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For me there are Four color wheels that function in different situations.
The "old" one is still the foundation - red, yellow, blue. Great for
learning the basic theories, but when you are dealing with real pigments,
there isn't a pigment that purely reflects one of those colors without
"leaning" one direction or another.
So, as a painter, I was taught to lay out what for me is color wheel #2: a
palette made up of, for example, Thalo blue and ultramarine blue // cadmium
yellow and lemon yellow // cadmium red and alizarin crimson. The reason
behind this is - for example - Thalo blue shifts to green. So that means it
visually includes yellow. Cadmium red shifts to orange (ever so slightly) so
that means it, too, visually includes yellow. Therefore, when I mix cadmium
red and thalo blue, sure, it's red and blue so it should make violet,
right? - and it does, but its a slightly muddy violet, because both pigments
include a little of yellow, which is violet's complement. Get it? But, if I
mix ultramarine blue, which is on the violet side, which means it visually
contains a bit of red, and alizarin crimson, which shifts also towards
violet, which means it contains a little blue, I get a much more vivid,
pure, violet, because it's just red and blue. See??? It's really based on
warms and cools. Learning this gave me so much more control over my colors,
I never mixed "mud" again. If that isn't clear, let me know. (as mud? -
somebody had to say it;)
The next color wheel is the "new" one you mentioned: cyan, magenta and
yellow. There is really nothing new about that since the advent of
commercial printing. Your local newspaper, if they have color photos, will
usually be happy to give you some leftover color photo separations so your
kids can see for themselves. They're on film so you can cut them up. It is
useful, in my own experience, primarily when using transparent inks, for the
same reasons I mention above. I would personally never teach it as the
"only" color wheel to someone wanting to paint, I think it would lead to
color mixing frustration.
The 4th color wheel is light, which acts completely differently. I realized
this as a kid on the school stage when I looked up and noticed all the stage
lightbulbs were alternately red and blue. I asked why, and was told red +
blue light = white. So that's fun to play with, like math equations: orange
light + green light = ?