color vision

D'Aguilon discussed the optics of the eye, linear perspective, surveying instruments, and the behavior of light and color, using practical demonstrations that would be of interest to painters. (For example, he describes partitive color mixing, a technique Rubens used with great skill.) When he touched on the origins of color and the rules of color mixing, d'Aguilon endorsed the medieval view that yellow, red and blue were the basic or "noble" hues from which all other colors derived.

françois d'aguilon's color mixing theory (1613)

the "primaries" are white and black (light and dark)

From a modern perspective, the most peculiar feature of d'Aguilon's theory is that these three "noble" hues were themselves created from the mysterious blending of white and black, or light and dark (upper curved lines in the figure), so that light and dark were the two "simple" or primary colors. The "composite" hues green, orange (gold) and purple (lower curved lines) were mixed from the "noble" triad colors. D'Aguilon's diagram was reprinted by the Jesuit scholar Athanasius Kircher in his optical treatise Ars magna lucis et umbrae (The Great Art of Light and Shadow, 1646). Both sources were widely read in the 17th century and shaped the explanation of color mixing dominant during the Baroque.

This light/dark color theory was inherited from medieval books on optics, which in matters of color borrowed from ancient Greek philosophical texts: an extended account in Plato's (c.390 BCE) creation poem Timaeus, scattered and inconsistent passages in the writings of Aristotle (c.350 BCE), and the De Coloribus (On Color), sometimes attributed to Theophrastus (c.330 BCE), Aristotle's commentator and Lyceum administrator.

Three points to keep in mind about the ancient texts. First, color was a quality of substances or the surfaces of things, or of surface appearance altered by transparent media such as smoke, haze or water. Second, many Greek texts assert that sight was a kind of touch, produced by rays emanating from the eye, and (like touch) responding to the qualities of physical bodies. The ancients seem to have made no distinction between sight and light, so that distance, darkness or disease produced an equivalent "weakening" of colors. Third, the ancients had by modern standards a very erratic color nomenclature. In particular, the Greek words for white seem to have covered meanings such as white, light valued, bright, saturated, glowing, transparent, metallic, reflective (as in a mirror or water) or smooth, while black referred to black, dark valued, shadow, darkness, dull, opaque or rough, more or less. The modern color terms white and black, or light and dark, introduce different or more restrictive meanings than the ancient authors intended.

Aristotle's On Sense and Sensible Things observes that color only appears in reflected or filtered light that is not as bright as the illumination but is brighter than darkness. From this came the "conceivable hypothesis" that all colors appear due to variations in a "common nature or power", translated in English as the translucent, which allows sight to reach across space or into transmissive bodies such as the air or ocean. The translucent facilitates sight (which is equated with light), and an absence of translucency is produced by haze, smoke, shadow or darkness. The opacity and color of objects might arise from a fixed blending of invisibly small particles of white and black. And the purest or most intense colors might result from a whole number mixture ratio of light and dark, similar to the whole number divisions of a vibrating string that produce musical notes or harmonics:

Such, then, is a possible way of conceiving the existence of a plurality of colors besides the white and black, and we may suppose that many are the result of a ratio; for they [white and black] may be juxtaposed in the ratio of 3 to 2, or 3 to 4, or in ratios expressible by other numbers; while some may be juxtaposed according to no numerically expressible ratio, but according to some incommensurable relation of excess or defect; and that those involving numerical ratios, like the concords in music, may be those generally regarded as most agreeable, as for example, violet, crimson and some few such colors, their fewness being due to the same causes which render the concords few. [439^b]

D'Aguilon's figure adopts this musical analogy as arcs over a straight line, like stops on a vibrating string. The clearest statement of the proportions of light and dark that produce colors appears in Aristotle's discussion of the rainbow (On Meterology):

White [bright, pure] color through a dark medium or on a dark surface (it makes no difference) looks red. We can see how red the flame of green wood is: this because so much smoke is mixed with the bright white firelight; so too, the sun appears red through smoke or mist. ... When the sight [light] is relatively strong the [color] change is to red; the next stage is green; and a further degree of weakness gives violet. No further change is visible, but three completes the series of colors. ... The appearance of yellow [in the rainbow] is due to contrast, for the red is whitened [lightened] by its juxtaposition with green. ... Bright dyes too show the effect of contrast. In woven and embroidered fabrics the appearance of colors is profoundly affected by their juxtaposition (purple, for instance, appears different on white and on black wool). [374^b]

These are almost the only colors which painters cannot manufacture, for there are colors which they create by mixing, but no mixing will give red, green or violet. [372^a]

Aristotle apparently preferred the rainbow or veiling media as examples of "natural" colors because they represented a "pure" display of color variation; paint mixtures just muddled different colors together. This preference was maintained by the later Peripatetic philosophers who taught at the Lyceum. Thus, the De Coloribus advises that:

We must not proceed in this inquiry by blending pigments as painters do, but rather by comparing the rays reflected from the aforesaid known colors [white, yellow, and black], this being the best way of investigating the true nature of color-blends. ... [Thus], the different shades of crimson and violet depend on differences in the strength of their constituents, while blending is exemplified by the mixture of white and black, which gives gray. [792^a-792^b]

And still later, the exclusion of subtractive mixture was expounded by Alexander of Aphrodisias (c.200CE), whose writings were well known in the Middle Ages and Renaissance. His commentary on Aristotle's On Meterology reveals a very fuzzy knowledge of painters' pigments and dye manufacture:

That the ... colors of the rainbow cannot be compounded or imitated by painters, and that red is closer to white than green and violet, is clear from the following. The natural [unmixed] red pigments are cinnebar and dragon's blood, which are made from the blood of animals; red is also made from the mixture [laking] of talc and purple, but this is much inferior to the natural colors. Natural green and violet are chrysocolla [malachite?] and tyrian purple, the one made from blood and the other from sea creatures. But the artificial [mixed] colors cannot match them: green can indeed be made [mixed] from blue and yellow, and violet from blue and red, for the contrasting energies of blue and yellow make green, but those of blue and red, violet; but in these cases the artificial colors are far inferior to the natural. ... That red is closer to white than to green and violet is evident from their origin. For red is made with [laked onto] talc, which is white, but green from ochre, which is a weaker [darker] white.

I've quoted these passages at length because they are the earliest descriptions of color change and color mixture. Yet they include several concepts that have been carried into modern color science. The quantity of light or luminance is the fundamental color attribute. Color must be explained as a balance between the light (reflectance or transmittance) and dark (the absorptance) of substances. The proportions of light and dark produce the visible range of hue and hue purity. There are two types of color mixture — light interacting with materials or the atmosphere, and the blending of substances. Color should be studied only in the behavior of light. The blending of substances is a separate color change process; subtractive mixtures of green and violet were known to the ancients. These blended colors are inferior (dulled, polluted, lower in saturation) than colors in their "natural" form. Rules of color change must be deduced from the behavior of light in materials or the atmosphere; the color change in blended pigments and dyes can violate these rules.

The content of this section, but not the conclusions, relies on John Gage, Color and Culture: Practice and Meaning from Antiquity to Abstraction (University of California, 1993).

The peripatetic texts are drawn from The Complete Works of Aristotle, edited by Jonathan Barnes (Princeton University Press: 1984).

the painter's primaries

The ancient texts were penned by scholars in leisure, not by working artists. We do not have testimony about color from artists until the early Renaissance, when writings by painters about painting methods first appear.

But color blending is no longer disparaged. Cennini describes a method devised to model forms that curve from light into shadow (right). This approach uses variations in black or white to model illumination intensity across the same hue. The pigment (the hue) only appears in its pure (darkest and most saturated) form next to the shadow terminator; the terminator is pigment grayed by the underpainting color (a pale green or brown).

Philosophical texts written in the 16th century paralleled this change in approach: the Italians Simon Portius and Girolamo Cardano advanced the idea of an inherent luminosity to characterize each pigment color at its most saturated preparation: yellow is a "white" or light valued color while blue or red is a "black" or dark valued color. Colors ranked in this way retained the ancient light/dark ordering of "natural" hues, so the ancient texts were not contradicted. However "broken colors" — pigment mixed with white or black — created the color scale or tonal gradation needed by painters. The blending of pigments, which the ancient and medieval artists disliked because it obscured decorative (and often expensive) "pure" color, was justified as a method of representation.

As the Greek texts show, awareness and use of subtractive primary mixtures based on red (scarlet or carmine), yellow and blue colorants was already well known to ancient painters and dyers. Yet this longstanding painter's knowledge was not the painter's common practice. The primary triad was an unsatisfactory method for mixing colors because traditional pigments were not saturated and light valued enough to span wide differences in hue effectively. If painters wanted a dull scarlet color, they did not mix carmine and yellow lake: they started with a bright iron oxide or vermilion, then "broke" it with white or black.

