## Color: odds & ends

A friend sent me an e-mail about the first color post. What started out as a few comments in reply has grown into a full-fledged post.

I think his questions boil down to:

• how are the terms additive and subtractive related to the primaries?
• Are the primaries unique?

Let me elaborate. Early on I’m going to mention a couple of things I think you should avoid.

## Additive & Subtractive

Here are two fairly standard drawings. I believe the one on the left should be called “subtractive mixing of CMY primaries”. I believe the one on the right should be called “additive mixing of RGB primaries”.

We can tell at a glance which one is subtractive: it has black in the center. The other one has white in the center, so it is additive.

(Here is the first no-no: I think it is a mistake to call these images “the subtractive primaries” and “the additive primaries”. Primaries are not unique. I realize that I’ve also just violated a standard rule of teaching: never write falsehoods on the blackboard. Well, that’s why I emphasized the word “mistake” instead of the mistake itself.)

One very interesting thing about these drawings is that they have six colors in common: red, green, blue, cyan, magenta, yellow. The drawing on the left takes CMY as primaries, and produces RGB as subtractive secondaries. The drawing on the right takes RGB as primaries, and produces CMY as additive secondaries.

Oh, using the RGB color space, these six colors are easily specified. What color you get is another issue, but the specs are easy: e.g. R = (1,0,0) and C = (0,1,1). (A refresher follows.)

Two consequences of these two drawings are blindingly obvious, but need to be stated: magenta and red are different colors; and blue and cyan are different colors.

(That’s the second no-no: I think it is a mistake to act as though blue and cyan are the same color, or as though magenta and red are the same color. I am more than okay with the idea that the subtractive mixing of magenta and yellow gives me red, and the subtractive mixing of red and yellow gives me orange.)

Life is complicated because the left-hand drawing could be, and often is, replaced by a different set of colors: subtractive mixing of RYB primaries. That is, the drawing on the left is often replaced by one showing how orange, green, and violet are obtained by subtractive mixing of the painters primary colors of red, yellow, blue.

(These secondaries are harder to specify. As before, I used the CYMK specs from the back of the Color Harmony Workbook.)

I would point out that the green we get by mixing blue and yellow should be different from the green we get by mixing cyan and yellow. The other two colors we get should be as different as their names would suggest: violet instead of blue, and orange instead of red.

We are seeing that a different set of primaries is giving us a different set of secondaries. If our subtractive primaries are CMY, our subtractive secondaries are RGB. If our primaries are RYB, then our subtractive secondaries are OGV.

Cabarga tells us in “The Designer’s Guide to Color Combinations” (p. 11) that “Before the CMYK system became firmly established, three- and four-color printing was often accomplished with a vermillion red, a royal blue and a golden yellow.” In other words, even printers used to use a different set of subtractive primaries.

Now that I’ve shown you two drawings with different subtractive primaries, let me rephrase my point. It seems unreasonable to say that RYB are “the” subtractive primaries, and also that CMY are “the” subtractive primaries, when we also have drawings showing that blue & cyan, red and magenta are different colors. What is true is that the most common subtractive primaries are reddish, bluish, and yellowish. Beyond that, printers and artists usually make different choices, as for example, cyan versus blue, respectively.

As for additive primaries, we will see when we study the CIE chromaticity chart that we have a great deal of freedom there, too. But our choices are not without limitations. We will see, for example, that RYB would not be a good choice for additive primaries: there would be a lot of colors which we could not produce additively from RYB. In the case of additive color, the key is: what are the characteristics of the RGB phosphors on the screen?

## Artists mixing colors in the real wold

I have three art books at hand which describe how to mix colors from three primaries.

But.

They all use 6 primaries.

The key is that one of each primary is biased toward one of the adjacent secondaries.

Wilcox, for example, chooses Cadmium Red Light, “a warm red, leaning towards orange”, and Quinacridone Violet, “a cool red, leaning towards violet.” The warm red, mixed with the appropriate of his two yellows, gives a nice orange; the cold red, mixed with the appropriate of his two blues, gives a nice violet. Mixing the cold red with his yellows would not give him a nice orange.

