# Color Theory Math

While I understand the basic concept of the 12-step color wheel, I've noticed that in many color palettes, the saturation (S) and brightness (B) values of the colors often do not remain constant, even though the colors fit together harmoniously.

This observation has led me to wonder: are there any established mathematical formulas or progression rules that designers follow when creating these palettes? I'm particularly interested in how colors are chosen or modified from a base color to create a harmonious palette.

For example, if I start with a specific hue, how should I systematically alter the saturation and brightness to create a visually appealing set of colors? Are there specific ratios or patterns that are commonly used in the industry for this purpose?

Any insights, references, or examples would be greatly appreciated, as I feel I might be missing a key piece of understanding in color theory and its application in design.

• There are many color theories. Commented Dec 21, 2023 at 17:56
• Yes, but are there any guiding principles that you recommend for creating harmonious color palette using simple calculations? e.g – H = Hue formula: H1 = |(H0 + 30 degrees) - 360 degrees| formula: H2 = |(H0 + 60 degrees) - 360 degrees| formula: H3 = |(H0 + 90 degrees) - 360 degrees| How do I determine the proper saturation and brightness? Or is this all just done adhoc and I'm thinking too deep about it? Commented Dec 21, 2023 at 18:16
• Using math like this is not probably too deep, but a bit simplistic. Commented Dec 21, 2023 at 18:31
• @Rafael a bit. Understatment of the year. Commented Dec 22, 2023 at 2:22
• "mathematical formulas or progression rules that designers follow when creating these palettes" - there are but most designers don't need to use them since colour choice is subjective. It's art, not maths. Commented Dec 22, 2023 at 10:53

This is not a real answer because there is no real answer.

What is the basic concept? That you have primary secondary tertiary? Let me just add some things to that concept:

Do you know color is not a wheel? It can be a cube, a cylinder, a cone, a double cone, a sphere, a spheroid? That there are many color models? That a complementary color can be different, depending on the model?

That even if you can think of some standardized primary colors, some people use other primary colors because they look a bit more harmonious.

That our eye has different sensitivity to different colors. That different hues have different lightness.

That harmony also depends on culture, fashion, trends, mood, psychology, feelings, messages, symbols, history, demographics, and a ton of things.

I can add things to the list.

Are there specific ratios or patterns that are commonly used in the industry for this purpose?

No. There are not.

I think we are tempted by the idea of using a mathematical formula to generate a palette instead of making decisions that we think are subjective. But they are.

To generate a pallet just decide what you need. There are tools that can help you visualize but the decision should be made by someone.

A tool that might help you is https://paletton.com That does not mean they are standard. That just means someone decided on assigning values.

Regarding the comment.

You can use a 30-degree value for an "adjacent" triad of hues. But you can change it to 15°, 20°? does that break any color theory rule? No. Because there is none.

A complementary color (180°) of red is cyan... in an RGB color wheel... but green in Lab and RYB. So that means that one is incorrect and the angle in RGB should be 120° but only when going counterclockwise?

I do not mean we do not use mathematics when dealing with color. There are a TON of them.

Gamma correction, matrix transforms of color spaces, bit depth, maximum values, color temperature, and so on. A ton of them.

But the question is:

How should I systematically alter the saturation and brightness to create a visually appealing set of colors?

And the answer is you need to decide.

1. For example, when using hexadecimal colors on a website I use the 3 letter notation. Instead of using #FFFFFF I only use #FFF. This reduces the possible palette orders of magnitude.

2. You can do the same in HSL. For example, use only .1 increments, so you have 10 times fewer options to choose from.

3. Or start dividing by half. 50%, 25%, 12%, 6%.

4. Or a kinda logaritmic scale: X2

1%, 4% 9% 16% 25% 36% 49% 64% 81% 100%

(This scales need to be used inverted, being the smaller difference the bright side of the scale)

Or the same scale but divided /2 and used from 50% to 100% and to 0 for the L axis.

Or explore similar scales on HWB or RGB notation for example.

But all these decisions depend on some other, even like size and proportion.

The point is, that color although is objective in many ways, wavelength, Lab values, energy... it is subjective in many others, specifically in terms of "harmony".

Edit 3

I am getting obsessed with this question: Why we do not have "pre-defined" values for saturation and lightness, in any model but we do have some in hue relations? Complementary, adjacent, triad, etc.

I was reviewing a framework I myself am working on (screen capture of one of the tests for opacity)

I think that this decision is a secondary one to the one defining the basic hues, and probably we need to leave the door open to have alternatives when the fine detail needs to be defined and tested.

Some color systems for UI design, (which I do not like) throw you a pallet so big that it is almost as useless as not having one. So I understand the need to narrow the alternatives.

One way I am trying to solve it, in CSS is by leaving the saturation and lightness as variables on a defined .colorSecondary so that when this color is applied to a specific situation, it can be fine-tuned.

