The Lyre Ensemble built a replica 4,500 year old instrument and made some songs based on Mesopotamian texts. They are, of course, best guesses though.
Some of the earliest written information on music that we have is from a "treatise" on Pythagorean tuning. Pythagoras determined what ratios of frequencies were consonant and dissonant. By the time of Plato people had begun mixing the consonances and dissonances to form "modes" which Plato writes about in The Republic