Nor were the primaries useful to figure out complex or near neutral mixtures: painters instead relied on their experience with conventional methods. To create a flesh tone, medieval painters began with an thin underpainting of green terre verte, then glazed it with carmine. The exceptional colors that might be created by paint mixture were (and still are) greens and purples, for which there were only dull or hopelessly impermanent pigments. These colors are furnished as convenience mixtures even today.

Thus, the common practice of painters (and presumably also dyers) from ancient times through the Baroque reflects a completely different understanding of colors and color mixtures. There is no historical source prior to the 18th century that starts with three "primary" or "primitive" colors and explains how to mix all other colors from them. Mention by Alberti or Leonardo of the "artists' primaries" (red, yellow, green and blue) is not applied to explain paint mixing. When mixing techniques are described, focus is on specific color effects that are possible with specific preparations of pigments or grounds.

As a scholar personally acquainted with Rubens, D'Aguilon adopted red, yellow and blue, the subtractive primary triad familiar to painters and dyers of the time, as the fundamental sequence of colors produced by ratios of white and black. But again, these primaries are not recommended for color mixture but are used to anchor the ratios that produce orange, green and violet. D'Aguilon uses the Aristotelian framework to systematize facts, a characteristic tactic of 17th century intellectual culture, but by doing so he discards Aristotle's concern for the causes of color as displayed in light mixtures.

These changes are consistent with color materialism: the natural and obvious interpretation of color as something fixed, unchanging and inherent in physical substances or objects. This is the universal experience of color in the human species because it is the organizing principle of our color vision. Material colors and paint mixtures are now identified as "real" colors; all other colors, including reflections, shadows and Aristotle's beloved rainbows, are only "apparent" or "accidental" — colors that are not material and that change inexplicably, like a peacock's tail, with point of view or angle of light. Abstract explanations of light and dark are displaced by practical explanations of pigment mixtures. D'Aguilon's account reflects these accumulated practical revisions to the ancient color theory.

Why then did the painters' primary colors become prominent? The broad answer is that the Scientific Revolution of the 17th century and the Enlightenment of the 18th century shifted the European understanding of "theory" away from airy concepts and logic and toward observable consequences and practical applications. By 1664, for example, early chemists were studying dyes and pigments in order to improve them, efforts motivated by the huge commercial importance of textile manufacture. This stimulated a a practical and scientific focus on artistic practice. Thus the Irish chemist Robert Boyle wrote in 1664 that the painter's "simple and primary colors" were black, white, red, yellow and blue, which could "imitate the hues (though not always the splendor) of those almost numberless differing colors" found in nature.

newtonian color confusions

The Baroque's abstract and muddled color theory was overturned in the late 17th century by the widely discussed researches of Isaac Newton (1642-1726), first made public in his lectures at Cambridge University in the 1670's and finally published in his Opticks of 1704.

cennini's rendering of illumination and shadow

C = pure color, W = white, Bk = black, U = underpainting; after Kemp (2000)

6. Finally, Newton aroused intense intellectual animosity. He confounded the dogma debating Aristotelian scholars of the time by his empirical demonstration that "white" light is not homogeneous but compounded of various hues; he was attacked by English partisans of a "wave" theory of light for seeming to endorse the competing theory of light particles; in his earlier masterpiece Principia Mathematica (1686) he had completely gutted the "vorticist" physical theory of René Descartes, inflaming the Cartesian scholars of France and Italy; and his prism experiments were difficult to replicate (especially by Continental naturalists) and in fact were not adequately verified until the 19th century. These partisans seized on any apparent falsehood or contradiction in Newton's theories as weapons to repudiate him.

So the 18th century public debate came down to this: Newton's hue circle was based on seven spectral "primary" lights, and made very specific predictions about the color that would result from any light mixture. For example, according to his hue circle, a mixture of "orange" light and "green" light would create the color yellow. Yet these prismatic mixtures were not easily verified by other naturalists, and it was quickly demonstrated that Newton's mixtures did not apply to paints, where orange and green make a cadaverous gray. Not realizing the difference between paint and light mixtures, and inflamed by partisan rivalries, many of Newton's adversaries used paint mixtures and the painter's primaries as proof that Newton's primaries (and hence his other color ideas) must be wrong. This wrangle went on for more than a century.

material trichromacy

Yet Newton's scientific authority, and the potential usefulness of his analytical hue circle, were too attractive to reject completely. So artists found the practical compromise, and inserted pigment primary colors into Newton's hue circle. Even though Newton had shown that all hues were equally "primary," Newton's theory (like the ancient Greek theory) was revised to fit experience with paint and dye mixtures.

Many of these practical problems were solved in the third statement of material trichromacy, by the entrepreneurial German printer Jakob Christoffel Le Blon (1667-1741), in his Coloritto: or the Harmony of Coloring in Painting Reduced to Mechanical Practice (1725).

Le Blon is one of those many fascinating 18th century characters who focused a deep intellectual curiosity on an unexplored capitalist opportunity. He was an artisan who had read and actually understood Newton's Opticks, and credited Newton for the idea of describing color mixtures as a circle. A decade before, while in Holland, Le Blon innovated a system for using three separate printing plates, each inked with one of the painter's primary colors red, yellow or blue (and sometimes a fourth plate inked with black) to create full color mezzotint prints (right) — the practical basis for today's multicolor process printing. But his key innovation was to join Newton's hue circle with the three primaries of material trichromacy — red, yellow, and blue. He was also among the first color writers to state explicitly the difference between additive and subtractive color mixing:

Painting can represent all visible Objects with three Colours, Yellow, Red and Blue; for all other Colours can be compos'd of these Three, which I call Primitive. ... And a mixture of those Three original Colours makes a Black, and all other Colours whatsoever. ... I am only speaking of Material Colours, or those used by Painters; for a mixture of all the primitive impalpable Colours [of light], that cannot be felt, will not produce Black, but the very Contrary, White, as the great Sir Isaac Newton has demonstrated in his Opticks. White, is a Concentration, or an Excess of Lights. Black is a deep Hiding, or Privation of Lights.

Unfortunately his business venture to provide "printed paintings" ultimately failed, but Le Blon established the circular arrangement of painters' primaries as the definitive representation of color behavior, and his system inspired many imitators.

Among these, the English entomologist Moses Harris (1731-1785) published an especially elegant color wheel in his slim Natural System of Colours (1766), dedicated to the Royal Academy's Sir Joshua Reynolds (and reprinted in 1811 with a dedication to Benjamin West).

the color wheels of moses harris (1766)

a 20th century reconstruction of (left) the Harris color wheel of mixtures between two "primitive" colors red, yellow and blue; (right) the Harris color wheel of mixtures between two "compound" colors orange, green and purple; note the imbalanced color distributions caused by the weak tinting strength of the blue pigment

Harris's book briefly summarizes the established dogma of material trichromacy as a kind of commentary to his color wheels, presented as full page, hand tinted engravings. Harris is notable for his explicit recommendation of complementary color contrast, which was a subject of keen interest to late 18th century color theorists:

"If a contrast is wanting to any colour or teint, look for the colour or teint in the system [wheel], and directly opposite you will find the contrast wanted. Suppose it is required what colour is most opposite, or contrary in hue to red, look directly opposite to that colour in the system and it will be found to be green, the most contrary to blue is orange, and opposite to yellow is purple."

The Harris wheel became quite influential — it was studied and used by many 19th century English painters, including Joseph Turner — and it spawned many related color wheel systems: by English naturalist James Sowerby in 1809, colormaker George Field in 1817, and artist Charles Hayter in 1826.

the second and final stages in Le Blon's four color printing method (c.1720)

The French Jesuit mathematician Louis Bertrand Castel (1688-1757) — who famously claimed that "Newton has reduced man to using only his eyes" — strongly opposed Newton's experimental approach to science in favor of the "deduction from first principles" method used by mathematicians and geometricians such as René Descartes. In 1739 Castel advocated a trichromatic color spiral, in imitation of the musical circle of fifths, accompanied by a prism diagram labeled Vrai ("Truth", right), to demonstrate that all colors, including the spectral series, could be explained by overlapping mixtures of red, yellow and blue light. Castel's color wheel was adopted by the Viennese entomologist Ignaz Schiffermüller in 1771, who attempted to expand it into a color system.

