Presumably, the z3100 is putting down red ink where it's called for in my image, no?
If so, can you tell me how the media can make such a great difference in the appearance of the color? I mean, the media *looks* white to me, so shouldn't the red ink color the media red?
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Yes, the Z3100 is putting down red ink where it's called for. The challenge is that some of the values in the red on your monitor are outside the values the printer can produce on that particular media. They're "out of gamut".
The white of the media isn't the only determining factor. The coating has a lot more to do with it. Also our brains are easily fooled. White is a relative term and you shouldn't trust your eyes all that much in this case. A spectro measurement is far more reliable a measure.
What rendering intent were you softproofing and printing with? I would think relative colorimetric would be a good choice and it will help fool the eye with the white.
The media doesn't have the gamut of your image. That's why we run gamut warning on certain images. Add that to the fact that your monitor has a much higher contrast ratio than the media and we're at the point where things won't print as you'd like them. Its all part of the fun of color management.
Softproofing gives as good a representation as possible today but its not perfect. We have to add some knowledge and experience to it to get the best possible prints.
Gamut was difficult for me to understand until I could "see" it. That's where the demo version of [a href=\"http://www2.chromix.com/colorthink/download/?-session=SessID:A9C8AD4E191fb149E3gYL1019176]Chromix ColorThink[/url] came in very handy.