HDMI/DVI Cable Question

Hi, I have an HD computer monitor but no HDMI cable, only DVI. I have read that you don't need an HDMI cable to get HD. That HDMI cable does both sound and video whereas DVI only does video. Is this true?
I have my DVI cable plugged into the DVI port on my graphics card and then into the DVI port on my monitor. Do I need an adapter to plug it into the HD port or will just strait DVI give me HD?

I hope you get what I mean. I'm kind of half asleep and new with the HD thing.