The music gets more curious. In the scene in the club, the first tune, a New Orleans style blues, was pretty timeless, yet the second song could just have been sung by the 1940's Andrews Sisters.
It's the saxophone which sound most out of era to me, though I didn't notice a high F# key on one, so if there weren't, that would be in keeping with the era.
In fact the production values seem fine and everything seems of the period. As a drama it has its merits with a good cast.