By contrast, the main color, is normally pre-defined, including the lightness and saturation, let's say as a corporation brand color, by the brand message it needs to convey.

• I understand your perspective on color theory being largely subjective, but I'm still confused. In digital color theory, mathematics plays a crucial role. For instance, digital colors can be defined with HSB values between 0, 100, and 360. When creating a color wheel, we can use mathematical divisions, like dividing 360º by 12 to create 30º intervals for hues. Tools like paletton.com also utilize mathematical formulas to generate palettes. This seems to contradict the idea that there's no math involved in color theory, especially in digital contexts. Commented Dec 21, 2023 at 18:47
• @DesignName color theory is not a singular thing. Its not a science thing either. its art. The thing is theres no underlying motivation to the hsb wheel. So if you choose mathematically on a hsb wheel great. The wheel is not motivated. You could have any other wheel... and youd get different colors. Great. So yes by choosing this way you get a big utilisation of colorspace. but is it harmonious, maybe. But other comboscan be equally harmonious, depends on what your after. So you can do that does not mean its super meaningful. Commented Dec 22, 2023 at 1:24
• I added some tips that you could use resembling a "systematic" approach. Commented Dec 22, 2023 at 4:00
• And I added more options. I am getting obsessed with this question. Commented Dec 22, 2023 at 23:12
• @Rafael After an extensive search, I've come across a pivotal resource that addresses our query. The book I found, 'Semiology of Graphics,' offers a comprehensive exploration into the art of color system development. It masterfully combines ratio theories and the principles underlying traditional rules. What's particularly intriguing is that this book predates the integration of computers in design, yet it demonstrates how fundamental mathematical concepts can be instrumental in crafting value variations and harmonies. Highly recommend delving into this book if you haven't already. Commented Dec 29, 2023 at 16:27

RGB color system and its 1:1 mathematical conversion called HSB have their roots in television engineering. RGB numbers were originally used to control electron beam intensities in color tv screens.

The television video signal didn't have R, G nor B components, but there were phase-amplitude modulated subcarrier which carried hue and saturation as measurable signal properties. In addition there were the luminance as added base voltage level.

Today RGB numbers control the brightnesses of red, green and blue pixels on a flat computer screen. The wavelengths of the pixel leds are selected to be about the same as the most sensitive wavelengths of our different eye light sensing cells. That makes possible to generate a very limited set of different light spectrums, but due the poor wavelength resolution of our eyes the light from RGB leds can still cover a somehow useful subset of all possible color sensations.

We can separate from watched light such quantities as hue, luminance and colorfulness. It happens in our brains. Scientists have made millions of tests with humans and found useful math color models which predict the perceived hue, colorfulness and luminance of the measured wavelength spectrum of a light sample. But the perceived color is not a physical property of the light which reaches the eye nor a property of the object which emits or reflects the light. Our brains create the impression of hue, colorfulness and lightness.

As said, RGB numbers control making of light and so do their equivalent HSB numbers. Of course, engineers have selected the math formulas between RGB and HSB so that

• the calculated HSB hue roughly follows the perceived hue
• the calculated HSB saturation roughly follows the perceived colorfulness
• the calculated HSB brightness roughly follows the perceived luminosity

The roughness comes from the simplicity of the used math formulas. They are a compromise which made at the same time possible to

• reach at least some correlation between the HSB numbers and perceived color and
• make the calculations with useful speed in computers.

The relation between HSB and RGB is (as math formulas) simpler than the relation between old color television video signal and RGB. The commonly used modulations for hue, saturation and luminosity in the video signal were designed to make commercially profitable analog circuit implementations possible. But as a presentation of perceived colors they were approximately as rough as the HSB system in computers.

The mentioned roughness means non-linearity and inter-dependence. Change for ex only HSB hue, the perceived hue, colorfulness and luminosity are all changed. Or change the HSB saturation say 10% higher, the perceived colorfulness can get changed only a little or it can get changed much depending on what's the starting point of the change.

I have thought (and used it, too) that making a set of colors where hue, colorfulness and luminosity vary with certain steps earns to be called harmonic. But the roughness of the HSB system causes that there's no point to try to write any usefully simple rules for HSB numbers (and even less for RGB numbers) how to generate such color sets.

To make such sets with numbers you can use HCL (a.k.a LCH, LHC) color system which follows the perceived hue, colorfulness and luminosity much better than HSB.

Of course, the conversion between HCL and RGB is mathematically complex, but it's available for ex. in program GIMP. And you find it easily as math rules and online converters in the web.

As a drawback I must tell that adjusting HCL numbers in GIMP or other computer program generates easily a color that RGB cannot produce. The selected colorfulness is too high. HSB saturation is only a relative percentage of the possible colorfulness in the RGB system. The absolute maximum colorfulness of an RGB color depends heavily on the hue and brightness.

Finally "what's a harmonic color set" depends on fashion, cultural and personal habits. It's an opinion and varies from time to time, from culture to culture and from individual to individual. The idea of making in HCL system a color sample set by stepping H, C and L components or only some of them from a starting color is only my opinion of "what's harmonic".

