They used the old Greek letter notation as well as Greek music theory. This was, as far as we can tell, a matter for the educated in theorising about music, rather than a tool for musicians to help remember and communicate musical ideas. One of the best preserved antique pieces of music is from the roman period, but it is culturally Greek rather than Roman. Seikilos Epitaph, which was inscribed on a tombstone found in what is now Turkey. As far as I am aware, we have no evidence in the form of written down music of how music may have sounded in the city of Rome, though it surely changed a lot over the centuries.

hi! here are a bunch of links I rounded up a few days ago for a similar question (what did ancient Roman music sound like, and did they have notation?); check 'em out ~

• Do we have any idea what Ancient Roman music sounded like?

• Is there any surviving sheet music from the Roman Republic/Empire? Is there somewhere I could hear it?

• Was Roman music different from Greek music?

• What was music like in the Roman Republic/Empire? Was there anything close to an orchestra in scale?

• What musical instruments were there in 0CE?

• What did popular music sound like in the Roman Empire?

• What type of music was common in ancient Roman and Greek societies?

• Did urban Romans and Greeks have a concept of folk music, dress, and so on?