If you’ve heard a great song recorded in New Orleans sometime in the second half of the 20th century, chances are pretty good Allen Toussaint had something to do with it. He’s a songwriter, producer, arranger, pianist, and (occasionally) singer, who’s worked with Elvis Costello, Paul Simon, the Meters, and many others. The Bright Mississippi is something else again — Toussaint’s first foray into (nearly) all-instrumental jazz. It’s an homage to New Orleans in pop, gospel, jazz, and blues, written in, about, or associated with the city. Mississippi is gloriously under-produced — you practically feel like you’re eavesdropping on a private session. Toussaint’s piano — elegant and blues-rich — is front and center; he certainly doesn’t skimp on the notes, but the result sounds opulent and uncluttered. Accompaniment is both sparing and superb: Nicholas Payton’s crackling, emotive trumpet, Marc Ribot’s delicate yet pointed guitar, Don Byron’s soulful clarinet, and on one track, the gorgeous, slightly rough, big-toned tenor sax of Joshua Redman.