Um, the Ancient Greeks did have a way of writing down melodies (albeit, rhythm notation was underdeveloped or entirely absent, and so we mostly have to conjecture their rhythms based on the verse structure in the lyrics). The Seikilos Epitaph is perhaps the best known example of an 'ancient' melody that has come down to us. It's true however that these notations were underused compared to what we might expect in a modern context, but that's because music was primarily engaged in as a practical activity: the idea of writing down a tune just wouldn't occur to them, except for very rare cases (e.g. we have a few fragments of melodies from Greek theatrical works and religious hymns). One other ancient culture that also had a notation for melodies is the Hurrians, albeit the decipherment of their notation is controversial.