I don't hear harmony, I see it.
At this point you may be scratching your head and saying, "You can't SEE music!" To some level I agree with you. All music is, after all, is the vibration of things in our world that create frequencies we can hear. Some frequencies are more enjoyable than others, but that is beside the point. Music is primarily a "hearing" art, and I do "hear" notes in my head, but really it is just my brain using aural recall of things I have heard other musicians do. It is like knowing correct grammar from hearing your parents use it while growing up or that pepperoni and pizza go very well together because you have eaten it for as long as you can remember. It works the same way with how I hear harmony. Surround yourself with it, and you begin to know what works and what doesn't. However, when I say that I "see" harmony, I am simply saying that when I hear, sing, or play music my mind is producing images that help me know what I am doing, where I have been, and where I am going.
Let me make one thing clear. I do NOT see specific images. I see stairs. I see horizontal lines that move up and down like a flight of stairs. Sometimes I see more than just lines, but whatever I see, it moves in a stair formation. I have tried to explain this to people before and it only partially gets the message across through words. I've thought over and over again, "If only I could get people in my brain so they could see what I am talking about, they could nail that harmony!" Finally, by way of Noel Piper I discovered The Music Animation Machine. Before you follow the link I want to show you a few videos that will hopefully show how I, and maybe others, "see" musical harmony. The first is easier to follow along with visually while the second and third are more complex. What is visually enjoyable about the first (Bach) and third (Mozart) videos is that both men were notorious in their compositions for being almost mathematical. Watch for parallel lines especially in Mozart. Very deliberate patterns going on here. Enjoy and welcome to my brain!
*Quick thought: If you want to help train your ear to hear some of what is going on, visually pick out a line (the top is easiest to follow) and just pay attention to that line. Then move to the lowest line, and so on. It will really help you hear the lines and develop an understanding of harmony if you are a visual learner.*