Wonderfully, there are some surviving examples of vocal music from the Mediterranean cultures in the classical era. This includes quite a few examples from Ancient Greece! MalPerMeCheMaffidai has already posted one here, but there are plenty more!
The nice thing about vocal music is that the human voice is still largely as it was several thousand years ago. So, while of course we need to worry about language, tone and colour, at least we don’t need to recreate the instrument!
Perhaps particularly relevant to this thread is Mesomedes of Crete. Minoh Workshop has already anticipated me in mentioning him – unsurprisingly! – but I’ll include my little spiel here anyway. After all, I’m focusing on vocals here specifically!
Mesomedes lived during the late 2nd century AD, which means he was broadly contemporary with the Seikilos epitaph. Several devotional hymns of his survive, complete with both music and text. These include hymns to the Calliope, Nemesis (see above) and the Sun (and so Apollo).
Somehow, I can see Asterion appreciating a hymn to Calliope, who was the muse related to epic poetry. It’s short, but beautiful:
Here’s his Hymn to the Sun, which is a fair bit longer:
These recordings are a touch over-interpreted – the original hymns are a single vocal line with no accompaniment, while the recordings add various instruments and changes in tone based on modern tastes. Not that there’s anything wrong with that! Just be aware that the original performances might have sounded quite different from what you’re hearing here. The same goes for the Hymn to Nemesis too. That being said, all these performances are played with real spirit, and are interesting interpretations. I enjoy them!
If you allow short pieces, you can go back much earlier. For instance, music survives for parts of Euripides’s plays, who was writing a good 500 years before the Seikilos epitaph and Mesomedes. (Although still 1000 years after Asterion’s day). Here’s some music from Orestes – κατολοφύρομαι – which was written in 408 BC. It’s particularly interesting, because it includes music for both voice and instrumental accompaniment. That’s quite unusual! This work is a lament on Orestes’s fate:
These are just a few examples. There’s quite a rabbit-hole of fine music from Ancient Greece if you go digging for it. I look forward to seeing what else you post, Minoh Workshop, and everyone else in this thread!