtures, because the visual chromaticity of a light predicts its mixing behavior with other lights. But in subtractive mixtures it is not the color of the substance but its reflectance profile and physical attributes that determine its behavior in physical mixture, and this information is simply discarded when we define substances indirectly, in terms of visual color categories such as blue or yellow.
Even if we do know all the important physical attributes of the colorants we mix, the prediction of their subtractive mixture from their separate reflectance curves is mathematically complex. A paint layer is essentially a physical object (it has thickness, surface, transparency and so on), so the prediction must significantly limit or simplify the paint's material characteristics. (See comments on the Kubelka-Munk theory below.) These arbitrary limitations and lost complexities mean there are usually more practical and reliable ways to guide color mixing decisions. As a color chemist in the automotive industry explained to me, you mix the two pigments and look at color you get. Or as I like to say, subtractive color mixing concepts are only useful as a compass to color improvisation.
I give the name substance uncertainty to these confusing connections among a colorant's reflectance curve, physical attributes, apparent color when prepared in a specific medium, and mixture color with other colorants, and I explore these issues in more detail below. For now, the essential point is that we can't reliably use the color appearance of two paints to predict the color appearance of their mixture. This is the most important point of difference with additive color theory, where the color of two lights of moderate brightness can predict the color of their mixture.
The Subtractive "Primary" Colors. Subtractive color mixture is fundamentally about substances, and for that reason it has also been the color mixing practice with the longest commercial application. Additive color mixing has been technically important only since the advent of color television 50 years ago. Subtractive mixtures have been recognized and used in dyers' and painters' trades since ancient Greece. That long trial and error practice fixed on blue, yellow and red as the best subtractive primary colors, which achieved the form of a published "theory" in the 18th century.
In fact, the historical choice of primary colors was limited by the historical availability of suitable pigments, which until the late 19th century were comparatively dull and dark. Color choices today have been greatly expanded by modern industrial chemistry, so that the modern subtractive "primary" colors are cyan, yellow and magenta (CYM), as shown in the figure below. The traditional and grade school subtractive primaries blue, yellow and red are relicts of 18th century color theory and are best forgotten.
subtractive color mixtures
as demonstrated in overlapping sheets of transparent colored plastic (transmission or filter mixture)
These primaries produce the mixtures familiar to us in paints. When we mix together a yellow and magenta paint, the resulting mixture is scarlet or orange; the mixture of magenta and cyan yields purples and blues, and yellow greens and blue greens result from the mixture of yellow and cyan.
What Is "Primary" About Subtractive Primary Colors? The subtractive cyan, yellow and magenta primaries are presented as the basic or elemental colors in subtractive color mixing, no matter what kinds of materials paints, inks, dyes, pigments or filters are used to embody those colors.
This substitutes apparent color, the color sensation in average or individual eyes, for the chemical and physical attributes of paints or inks that determines the reflectance curve. Apparent color sweeps under the rug all the material qualities of paint pigments, especially mineral and opaque ones.
The "Black" Color Theory. The first question to address is: what is the universal visual effect on color that happens when we mix material substances?
The answer: when we combine paints, dyes or filters, we do not increase their light reflecting (or transmitting) behavior but their light absorbing behavior. A subtractive mixture absorbs light wavelengths that either colorant absorbs by itself. Subtractive mixture always increases darkness in material colors.
This makes subtractive color mixing the "black" color theory. Mixing all three subtractive primaries produces a dark neutral, the opposite of white, because each paint subtracts or absorbs light that might be reflected by the other. Subtractive color mixtures can only be made lighter by diluting the amount of pigment in the mixture with white paint or water; either remedy weakens the color saturation. So subtractive mixture typically also reduces the hue purity (increases the grayness) in the color of mixed substances.
For the same reason, a subtractive mixture reflects only those light wavelengths that both colorants reflect by temselves. A mixture can only reflect in the "blue" wavelengths if both colorants reflect significant amounts of "blue" light. Yellow and blue make green only because both yellow and blue paints reflect significant amounts of "green" light. Ultramarine blue mixed with cadmium red makes a dark, dull purple, but mixed with quinacridone red (which is the same hue as cadmium red) makes a brighter purple, because cadmium red reflects no "blue" light, but quinacridone red does.
