Branford Marsalis knows that he knows more about jazz than most people: Video, Photos

The premier saxophonist is also convinced that the majority of those coming to see his kinetic Branford Marsalis Quartet at the SFJazz Center this week won’t even recognize much of the music.

“Most people that go to jazz concerts are not jazz fans, so it’s all new to them,” says Marsalis. “They come to the concert as an ‘eclectic night out’ and have a great time. (But) they don’t run out after the concert and download a pile of music. So almost anything we play — they are hearing it for the first time or for the first time in a long time. Either way, it’s not familiar to them.”

Perhaps best known to dilettantes as the former “Tonight Show with Jay Leno” bandleader, the New Orleans native (the eldest son of the late pianist and educator Ellis Marsalis and big brother to trumpeter Wynton, trombonist Delfeayo and drummer Jason) first honed his craft as part of Art Blakey’s Jazz Messengers and Wynton’s band before starting the Branford Marsalis Quartet in 1986.

Since then, the 2011 National Endowment for the Arts Jazz Master and three-time Grammy winner has released close to 30 albums, including 2014’s “In My Solitude,” a live solo recital recorded the previous year at Grace Cathedral as part of the 30th San Francisco Jazz Festival.

“There was a lot of mental pressure for me because I understand how difficult it is to keep an audience engaged for an hour and a half with one instrument, particularly because a lot of people think that the strength of jazz is improvisation,” says Marsalis. “Well, improvisation starts to sound all the same after a while if that’s all you’re doing, so I’ve always thought that the strength of jazz was jazz itself, melodically, rhythmically, harmonically.”

To sustain the attendees’ interest, Marsalis created a program that was a mix of improvised and scripted music. The fact that half the audience didn’t leave during intermission, says the saxophonist, was evidence enough that the show was a resounding success.

While Marsalis is mostly known as a contemporary jazz performer, he is equally adept at playing the music of the post-war jazz pioneers who preceded him.

He’s worked with Miles Davis and Dizzy Gillespie and successfully replicated saxophonist John Coltrane’s highly challenging four-part masterpiece “A Love Supreme” — both on his 2002 album, “Footsteps of Our Fathers,” and live onstage in Amsterdam the following year.

Marsalis has a deep understanding of the mechanics behind the avant-garde album, which inspired a San Francisco couple to found a house of worship in its maker’s honor — St. John Coltrane Church.

According to Marsalis, it’s the harmonic simplicity of “A Love Supreme” that makes it so challenging to play.

“The songs with complex forms, more often than not, play themselves,” says Marsalis. “It doesn’t require imagination, but harmonic understanding and technical mastery. But in the John Coltrane Quartet years of the ’60s, you had to bring something to the music to play it. One of the things that [they] did better than anybody else is to create maximum amounts of intensity on one-chord songs.”

Outside of jazz, Marsalis has delved into hip-hop with a feature on Public Enemy’s 1989 “Fight the Power” single and his mid-’90s Buckshot LeFonque project; played classical music with the New York Philharmonic; and worked in the pop and rock realms with artists like Sting, Bruce Hornsby, Dave Matthews Band and San Francisco’s own Grateful Dead.

A wiz at improvisation, himself, Marsalis was a natural to play with the exploratory Bay Area jam band.

“It was fun,” he says. “Their music is more wide open and less controlled. With the Grateful Dead, you had a beginning, and then everything after the beginning is negotiable.”

Amid all the artists he’s collaborated with over his decades-long career and all the stage shows and films he’s scored, most recently Netflix’s “Ma Rainey’s Black Bottom,” Marsalis continues to prioritize his quartet, which he’s excited to bring to The City — aka the Harlem of the West — for a four-night engagement, starting Thursday.

“San Francisco is more of a jazz town than any other city in California,” says Marsalis, who conducted master classes in the genre at San Francisco State University in the early 2000s. “San Francisco is the cultural capital on the West Coast, and I always have a great time playing there because the audiences are more adventurous. I know you can challenge them and they’ll understand what you’re trying to do.”