Mark Twain may not be one of the great musicologists, but he is responsible for one of the most frequently quoted aphorisms in music criticism: “Wagner’s music is better than it sounds.”

With a little tweaking, the line could also apply to the music of the avant-garde jazz composer and multi-instrumentalist Anthony Braxton, who is making a rare appearance in a mainstream Manhattan club this week, at the Iridium. (The music he played a year ago at the club is about to be released in “9 Compositions (Iridium) 2006” an ambitious 10-disc boxed set from Firehouse 12 Records.)

The 61-year-old Mr. Braxton seems to do everything possible to present his music in a way that makes it sound serious and artsy, which is to say foreboding and inaccessible. Even his physical appearance —bespectacled, sweater-wearing, pipesmoking — is outwardly academic. But when you open your ears to what he’s playing, Mr. Braxton’s compositions are surprisingly listenable. Granted, there are long interludes of screeching and shrieking on his various horns that naught but a hardcore avant-garde admirer would relish, but for all his academic posturing, much of his music is playful, swinging, witty, and — dare I say it? — fun.

The first thing you need to know about Mr. Braxton’s music — much in the way you need to know that Django Reinhardt played his guitar with only three fingers or that Roland Kirk played three saxophones at once — is that the composer considers it too conventional to give his works traditional grammatical titles. He not only assigns numbers to the pieces, as a classical composer or even a mathematician might, but differentiates one work from the other with a set of schematics. That’s right: He doesn’t call his tunes “Salty Mama Blues” or “My Love Song for You.” Instead, he uses what looks like a diagram of electrical circuits in a fuse box — a set of lines, forms, and shapes intersecting in geometric patterns.

It follows, then, that the Chicago-born Mr. Braxton has spent much of his career on campus, studying at the Chicago School of Music as a teenager and serving on the faculty of Wesleyan College for the past two decades. He was in the right time and place to participate in the first wave of free jazz in Chicago and credits the pianist and composer Muhal Richard Abrams, founder of that city’s Association for the Advancement of Creative Musicians, as one of his key inspirations. He recorded for the first time three days after his 22nd birthday as a sideman on Mr. Abrams’s “Levels and Degrees of Light.”

In retrospect, the striking thing about Mr. Braxton’s relationship with the AACM is that few of those players came from what could be called an academic background. Many of these musicians, such as the members of the organization’s most famous band, the Art Ensemble of Chicago, would have been playing Chicago-style blues or soul music had they not discovered free jazz. Contrastingly, there is little blues relevance in Mr. Braxton’s work: He has named Charlie Parker as a major influence, but focuses on more on the sax master’s intellectual side than his mastery of 12-bar blues. Otherwise, the precedents he has cited are a pair of piano-alto saxophone combinations — Lennie Tristano and Lee Konitz, and Dave Brubeck and Paul Desmond, though he also acknowledged a debt to such contemporary classicists as John Cage and Karlheinz Stockhausen — that were perceived as suspect in the eyes of certain extreme members of the free-jazz fringe.

In 1968, Mr. Braxton recorded his first album under his own name, “Three Compositions of New Jazz,” which introduced his unique compositions through an all-star quartet of multi-instrumentalists, including Mr. Abrams on piano, Leroy Jenkins primarily on violin, Leo “Wadada” Smith on trumpet, and the star on alto. The four players constantly switched instruments, passing around whatever they could find and cart into the studio, including harmonica and xylophone. The album introduced Mr. Braxton’s numbered and diagramed compositions and, in a typical move, begins atypically, with the foursome humming and chanting like some kind of post-nuclear barbershop quartet.

Mr. Braxton’s second album, “For Alto,” cut a few months later, represented another major breakthrough: It was the first full-length unaccompanied solo album by a horn player in jazz. By the time he returned to the solo format in 1972 on the album “Saxophone Improvisations Series F,” he had refined his approach considerably: The opener, “8F,” dedicated to the chess champion Bobby Fischer, is a series of highly coherent squeaks and squawks, whimpers, and purrs. The second piece, contrastingly, is a surprisingly lyrical melody rendered with a luxurious vibrato-rich tone that could almost be Benny Carter; I can almost hear a pair of lovers whispering to each other, “Oh darling, they’re playing “26A” — our song!”

Apart from Mr. Braxton, much of free jazz is rooted in the traditional concept of horns in the front line and a rhythm section in the back. But he starts each project with the apparent idea that an ensemble can consist of any possible combination, from his solo alto to his enormously ambitious music for multiple orchestras, which involve more than 100 musicians. One of his warmest pieces is a 1971 quintet played by the London Tuba ensemble — four tubas in E-flat and one in C. Equally witty is “6C,” which costars the like-minded trombonist George Lewis, in which the two horns simulate everything from a gaggle of geese gargling at a dentists’ convention to bouncy, buoyant, almost bluesy runs that Louis Jordan could have played.

Elsewhere, Mr. Braxton has dueted with Mr. Lewis on the enormous contrabass saxophone, written two compositions (one in two parts) for the 30-piece Ensemble Modern Frankfurt, and composed a set of tunes for a unit of brass, reeds, and rhythm, more like a jazz big band. Indeed, because he varies between chordalbased bop underpinnings and completely free playing without stopping at such halfway measures as modal playing, it is sometimes easier to list what Mr. Braxton doesn’t do in his music, such as allude to international styles, whether from Asia, the Caribbean, or South America. Still, it seems the only thing he hasn’t done after 40 years in the business is play a clichéé or a predictable note.

wfriedwald@nysun.com