Multiplicative Darkness Mixture. The second point of difference with additive color mixing has to do with how the colors combine in subtractive mixtures. This is some form of multiplication or product of the separate reflectance curves, assuming that the two paints have identical tinting strength, particle size, refractive index and hiding power and are mixed in equal proportions. An example is shown below for two common paint colors, categorically labeled magenta and yellow.
subtractive color mixing of yellow and magenta
white line shows reflectance curve of subtractive mixture; high reflectance remains only where both paints reflect light
In this mixture, the yellow absorbance subtracts light from the "blue" reflectance in magenta, and the magenta absorbance destroys the "green" reflectance in yellow. The common reflectance, the light reflected by both paints, is largely in the "red orange" and "red" part of the spectrum, which is the approximate hue of the mixture. It is specifically this mutual antagonism among light absorbing substances that subtractive color mixing tries to explain.
As explained above, this mutual antagonism depends on many physical attributes of the colorants, so there are no "Grassmann's Laws" for material color mixing. However, for most paints and dyes in most applications, the reflectance resulting from a physical mixture of pigments is usually close to the geometric mean of the separate paint reflectance curves across each wavelength in the spectrum. (The geometric mean of two numbers is the square root of their product.) For example, if a white paint reflects 98% of the light at 452nm, and a black paint reflects 10% of the light, their mixture (in equal proportions at equal tinting strength) will reflect approximately 31% of the light at that wavelength.
However, because subtractive mixing behaves differently in different substances, we have to use a different mixing rule for filters, or for pigments in solution, where the mixture color is usually equal to the product of the separate transmission profiles. That is, two filters that separately transmit 98% and 10% of a wavelength will transmit about 9.8% of the light when they are combined.
When we apply these mixing calculations to the reflectance or transmission profiles, we find that the mixture profile is always closer to the darker profile in the combined total reflectance curve, or darker than the darkest profile in the combined total transmission curve. Mixing white and black in equal proportions does not reduce the luminance of white by half, but by at least two thirds. As a result, sequentially (transmissively) combining any two colorants always results in a darker mixture than physically mixing the two same two colorants; and physically (subtractively) mixing two colorants always results in a duller, darker color than visually (additively) mixing the same colorants, for example on a color top!
Double Cone Stimulation. We've identified the multiplicative combination of light darkening (absorbing) qualities as the two universal traits of subtractive color mixture. But we haven't identified the attributes that define the subtractive primary colors cyan, yellow and magenta. What is the material attribute of "yellowness" that occurs in all yellow colored substances? Why do we choose those visual colors, and not some others?
The answer begins with the fact that subtractive mixtures always destroy ("subtract") the material luminance, making color both darker and duller. To compensate for this, painters should start with colors that are both light and bright (that is, light valued and highly saturated).
However, if we play around with various light valued, high chroma paints, as the ancient painters and dyers did, we discover that some do much better as subtractive primary colors than others. Why? Because the key to subtractive primaries is not in their light value or high chroma alone. It's in how that color intensity affects the eye:
An ideal subtractive primary color must stimulate two types of receptor cones (L and M, or M and S, or L and S) as strongly and equally as possible, and stimulate the third type of cone as little as possible.
In other words, the subtractive primaries are only an indirect way to specify the L, M and S cone responses of additive color mixing! Once again, these cone outputs are the "true" color mixing primaries.
Some texts express this point in negative terms, saying that each subtractive primary absorbs or "subtracts" from "white" light the wavelengths representing a single additive primary. These are often written (or diagrammed) as subtractive formulas, including both white (W) and black (K):
C = W R
Thus cyan subtracts "red" light from the total "white" light spectrum; magenta subtracts "green" light from the spectrum, yellow subtracts "blue" light; black subtracts all light from the spectrum.
This way of defining subtractive primaries is helpful to remember their complementary hues, but it is essentially a definition that allows for dull "primary" hues. Thus, raw umber almost completely absorbs "blue" light, and iron (prussian) blue almost completely absorbs "red" light, so they can be used as a primary yellow and blue, even though they also absorb light from other parts of the spectrum and therefore appear relatively dull or dark.
