Complementary colors

1Lights add, paints subtract

White light is every wavelength at once. You can get the look of white without all of them, though. Your eye has only three kinds of cone cells, and two well-chosen lights can excite them in the same proportions as white does. Those two lights are complements.

Until the 1850s, people assumed mixed lights behave like mixed pigments. Hermann von Helmholtz showed in 1852 that they are two different processes [6]. Blue and yellow lights, added together, give white. Blue and yellow pigments, mixed together, give green. The lights pool their wavelengths. The pigments each take wavelengths away, and the mix keeps only what neither one takes.

Hermann Grassmann worked out the arithmetic of adding lights in 1853 [5]. On the CIE chromaticity diagram, every mixture of two lights lies on the straight line between them. Two lights are complements when that line passes through white. Follow the line from a color, through white, to the other side, and you reach its complement.

Computed with the CIE 1931 observer and daylight white [2], 450 nm blue pairs with 568 nm yellow, 480 nm with 578 nm, and 490 nm blue-green with 601 nm orange. Helmholtz measured pairs like these by eye in 1855 [7]. The two lights don’t need equal brightness. White needs about 40 times more brightness from the 568 nm yellow than from the 450 nm blue, because the eye is far more sensitive to yellow.

Greens between 494 and 567 nm have no partner in the spectrum. The line from a green through white lands on the purple line, so a green’s complement is a purple: red light plus violet light, a color that isn’t in the rainbow at all.

2What a mix of paints keeps

A pigment doesn’t add light. It removes a share of each wavelength. In one wash, two pigments’ absorptions add up, wavelength by wavelength [4][9]. The light that makes it back from the paper has had to get past both, so what comes back is roughly the product of what each lets through. A wavelength survives only if both pigments let it pass.

Here’s what each paint returns at a few wavelengths, alone and mixed, at the mixer’s starting settings:

Each paint is bright where the other is dark, so the mix is dark everywhere, and about equally dark everywhere. A flat, low curve is what a dark gray is. Swap the orange for hansa yellow and the picture changes. Hansa yellow absorbs only below about 490 nm, and phthalo blue only above about 540 nm. Between them is a window, around 500–530 nm, that neither pigment absorbs much. At 500 nm the yellow returns 26% and the blue 31%, and their 50:50 mix still returns 29%. That window is green.

This is why a color wheel can’t settle what a paint’s mixing complement is. Bruce MacEvoy, writing on watercolor paints, points out that different paints mix to gray with the same partner, so “we often end up with several different hues as mixing complements for a single paint” [10]. What matters is where each pigment’s absorption band sits, and two paints of the same hue can have different bands. Digital painting tools run into the same thing. Mixing colors as RGB numbers treats them like lights, so blue and yellow average to gray. Sochorová and Jamriška built a digital paint mixer on Kubelka–Munk theory so that blue and yellow make green, as paint does [12].

Mix hansa yellow and phthalo blue Mix pyrrole orange and phthalo blue

3Why the paint gray is so dark

Look at the specks swatch in the mixer. It holds the same pigment as the wash, just not mixed: each speck is one paint or the other. Seen from far enough away that the specks blur, your eye averages their light, the way it blends specks of bare paper in a broken glaze. Averaging light can never make anything darker than its darkest part.

Mixing in one wash is different: every bit of light has to get past both pigments. At the grayest mix of pyrrole orange and phthalo blue, the wash returns 1.7% of the light (L* 14). The specks return 20% (L* 52). That’s why two strong complements make a near-black, and why, as lights, these two don’t cancel at all. Averaged as specks, orange and phthalo blue light give a pink (C* 56), because phthalo blue isn’t orange’s complement in light.

In watercolor, the white comes from the paper, so a lighter gray is a thinner wash. At the same 60:40 mix, cutting the pigment from 0.6 to 0.15 raises the gray from L* 14 to L* 57. Try the strength slider.

4Finding the neutral

The grayest mix is rarely half and half. Pigments differ in tinting strength, how much absorbing a little of them does. Phthalo blue is one of the strongest. MacEvoy puts its tinting strength at about 40 times ultramarine’s [11]. Here, pyrrole orange and phthalo blue reach their grayest at 40% phthalo, and the gray drifts warm or cool as you move away from it.

The figure below plots the mixer’s pair on the CIELAB a*b* plane: hue around the center, chroma C* outward from it, gray at the center. As the share of the second paint goes from 0 to 100%, the mix traces a path from one paint to the other.

The specks path is close to a straight line, since lights add along straight lines (section 1). The one-wash path curves. Mixing complements bend toward the center and pass close to it. Hansa yellow with phthalo blue bends the other way, out through green, and never comes within C* 42 of gray. As specks, the same two paints pass within C* 3 of gray: they are complements in light but not in paint. Pyrrole orange and phthalo blue are the reverse.

Mix burnt sienna and ultramarine Mix quinacridone rose and phthalo green Mix hansa yellow and dioxazine violet

5Complements side by side

There’s a third meaning of complementary, and it lives in the eye. Stare at a blue patch and then at white paper, and you see a yellowish-orange afterimage: the afterimage complement [3]. Your visual system codes color as opposing pairs, red against green and yellow against blue. Hurvich and Jameson measured these opponent channels in 1957 by canceling one hue with its opponent [8].

Opposition shows up when complements sit next to each other, too. The chemist Michel-Eugène Chevreul described in 1839 how two colors side by side push each other apart in appearance, so complements placed together each look more intense [1]. That’s a reason painters set an orange against a blue. It is about how the colors look next to each other, not how they mix, and none of the numbers on this page model it.

The Colour Literacy Project sums up the tangle: depending on which definition you use, the complement of blue can be red-orange (mixing paint), yellow (adding light) or yellowish-orange (afterimage) [3]. MacEvoy makes the same point: “the mixing complement will often be very different from the visual complement” [10].

6What this model leaves out

• Illustrative pigments. The curves have their absorption bands in about the right places, but they aren’t measurements of real tubes. Your phthalo blue and your orange will find their gray at a different ratio.
• Mixes that separate. Some pairs don’t stay mixed on the paper. Heavy pigments such as ultramarine settle into the paper’s hollows (granulation) [11], so an ultramarine–burnt sienna gray dries mottled, part mix and part specks.
• One layer. Complements glazed one over the other behave much like a mix if both are transparent, and differently if either scatters light. See Glaze vs. mix.
• One viewer, one light. Colors are computed for the CIE standard observer under daylight. Other lights, and other eyes, shift both kinds of complement.