• Even LHc has problems. Although its better than rgb based representations. It does not spread the colors well on the polar axis. Which is kind of the point though we do not know how to make a perfect wheel. But yes i was considering changing my answer so that it would be clearer that color harmony is so badly defined that that is the reason you cant have an answer. Commented Dec 22, 2023 at 7:54
• HSB (often called HSV) does not come from television engineering. It comes from the early days of computer graphics, and is intended to provide a more intuitive way to select a color. It was never meant to be perceptually uniform, so any computations in this color space are meaningless. What you call HCL might be the polar version of CIE Lab, which attempts to be perceptually uniform (it fails most strongly in the blues). CIE Luv tends to be better for color computations, though it's not perfect either. Commented Dec 22, 2023 at 21:22
• OKLab is a good alternative for color computation. Commented Dec 22, 2023 at 21:23
• @CrisLuengo Check the color coding in PAL or NTSC (a.k.a never the same color) analog video signal as phase and amplitude modulation of the color subcarrier. It's the ancestor form of conversion between RGB and concepts hue and colorfulness. The luminosity was there already for BW-only image in the same signal. HSB is a computer program version of the same idea except it used piecewise linearized functions instead of computation intensive trigonometric functions. (which surprisingly are produced easily in analog circuits ) Commented Dec 22, 2023 at 21:58
• PAL uses YUV (also used in e.g. JPEG), which does not involve a hue (angle) computation. Y is the intensity (B&W picture), a weighted addition of R, G and B. U and V encode color, U = B-Y, V = R-Y. NTSC uses YIQ, where IQ is rotated 33 degrees from UV (ie also no hue computed). The way IQ is encoded into a carrier wave causes the phase of the carrier wave to represent hue, but this phase is not directly used, it is all about the amplitudes of two equal-frequency carrier waves with a 90 degree phase difference. Commented Dec 22, 2023 at 22:52

Its a bit hard to put color theory in perspective when you start out asking about a mathematical basis and rigor. This gets you in trouble really fast.*

So the color wheels you use for picking arent super motivated from a scientific standpoint. So you could have different color wheels that would have different basis. So if you use absolute angles for your color choices its a random pick of sorts. Its not super meaningful since any number of different color wheels would get you a different choice.

So yes you can choose colors with mathematical precission on any color wheel and call it a day. Or you can spend your time in trying to make it more mathematically and physically rigorous. Only to discover that its a very very deep rabbit hole that makes Alice in wonderland seem like a somber description of reality.

We know a lot about this but nearly no application you use for palette picking do use any fraction of this since it does not need a motivation as such. This said we dont know enough to motivate things very well. Nor do we know of any rigorious method that is correct.

So if you dig into the lore youll find that its not a super meaningful question. And a lot of people want it to be super meaningful. There is no right or wrong way, just ways.

So what value does any of this have? Well it seems to aleviate analysis paralysis. If you have a reasonable spread of color by any means then you get a reasonable way of choosing more distictive colors, just as long as you also play with saturation and brightness too your probably good to go. (or not you might also want less clarity)

Nearly anything would work just as well. But sometimes it does not work. Or you have some weird priorities or something.

PS: So we know for a fact that a HSB wheel chosen from RGB is not in any substantial way correct and has many issues with not being linear and having broken visual uniformity. So its a bit of a ask for it to give clarity. But it works, because its not super meaningful.

* try using LCh instead of HSB for starters. Random sure but atleast somewhat motivated though not for palette picking.

• I see where you're coming from, but I believe there's a bit of a misunderstanding regarding the application of mathematics to color theory. Incorporating mathematics into color theory, as Karl Gerstner observed in 1986, offers a structured approach to understanding color. It systematically arranges the vast spectrum of colors using objective criteria and constructs models where every single shade has a specific position. Far from being troublesome, it's a beneficial, systematic method to master the complexity of color. Commented Jan 1 at 21:27
• @DesignName Well, i didnt say its not useful. Im just disputing the validity of using RGB based models with mathematical accuracy. I think there is a alternate argument to be made that does not rely on a circle thats equally feasible. The circle is a way for you to reduce complexity in thinking sure. But not the whole picture. Anyway, there is a difference between a science and academic work. Theres lots of academic literature sure. But yes theres some work that may in fact solve the issue but thats yet to be seen. Problem as i see it that you cant approach colors as vectors. Commented Jan 1 at 21:56
• The debate here isn't about the precision of RGB models but the structural organization of color. Karl Gerstner's theory, which pre-dates digital design, offers a timeless framework for systematically arranging hues—akin to vectors in space—which is still relevant in the design process today. This approach is rooted in the practical insights of a renowned graphic designer, not just academic literature. Commented Jan 2 at 0:51
• @DesignName Well then you should make a color model that follows gerstners work. Commented Jan 2 at 5:30