We present a palette-based framework for color composition for visual applications. Color composition is a critical aspect of visual applications in art, design, and visualization. The color wheel is often used to explain pleasing color combinations in geometric terms, and, in digital design, to provide a user interface to visualize and manipulate colors. We abstract relationships between palette colors as a compact set of axes describing harmonic templates over perceptually uniform color wheels. Our framework provides a basis for a variety of color-aware image operations, such as color harmonization and color transfer, and can be applied to videos. To enable our approach, we introduce an extremely scalable and efficient yet simple palette-based image decomposition algorithm. Our approach is based on the geometry of images in RGBXY-space. This new geometric approach is orders of magnitude more efficient than previous work and requires no numerical optimization. We demonstrate a real-time layer decomposition tool. After preprocessing, our algorithm can decompose 6 MP images into layers in 20 milliseconds. We also conducted three large-scale, wide-ranging perceptual studies on the perception of harmonic colors and harmonization algorithms.