I think I brought this up before but I thought I'd revisit the idea.

As far as I know,every culture,everywhere and everywhen, has music to represent itself.

What music would Caesar find acceptable, examples? Would he be one of the first wastelanders to actually encourage entirely new music not connected to the old world?

Not really an expert on music but looking at how the armies treated music during the ancient times, it would probably revolve around drums and horns.

This would be a good way to let troops know if they would advance or retreat and soldiers could use these instruments as a form of recreation.

Yeah it occurs to me if they aren't using much in the way of radio, they'd fall back on old systems.

Drums and horns, as @ahyuser001 are most likely the best bet.

The Legion is a military culture, and it seems devoid of art. It's fairly likely those who live in Legion controlled regions, "citizens" if you can call them that, listen to whatever old holotapes they can scavange, or write their own folk songs like we see in New Vegas/California, but again, since we have little idea what is contraband/allowed in the Legion's lands, it's hard to say.

For example we know chems are banned, but certain heavy weapons and ammo types are also banned, considering we can find them among the legion contraband caches we have access to for siding with them. Yet this may just be an example of things that aren't allowed without Caesar's permission. So maybe it works the same way with music? I'd guess the music allowed by the legion would have to comply with it's values, and they're pretty rigid. So most of the patriotic old world songs, or songs which promote things like freedom, excess or sexuality would be pretty much no gos.

Maybe it's mainly instrumental music. Again, I'm sure most music in the legion is pretty small scale anyway. Legion music would be a mix of self composed songs. Legion aren't too big on tech. There are 0 radios in Legion camps that are broadcasting any stations. Legion members didn't grow up listening to music, aside from marching drums that is. It probably doesn't factor into a legionary's life. The important thing to note here is that men of the Legion aren't people. They are tools for Caesar to use as pleases. Thus is his right as son of Mars. Or at least, that how a legionary is instructed to act.

The legion we saw was just the army... the actual people we were, sadly, never shown. They have cities, as Caesar said, and must have an economy since they're minting money and have caravans in their land. Even though the legion soldiers might not practice art, I have to wonder if the regular people did, and what they might be able to get away with in that regard. Would old world music be banned, I wonder?

There's no real evidence that the roman army while marching or in battle they used any type of "music" only as my partner ahyuser001 here says maybe only drums and maybe horn trumpet "buccina" or even bagpipes.

In fallout caesar's legion i think music would be based on roman chants or norman music.