Is there any composer on the planet more wonderfully enigmatic than Anthony Braxton? Jazz saxophonist, ecstatic mystic, academic theorist, certified MacArthur “genius,” all-around American original — call him what you want, Braxton brought his Diamond Curtain Wall Quartet to the Terrace Theater on Saturday night for an extended set of some of the most inventive, maddening and impossible-to-pigeonhole music heard at the Kennedy Center in some time.

Braxton, of course, has been punching at the edges of new music since the late 1960s, and at age 67 he shows no sign of letting up. His music makes few concessions to comfort; neither jazz nor classical, it’s rooted in an almost impenetrable theoretical framework, and even the cryptic titles — such as “Composition #367F plus #241,” which made up Saturday’s entire 80-minute program — make the average ear curl up in alarm. But it’s also music of extraordinary vitality and a kind of seductive, cerebral beauty, marrying the anything-goes language of free jazz with the complex structures of the more — forgive the term — “serious” world of classical music.

And, demanding as it was, “Composition” was a gorgeous work. Wildly colorful if often daunting on the surface, it held together with seamless musical logic, a cascade of ideas that seemed to well up and evolve with effortless spontaneity. Over a base of sustained electronic tones, Braxton and the three young virtuosos of his quartet (joined by the Kennedy Center’s Jason Moran at the piano) built fragments of intensely detailed sound — animalistic roars, delicate filigrees, sudden driving attacks, breathy fragments of melody, amplified scratches — into ever more complex textures, building layer upon layer until the momentum of the piece kicked in and it began to flow with the unstoppable fluidity and power of a river. Was it enigmatic? Absolutely — and beautifully so. Forget pigeonholes: This was music as complex and un-categorizable as life itself.

Brookes is a freelance writer.

Anthony Braxton performs in the Kennedy Center Terrace Theater on Dec. 15, 2012. (Yassine El Mansouri)