Well into the 19th century, color researchers such as the German poet and bureaucrat J.W. von Goethe (1749-1832, diagram right) or the English color chemist George Field (1777-1854) were still espousing the ancient epic poetry of light and dark, and rejecting Newton, in essentially the same terms as Castel had decades earlier. The Scottish physicist David Brewster (1781-1868) was an especially pugnacious holdout, arguing as late as the 1840's that all spectral hues could be explained by red, yellow and blue fundamental colors of light, which Brewster equated with three colored filters or transmittance curves that could reproduce the entire spectrum (bottom diagram, right). Brewster's theory was dislodged only after Hermann von Helmholtz experimentally demonstrated that mixtures of "yellow" and "blue" light do not make a green mixture: the color is always either a reddish or yellowish gray.

castel's color theory

hue circle and explanation of prismatic hues from overlapping light and dark, color added for emphasis; after Kemp (1990)

Artists focused on the practical guidance provided by the color wheel while ignoring the scientific debate. Perhaps the last artist to subscribe to the antinewtonian theory was the Romantic painter J.M.W. Turner (1775-1851) who, following Goethe, used yellow and blue in his paintings as symbols of the spiritual nature of light and dark. It wasn't until Turner's death, and the publication a year later of Helmholtz's early color researches, that the Aristotelian theory and its classical poetics lost its grip on artistic thinking.

So what became of the ancients' light and dark? Newton's hue circle dealt with changes in hue and saturation only, and left the paint wheel designers with little guidance as to how to represent changes in the brightness or lightness of colors. All the 18th century color wheel authors resolved this issue in terms of tonal gradation, placing either pure black or white paint at the center of the color wheel, mixtures of the primary colors around the circumference, and steps from the primary mixtures toward the center as increasing quantities of white or black paint.

Unfortunately these different mixtures have different effects on a paint's saturation or lightness. Mixing with white changes both saturation and lightness; mixing with black affects lightness only. And, as Newton showed, mixing two different hues of paint also reduces saturation and typically makes a paint mixture darker valued as well. As a result, the 18th century artists' color wheels prolonged confusion about the difference between lightness and saturation, a confusion that is still unresolved in the color hemisphere proposed by Michel-Eugène Chevreul in 1839.

J.B. Mollon's chapter "The Origins of Color Science" in The Science of Color (2nd ed.) (Elsevier, 2005) provides a fine account of color science since Newton, and distinguishes between physical and physiological trichromacy. Martin Kemp, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat (Yale, 1990) is an astonishingly detailed review of the relationship between science and art, focusing on linear perspective but with three chapters on color theory.

comprehensive color models

During the same period in which the "primary" color wheel was becoming entrenched in painters' lore, and scientific factions were wrangling over the fundamental causes of color, a few 18th century color enthusiasts focused on a specific practical problem: how to define and represent all possible colors as a single color order system. These early color models were motivated by a scientific interest in summarizing color perception, by the need for a standardized system of color identification for use in science and industry, and by an artist's mystical enthusiasm for color.

19th century color theories that rejected Newton's Opticks

(top) J.W. Goethe's Aristotelian theory of prism colors as overlapping light and dark (1810); (bottom) George Brewster's three primary explanation of spectral hues as overlapping red, yellow and blue light (1831)

Finally, Mayer made the triangle into a multilevel color solid by mixing the color locations with either white or black paint to produce compounds of four primaries, for example, r₃g₂b₄w₃ or r₆g₁b₃k₂. (He did not require mixtures of both white and black, as gray is obtained at alternate levels by equal mixture of the three primaries.) Because the proportions still add up to 12, each added increment of white or black reduces the number of combinations remaining for the three colored primaries and therefore reduces the edge dimensions of the mixture triangles. By stacking these triangles in tonal order above or below the middle, pure hue triangle, Mayer created a dark to light color hexahedron comprising 819 unique color mixtures (right). Using this system he described the color location of several common pigments — the first instance of a pigment color analysis using an abstract color system.

The breakthrough in Mayer's concept is the equivalence between the primary color mixtures, the geometrical framework, and the color notation system. Any color can be analyzed into some mixture of the three primaries plus white or black. This "chemical" color analysis, which exactly anticipates the modern system of tristimulus values, also serves as the color's name; the name is the color's location within the easily visualized color solid. This was a great advance over Le Blon's industrial color mixing system because Le Blon left mixture proportions to the relative ink densities on the handmade mezzotint plates used to print them. Mayer embraced all visible colors in a single abstract measurement framework that (in principle) would apply equally to the natural colors of flowers or stars or the manufactured colors of pigments or dyes.

However, Mayer stumbled over four practical problems:

(1) Any set of three "primary" paints or lights cannot mix all possible colors, so his system is incomplete.

(2) Mixtures of lights or pigments, notated the same way in his system, can produce very different colors — r₀g₆b₆ is approximately a "white" light mixture but a green paint mixture — so his system is not applicable to both colored lights and paints.

(3) The geometry does not clearly represent the three colormaking attributes; there is no continuous white to black grayscale, or an explicit dimension of saturation, so color differences in his system are difficult to interpret.

(4) Finally, Mayer did not solve the practical problem of translating his abstract colors into physical color exemplars: in particular, the tinting strengths of his primary pigments or lights must be exactly equated to produce balanced color mixtures. Failing that, the "gray" exemplar in his handpainted color triangle (marked by a dot, above) is not at the center but is next to pure blue, because his red pigment had an overwhelming tinting strength.

These practical problems seem to have especially occupied the Prussian astronomer and perspective theorist Johann Heinrich Lambert (1728-1777), who published an account of his Farbenpyramide (Color Pyramid) in 1772. Lambert knew of Mayer's system and adopted its triangular geometry, but limited to 7 step mixing scales between the pure primary colors. Lambert only expanded colors by mixtures with white, stating that black and gradations of gray could be found at the center of each primary mixture triangle. He tasked a Prussian court painter with transforming his conceptual color pyramid into encaustic color exemplars (wax providing both durability and the most brilliant colors), and he discussed at length the procedures necessary to equate the tinting strengths of his primary colors (carmine, gamboge and iron blue). Despite this, his gray sample, like Mayer's, is pushed too far towards the blue primary. Lambert offered his color model as a way for tradesmen (dyers, clothiers) and their customers to reliably identify and compare colors, and perhaps for this reason he rejected Mayer's intuitive but abstract notational scheme for a horridly hopscotch numbering system — from top to bottom down each diagonal, from the left triangle side to the bottom right corner — supplemented by compound color names ("chestnut red brown," "bluish reddish blue") that were better suited to the tradesman's inventory system and fashion patter.

tobias mayer's hexahedric color solid

after Grusser (1989)

Lambert's model again treats hue, saturation and lightness unequally: the most unsatisfactory element in all 18th century color models is the color geometry. But a truly modern representation of colors, all correctly ordered around the three colormaking attributes was first achieved in the Farbenkugel or "Color Sphere", published in 1810 by the German Romantic painter and color mystic Philipp Otto Runge.

As his color specification, Runge derived all colors from mixtures of the primaries red, yellow and blue, plus white and black. But by interpreting Newton's hue circle as representing a slice through a sphere, Runge hit on the geometrical framework that is common to all modern color order systems. Lightness is measured by the polar axis (or latitude of the sphere), hue is represented by longitude around the equator, and saturation by the distance from center to surface. In his model, as he proudly (and correctly) announced, "Every color is placed in its proper relation to all pure elements as well as all mixtures." Unfortunately, the Color Sphere lacked a color notation system or exemplification; and Runge died before he could define a system of color labels or develop physical color exemplars. But his hand tinted engravings of the model's exterior surface and interior sections showed that a physical model was entirely feasible.

The color systems of Mayer, Lambert and Runge show that 18th century color theorists grappled with the problems of color models piecemeal and from different perspectives, without achieving a comprehensive approach that unified color specification, color labeling and color exemplification within a geometrical framework consistent with the three colormaking attributes. (This story is continued in the page modern color models.)

perceptual trichromacy

Let's now turn to the perceptual side of the Enlightenment march of theory. Newton's proof that colors were not in the light forced the conclusion that they must be in the eye. However, the proposal that color sensations might arise from three different "particles" or "membranes" in the retina (the trichromatic theory of color vision) emerged only after the usage of three primary colors had been evangelized by 18th century painting and printing treatises and in the color models just described.

The Trichromatic Theory. A trichromatic theory was suggested several times in the century after Newton's Opticks appeared — by Mikhail Lomonosov in 1757, George Palmer in 1777, and the English physicist Thomas Young (1773-1829) in 1802 — but 18th century scientists lacked the optical tools to pursue the idea in depth. The hypothesis was revived, definitively stated and demonstrated by the renowned German physicist and physiologist Hermann von Helmholtz (1821-1894) in his On the Theory of Compound Colors (1850) and Handbook of Physiological Optics (1856). Helmholtz argued that just three types of "nervous fibers" or receptors were sufficient to produce the fundamental sensations of color, and he proposed three stretched out and overlapping sensitivity functions (right) to describe the response of these light receptors to different wavelengths of light. This is the core of what is now called the Young-Helmholtz three component theory.

