Musical archeology has revealed a few tunes going back to ancient Greek times and even earlier. The oldest Jewish music are neumes in the Hebrew text, dating back 1000 years. How far back as oral tradition, this might go, is anyone's guess. But here is a fine example of musical reconstruction ...
https://www.youtube.com/watch?v=8A7SPcdU03w&index=5&list=PLCzCJ_p0zXyIYqgszZEDaIYGi9D3zTXG3