The most far-reaching addition to music during the Middle Ages was the invention of polyphony—music in more than one part—an aspect of Western art that is not duplicated in any other culture. The idea itself undoubtedly originated centuries earlier than the earliest written evidence or even the first mention in theoretical treatises. In its simplest forms polyphony can easily be improvised as, for example, when two or more performers simultaneously sing the same song at different pitches, and it still exists in that form in a number of cultures. But the musicians of Europe.....