Chucho Valdés, now 71, is one of pillars of the Cuban jazz-piano tradition. Founder of the seminal Afro-Cuban band Irakere in 1972, he’s made that group’s fusion of multiple genres his mission ever since. On the new “Border-Free,” his mastery as composer and player is on full display: an encyclopedic vocabulary of Afro-Cuban rhythms and an expansive palette that includes all manner of modern jazz as well as traditional classical repertoire. So the lovely tribute to his late mother, the singer and pianist Pilar Rodríguez, combines elements of Miles Davis and Bill Evans’s “Blue in Green” with a smattering of her beloved Bach, and the piece named after his grandmother, “Caridad Amaro,” quotes the Rachmaninoff he used to play for her. Again Valdés challenges himself by exploring other traditions that he finds “compatible with Afro-Cuban roots” in pieces like “Afro-Comanche” and the Moroccan-flavored “Abdel.” Saxophonist Branford Marsalis is a guest on three tracks, but it’s the leader and his band who deliver the message here. You’ll hear it in “Pilar” as the pianist plays extended silky lines in his right hand over a chordal melody in his left, and in the deathless medium-tempo groove of piano, soft hand drums, and bass in “Tabú.” (Out Tuesday) Jon Garelick

Essential “Tabú”