Young is famous for stating and experimentally confirming the wave theory of light, based on his observation of interference bands created when a single beam of light is passed through two very narrow, closely parallel slits. His ideas were verified and extended by the French civil servant and amateur physicist Augustin Fresnel and the German optics manufacturer Joseph von Fraunhofer in the early 1820's. These wave experiments affirmed Newton's conjecture that each spectral color had a unique physical attribute (its wavelength or frequency) that determined the perceptual response of the eye's color receptors.

The logical crux of the trichromatic theory, nicely stated by Young, is that the almost infinite number of perceived colors could not possibly arise from an almost infinite number of different wavelength specific receptors in the eye: they couldn't all fit on the retina. So color perception must originate a small number of primary receptor cells, each tuned to a different part of the spectrum and present in every part of the retina. The blending of these different receptor outputs would create the perceived color variety. The painter's trick suggested this small number was three, tuned to colors that Young first suggested were red, yellow and blue (as propoosed by earlier authors, such as Palmer). By 1807, however, Young had changed his primaries to red, green and blue violet.

Nearly a century of research was necessary to confirm the existence of these color receptors in the retina — by microscopic dissection of retinas in 1828, the isolation of rod photopigment ("visual purple") in 1877, and exhaustive color matching experiments in the late 19th and early 20th centuries. These confirmed the essentials of Helmholtz's theory and Young's hypothesis that additive mixtures are best modeled by the additive primary lights red, green and violet (although it turned out that the human light receptors are actually most sensitive to yellow green, green and blue violet wavelengths).

three receptor sensitivity curves proposed by Helmholtz in 1856

letters indicate spectral colors; diagram reversed to match modern convention for spectral diagrams

Maxwell's Trichromatic Measurements. The Young-Helmholtz primaries were used in the first precise, quantitative color matching experiments, conducted in the 1850's by both Helmholtz and the brilliant Scottish physicist James Clerk Maxwell (1831-1879, right). Maxwell created additive color mixtures in two ways: by mixing colored lights through a system of prisms, mirrors and neutral filters enclosed in a long, flat box (a Maxwell box, the first modern colorimeter), or — the method he found more practical and accurate — by visually mixing circular wedges of colored papers on a rapidly spinning disk (a color top or Maxwell disk, shown at right). He demonstrated that any color of light could be matched by a specific combination of just three primaries, which he represented on his color top with the pigments vermilion (scarlet, PR106), emerald green (bluish green, PR21), and ultramarine blue (blue violet, PB29), and in his prism box by the wavelengths 650 nm ("scarlet"), 510 nm ("green") and 480 nm ("blue"). His colorimeter became the most widely used apparatus in modern color research.

In the original version of Maxwell's color matching experiments, the viewer looked through a small lens or eyepiece to see a round patch of illumination surrounded by a darkened field (right). This patch was divided into matching semicircles. The upper half contained a constant target color of "white" light. This was the color to be matched. (Maxwell found that "white" light was a convenient color target because it minimized the errors introduced by chromatic adaptation and the Helmholtz-Kohlrausch effect; and it allowed easy identification of the missing primary in colorblindness.) The lower half of the field contained the additive mixture of two monochromatic "primary" lights (R, G or B) and the test light (T), the light whose chromaticity he wanted to measure. The viewer was asked to turn a small knob to adjust a neutral density wedge filter, placed across the beam of each "primary" light, until their mixture, with the target light, produced a color match to the "white" upper half.

Maxwell's method depends on the fact that a "white" light mixture can always be produced by the mixture of any spectral color with two of the three additive primaries. (Which two depends on the hue of the target color: G and B must be used with "yellow" to "red" wavelengths, R and B with "green" wavelengths, and R and G with "blue" wavelengths.) First, the brightnesses of the three primaries are adjusted until they match the white standard: this identifies their relative proportions in a white mixture. Then one of the three primary lights is replaced by the test color, such as an orange light or colored paper disk, and the matching is repeated. By subtracting and renormalizing the contribution of the two primaries in the second white mixture from their contribution in the three primary mixture, Maxwell was able to define the test color in terms of the quantity of three primary values it replaced.

Maxwell organized his results as a "diagram of colors" — an equilateral triangle now more often called a Maxwell triangle. Any color can be specified in this triangle as the relative proportions of the scarlet, emerald green and blue violet primaries necessary to match it. The red color represented by the pure scarlet primary is located at the red corner of the triangle; the yellow color matched by equal mixtures of the scarlet and green primaries is located midway between the red and green corners; and the white color matched by an equal mixture of all three colors is located at the center — but only if the primaries have been adjusted to have equal luminance or tinting strength.

james clerk maxwell's "diagram of colors"

after Maxwell (1857); the proportions of the additive primaries red, green, and blue violet (exemplified by the pigments vermilion, emerald green, and ultramarine blue) always add up to 1; the approximate location of cerulean blue is shown as an example

Maxwell's Imaginary Primaries. But there was a catch. Maxwell used his color triangle to analyze the primary color composition of many common artists' pigments, only to discover that some pigments were more saturated than any mixtures of his three primaries could match. Thus, the artists' pigment natural gamboge (NY24) was a more intense yellow than any additive mixture of vermilion and emerald green on his color top.

the young James Clerk Maxwell holding his color top (c.1860)

Maxwell method of color matching

the intensities of two monochromatic primary lights are adjusted in mixture with a monochromatic test light (T) until the mixture matches the "white" target field

Rather than add a fourth, (yellow) primary to his system, Maxwell chose to subtract chroma from the gamboge. He did this by visually mixing it either with a gray of equal lightness or with a desaturating quantity of the complementary blue violet primary. This shifted the gamboge color toward gray and brought the color back within the triangle, where it could be matched by a mixture of the remaining two primary colors. The amount of desaturating color required to make this match was used to estimate how far the chroma of the gamboge exceeded the gamut of the three primary mixing triangle. This method was extended and carefully explained by the American physicist Ogden Rood, who showed that this "subtractive" method permitted accurate measurements of pigment chroma even if the color was more intense than the visual primaries used in the analysis.

ogden rood's analysis of pigment chroma

showing the location of pigments more saturated than any visual mixture of the three primary colors; adapted from Modern Chromatics (1879) with numerical values omitted and pigment names modernized

In effect, Maxwell defined "primary colors" as mathematical or imaginary concepts, because the true primaries that could match the undiluted color of gamboge yellow would have to be much more saturated than the actual paint primaries used to match the dulled gamboge yellow on a color top!

This was a crucial step in the development of color science, because primary colors no longer had to be real colors, that is, paints you can actually spin on a color top or lights you can actually extract from the spectrum. Even though this seems to make no physical or perceptual sense, it reflects the fact that the mind never sees the cone outputs and therefore our visual primaries are imaginary colors to begin with. Maxwell's system of imaginary, mathematically defined primaries is so useful that it has become the standard method for specifying the appearance and mixtures of all colors.

the dudeen subtractive color mixing triangle

from John Sloan's The Gist of Art (1939); the primaries are cadmium lemon, alizarin crimson, and phthalocyanine blue [winsor blue]

Rood's work was imitated in many subsequent pigment plot color triangles and color wheels, and for several decades of the early 1900's the triangle replaced the wheel as the trendy representation of artists' color mixing. The Dudeen color triangle, introduced around 1910 by the American artist Charles Winter (1869-1942), exports Rood's plot of isolated pigment colors into subtractive color mixing by replacing Maxwell's green primary with a yellow paint. Unfortunately, the triangle does not accurately represent paint chroma or saturation because Winter adhered to the 18th century artists' dogma that all colors can be mixed from three "primary" paints. For example, cadmium red light is more saturated than any mixture of cadmium lemon and alizarin crimson, yet the color is shown inside the triangle, meaning it is a color that can be mixed directly by those two primaries. Winter's triangle also ignores the many differences between light mixing on a hue circle and paint mixing on a color wheel (the "color wheel fallacy" caused by substance uncertainty), which set the precedent for today's mixing color wheels that reduce pigment measurements to the 18th century color wheel format.

Sources of Color Science edited by David L. MacAdam (Cambridge: MIT Press, 1970) conveniently gathers the important original writings of Grassmann, Palmer, Young, Helmholtz, Maxwell and others in a single volume.

colorimetry

But faulty artists' ideas were only a sideshow in the history of primary colors. Maxwell's color measurement techniques and approach to color specification, as presented to the Royal Society in the 1860's, are among the founding documents of modern colorimetry, the final step in our pursuit of "primary" colors.

