There is very little surviving music from Rome, so this is a tricky question to answer. Douglass Seaton in is book Ideas and Styles in the Western Musical Tradition very briefly addresses Roman Military music.

As one would expect, the military conquests of the Roman armies provided one special field for musical development, the field of battle. It is not surprising that this period produced notable developments in brass instruments.

So we do know that brass instruments were developed and used for the army. We also know that music was fairly common in Rome and that Romans used music for pleasure more than the Ancients Greeks did. We also know that there were professional musicians in Rome who enjoyed a fair amount of popularity.

Those last few items don't directly relate to you question, but it provides some context for music in Roman society.

Source: Douglass Seaton Ideas and Styles in the Western Musical Tradition

Check out Reconstructing Roman Marching Tunes by Paul Geddes. The long and short of it is that only one genuine roman marching song survives: Urbane (called Urbani by other sources). Even then, although we know the lyrics the actual tune has been lost to time. We only know the lyrics because they were recorded by Suetonius as having been sung during Julius Caesar's triumph.

That said, we do have knowledge about what the syllabic structure of other marches likely was and can speculate on what they might have sounded like based on other music from the period. The article I linked goes it to more detail.

Anyways, here's wonderwall Urbani. It's quite... colourful:

Urbani, servate uxores, moechum calvum adducimus. Aurum in gallia effutuisti, hic sumpsisti mutuum.

(Citizens, keep an eye on your wives, we’re bringing back the bald adulterer. He’s fucked away the gold in Gaul that you loaned him here in Rome.)

Gallias caesar subegit, nicomedes caesarem, ecce caesar nunc triumphat qui subegit gallias. Nicomedes non triumphat, qui subegit caesarem.

(Caesar vanquished the Gauls, Nicomedes Caesar, Caesar who vanquished the Gauls now triumphs. Nicomedes does not triumph, who vanquished Caesar.)

Gallos caesar in triumphum ducit, idem in curiam, galli bracas deposuerunt, latum clavum sumperunt.

(Caesar leads the Gauls in triumph, likewise into the Senate House. The Gauls have laid aside their trousers and put on the broad purple stripe.)