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Harmonies theory and math

Color harmonies can be easily calculated looking at the "color wheel". A color wheel (also referred to as a color circle) is a visual representation of colors arranged according to their chromatic relationship.

A good way to represent a color wheel is using the CIE-L*CH color space. Each color is characterized by Hue, Saturation and Lightness, where the Hue value represent the angular position of the color in the color wheel.
For practical purposes also the HSV and HSL color spaces (popular in computer applications) can be used but they introduce a much higher degree of non-linearity in calculating each color positions.

These are the more common color harmonies and the way to calculate them:

Type of Harmony

CIE-H

Color relation

Triadic
This is the typical configuration of three colors that are equally spaced from each other on the color wheel.

 H +/- 120

Split complements
This color scheme combines the two colors on either side of a colors complement.

H +/- 150

Analogous
Uses the colors of the same color temperature near each other on the wheel.

H +/- 30

Monochromatic
Colors from the same family on the wheel. This will include lighter, darker and differently saturated versions of the color.

H + 0
(C* will change)

Complement
This is the color opposite on the color wheel.

H + 180

In theory the L* and C* values off all harmonies (the monochromatic is the only exception) should not change. In practical terms these values should be adjusted to obtain a more pleasant color combination or to remain inside the color space gamut. This correction unfortunately does not come with any fixed mathematical rule.