Two years ago, I asked the free jazz pianist Matthew Shipp if he would take part in a concert I was organising in remembrance of Cecil Taylor, who had just died. He said he’d be willing to give a talk, but not to perform. Taylor hadn’t influenced his work, and he didn’t want to encourage the notion that he had. I wasn’t surprised (I’ve known Shipp for more than twenty years). His feelings about Taylor were complicated, and the two men often jousted, especially on the subject of Bill Evans, whom Taylor disparaged as the great white hope of jazz piano, and Shipp reveres. Shipp had also been saddled with the ‘heir of Cecil Taylor’ label for three decades, even though the resemblances in their playing are superficial. The only comparison with Taylor that Shipp ever welcomed was made by a mutual friend, the saxophonist Jemeel Moondoc, who told him: ‘You’re just like Cecil Taylor – you’re both bad motherfuckers.’

Shipp eventually agreed to play a solo in Taylor’s memory, but on one condition: he would preface it with a lecture. The talk was erudite, probing and surprisingly warm: a concise and admiring primer on Taylor’s work. Then he sat down at the piano and played a gorgeous solo of about six minutes. It was a typical Shipp improvisation: dense, rumbling figures, leavened by boppish quavers; dark, ringing overtones alternating with impressionistic chords suggestive of an Evans ballad. It included none of the signature Taylor gestures evoked by the other pianists on the programme that evening, such as his percussive pummelling of the keys. In moving from the lectern to the piano, Shipp had moved from Taylor’s world back to his own.

At first this struck me as a proud, if not stubborn, assertion of their differences, but I came to see it as a peculiarly moving tribute. By steering clear of Taylorisms – and even invoking his nemesis – he wasn’t taunting Taylor so much as continuing their argument after his death. Shipp seemed to be saying that their long-running quarrel had helped shape him as a pianist – that it remained alive to him. It also underscored Taylor’s central lesson for artists, the insistence on independence.

Shipp, who turns 60 in December, came to prominence in the quartet led by the late saxophonist David S. Ware, one of the great bands of the 1990s. He went on to become one of the most original solo improvisers in free jazz, and a leader of some fine ensembles, including his current trio with the bassist Michael Bisio and the drummer Newman Taylor Baker. But it’s been easy to lose sight of him in recent years. He’s made far more records than even the most ardent fan could absorb, even as he’s threatened to retire from recording and vanish.

Having listened obsessively to Shipp’s new solo recording, The Piano Equation, I’m glad that he hasn’t. The album has made me realise that what I mistook for monotony in some of his recent work was in fact focus. It links him not only to Taylor, but to Thelonious Monk, Mal Waldron, Andrew Hill, Paul Bley and Ran Blake. He calls these pianists ‘crunchy’, by which he means their work requires some effort, even a bit of struggle, on the listener’s part. There are moments of lyricism in his own crunchy work, but they’re hard-won and ephemeral, often overwhelmed by what he calls ‘clouds of sound’ – otherworldly effects that he creates through overtones and pedalling.

As a child Shipp played the organ at the Episcopalian church in Delaware that his family attended. ‘Religious music is a massive part of my background,’ he told me. Gregorian chant and hymns have seeped into his playing and fused with stride piano and bebop, as well as Scriabin, Satie, Bartok and Stockhausen. Where Taylor tried to make the piano sound like ‘88 tuned drums’, Shipp would rather suggest the toll of a church bell. ‘It comes down to touch,’ the pianist Craig Taborn told me. ‘I just do not know how he gets the piano to resonate that way.’

The Piano Equation is a sequence of 11 short pieces, slow to mid-tempo, and best heard as a suite. They aren’t tunes, since Shipp improvised them in the studio, after working out his ideas in long practice sessions. Nor are they studies in the narrow sense of technical exercises. But they’re rich in melodic content, and rigorous in their exploration of specific gestures and figures. You can hear echoes of the jazz piano tradition – a Monkish fragment opens ‘Swing Note from Deep Space’; the bass line in ‘Vortex Factor’ suggests Lenny Tristano; the plaintive notes at the beginning of ‘Land of the Secrets’ are, in Shipp’s words, ‘like cold Bill Evans’; ‘Radio Signals Emission’ is almost a blues. But their sonic architecture is more gnarled, since Shipp dispenses with the usual signposts of jazz piano: chord changes, an obvious melody or consistent theme, a fixed pulse. At times, he creates what he calls the ‘perception or illusion’ of these elements, but only to disrupt any sense of flow or ease. Instead of the inexorable logic of a song unfolding, we hear the act of decision-making in real time, like action painting in sound.

Critics often describe his work as ‘brooding’ – an adjective he often mentions in conversation, with self-deprecating humour. But Shipp’s music has never struck me as ‘brooding’ so much as combative. He’s a boxing fan – he calls the sport a ‘neurological dance’ – and when he sits down to play, he enters the ring. His playing is physical and relentless.

It’s also strategic. He’s there to honour the great pianists of the past, as he was at that Cecil Taylor memorial, but also to fight them. ‘I’m trying to find an apocalyptic, William Blake imaginary realm,’ he told me, ‘between Bud Powell and Monk.’ And in the last moments of The Piano Equation, we arrive there, in a piece called ‘Cosmic Juice’ that Shipp compares to the Book of Revelation. He plays a gloomy, shimmering series of poly-chords over and again, with long, enigmatic cadences. Those notes, he says, are intended to ‘sound like chimes. I’m ringing a bell to a new world.’ I have listened to ‘Cosmic Juice’ repeatedly in the last two months, enchanted by its terrifying beauty, and strangely comforted by it, too, since it evokes the journey of someone who has found a hidden door, glimpsed another world, and realised that the waiting is finally over.