The Metameric Foundation. Colorimetry is built on the principle of additive metameric colors. As mentioned earlier, metamers are any two different spectral emittance, transmittance or reflectance curves that appear to be the same color — that is, different spectral profiles that produce exactly the same relative stimulation to the L, M and S cones (assuming that differences in luminance can be adjusted away). Metamers arise in three situations:

• light metamers, different spectral emittance curves perceived as the same color

• material metamers, two different surface reflectance curves or filter transmittance curves perceived as the same color when each is viewed with the same light source (right)

• observer metamers, different spectral profiles perceived as the same color due to limitations in the viewer's visual responses (colorblindness or dark adapted vision).

Light metamers are extremely common, for two reasons: (1) lights may comprise any combination of wavelengths, from monochromatic to full spectrum; and (2) all lights are summarized by the eye as just three cone outputs. There is great flexibility in mixing lights, and great limitation in perceiving lights, so that the same color sensation can be produced in many very different ways.

The basis for color matching through the additive mixture of three "primary" lights is that light metamers can be generated by a specific combination of a small number of monochromatic lights. This is summarized as the additive metameric generalization: provided they are arranged correctly, three "primary" lights can always produce a metameric color match to any color of light — provided that none of the primary lights can be matched by a mixture of the other two, and the luminance (brightness) of the lights can be freely adjusted.

The primary lights provide a common currency in which any spectral profile can be summarized as the relative power or luminance of just three monochromatic lights. This means, in turn, that two spectral profiles can be said to produce a visual color match if their trichromatic metamers are exactly the same. In this way, metameric color matching provides a procedure by which any color can be objectively compared with any other color — the foundation of the colorimetric framework.

the colorimetric framework

two spectral profiles form a colorimetric match if they are metamers for the same trichromatic mixture

Grassmann's Laws. The usefulness of these "primary" color matches might be trivial — if they only applied in situations that were "arranged correctly". But a few principles known as Grassmann's Laws greatly increase the importance of additive metameric matches by greatly expanding the range of circumstances in which the test light and its equivalent "primary" mixture will appear to be the same color. In modern colorimetry, Grassmann's original statements are summarized as three algebraic principles:

• Additivity: if a third light is mixed in equal amounts with both the test light and the metameric mixture of three "primary" lights, the color match (metamerism) remains unchanged. (In algebra: if x = y [the colors match], then x+z = y+z [the match is unchanged].)

• Proportionality: if the luminance of both the test light and the three "primary" lights in the matching mixture are increased or decreased by an equal proportional amount (such as 10%), the color match remains unchanged. (If x = y then x*z = y*z or x/z = y/z.);

• Transitivity: If either the test light or the "primary" mixture is metameric with any third light or mixture, then either (a) the test light or (b) the "primary" mixture can be replaced by this third light, and both the additivity and proportionality laws will still govern the new color match. (If x = y and x = a, then a = y; and if also y = b, then x = b and a = b.)

Grassmann's Laws mean that a color match persists despite a change in color appearance. The test light and its matching primary mixture can be made dimmer or brighter, or mixed equally with another color of light, or replaced by a third matching light, and the two lights still appear to match, even though the color or brightness of the lights has visibly changed.

The last point means that all the color matches produced by mixing monochromatic (single wavelength) primary lights can be duplicated by passing "white" light through narrowpass (multiple wavelength) color filters, or even by changing the color of the "primary" lights themselves: one choice of "primary" lights can be replaced by another. Colorimetry builds on the fact that additive color mixing is both promiscuous and precise.

As an aside: it turns out that Grassmann's Laws are not true for many additive mixtures, most famously in the Helmholtz-Kohlrausch effect, and do not hold across changes in luminance adaptation, but they are true enough often enough so that color matches by themselves can unlock the basic perceptual structure of color.

reflectance profiles for three metameric grays

these appear to be identical middle gray surfaces when viewed under illuminant C; after Wyszecki & Stiles (1982)

Color Matching Experiments. We can motivate an explanation of the colorimetric system by looking at the specific problems that must be solved to measure the chromaticity of the spectrum locus — the single wavelength lights or monochromatic lights that define the physical limits of color vision — using the colorimetric method of trichromatic color matching.

First, it was discovered that Maxwell's color matching method, in which a trichromatic mixture is adjusted to match a "white" standard, tends to produce errors in the trichromatic specification of highly saturated colors such as monochromatic lights. So an alternative method of matching colors with a mixture of three "primaries," known as the maximum saturation method (right), was used instead.

The target color in the match is no longer white, as in Maxwell's method, but any single wavelength or complex (broadband) light mixture we choose. The viewer attempts to match this target color by a mixture of three real (visible) RGB "primaries" of monochromatic light, typically at wavelengths around 645 nm (R), 526 nm (G) and 444 nm (B). For all moderately saturated and near white colors, this arrangement leads to a direct color match; for all wavelengths above "yellow" (~570 nm), only the R and G monochromatic primaries are needed to match a spectral hue.

However, as Maxwell discovered, there are many colors, including many monochromatic lights, that are more saturated than can be matched by any mixture of three standard primary lights. In the conventional choice of RGB primaries, these hues include "green", "cyan", "blue" and "violet" lights on or near the spectrum locus, and highly saturated extraspectral "magenta" and "purple" mixtures. In these cases, the third primary is mixed with the out of gamut target color (shown in the diagram as R added to a "blue green" monochromatic light in the upper semicircular field), to desaturate the target color — move it toward "white" — until it can be matched by a mixture of the remaining two primaries (shown in the diagram as G and B mixed in the lower semicircular field).

maximum saturation method
of color matching

the intensity of a desaturating primary is adjusted until its mixture with an out of gamut target color matches a mixture possible with the remaining two primaries

The diagram (right) shows this solution in a generic chromaticity diagram. The RGB lights, located on the spectrum locus, define a triangular "real" gamut that contains all light colors that can be matched by a direct mixture of the RGB lights. The "out of gamut" colors that are within the chromaticity diagram but outside the RGB mixing triangle (shown as squares) must be matched by mixing them with one of the RGB primaries, to desaturate them. The quantity of the added primary, as shown by the arrows, measures how far the color's chroma exceeds the gamut of the RGB mixing triangle. Thus the RGB primaries are made to match the color — even though they can't match the color!

All colors that must be desaturated in this way are "out of gamut" or unmixable colors. They arise because of the asymmetrical curved or horseshoe shape of the chromaticity space. That is, the desaturating measurement technique is a way to get past a physiological obstacle to the measurement of color vision: the overlap between the L, M and S cone fundamentals.

Individual Differences. After color matching mixtures were defined by several viewers for all the spectral hues, large individual differences were discovered in the results. These were determined to arise from (1) the prereceptoral filtering of "blue" and "violet" light by the lens and macular pigment, and from (2) individual differences in the cone sensitivity curves, caused by genetic differences in the cone photopigments and in the proportional numbers of L, M and S cones in the retina. These individual differences constitute a physiological complication in the measurement of color vision.

In the late 1920's W.D. Wright realized that most of these individual differences or "errors" could be eliminated mathematically. The saving resource had been described by Newton: passing a monochromatic light through a colored filter does not change the color of the light, it only makes it dimmer. So the logical fix to the "yellow" filtering of light was to eliminate differences in the perceived brightness and tinting strength of the monochromatic RGB primary lights.

wright's WDW chromaticity coefficients

his rescaling of the Stiles-Burch 1959 color matching data
(adapted from Wyszecki & Stiles, 1982)

Wright's adjustment involves two steps: (1) mathematically rescale the relative luminances of the three primaries for each viewer so that the R and G proportions are exactly equal at 580 nm ("yellow") and the G and B proportions are equal at 488 nm ("green blue") (equal tinting strength); then (2) normalize the luminances across all viewers so that the maximum value of each pure primary hue is 1.0 (equal brightness). This eliminates nearly all of the color matching discrepancies between observers, as shown above in the WDW diagram of mixture curves for 50 subjects with normal color vision.

The White Point. In order to revert these WDW corrected color matching proportions back to the relative proportions of monochromatic light required in the original metameric mixtures, or equate the colormatching results from different colored "primary" lights, the proportional radiance (radiant power) or luminance of the three lights must also be specified.

Either choice determines a fourth primary color, the pure white that results when the three monochromatic lights are mixed. The location of this mixture in a chromaticity diagram is called the white point — the only color with exactly zero chroma. Thus, all trichromatic systems are defined by four primary lights — referred to as the cardinal stimuli.

real (RGB) and imaginary (XYZ) primaries in relation to a chromaticity diagram

The abstract "white" standard is the equal energy illuminant (EE), defined as a perfectly flat spectral power distribution. This is the psychophysical standard implicit in the publication of all visual sensitivity curves because it produces exactly equal outputs from equal area cone fundamentals; these cone fundamentals must be multiplied by (adapted to) the spectral profile of any physical light source in order to define color appearance under real world viewing conditions. The EE white balance requires radiant power ratios between the three "primary" lights of roughly 72.1 : 1.4 : 1.0 (R:G:B), which produce relative luminances of about 10 : 46 : 1 (diagram, right).