Ideal Subtractive Primaries. Once we have determined that the best subtractive primary colors will produce the maximum possible stimulation in two types of cones and the minimum possible stimulation to the third type of cone, we simply choose the colorants that achieve those receptor effects as far as possible in a physical color stimulus.
To guide our search, it turns out that we can match those criteria by means of an ideal reflectance profile. The ideal profiles we look to are optimal colors, which define the theoretically brightest possible colors in a nonfluorescent physical surface. These colors always have the maximum possible saturation or hue purity of any surface color at a given hue and lightness, and the maximum possible lightness of any surface color of a given hue and saturation.
ideal spectral reflectance curves for subtractive primary colors
each subtractive primary reflects or transmits the light representing two additive primary colors
To qualify as an optimal color, a reflectance profile must have two attributes: (1) the reflectance (or transmittance in filters or dissolved dyes) must be at 100% or 0% at every wavelength across the entire spectrum, and (2) the reflectance curve may change from 100% to 0%, or from 0% to 100%, no more than two times within the spectrum (arbitrarily, 400 nm to 700 nm).
By these criteria, there are roughly 136,532 unique (but not necessarily perceptibly different) optimal colors, distributed equally across all levels of lightness and chroma. If we sort through these until we find the reflectance profiles that stimulate two of the three types of cones as much and as equally as possible, while reducing stimulation to the third type of cone as much as possible, we have the reflectance curves for optimal primary paints (top row in the figure).
The cone response profiles (middle row) show how these optimal subtractive primaries affect the eye. For example, the ideal yellow colorant transmits or reflects all the "red," "yellow" and "green" light and omits all light in the "blue" and "violet" wavelengths; this spectral profile results in high stimulation to the L and M cones, with low stimulation to the S cones. The bottom row shows the perceived colors that result from the cone responses in additive color mixing: "red" and "green" reflectance, with no "blue" reflectance, appears as a bright yellow (Y). The idealized profiles, cone responses and perceived color for magenta and cyan are presented in the same way.
reflectance curve changes
the masstone reflectance curve of quinacridone violet (PV19) changes shape, not just overall level, when it is diluted into a tint
These optimal colors are the physical ideal case. So it is instructive to see how the chromaticity of these ideal subtractive primaries compares with the chromaticity of historical and contemporary pigment choices for subtractive primary colors in watercolor paints or printing inks. In general the preferred yellow primaries are too red, the red primaries are too yellow, and the blue primaries are too red, compared to the theoretical ideal. In fact, the theoretically optimal pigments underlined in the diagram bismuth yellow (PY184), cobalt teal blue (PG50) and cobalt violet (PV49) have never been used as "primary" paint or ink pigments!
The main reason for these preferences is that the phalocyanine and quinacridone pigments actually work very well as "primary" colorants. They have a much smaller particle size, higher tinting strength, higher transparency and greater chroma in tints than the mineral cobalt pigments, especially when used as inks. More important, however: historical visual preferences have always required the use of primary colors that can mix intense yellow to red colors, at the cost of relatively dull green and purple mixtures.
Finally, in the ideal cone response profiles above, all three physically ideal subtractive primaries stimulate to a significant degree the third or "unwanted" L, M or S cone. (Note in particular the M response in magenta.) In each case we cannot achieve a visually pure primary hue of paint, because of a physiological limitation: the overlap between the M cone and L cone fundamentals. We just can't stimulate the L cone with "orange red" light, or the S cone with "violet blue" light, without also stimulating the M cone, just as if we stimulated it with "green" light. Paradoxically, the "invisible" quality of the true additive primaries is partly responsible for the "impure" quality of the material subtractive primaries.
Mixing Subtractive Primaries. What happens when these ideal primary paints are mixed? Because any two subtractive primaries will share reflectance in either the "red," "green" or "blue" wavelengths associated with a single additive primary color, the mixture of two subtractive primaries holds constant the response of a single photoreceptor. Yellow and magenta share "red" reflectance that stimulates the L cone, yellow and cyan share "green" reflectance that stimulates the M cone, and magenta and cyan share "blue" reflectance that stimulates the S cone.
