But three part harmony is only one way of singing together. Another way to go is to have two or more melody lines, or part melody lines, going alongside each other. It's a different way of thinking, and probably more rooted in a lot of traditions. Genie's example with Amazing Grace illustrates the way that thinking in terms of chords can block of good things sometimes.

The arrangements used in West Gallery singing (a not too distant relative/ancestor of Sacred Harp) include lots of examples of harmonies that really got frowned on by the respectable music professionals.