Despite the fact that short wavelength "primary" light is both visually dim and radiantly weak, it is able to counterbalance the high saturation (tinting strength) of the L cones and the high luminance of the M cones and the L+M mixtures. If we apply Newton's method of geometrical weighting to the luminances alone, then the centroid (C) is located substantially toward the G primary to give the S cones a compensating leverage. (Note the locations of the white point in this chromaticity diagram, which shows the relative contribution of foveal and wide field cones to a "white" color sensation.) This is the sense in which the S cone outputs are heavily weighted in color perception.

As just mentioned, the equal energy illuminant is usually not used to define the white point in practical colormatching tasks, such as industrial color control: compared with natural daylight, the EE illuminant has a faint magenta tint. Illuminants that approximate the chromaticity of artificial or natural radiant power distributions, especially CIE D65, D50 or A, are used instead.

Real Color Matching Functions. The WDW averaging of color matching curves and the luminance adjustments to a standard white result in the RGB color matching functions of three real, monochromatic primary lights. The r(λ) curve is more than 3 times higher than the others because "red" wavelength primary has a low luminance and moderate tinting strength, so more of the R primary must be used to match the high luminance of the G primary and the high tinting strength of the B primary.

RGB color matching functions

Stiles-Burch 10° color matching functions averaged across 37 observers (adapted from Wyszecki & Stiles, 1982)

These are not the spectral emittance curves for the RGB primary lights used in the color matching. They trace the tristimulus values or proportional quantities of each R, G or B primary light in the mixture that matches the spectral hue at each wavelength. The negative tristimulus values indicate that the primary must be mixed with the target light — primarily R with "blue green" and most "blue" spectral hues and G with extreme "violet" hues. Each pair of curves intersects the zero line at the location of the third primary.

The tristimulus values are stimulus quantities determined by the physical properties of the "primary" lights. As a result, the curves are specific to the actual choice of primaries used: different primary lights result in different color matching proportions, which produce different tristimulus values.

the fourth primary

circle areas are proportional to the luminance or radiance of the trichromatic "primaries" that match an equal energy illuminant (E); the centroid of the luminances is at C

To avoid confusion, the primary lights are denoted with a capital R, and the tristimulus value of the light, the relative quantity required in a color match at each wavelength, with a lowercase r(λ). Thus, in the example (right), 2.8 parts of the R primary light (read from the r(λ) curve), 0.6 parts of the G primary light (read from the g(λ) curve) and zero parts of the B primary light (read from the b(λ) curve) will produce an exact color match for a "yellow" spectral hue.

If the light is actually a mixture of many different wavelengths, then the tristimulus values for each wavelength are multiplied by the intensity of light at that wavelength, and the products are summed across all wavelengths to get the three tristimulus values. (The color of the white point light is simply the sum of all three curves across all wavelengths.)

Size of Color Stimulus. Now color researchers encountered yet another physiological complication to the measurement of color vision. Because the cones are unequally distributed across the retina, it matters how large and where the color stimulus appears in the visual field — that is, how it is imaged on the retina. This motivated the development of two different standards for color matching:

• The earliest color matching curves (circa 1890 to 1930) were based on a 2° or foveal presentation of color stimuli. (A 2° visual field is about the size of 1 inch viewed from a distance of 29 inches, or 10 cm viewed from 2.9 meters.) This was done because it was not then technically feasible to produce bright and homogeneous color stimuli across large visual areas, and because it minimized rod intrusion, or the mixture of rod and cone responses, in color matching of dim lights. As a result, the 2° mixtures are more robust across moderate ("reading light") to bright (noon daylight) levels of illumination. These curves are denoted by the date they were adopted (1931) or the field size (2°). Unfortunately, these early studies combined data using different primary lights (including filtered "white" lights), and estimated the relative luminances of the primary lights from the 1924 luminous efficiency function, which was later found to underestimate the luminance of short wavelengths. Judgmental revisions were imposed in 1951 to correct for this, but the uncorrected curves are still occasionally used.

• Later (circa 1960), color matching curves were measured in a 10° or wide field presentation of color areas that extended well outside the fovea and filtering by the macular pigment. (A 10° visual field is about the size of 1 inch viewed from 5 3/4 inches, or 10 cm viewed from 57 cm.) These curves are based on two large, carefully screened samples of subjects who viewed monochromatic "primary" light mixtures at radiances well above rod saturation (except for extreme "red" mixtures, which were corrected for rod intrusion). These curves are also denoted by the publication date (1964) or the field size (10°). I use the wide field data throughout this site because human wide field color discrimination is about 2 to 3 times more accurate than foveal color discrimination, and because the "primary" light radiances were directly measured rather than inferred from a flawed luminous efficiency function; the 10° curves are also preferred for industrial colorimetry. The drawback is that the curves are most accurate at daylight levels of illumination and can succumb to additivity failures under mesopic light levels.

Imaginary Color Matching Functions. But we're still not done. The tristimulus values contain negative (subtracted) r(λ) and g(λ) quantities for the out of gamut wavelengths in the original color matching mixtures. As these negative values conceptually amount to saying that the photoreceptor pigments sometimes emit rather than absorb light, and as they are both an artifact of the maximum saturation method with real primary lights and of the overlap in the L, M and S cone fundamentals, a mathematical manipulation is applied to create three new imaginary XYZ primaries (as shown in the diagram above right).

The new mixture proportions x(λ), y(λ) and z(λ) are found by multiplying the RGB tristimulus values at each wavelength by a transformation matrix to get the XYZ color matching functions:

x₁₀(λ) = 0.341r₁₀(λ) + 0.189g₁₀(λ) + 0.388b₁₀(λ)

y₁₀(λ) = 0.139r₁₀(λ) + 0.837g₁₀(λ) + 0.073b₁₀(λ)

z₁₀(λ) = 0.000r₁₀(λ) + 0.040g₁₀(λ) + 2.026b₁₀(λ)

where as before λ denotes a specific wavelength.

This transformation does not undo the desaturating physiological obstacle to the measurement of color vision — it just turns it on its head as a supersaturated definition of the primary lights! That is, the XYZ primaries are outside the gamut of all real colors and are therefore invisible. No color of light or surface can reproduce them. However, their imaginary mixing triangle completely contains the chromaticity space of all real, visible colors, so all colors can be described as the positive mixture of the XYZ imaginary lights.

So here, at last, are the 10° (wide field) XYZ color matching functions:

1964 XYZ color matching functions

CIE 1964 color matching functions for the 10° standard observer and the imaginary XYZ primaries (from Wyszecki & Stiles, 1982)

All negative values have been removed, and the white point or achromatic standard is still defined by an equal energy illuminant. The XYZ tristimulus values are the fundamental stimulus metric used in colorimetry.

The Chromaticity Diagram. Each of the tristimulus values combines information about chromaticity (radiance in its part of the spectrum) and luminance (its part of the radiance across the whole spectrum). To obtain coordinates for a two dimensional chromaticity diagram, the effect of luminance on the tristimulus values is made constant, which is done by normalizing the values. That is, X and Y (the x₁₀(λ) and y₁₀(λ) values summed across all wavelengths) are divided by the sum of all three tristimulus values:

The normalized z value is redundant, since the normalized weights sum to 1.0, so z can be recovered by subtraction:

z = 1.0 – x – y.

rgb color matching functions

CIE 1964 color matching functions for the 10° standard observer and the real (RGB) primaries

This leaves the normalized x and y values to indicate the chromaticity of the color. The curves are never displayed in the format we have used so far — value as a function of wavelength (right) — because hue (dominant wavelength) is what we want to describe. Instead, the x and y values are displayed as a two dimensional rectangular plot, which creates the CIE 1964 xy chromaticity diagram.

CIE 1964 xy chromaticity diagram

the chromaticity diagram for the 10° standard observer, represented as the imaginary XYZ primaries normalized to give all colors equal brightness; colors are illustrative only (adapted from Wyszecki & Stiles, 1982)

In this diagram the x,y values at each wavelength define the spectrum locus or the chromaticity of monochromatic light and the perceptual limits of color vision. All additive color mixing can be represented as the geometric mean of all chromaticities in the mixture, and two colors with the same chromaticity (x,y values), viewed in standard display and lighting conditions, will appear identical to normal viewers.

There is one last twist. Thanks to the specific transformation matrix used, the unnormalized Y primary (the y₁₀(λ) color matching function) is identical to a 10° photopic sensitivity function, so the y₁₀(λ) values define the apparent brightness of each wavelength. As a result, colors are commonly specified using Y for luminance and x,y for chromaticity. This creates a three dimensional color solid or color pyramid, as shown below.

