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Ancient Greece had a sophisticated musical system that allowed for transposable modes and flexible instrument tunings. Known as the Greater Perfect System, it is discussed in detail in What are the greek modes, and how do they differ from modern modes?
But how was their music notated? What symbols were used, and what did they represent?
The pitch of the melody tones was notated by letters, the rhythm was corresponding to the lyric modes: jambus, trochaeus, anapaest etc.
https://www.musicportal.gr/ancient_greek_music_system/?lang=en
Edit: The letters assigning the notes are referring to the Greek modes and descending tetrachords, described by all the antique music theorists.
According to Howard Goodall on page 9 paragraph 3 of his book Big Bangs, the earliest surviving Greek musical notation is the Seikilos epitaph, which is an engraved tombstone.
I think the question borders on being too broad but one important part of the answer is we’re still not certain about most of Greek notation and what it means. Reconstructions of Ancient Greek music are still somewhat speculative.
What is really exciting in this area is in the last couple decades there have been some innovations and discoveries in understanding and reconstructing the instruments of Ancient Greece, mainly the kithara and the aulos.
Reconstructed aulos give us some great clues to Ancient Greek music because they have somewhat fixed tuning and we can extract the dimensions and therefore the tuning from actual specimens. But there are still gaps because even those a pair of aulos as carved are fixed, we believe they were not played as carved but instead were stopped with wax to create scales, and we’re not totally sure exactly what combinations of stops were used. So again there’s a lot of educated guessing that uses some logic and correlation between the instruments and the notation but also uses aesthetics and a musical ear to reconstruct music that seems to sound good or at least appropriate to the theme of the piece.
I can come back to this answer and edit in some links and details of what music exists and some of the more popular interpretations are. And/or others can edit this or contribute their own answers. The final results are at least partly guesswork, similar to how we don’t know what color dinosaur skin might have been.