Wilcox’s book is focused on the mixing of primaries. Other books show two ideas. If you can afford to buy more than 6 primaries, go ahead and buy additional colors, secondaries and tertiaries, etc. Oh, and buy fewer primaries, since you’re no longer planning to produce your secondaries from them.

Quiller chooses alizarin crimson, cadmium lemon, and viridian green as his RYB primaries. He is prepared to purchase 68 individual paint colors. A few pages later, however, he describes the same scheme as Wilcox — but with different specific choices — of using two of each of the three primaries, in order to produce additional colors rather than buy them.

Kessler (Kessler, Margaret. Color Harmony In Your paintings. North Light Books, 2004. ISBN 1 58180 401 6) also shows both ideas. On the one hand, she shows two of each primary for mixing other colors – but she uses different reds, for example, than Wilcox. On the other hand, her personal palette is only 15 colors specified by brand and name, plus Titanium Zinc-White.

## Additive RGB Space

One of the reasons my friend asked about the relationship between additive & subtractive and the primaries, was my reference to “the additive RGB space”.

The additive RGB space, as we saw before, starts by defining 5 colors:
black (0,0,0)
red (1,0,0)
green (0,1,0)
blue (0,0,1)
white (1,1,1)

In a sense, we have three basis vectors; the challenge is that we cannot multiply them by arbitrary numbers, and if we add these “vectors”, we cannot end up with components greater than one. (What we have is reminiscent of a simplex, and what we can get by mixing is the convex hull of the vertices defined by red, green, blue.)

To put it another way. I can add the colors (1,0,0) and (0,1,0) to get (1,1,0). But then I have no sure idea how to add (1,0,0) and (1,1,0). (2,1,0) is not a legal answer. But is the answer (1,1,0) — truncating values above one? Or is it (1,.05,0) — scaling by the largest value? Or something else?

Maybe I shouldn’t say this, but it’s not all that important to me. I don’t need to mix colors in the additive RGB space. But if I do end up tracking this down, it will be to see if the RGB coordinates are “barycentric coordinates” in a simplex. (I may regret that conjecture, but hey!)

What I like about this space is that it defines three other colors for me:
cyan (0,1,1)
magenta (1,0,1)
yellow (1,1,0)

That is, it has defined CMY so that, for example, the mixing of cyan and red gives us white.

White. Additive mixing. Hence: the additive RGB space. This also corresponds exactly to the right-hand drawing that began this post, which said that the mixing of cyan and red would give us white.

When I think of additive or subtractive, I think of them specifically for the mixing of lights, or the mixing of paints, respectively. But what about the equation C = W – R, which says that cyan is what remains after we subtract red light from white?

In a very real sense, that must be a subtractive process. And yet, we could rearrange it to say that cyan plus red would give us white.

Yes, but we need to be more precise: cyan light plus red light would give us white light. But cyan ink plus red ink would give us black ink (in principle, if we had perfect inks). And I believe that the distinction between additive and subtractive must be that: do colors combine to get us closer to white or closer to black? And that’s the question: what is the process?

The digital image processing book phrases it as: a surface painted with cyan does not reflect red. And that is a subtractive process, even though it’s not the “mixing” of pigments, but the reflection of light by a material.

## Color Wheels

Let me elaborate on another question. Why do I emphasize two color wheels?

Because after trying to reconcile them for quite some time, I made a command decision that “the opposite of red” was a crucial distinguishing fact. In a very real sense, this was a very personal choice on my part.

We can find other color wheels out there. In particular, we can find color wheels which have green at 5 PM and cyan at 7 PM (with red at noon), and a mixture of green and cyan at 6 PM. I dislike that compromise, but it may be only a personal choice.

Oh, I have recently observed (since the previous post!) that the Munsell color system has five primaries: red, yellow, green, blue, violet. With them spaced evenly around the circle, green cannot be opposite red. But exactly what blue-green mixture do they put there? Beyond that, I’m not familiar with the Munsell color wheel.

But now we have examples using 3, 4, or 5 primaries.

What about the use of the artists color wheel for color harmony? My statement that green is the complement of red because our eye transmits, among other things, the difference R – G, is only a plausibility argument.