CIE 1964 Yxy color solid

adapted from a wireframe animation © 2007 Bruce Lindbloom

Here also there is a muddle. The Judd-Vos luminosity function V_M(λ), which is in widespread use, is derived from 2° color matching functions that give unrealistically low weight to the "blue" and "violet" wavelengths. The recently published Stockman & Sharpe 2° luminosity function V*(λ), which is not in standard use, gives a more accurate weight to the short wavelengths. It is also closer to the CIE 1964 y₁₀(λ) values.

The CIE Yxy Standard Observer. The WDW corrected XYZ values, in the form of the x,y chromaticity diagram and Y luminous efficiency function, represent the CIE Yxy standard observer — first published from 2° color matching data in 1931 and supplemented by 10° data in 1964. The standard observer is an idealized human retina that does very well at the limited task of predicting additive color mixtures and color matches that are viewed in isolation and at mid mesopic to photopic luminance levels.

Color matching data, repeated and varied across hundreds of different color samples and viewing situations for both normal and color deficient viewers, and archived in the color vision literature with light sensitivity and hue cancellation data, are the foundation texts of color research. They provide a quantitative basis for evaluating theories of color vision, and they stand for all the practical situations in which people say two different color stimuli do or do not create the same color sensation.

In addition, different transformation matrices can be used to convert the XYZ color matching curves into cone sensitivity curves, or to define the L*, u* and v* uniform perceptual dimensions of the CIELUV color model, or to define the L*, a* and b* opponent dimensions of the CIELAB or CIECAM02 color appearance models. Here, for example, are the CIE transformation equations that define the equal area cone fundamentals:

L(λ) =  0.390X(λ)+0.690Y(λ)–0.079Z(λ)

S(λ) =  1.000Z(λ).

Smith & Pokorny devised population weighted curves from the 2° standard observer so that the L and M functions sum to the photopic sensitivity function V(λ):

L(λ) =  0.15516X(λ)+0.54308Y(λ)–0.03287Z(λ)

S(λ) =  1.000Z(λ).

And here are Stockman, MacLeod & Johnson (1993) 10° cone fundamentals, calculated from the 1964 10° standard observer:

L(λ) =  0.23616X(λ)+0.82643Y(λ)–0.04571Z(λ)

S(λ) =  0.04056X(λ)–0.01968Y(λ)+0.48619Z(λ)

We might pity an observer who, like Lieutenant Kije, exists only in official documents, but he has had an unusually productive career. The XYZ color matching functions enable electronic color measurement in thousands of practical applications. Light intensities are measured through three colored filters or photometric diodes with transmission profiles that match each function, or light intensities are measured at regular (usually 1 nm to 5 nm) intervals across the spectrum, then multiplied by the x₁₀, y₁₀ and z₁₀ weights at each point and summed to get the total XYZ tristimulus values. These estimate the color's brightness and chromaticity as it appears to an average viewer with "normal" color vision.

The chromaticity diagram, and the trichromatic or trilinear system it is based on, have several cool and useful properties — as backround, you may want to refer to the discussion of the trilinear mixing triangle:

• Three Number Color Specification. Every visible color must lie within the chromaticity diagram, which means all possible colors can be defined as a proportional mixture of the XYZ primary lights.

• Illuminant Adaptation. The definition of the white point in the chromaticity diagram can be adjusted in relation to a second set of XYZ values, which define the brightness and chromaticity of the light source. In this way the standard observer can "adapt" to a wide range of viewing situations and predict illuminant based metamers.

• Luminosity Specification. The Yxy system defines the brightness of a color as its total Y value; its lightness is the ratio between the Y values of the color and a white surface under the same illumination.

• Hue Specification. The dominant wavelength of a color is defined as the point where the spectrum locus intercepts a line drawn from the white point through the color's x,y location in a chromaticity diagram.

• Chroma Specification. The hue purity of a color is approximately defined by its chromaticity distance from the white point. (In the original x,y chromaticity diagram, the relationship between chromaticity distance and chroma varied across hue, a problem largely fixed in CIELAB.)

• Straight Mixing Lines. The mixtures between any two colors of light, defined as points within the chromaticity diagram, are described by a straight line between them. (Beware! Mixtures of colors of paint, because they are surface colors, do not make straight lines in a chromaticity diagram or color space!)

• Visual Complements. The visual complement of any hue is defined by a straight line from that hue through the white point to the opposite side of the diagram. Thus, the visual complement of a "deep yellow" (580 nm) is a "greenish blue" (480 nm).

• Perceived Color Differences. The distance between any two colors on the chromaticity diagram approximates the perceived difference between the colors. Unfortunately, this approximation is very poor in the original x,y chromaticity diagram and Y brightness measure, but was substantially improved in the CIELUV u,v chromaticity diagram (for emitting colors) and in CIELAB and CIECAM (for reflecting colors).

The intellectual elegance of this colorimetric edifice, built over a century of continuous work, is that all the essential information about an unrelated color — any color seen in isolation, from a single wavelength light to the most complex surface reflectance curve — can be captured, compared and contrasted with other colors through the mechanism of three numbers.

imaginary or imperfect primaries

This completes my historical survey of "primary" colors. We began with the rudimentary color concepts of Greek philosophers and ended with the complex color analysis of 20th century colorimetry. I have tried to show how the idea of "primary" colors has evolved from vague abstract notions to specific technical concepts. Now I want to summarize this review as principles for artistic practice.

normalized tristimulus values

compare with the WDW diagram above

The first primary paradox is:

Primary colors are either imaginary, invisible "lights" that can describe all colors, or they are imperfect, real colorants that reproduce only some colors.

This double impossibility — you can't mix all colors with the primary colors you can see, and you can't see the primary colors that can mix all colors — arises from the physiology of color vision, the way the human eye is structured.

The sensitivity curves of the L, M and S cones overlap each other: every monochromatic (single wavelength) hue must stimulate two or even three cones simultaneously. As a result, the boundary of visible colors curves away from the "pure" primary corners of a mixing triangle, creating the horseshoe shaped chromaticity space of visible colors (right).

Because of its curved borders, the chromaticity space cannot be completely enclosed by any triangle defined by three monochromatic lights RGB around its border, and therefore all visible primaries cannot mix all possible colors — which makes them imperfect. Any three primary colors XYZ that completely enclose the chromaticity space, and therefore define all visible colors, must located outside the chromaticity space of real colors — which makes them imaginary.

the first primary paradox

spectral primaries RGB, which are visible, can't mix all colors; mathematical primaries XYZ, which describe all colors, are invisible

The second primary paradox is:

All choices of imaginary primary colors are arbitrary; they are only measurement units. All choices of "real" primary colors are arbitrary; colorant selections depend on cost, availability, convenience, medium and image quality.

The imaginary primaries used in colorimetry are simply standardized units of measurement, like the meter, joule or yen. Just as the imaginary foot used in distance measurement does not represent a real human foot, the imaginary primaries used in color measurement do not represent real lights. Just as the meter could be longer or shorter and still work just fine as a standard unit of measurement, there is an infinite number of triangles of different sizes or shapes that would completely enclose the chromaticity diagram (right) and therefore would work just fine as a standard color gamut of imaginary primaries.

As with most measurement units, the imaginary XYZ primaries have been adopted in part for reasons of convenience. The transformation matrix used to define the imaginary primaries was chosen to reproduce the luminous efficiency function in the Y primary, but this was an arbitrary decision. All imaginary primary colors are arbitrary.

The material primaries are also arbitrary standards, but of a different sort. There is only one necessary restriction on the "primary" lights used in color matching experiments: they must form a triangle with all three corners inside the chromaticity diagram. Otherwise, the shape and location of the triangle doesn't really matter. In fact, many different monochromatic lights and "white" lights tinted with color filters have been used color matching experiments: there has never been a standard or "best" set of RGB lights. They were chosen for a variety of operational reasons, then were transformed into the same XYZ system by using different transformation matrices.

The material primaries used in color reproduction (including painting, photography and video) are the outcome of a long and painstaking development of physical colorants (dyes, pigments, phosphors, diodes and lights) in chemistry, physics and engineering over the past three centuries. This development was marked at each step by tradeoffs or compromises. The colorants have not been standardized by laws of nature but by government codes and industrial standards, the envelope of feasible manufacturing costs, consumer expectations of longevity or durability, and accepted practices of color reproduction or existing color technologies. They also accommodate subjective standards of "good" and "bad" color reproduction for a given purpose in a given viewing situation. Compromise always occurs in color reproduction, which means all real primary colors are arbitrary.

the second primary paradox

all real primary colors RGB, and all mathematical primaries XYZ, are arbitrary choices

The diagram (right) shows the location of the subtractive magenta, yellow and cyan primaries, as defined by optimal (maximally saturated) colors in the CIELAB a*b* plane. Painters who wanted to use a primary palette historically have chosen from among the labeled pigments. It is clear that throughout the history of painting, painters have used a hue of "primary" yellow that is too red, a "primary" red that is much too yellow, and a "primary" blue that is too red. They specifically have not used the underlined pigments, which are closest to the optimal color choices — cadmium lemon (PY35), cobalt teal blue (PG50) and cobalt violet (PV49).