mixing two ideal subtractive primary colors
reflectance representing a single additive primary remains high; other parts of the spectrum also reflect light (white line shows cone response to a 50:50 paint mixture), and this flatter cone response profile is perceived as a grayer color
What about the other two photoreceptors? In any subtractive mixture, the remaining two additive primaries must compete with each other. As shown above for the mixture of yellow and cyan, the "red" light that primarily stimulates the L cone is reflected by yellow but absorbed by cyan; the "blue" light that stimulates the S cone is reflected by cyan but absorbed by yellow. So both are substantially darkened.
subtractive primary colors
defined as optimal stimuli, with real CYM pigments in the CIELAB a*b* plane
The common or shared additive primary (that is, the eye's M response, in the case of cyan and yellow) remains roughly the same, but the other two additive primaries (the eye's L and S responses) work against each other: like a seesaw, as "blue" reflectance goes up, "red" reflectance goes down, and vice versa. The resulting light mixture is interpreted according to additive principles as containing mostly "green" reflectance, but ranging from a blue green (when "blue" reflectance greatly exceeds the "red" reflectance) to yellow green (when "red" exceeds "blue").
These tradeoffs also mean that mixtures of two subtractive primaries reflect light from all parts of the spectrum. The result is a flatter cone response profile (shown in the middle diagram of the figure), which creates the perception of a less saturated color mixture a color closer to gray. This explains the saturation costs in subtractive mixtures the tendency of paint mixtures to be darker and grayer than the original paints.
These saturation costs the unwanted third cone stimulation in ideal colors and the added "white" reflectance in real colors are the fundamental reason why primary colors are either imaginary or imperfect, as explained here. There is no combination of three real primary colors in a specific medium (dyes, paints, phosphors, filters) that can mix every possible color in that medium. And any set of primary colors that can mix every possible (visible) color must be imaginary they cannot be embodied in any material substance or light source, so they cannot be experienced as colors by the eye.
Don't Confuse Additive & Subtractive Mixtures. I hope you now understand why all color mixing involves the retinal response to light; the only issue is whether or how we let the physical properties of substances muck with the behavior of the light stimulus.
Because subtractive color mixing (in materials) is actually an indirect manipulation of additive color mixing (in cone responses), the two types of color mixture can be demonstrated in superficially similar ways. To avoid confusion, remember that the fundamental difference is whether light wavelengths are excluded by the colored substances before the light reaches the eye (the light mixing occurs in the external world), or light wavelengths are separately able to reach the receptor cones (the light mixing occurs in the eye):
colored transmission filters in the additive mixing demonstration, a colored yellow filter is placed over one beam of white light, and a blue filter over a second beam of white light, and the two colored beams are overlapped on a reflective surface. Because each filter is placed over a separate beam of light, the blue and yellow lights are separately reflected to the eye, where they both affect the receptor cones to create the sensation of "white" light. In the subtractive color mixing demonstration, the same blue and yellow filters are both placed over a single beam of light. Then the two filters act in combination before light ever reaches the eye; the only wavelengths that can pass through both filters at the same time are in the "green" section of the spectrum, so green is the color we see.
mixing paints in the additive mixing demonstration the two paints can still separately reflect light to eye when they are visually mixed on a spinning surface (a color top) or as closely spaced dots of color (in visual fusion); but they cancel reflectance in each other when they are materially mixed as paints.
Finally, it should be clear why red and blue are not subtractive primary colors. A red paint reflects light only from the "red" end of the spectrum; it stimulates primarily the L cones, but not the M or S. Most blue paints reflect mostly "blue" and some "green" light, stimulating the S and M cones, but not the L. So their mixture creates a very dull purple, because the two colors have no reflectance in common: most wavelengths reflected by one color are absorbed by the other.
The same considerations explain why the RGB additive primaries are effective only in light stimuli, such as televisions or computer monitors, but not in paints or inks. There is no shared reflectance in the reflectance curves of red orange, green and blue violet paints, so these produce very dull, dark colors when mixed subtractively. The additive primaries are only effective when the mixing occurs in the retina.
By the same token, the CYM primaries are ineffective in televisions or computer monitors. There is a large overlap in the emittance curves of cyan, yellow and magenta lights, so that their additive light mixtures appear whitened and bright the equivalent of dark and dull in subtractive mixing. The subtractive primaries are only effective when the mixing occurs in materials.
