Allen Toussaint is to American music what Commander’s Palace is to American restaurants. Both have distilled the essence of New Orleans into something refined, elegant, stately. And both have become institutions in the process, the standard by which others are measured. The pianist and producer Joe Henry could hardly have known that these would be Toussaint’s last sessions, before he died after a performance last year at the age of 77, but if they had, they could hardly have scripted it better. And the masterful American Tunes (out June 10 on Nonesuch) feels scripted, both in performance and sequencing, beginning with the precision and touch of the solo piano on “Delores’ Boyfriend,” adding the empathic interplay of a rhythm section as the album progresses, casting the operatic voice of Rihanna Giddens on a couple of tunes, a blues (“Rocks in My Bed”) and a spiritual (“Come Sunday”), punctuating arrangements with the guitar of Bill Frisell, the saxophone of Charles Lloyd, the second piano of Van Dyke Parks. As he did on his earlier transformation of “Tipitina and Me,” Toussaint extends the legacy of Professor Longhair through reflective, meditative readings of “Mardi Gras in New Orleans” (echoed later in “Hey Little Girl”) and “Big Chief,” acknowledging his seminal influence but taking this music where ‘Fess never would. The album builds toward a majestic conclusion with the penultimate “Southern Nights,” a jaunty hit in the Glen Campbell version and a reverie in Toussaint’s earlier vocal version, but here transformed into something more intricate. It seems like the only thing missing from the album is Toussaint’s own voice, and then it appears on the very last cut, his conversational warmth bringing comfort to Paul Simon’s title tune, as if singing from beyond the grave. It’s all right, all right.