A complementary harmony is one that juxtaposes two colors that are opposite one another on the color wheel. The interesting aspect of this harmony is that because it combines a primary color (red) and a secondary color (green), and the secondary color contains both other primary colors (yellow and blue), all three primary colors are always present in a complementary harmony even though we see only two colors. Note that these colors are always found across from one another on the color wheel. Although there can be variations, the three complementary harmonies are red/green, yellow/violet, and blue orange. Here is a painting by Jan van Eyck that utilizes the red/green complementary harmony. Also notice that these are the Christmas colors as well.
A complementary harmony would best correlate to music that featured contrasting timbres such as a percussion section and a brass section, as those families have very different colors.