Albert Ayler’s body was retrieved from the East River, in Brooklyn, on November 25, 1970, a few months after his 34th birthday. The saxophonist grew up in Cleveland, left to find work as a musician playing in restaurants in Sweden in 1962, and returned to New York, changed and inspired, to take the American avant-garde jazz of the period further from its roots than anyone thought it could, or perhaps should, go.

Ayler played at John Coltrane’s funeral, as if the baton were being handed on, but his mysterious death means we’ll never know how he might have developed. For many, the apparently uncompromising aggression of the raucous free-jazz movement dubbed the New Thing encapsulated black anger. But Ayler’s music resisted definition, suggesting euphoric celebration and revolutionary fervour in equal measure.

Ayler’s obituary in the jazz magazine Downbeat struggled to categorise the saxophonist. Was the music he made — a mix of nursery-rhyme melodies, military bugle blasts, raging spirituals, funereal dirges and unrelenting improvisations of the harshest quality — really jazz at all? Faced with confusion in his lifetime, Ayler claimed that history would be his judge. “One day the people will understand” was his oft- repeated mantra.

This year’s London Jazz Festival features two Ayler-related performances: a concert of his music by the guitarist Marc Ribot and Ayler’s bassist, Henry Grimes, and a free afternoon show by the American saxophonist Caroline Kraabel. This is preceded by a screening of a new documentary, My Name Is Albert Ayler, by the young Swedish director Kasper Collin. Does this flurry of officially sanctioned South Bank approval mean that the people do now, finally, understand Ayler? Collin’s film is a haunting mesh of old cine footage, paint-stripping live performances and interviews with surviving friends and family. A strange shot of a semi-naked Ayler, staring silently into the camera, threads through the film, as if the subject is daring you to dismiss him. “I didn’t want to speculate about things too much,” says the director. “I wanted to leave it up to the audience to decide.”

He avoids commentary and frames Ayler’s life with impressionistic images. On his first visit to Sweden, we see footage of the midnight sun that fascinated him. His closing months in Brooklyn see him again obsessed with the sun, staring into it across the East River. And when Collin goes to Cleveland to meet relatives — his brother and collaborator, the trumpeter Donald, and his sprightly father, Edward — they get lost in a cemetery looking for his grave.

“The film was produced over a long time,” says Collin. “I knew about Albert Ayler seeing the sun in Sweden maybe two years into the project. The film wasn’t really scripted. I built it around recordings of Ayler’s own voice. The contrast between his music and his soft, gentle voice was fantastic, because it is not the voice you are expecting.”

For Collin, Sweden is crucial to Ayler’s career. “Scandinavia was important for the development of American free jazz and avant-garde music,” he says. “Ayler felt more relaxed in Sweden. Probably there were some people here who believed in him. It helped him get confident. One big event was in the spring of 1962, when the jazz club The Golden Circle opened, and they had really great acts like Cecil Taylor and Sunny Murray.”

Murray played drums with Ayler throughout the 1960s. A bear of a man whose memories of Ayler often trail off into happy, helpless laughter, he toured here last month fronting a trio that included two British musicians, the saxophonist Tony Bevan and the double bass-player John Edwards. At the Red Rose, in Finsbury Park, London, the response to their opening set, and the demographic diversity of the crowd, would have delighted Ayler.

Does it seem strange to Murray that the saxophonist should leave his native land, only to discover the kindred spirits of the American avant-garde in Sweden? “No, not really,” he says, sitting at the side of the stage, rolling a cigarette. “I haven’t figured out yet how me and Cecil Taylor ended up in Sweden, but I met Albert when he came over to the club, wearing a very handsome cap, dressed very nice in his leather suit. He said he had been playing there in Sweden since he left the army. He said he had been playing his music alone in the forest by himself for a year. He asked if he could play with us. Back then, Cecil just wasn’t outgoing. It was such a weight having to carry the New Thing. Cecil said, ‘I don’t know. I don’t think so.’

“So me and Jimmy Lyons, we said, ‘Yeah, go home and get your horn.’ He came back with this beautiful new sax, so we told him, wait until we give him a signal. Then, in the middle of the gig, we said, ‘Come up’ — and it was beautiful. Albert was like a magic streak of light in the air.”

Collin suggests that Ayler’s time may have come: “Albert was always saying, ‘One day people will understand.’ He was right. He would have been very glad that his music is appreciated by a younger audience, coming to it from alternative rock. It’s not a real jazz thing any more.”

Marc Ribot’s Spiritual Unity, featuring Henry Grimes, QEH, SE1, Fri at 7.30pm; free gig, inspired by Ayler, by Caroline Kraabel, QEH Front Room, Sat at 4pm, after the film at 2pm