Partitive Mixture. A special case of additive color mixture presents a confusing paradox for some readers. In partitive mixture, an image composed of small, separate but closely crowded color dots or pixels are fused by the eye into a visually smooth or continuous color area. Thus, the text and every image in this web page are generated on your color monitor as thousands of tiny RGB lights that are blended into color by partitive mixture.
Visual fusion results when a surface texture, such as the spacing between the tiny lights in your computer monitor, is too small for the eye to resolve optically or retinally; this is the process that makes color areas appear from a field of halftone or overlapping colored dots in printed books and magazines or in the tiny dye molecules of color photographic papers. Additive (retinal) color mixing then resolves the differences in light stimulation between adjacent RGB cones in the eye. Yet all photographs and printed color images use the CYM subtractive primaries. So the question arises: why aren't the additive RGB primaries used instead?
To grasp the answer, it will help first to print the diagram below on your color printer.
subtractive color primaries
as subtractive colors and as additive RGB pixels
In this image, the CYM color areas in the upper row are actually created on the computer monitor by the visual fusion and additive mixture of two of the three RGB monitor lights. These are physically distinct but barely too small for the eye to resolve into distinct dots. The color areas in the lower row are created by the visual fusion of alternating RGB pixels. Each pixel contains only one monitor light, which doubles the amount of black (unilluminated) area within each color. (Examine the two areas with a magnifying glass.) This doubled black spacing between lights, and visual fusion between the darker pixel and the black background, coarsens the screen texture enough to make it visible.
the subtractive inks and additive mixtures printed on paper
The printed copy looks quite different (image above), first of all because the printer silently substitutes a pure yellow ink for the "yellow" R+G monitor light mixture. However your computer screen is fundamentally a light source, despite the illusion (created by the subdued "white" luminance and the slight blackening effect of the monitor light interstices) that it is a surface. The printed paper is a true surface, and therefore the inks printed on it have the absorptive grayness that characterizes surface color perception. If you hold the printed diagram next to your computer monitor and illuminate the paper to a matching brightness, you will see that the inks appear to be darker and less saturated than the monitor colors especially in the cyan and magenta. Absorbing inks are inherently a less effective source of luminance than emitting lights.
If you next look at the printout by itself, you see that the yellow created from the pure Y ink (top row) is much brighter than the yellow created from the visual fusion of alternating, printed R and G dots. Your printer renders the pixels without black space between them, so the darkening is not the same as on your monitor; rather, visual fusion averages the luminance (reflectance) of adjacent dots; it does not add them together as it does in blended light mixtures. The average lightness of red or green inks is far lower than a pure yellow ink, so the visually fused and additively interpreted yellow appears much darker and, therefore, closer to a dull ochre or brown. A similar dulling and darkening occurs in the cyan and magenta mixtures.
Thus, the RGB primaries suffer from three handicaps when applied to surfaces: (1) they lose the inherent brightness of light sources, and (2) RGB inks are much darker (lower luminance) surface colors than pure yellow, cyan or magenta inks. This severely compromises their effectiveness in the additive color mixing induced by visual fusion. Since RGB inks make drastically dark subtractive mixtures think of mixing a yellow color from a red and green paint (3) they would have to be printed as separate, nonoverlapping dots, which would double the visual texture of a printed image and greatly increase the registration (dot alignment) precision necessary for a clear image.
Because subtractive colors can be overprinted in a single dot or pixel location, to produce subtractive mixture with each other and with the white paper, they produce a much finer visual texture with less registration precision. The overprinting also subtractively creates the span of orange, green and violet colors necessary to complete the hue circle. These dots of subtractive mixture are effaced by visual fusion, and averaged together by additive color mixture. This provides an acceptable simulation in printed surfaces and photographic papers of the brightness and contrast experienced in the light images of monitor phosphors, projective transparencies, and the surfaces of the real world.
The Computer Sciences department at Brown University hosts a Color Theory Library with several Java applets that allow you to explore additive and subtractive color mixing, color metamers and more.
So far we've explored subtractive color mixing by looking at idealized reflectance curves. But, as we've just seen, there are differences between subtractive mixing using idealized spectral profiles and actual color mixing using paints. As an artist, you will forever be confused by paint mixtures until you understand this difference in depth.
