The American saxophonist Sonny Rollins has always been an uncompromising and iconoclastic improviser wired into the public mood, though rarely as directly as on his current round of gigs. His recently released live CD Without a Song (the 9/11 Concert) was recorded just four days after Rollins witnessed the attacks on the World Trade Center from his apartment nearby. To add extra poignancy, these are Rollins’ first public performances since his wife of more than 40 years died 18 months ago. And although Rollins’ solos are unlikely to explode into the stamina-sapping streams of consciousness that were his trademark, at 75 he still delivers extended, grainy-toned improvisations that are architectural in scope and complexity.

Rollins is far more than a great jazz performer making the most of special circumstances. He is an authentic innovator who assured his place in the jazz canon 50 years ago. By the late 1950s, Rollins had developed a highly personal, rhythmically obtuse and melodically wayward style that remained founded on rock-solid harmonic understanding. Seemingly abstract slashes of sound would suddenly resolve into harmonic consonance, at terrifyingly fast tempos. And, like Charlie Parker, his improvisations had recurring motifs that would pop up when least expected, giving his improvisations a compositional sense of structure.

All this might sound intimidating – yet Rollins has an affectionate rapport with his audience that is currently unique in jazz. Abstraction is never total, and his strong sense of melody and rhythmic placement is masterful.

Rollins also has an uncanny knack of transforming mundane material into significant statements on the human condition. An entire album – the delightful Way Out West – was devoted to cowboy songs. “St. Thomas”, his best-known composition, is a calypso, a form noted more for its lyrical content than its musical sophistication. And then there are those cheesy showtunes he breaks into fragments, reconstituting them into profound musical statements.

Rollins, christened Theodore Walter, was born in New York in 1930; by his teens he was immersed in the city’s hot-house jazz scene. But despite regular performances with Thelonious Monk, Miles Davis and Art Blakey, it was not until the mid-1950s that his career finally blossomed, and it took an enforced lay-off and advice from Charlie Parker for this to happen.

Like many around him, Rollins was addicted to heroin, eventually serving a year at Ryker’s Island prison. Urged by Parker to break out of his circle, in 1954 he relocated to Chicago, supporting himself by working as a janitor and in a transport yard, all the time developing his music by rigorous practice routines. He returned to New York the following year a healthier man. Even then he had to be coaxed to join the groundbreaking Max Roach/Clifford Brown Quintet in place of saxophonist Harold Land.

There followed a hugely fertile period for Rollins. Roach/Brown Incorporated became one of the most revered bands in modern jazz, assuming near-mythic status when Brown was killed in a car crash in 1955. By then Rollins was in demand as a freelance, working with jazz’s biggest names. This period is richly documented, as Rollins recorded promiscuously – majestic studio jams for Prestige; luxurious Blue Note sessions; the highly structured Freedom Suite on Riverside; and a stunning Live at the Village Vanguard that captured him in magnificent full flight, with only bass and drums in accompaniment.

Although the essential features of his style were now in place, Rollins continued to search for new avenues of expression, and found that the demands of continuous working got in the way. In summer 1959, at the height of his popularity, he embarked on a two-year sabbatical (legendarily practising on the upper walkway of New York’s Williamsburg Bridge).

When he re-emerged, some were disappointed that Rollins hadn’t come up with anything as revolutionary as free jazz. In fact he had considerably developed his existing style. His newly refined grasp of the outer limits of harmonic knowledge enabled him to sustain improvisations of even more fearsome complexity. He extended massively the textures, tonalities and range of the saxophone, adding circular breathing to his repertoire, documented by a mixed bag of recordings for RCA Victor. The Bridge highlights Rollins’ control of abstraction, while the quirky Sonny Meets Hawk neatly juxtaposes Rollins’s tonal experiments with the traditional timbres of the saxophone pioneer Coleman Hawkins.

Ten years on and Rollins had absorbed the rhythms of urban dance, using his knowledge of harmonic possibility to weave rich musical tapestries out of seemingly simple harmonic ingredients. Early 1970s recordings such as the atmospheric Horn Culture and the ebullient Nucleus set the tone for subsequent albums and performances – an obscure vaudeville melody, a calypso, some funk and an extended workout on a jazz standard. But in many ways the interactive pulse of Rollins’ music ebbed, his band acting more as a foundation for his rococo imagination than as equal partners in the architectural commission.

But it would be wrong to suppose that Rollins has succumbed to egotism. Such is the strength of his personal vision, and the richness of his musical vocabulary, that only a handful of musicians are capable of sustaining a truly equal dialogue.

Away from his instrument, Rollins remains remarkably unassuming. Asked once about his main influences, he credited a nightworking neighbour who, rather than complaining about the unholy racket of a beginner practising obsessively while he tried to sleep, remarked that Rollins was getting better and should keep at it.

Sonny Rollins plays at the Barbican, London, on May 13. Tel 845 120 7536

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