Are artists right when they say that red is the complement of green? Better, are they right when they say violet, rather than blue, is the complement of yellow? I don’t know.

I can think of two things we could do to investigate physiological opposites: one, study after-images; two, remove bands of light from white and see what colors we get (using the CIE chromaticity chart). Removing bands of light is something I can do. For the investigation of after-images, I’ll have to keep my eyes open for experimental results. (Did I really write that? Yes. Was I aware of the play on words? Hell, no!)

## Grayscale & lightness

Let me close with the results of a set of calculations. Sooner or later we realize that the brightness B (or value V) does not correspond to the “luminosity”. Okay, I’m being vague. I’ll fix that. In the HSB model, fully saturated yellow at full brightness is considerably brighter than fully saturated blue at full brightness. That is, of those two hues both with S = B = 1, yellow is considerably lighter to our eyes. There is more to color than HSB. Whatever this property is, let’s call it lightness for now.

In something like poster design, that yellow is considerably brighter than blue shows up as a suggestion that — in a digital world — you should check your design by converting it to grayscale. It is a good arrangement of colors if it still looks readable in grayscale.

In something like interior decorating, one might be told that one should vary exactly one of hue, saturation, brightness in a color scheme. When they say brightness, they mean what I’m calling lightness, not the value of B.

So how do we compute this lightness?

Here’s what I know. Assign grayscale values of 30, 59, 11, 100% to CMYK respectively. Given any CMYK specification, weight it by those numbers. That is, if our CMYK spec is…

(0, 0.6, 1, 0)

we multiply termwise by

(30, 59, 11, 100)

to get

(0, 35.4, 11, 0)

and we sum those to get

46.4 .

that is, .6 * 59 + 1 *11 = 46.4

Or, if our CMYK spec is (0, 1, 0, .2) then our grayscale is 1 * 59 + 2 * 1 = 79.

The grayscale value of a color uses its CMYK spec as coefficients in a linear combination of the individual values 30, 59, 11, 100%.

Yellow is the lightest at 11%, magenta the darkest at 59. It’s not right to say that yellow is 11% gray, because this is a pure, fully saturated and fully bright yellow we’re talking about, with S = B = 1 in HSB. But the translation to grayscale is saying that an 11% gray is the same lightness as pure yellow.

What’s blue? That’s CMY = (1, 1, 0), and we multiply termwise by the weights (30, 59, 11) to get (30, 59, 0), and we add them up to get 89, which is considerably darker than yellow, as we expect.

(I did that in Adobe Illustrator version 10, with some near-default setting of color spaces. The grayscale values assigned CMYK may vary with other color pickers on computers; and it’s possible that I just happened to use a color picker that had the simplest possible algorithm, namely linear. But I like this algorithm, and even if it’s not universal, it’s a handy touchstone.)

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Posted in color. 3 Comments »

### 3 Responses to “Color: odds & ends”

1. […] Applied Mathematics Blog, Color: odds and ends. Additive and subtractive colors, and how they can be manipulated like […]

2. drj11 Says:

You say, on additive RGB space, “But then I have no sure idea how to add (1,0,0) and (1,1,0). (2,1,0) is not a legal answer”.

That’s because you’re hampered by thinking of typical computer monitors. Instead, think of (1,0,0) as a standard red light shining onto a white card. (1,1,0) represents 1 standard red light, and 1 standard green light. Clearly you can shine as many lights onto the card as you like. (2,1,0) represents shining 2 standard red lights and 1 standard green light onto the same card.

You can think of it as photons if you like. (1,0,0) represents some number of red photons per second landing on the white card. You can have as many or as few photons as you like, so clearly the numbers you use to represent these colours can go as high as you like.

Of course, any particular monitor can only produce a finite number of photons per second. And therefore sucks.

3. rip Says:

Then we would need to change the algorithm for converting between HSB and RGB.

I prefer to distinguish luminosity (photons per second) from RGB coordinates.

That said, I still don’t know much about the effects of luminosity. I have read somewhere that psychological primaries can be characterized as those colors that appear to be the same hue under changes of luminosity.