A heterophonic musical texture is one in which several instruments play (or several voices sing) a different version of the same melody simultaneously. Usually, this occurs while one instrument plays the simplest version of the melody. Heterophony is a defining characteristic of much of the world's music, including most traditional East Asian music (notably in Japanese Gagaku), which did not have harmony or polyphony before contact with Western music. It was also a characteristic of one tradition of Hymnody in the 17th and 18th centuries, a tradition called "lining out" in which a song leader would sing a line of a hymn, and the congregation would answer, every member singing in the speed or style of their choice. Most notably, this style was adopted in early New Orleans jazz.
Garland Encyclopedia of World Music: Volume 7: East Asia: China, Japan, and Korea, 2003.
New Grove Dictionary of Jazz, 2001.