Why? Because the "optimal" pigments in practice produce unsatisfactory mixtures; because the alternative selections are less granulating, more transparent, and mix darker values; and because visual preferences have demanded relatively saturated yellow to red mixtures, obtained at the expense of relatively dull green and purple mixtures. Artists jettisoned "theory" to obtain the best color mixtures in practice.

Similar tradeoffs have determined the selection of commercial standards for video and photography (shown below on the CIELUV u*v* plane), and for commercial printing (shown here). Artists' pigments, film dyes and video phosphors can mix only about half the total range of visible colors , but this restriction avoids problems of impermanence (low lightfastness or chemical stability), high manufacturing cost, quality control and visual standards of image acceptability — especially in the imaging of flesh tones.

gamuts used in painting, photography and video

on the CIELUV chromaticity diagram; adapted from Hunt (2004)

The arbitrary nature of real "primary" color choices appears across history as it does across modern media. Because of advances in chemistry and industrial manufacture, pigment selections have changed considerably over the past three centuries (for a history, see Bright Earth: The Art and Invention of Color by Philip Ball). In the earliest industrial color mixing system devised by Le Blon, the primary colors were a mixture of carmine, madder lake and vermilion (mixed to make red), yellow lake, and prussian blue — the best pigments for the job available in the early 18th century. The artists' color wheel proposed by Moses Harris used vermilion, orpiment and natural ultramarine. Charles Winter's 20th century mixing triangle used alizarin crimson, cadmium lemon and phthalo blue. Contemporary painters would probably prefer hansa yellow, quinacridone rose, and phthalo blue.

Pigment innovation has even created entirely new primaries and new systems of color mixing. Thus, magenta was not identified as a subtractive primary color until the CMYK system was invented by Alexander Murray in 1934, because suitably lightfast magenta inks were unavailable before then. The CMYK system, in turn, cannot reproduce many saturated oranges, violets, blues and yellow greens, and in specific applications where brighter colors are required, newer printing systems with larger gamuts — Hexachrome™ (six primary colors of ink) or Heptatone™ (seven primary colors) — can be used instead.

What makes these tradeoffs feasible — and often unnoticeable in practice — is the remarkable ability of our color vision to accept different color images as equivalent or identical, provided the gamuts used to reproduce the images retain the relative relationships between all the colors in the image, especially relative lightness and hue. Thus, we can happily watch a Discovery Channel documentary on tropical birds, coastal reefs or erupting volcanos without noticing that the leafy greens, littoral cyans and magma reds are really much duller than they appear in life.

The reason we don't notice the color difference is that color vision treats all colors as arbitrary, in the sense that the accuracy of a color choice depends on the image context, viewing situation and viewer expectations. Video technology reproduces acceptable color relationships in context, so the image as a whole appears accurate even though separate image colors do not match the actual colors of the represented objects. (This topic is explored in the sections on color constancy and gamut mapping.)

Three Artists' Misconceptions. The primary paradoxes were already known in the 18th century, at least in the recognition that paints and dyes could only "imitate the hues (though not always the splendor) of those almost numberless differing colors" of nature, and that different colors (different pigments or dyes) could equally serve as a "primary" red, yellow or blue.

Unfortunately, the recognition came long before scientific explanations of overlapping photoreceptor sensitivities, additive mixtures and gamut mapping were available. As a result, four misconceptions developed in the 18th century to explain the problems with primary color mixtures ... and many artists repeat them even today.

The first misconception is that "primary" colors must be visible colors, in the sense that an artist can pick a color of sticky paint or a wavelength of visible light and say, "there, that is the primary yellow". But the first primary paradox shows this belief is false. The "primary" colors that describe color vision are imaginary colors — and the mind never has direct experience of the "primary" signals from our three photoreceptors. The choice is not between one shade of color and another but between a visible color and an imaginary color. It is no more possible to find a paint that matches "primary" yellow than it is to find a horse that matches Pegasus.

The second misconception is that "primary" colors must be specific colors, in the sense that an artist can pick one color of primary yellow paint as the "nearest match" to the "true" primary yellow color, or that one primary yellow paint is the "best" primary yellow. (See for example the color theory book by Jim Ames.) But, as we have seen, the selection of real colorants is always arbitrary: it has much more to do with perceived image quality than with "objective" color characteristics. Almost any three colors can serve as primary colors, depending on how you want to use them. The only relevant issues are (1) the actual range of mixtures (gamut) you are able make with the colorants, and (2) whether this gamut produces the desired visual effect in the images you want to represent.

The third misconception is that none of the various colors of primary paints are the "true" primary color because all paint colors are "impure" or polluted by added light from the other two primaries. (See for example the justification offered for the split "primary" palette.) This is an especially quaint anachronism from the 18th century, and it is wrong from several points of view. If a paint really were "pure" and only reflected a single wavelength of light (which is the "purest" possible color stimulus), the paint would have a luminance factor near zero and would appear blacker than the "purest" black paint! And that monochromatic "yellow" light is no more saturated than a mixture of a monochromatic "orange" and "yellow green" light, so light purity is not the cause of hue purity. Finally, even if our primaries were three "pure" colors of light (regardless of their hue), we still couldn't mix all other colors. "Purity" or pollution has absolutely nothing to do with the limitations of primary colors of paints or dyes.

optimal and actual primary paints

as located on the CIELAB a*b* plane

All three misconceptions are reification — a belief that primary colors are real. But if they are real, what do they consist of? The schematic (right) shows a sequence of physical or perceptual components that are currently used to explain the experience of color. The task for a "color theorist" is to point to a step in the diagram and say, "here is where primary colors are located". Are "primary" colors actually memory colors? Are they specific colorants? Are they produced by subtractive mixture, or additive perception? Are they cone outputs, or opponent codes?

There is no fixed location for a reified concept because it must apply to many different attributes of things in many different situations. The "primary colors" in the retina, in consciousness, in colorimetry, in color printing or in color television have nothing in common, either. All "primary colors" are a figure of speech, a fiction treated as a reality, not a feature of the world. If "color theorists" want to claim that primary colors are real, they have some fast talking to do!

These intellectual issues are inconsequential, however, because talk has little to do with procedure. From the Greeks through the Baroque, painters simply did what they did without an intellectual color theory to rely on. If they talked about "primary" colors, they meant the term to refer to types of colorants, not to recipes of light and dark, or yellow and green. Abstractions were used by the scholars, the philosophers and naturalists, not by the artisans who actually worked with color. The same is true today. The "primaries" in color television or color printing evolved by research trial and error, manufacturing limitations and consumer acceptance, not by turning a crank on a color theory calculator. Then as now, there is a large gap between theory and experience.

For artists, any attempt to fuse color mixing with paints with an abstract, rigid system of "primary" colors runs up against an additional problem: the system cannot accurately describe actual paint mixtures. It also idealizes color relationships, divorcing them from the visual context of specific objects in space, or pigmented materials on a specific type of ground, as viewed under a specific intensity and color of illumination.

The conclusion of this historical excursion is that "primary" colors are only useful fictions. They are either imaginary variables adopted by mathematical models of color vision, or they are imperfect but economical compromises adopted for specific color mixing purposes with lights, paints, dyes or inks.

Primary colors are sometimes defended as a pedagogical simplification to teach elementary color mixing. But, as I propose elsewhere, there are better frameworks for that purpose, too. "Useful fictions" should be employed only when they are useful.

At bottom, the only justification for primary colors is to minimize the number of components required to mix all colors. This limitation makes biological sense if you are evolving a color sensing eye (and need to minimize the number of photoreceptor cells), mathematical sense if you want to model how that eye works (and want to do it with the fewest variables), economic sense if you are printing a color job (where each color requires a separate printing plate, ink, and pass with the printing press), or technological sense if you are manufacturing color televisions or computer monitors or color film (where each color requires a separate phosphor or dye).

But in every case the choice of primary colors is either arbitrary or imperfect. And if you are not building eyes or modeling color vision responses or running a printing press or designing a computer monitor, and can inexpensively "expand your gamut" with four colors — or six, or twelve, or twenty — on your palette, then "primary" colors are irrelevant to the task before you.

N E X T :   modern color models

Last revised 08.I.2015 • © 2015 Bruce MacEvoy