At the time of the Romans and Greeks, all voices and instruments sang and played a single musical line. They had no idea of creating harmonies.
Historian Rodney Stark reveals that,
“It was medieval musicians who invented polyphony, the simultaneous sounding of two or more musical lines, hence harmonies. Just when this occurred is uncertain, but it was already well known when described in a manual published around 900. Moreover, it was during the Dark Ages that the instruments needed to fully exploit harmonies were perfected: the pipe organ, the clavichord and harpsichord, the violin and bass fiddle among others. And in about the tenth century, an adequate system of musical notation was invented and popularized so that music could be accurately performed by musicians who had never heard it.”
(Rodney Stark, The Victory of Reason: How Christianity Led to Freedom, Capitalism, and Western Success, New York, Random House, 2005, p. 51.)