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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]