Western poetry, philosophy, visual arts etc. all date back to Ancient Greece, but although music existed then, a continuous tradition only starts with Gregorian chant, roughly a thousand years later (and when I listen to the Seikilos song, I'm tempted to believe music wasn't as highly developed then as the other areas of culture). What is the reason for this?
More can be said about this particular topic, but your question reminds me of a recent post by the excellent /u/nmitchell076 about Ancient Rome's contributions to music, which clearly implies that, as with poetry and philosophy, the Western tradition does go back in plenty of ways to Ancient Greece (via Ancient Rome).