Material Colors and Visual Colors. The kernel of the problem lies in the distinction between material color, the light wavelengths that a paint actually reflects, and visual color, the color we see with our eyes.
The material light absorbing and light reflecting attributes of a pigment are exactly described by its spectral reflectance curve, and for that reason the guide to watercolor pigments provides the reflectance curve of all major pigments, linked from the spectrum icon .
Using the methods of colorimetry, the reflectance curve can be translated into three colormaking attributes that describe our visual color perception under normal conditions of lighting and display. These colormaking attributes fit the way we naturally think about colors, and are much simpler to interpret than reflectance curves. So the reflectance curve also exactly specifies the visual color at least in standard setups of display and lighting, and excluding all surface attributes of the color such as texture, gloss or depth.
However different reflectance curves can produce exactly the same color appearance, and this metamerism means we cannot identify the material reflectance curve of the pigment from its visual color alone.
And there is the problem. As explained above, the color of a paint mixture depends on the combined reflectance profiles of the paints being mixed. When the material reflectance curves of two visually identical colors of paint are different, they will produce different reflectances, and appear as visually different colors, when mixed with a third paint. But we can't tell, just by looking at the color, which wavelengths a paint absorbs or reflects. So we can't tell, just by looking at the color, that two paints with the same color will mix with other paints in the same way.
Visual Color Can't Predict Material Mixture. For painters, metamerism is the single most important cause of substance uncertainty, because the color of a paint does not define the color of mixtures made with the paint.
There is an idealized and a practical way to demonstrate the depth of the metameric problem. Let's start with idealized photographic gel (transparent) filters, which we design to pass either 100% or 0% of the light at each wavelength. In these examples, there are no constraints on the wavelengths we are able to filter, and two filters are placed in front of a single beam of "white" light. Then the apparent color of the transmitted light is the additive (retinal) mixture of all the wavelengths passed by the subtractive (material) mixture of the separate spectral transmittance profiles.
subtractive mixtures of different yellow and orange filters
The example above shows five pairs of ideal filters that appear yellow and orange to the eye they would all have the same "color" (that is, hue), though they would differ somewhat in lightness or chroma. Yet, as the examples show, the same "yellow plus orange" mixture can produce very different mixing results depending on the specific overlap in their transmittance profiles. Yellow and orange can combine to make yellow, orange, red or black ... yellow and orange filters could even mix to make green!
And in principle (though I have not worked through every variation), it is possible for two hypothetical transmission filters of any apparent color to create by mixture any other color. The example below shows how two "neutral" gray filters can mix an intense red (or green, or blue...).
how the subtractive mixture of two gray filters can produce an intense "red" color
the first gray filter passes all even numbered wavelengths in the spectrum; the second gray filter passes all odd numbered wavelengths below 600nm, and all even numbered wavelengths above 600nm
The point of these abstract examples is to show you that there is absolutely no logical or necessary connection between the visual color of two substances and the color of their subtractive mixture. If the only thing we know about two substances is their visual color (not as spectral reflectance curves, but as the way they appear to our eyes or as the lightness, chroma and hue measured by a spectrophotometer), then the material mixture of those two substances can potentially make any other color. There can never be universal or invariable color mixing rules in subtractive color mixing they simply don't exist.
Substance Uncertainty in Paints. But let's get practical. The extreme, idealized variations I've described are implausible. We certainly can't mix a red color from two gray paints! In fact, regularities or patterns often appear in the way colored substances mix. Why is that?
Because we live in a real world of atomic substances, and the atomic causes of color follow the organizing patterns of chemistry and physics. These tend to produce transmission or reflectance curves in most substances that follow more regular patterns, such as the "warm cliff" profile typical of saturated red, orange and yellow paints and filters. In addition, painters work with a very limited range of colored substances their paints and modern colorants produce a fairly predictable domain of reflectance profiles.
So I have to turn to paint color mixing demonstrations to assess the practical extent of the metameric problem for painters (or anyone else mixing paints, dyes or inks). To do that, let's see what happens with the most explicit paint mixing test possible (and the one most beloved in color theory): making a pure gray mixture from two complementary paint colors.
