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I know royal blue isn't truly achievable in the CMYK gamut, but I have a customer that is VERY particular about their banner and so I need it as close as possible. What are some color values I can try?
Do you have a sample of a rgb color they use?
Bright and Blue don't really go well together in CMYK - here's a rough approximation of the gamuts (the extent of color that can be represented) that shows why.
You can get a pretty "rich" blue, but it won't be bright. Try c=100 M=50 Y=0 K=25. (In the Pantone CMYK Coated reference, it's P 107-8 C).
Depending on the banner printer you use, they may in fact be using 6-ink printing rather than 4 (in most cases, I've found), which can sometimes help in eking out a bit more color (but usually just means smoother gradations across light-colored areas). You might contact them to get their advice, as well, they're likely to know the limits of their machine better than us. :-)
-[odmg]
Yes, the CMYK colour space is smaller than the RGB space. So when you move from RGB to CMYK your colours are mapped to the nearest printable colour. To generalize, the RGB space has more very saturated "bright" colours. For in depth info, read up on 'additive' (RGB) and 'subtractive' (CMYK) colour.
This might help you understand it better: