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I've been reading about colour theory and want to understand how it relates to rainbow palettes like the attached, also a palette type commonly used by the designer Paul Smith as his signature.
How are the colours related? I have a suspicion they are being mixed from a main set to get all the varieties?
How would I go about creating a similar palette?
If you want to understand more about color theory, look to the masters: Josef Albers and Johannes Itten. Read the texts and experiment with their theories and you will develop an eye for combinations like the one you reference and many more sophisticated examples.
Beyond the aesthetics of it, color is math. If you really want to get into generating palettes instead of choosing palettes you'll have to experiment with that. With an understanding of the math behind color harmonies (complimentary, analogous, split compliment, triad, pentagram, etc) and a little javascript, you could create an endless variety of palettes.
Consider the possibility that they were picked arbitrarily by some individual without any sort of theory. And no, "what looks good" does not qualify as a theory.
As to how one might develop a similar palette:
1) pick 10 colors at random (dice rolls, dart board, etc) we'll call it a set 2) repeat 100 times 3) pick your favorite 10 sets 4) put them in a drawer for a week 5) pick favorite 5 sets 6) show them to five people, ask them to pick favorite and a runner up. Eliminate the 2 with the fewest "likes" 7) pick your favorite one from the remainder