The test is simple (though tedious) to do. First, using the visual color only, arrange all the available "warm" colored paints in a series, from greenish yellow to purple, on one side of a page. Then arrange all the complementary "cool" colored paints, from blue violet to yellow green, on the opposite side. Align the paints so that they are approximately matched as mixing complement pairs blue violet across from yellow, blue across from deep yellow, and so on. Then mix all possible combinations of these warm and cool paints to identify the pairs that produce a neutral (gray or black) mixture. Finally, connect these mixing complement paints with a dark line.
If complementary paint mixtures are determined by the visual color of the paints, and if all the paints have regular, simple reflectance curves, then lines connecting these complementary pairs should be roughly parallel (diagram, right). As the hue of the warm paint changes from deep yellow to violet, the hue of its mixing complement should change from blue violet to green by an equal amount.
This is precisely what does not happen, as shown below!
substance uncertainty in watercolor paint mixtures
mixing complementary colors as measured on the a*b* plane in CIELAB: pigments that make "pure gray" mixtures are joined by dark lines, "near gray" mixtures by light lines (see this page for more information)
Instead, the mixing lines from each pigment fan apart, or skew up or down haphazardly. The mixing complement of teal blue can be anything from a bright scarlet red to a dull maroon and there are even paints between scarlet and maroon (such as quinacridone red, PR209) that do not mix a close gray with teal blue! And that is obviously the typical case for every pigment shown in the diagram.
The limiting rules of physics and chemistry take us out of those idealized transmission filter examples, where subtractive color theory doesn't exist, but they don't take us all the way to a perfect world where all pigment "colors" mix in a consistent way or where the reflectance curve of the paints is apparent in the visual color. So we end up in the middle, in a fuzzy, messy real world where subtractive color mixing is a real fuzzy mess.
This diagram also shows why a subtractive or mixing color wheel can never be defined precisely: the mixing behavior of pigments or paints is only weakly predicted by their hue relationships, or the relationships are too complex to be summarized as a simple wheel. Substance uncertainty is an important reason for the color wheel fallacy.
Finally, the diagram shows plainly the difficulty in using "primary" colors to explain color mixing. For example, cobalt teal blue (PG50) mixes a pure gray with both naphthol scarlet (PR188) and cadmium red deep (PR108). However, scarlet has substantially more "yellow" in it than deep red: how then can it mix the same gray with the same greenish blue paint? Simply because subtractive "primary" colors have no logical or necessary connection to the visual color of two substances or to the color of their subtractive mixture.
Couldn't we avoid substance uncertainty if we used real world colorants that had regular and simple reflectance curves? The answer is no: because the second reason for substance uncertainty arises from the many invisible differences in pigment material attributes: refractive index, particle size, crystal form, hiding power and tinting strength. Thus, the pigments cadmium yellow medium (PY35) and hansa yellow medium (PY97) have almost identical reflectance curves, yet they produce visibly different mixtures with other paints because their refractive indices (appearance in paint vehicle) are so different.
Suppose we could somehow make paints so that every apparent color had the same reflectance curve, and made sure every paint had exactly the same material attributes wouldn't that solve the problem? Again, the answer is no: because the third reason for substance uncertainty arises in the material attributes of the support and the paint application methods. The qualities of different papers or canvas supports have a significant impact. A glossy, highly reflective white paper can show up to 24,000 distinct color mixtures using modern process inks. The same inks, printed on ordinary newsprint, generate a much smaller range of perhaps only 2,000 distinct colors. In watercolors, a highly absorbent paper (which pulls the pigment particles into the cellulose mat) will produce duller color mixtures than a heavily sized, nonabsorbent paper. These effects occur because applying paint to paper effectively mixes three light reflecting substances the two paints in the mixture and the paper which means the material attributes of all three will determine the apparent color.
Color mixtures also depend on how the paints are applied. A familiar example for watercolorists occurs when paints are mixed by glazing or layering one color over another: cadmium yellow over phthalo green is a lighter and less saturated mixture than phthalo green glazed over cadmium yellow, even when the two mixtures have exactly the same hue.
These problems of metamerism, physical composition, support attributes and application methods all contribute to a single result: paint color appearance cannot predict mixture color appearance. And this happens because subtractive color mixing "theory," by trying to imitate additive color mixing theory, bites off much more than it can chew.
True, a complex mathematical model called the Kubelka-Munk theory has been used with some success to anticipate color mixtures in manufacturing and printing industries, but even there it has not replaced the empirical methods of mixing by eye or according to predefined recipes (as found in the Pantone or process color systems used in printing). The Kubelka-Munk equations require information about paint reflectance, the scattering power and hiding power of pigments across all wavelengths (which is often hard to come by for the specific pigments being used for a particular job), and they assume a homogeneous layer of paint vehicle containing very small pigment particles of identical tinting strength that do not affect the color reflectance of the other pigments restrictions that are often unrealistic.
In the situations that matter to watercolor painters, watercolor paint on paper does not form a continuous paint "layer" at all, and does not distribute pigment on the support in the same way that acrylic or oil mediums do, so theoretical models are even less applicable. There is no way to escape the real and practical in order to reach the abstract and ideal.
The Color Is In The Mixture. Substance uncertainty is such an insurmountable problem for popular color theory books and color wheels that they deal with it the only way they can they ignore it! (Or, even worse, they assertively deny it by presenting their color mixing explanations within perfect, idealized triangles and circles.) The artist tries to mix paints according to these idealized, perfect color theory rules, and is only confused by the many messy exceptions that result.
Experienced painters work with a more intuitive rule: the material color of paints appears in their mixtures. They put emphasis on the mixture colors that a paint produces with all other paints on the palette. They pay more attention to the chroma and lightness of paints, not just their hues whether they are intense or dull, light or dark. They learn, by trial and error, how the paints behave when they are diluted, and mixed with every other paint on the palette, and they prefer paints that are versatile, rather than pretty by themselves.
This is an important reason for the greater mixing skill in experienced compared to novice painters. Novices think about the pure color of the paint as it is painted on the paper, and attempt to judge mixtures in terms of the visual color. These painters, who often learn color theory in terms of the visual colors of paints arranged on a color wheel, learn how to mix hues "yellow and blue make green".
"theory" vs. experience
Where does this leave us? The fundamental issue is this: the artistic control of "color" can be pursued through experience or theory. The diagram shows the basic process tension.
approaching color through painting experience
If we take a specific part of a painting, such as a green color patch in a landscape, then we can either view it as the outcome of a material process that involves the manipulation of paints or pigments on specific supports using specific mixing and application techniques, or as the outcome of an intellectual process that assumes planning based on traditional principles of contrast and harmony based on abstract color ideas and fictional primary colors, or geometrical color models used to "predict" color mixtures.
The material process is guided by painting experience, which tends to be a very long, trial and error labor of exploration and imitative training that develops the painter's understanding of the best materials and techniques to produce a specific finished appearance though why that result occurred is often a mystery. The intellectual process is guided by color theory dogma, which provides sometimes accurate and sometimes inaccurate explanations for why colors appear a certain way, though it does not explain any of the process techniques necessary to produce the desired effect.
I have tried hard to assure you that substance uncertainty renders all the abstract rules based on artists' color wheels or primary colors imprecise or arbitrary. There is no "there" there. At the same time, color mixing is so inherently complicated that some kind of conceptual framework is essential to keep one's bearings and learn efficiently.
The key is striking a personal balance between practical experience and abstract rules, which every student can achieve by studying and painting with the following in mind:
color is fundamentally a subjective experience that differs considerably from one person to the next;
color theory rules consist of limited preconceptions based on past experience with specific artistic media, not predictive rules based on abstract scientific principles;
experience with materials the ultimate standard of best painting practice and the preconceptions you have about how materials behave;
color theory principles are useful when they accurately summarize your painting and color experience;
always consider color theory guidance when you are confused by a basic painting problem or design difficulty;
avoid the empty intellectual game of talking about color theory separate from a specific design or painting problem in relation to a specific design or painting; and
keep your eyes always open to the effects your materials create, so that theory does not become a limitation to your artistic growth.
Eventually, accumulated experience makes futher experience easier to acquire and understand, and makes theory less of an issue. Intimate knowledge of your paints, supports and technique is the key to effective color control. It is always more difficult to look at the actual behavior of the paints you use than to memorize the simplifed rules of "color theory." But experience, not memorization, is where learning actually takes place.