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A member of gallery staff poses for a photograph as she stands in front of paintings by Cy Twombly during a press preview for an exhibition of the works of artists Claude Monet, J M W Turner and Cy Twombly at the Tate Liverpool in Liverpool, northern England. Image Credit: Reuters

JMW Turner may be the most familiar of all British artists, but his allure remains so great that curators are on a permanent mission to find new angles from which to view him. The present trend is compare and contrast. In 2009 Tate Britain staged “Turner and the Masters”, which looked at the way Turner measured himself against earlier painters.

“Turner in the Light of Claude”, which has just closed at the National Gallery, examined his engagement with the great 17th-century landscapist Claude Lorrain. The latest manifestation of our obsession with the man is “Turner, Monet, Twombly: Later Paintings”, at Tate Liverpool. This exhibition, however, scrolls forward and looks at the artist in company with his successors rather than predecessors.

It is at first sight an unlikely grouping: while the links between the romantic Turner and the impressionist Monet are well documented, Cy Twombly, the 20th/21st-century American painter of pale, abstract calligraphic canvases, seems to have little affinity with either of them.

The exhibition, though, reveals a web of affinities that gives a new aspect to each. This is not a study of master and pupils or indeed of direct painterly influences but of shared themes and sensibilities. It is also about a long and unbroken painterly tradition: between them, Turner (1775-1851), Monet (1840-1926) and Twombly (1928-2011) form a three-generational strand that runs through nearly 250 years of Western art.

At one point the third painter of the trio was going to be Mark Rothko, until the full extent of Twombly's links with the older artists became clear. Before Twombly died last year, the exhibition's curator, Jeremy Lewison, had time to meet him just once in the planning stages of the show.

Despite neither Turner or Monet featuring in the artist's previous interviews or writings, it transpired that he owned an autographed letter from each of them as part of his collection of artefacts from artists he particularly admired. Twombly had already long identified himself with them.

The similarities Twombly saw and that this exhibition makes explicit include what Lewison lists as: "An interest in allusion and metaphor, a preoccupation with mortality, a liking for atmospheric effects and an engagement with the tradition of the sublime." On a less elevated note, all three painters were also the victims of vituperative reviews and critical miscomprehension during their careers.

If these correspondences suggest that the resulting pictures are gloomy, the opposite is true. In later life, all three painters had the self-confidence of old age and were not only still experimenting but producing some of the most radical work of their careers. They may have revisited the subjects of their earlier paintings — landscape, fire, water, the seasons — but they did so with urgent vigour.

As Twombly put it: "I've found when you get old you must return to certain things in the beginning, or things you have a sentiment for or something. Because your life closes up in so many ways or doesn't become as flexible or exciting or whatever you want to call it." As age took its toll on their physical power all three men found their flexibility and excitement in paint instead.

Indeed the proddings of mortality, of time and loss, memory and desire, spurred each of them on: between 1829 and his death in 1851 Turner produced 240 paintings; from 1897 to 1926 Monet made 482; and in the last 12 years of his life Twombly painted more than 70 (compared with 58 in the previous 18 years). The length of the past and the shortness of the future hit all of them in a rush. It was not enough though: Monet wrote at 78 that "I think I shall die without ever having arrived at something to my liking".

For all the airiness of their themes, however, the three men were never painters of nothingness. They each remained faithful to their chosen motifs. Turner may have complained that "Atmosphere is my style, indistinctness my fault" but, for example, the nebulous late canvases that so mystified and sometimes outraged his contemporaries, were not just experimental washes of sky, water and land but paintings that were not yet paintings and which often became the basis for fully formed works.

Because there was a large dose of the showman in his nature, he would arrive at the Varnishing Days before the annual Royal Academy Summer Exhibition with one of these rudimentary pictures and totally transform it.

The days had been instituted so that exhibiting artists could make minor tweaks to their pictures to take into account the rooms, light and paintings surrounding them. Turner called them "painting days", however, and used them not to adjust but to transform his pictures and also to prove to himself and younger artists that he could still do it. It was, said one contemporary, like watching "a magician, performing his incantations in public". The results would be recognisably Turnerean, although arrived at in a new way.

Because he was incapable of working from imagination, the huge scale and near-abstract qualities of Monet's waterlilies were a case of the painter settling on a motif through which to work out feelings of grief. He had first painted waterlilies in 1899 after the death of his friend Alfred Sisley and his own step-daughter Suzanne, and he returned to them later in response to emotional hardship.

The death of his son Jean; the growth of cataracts in his eyes; the death of his second wife, Alice; the outbreak of the First World War, all were dealt with by painting these watery scenes traditionally associated with mourning and calm. So fixated had he become that when he left for a painting trip to Venice, Alice wrote "What a miracle that he has left his garden! How happy I am!"

Twombly too turned to arcadia with a series of huge paintings of peonies sharing the title “Blooming”. These rich, blowsy flowers from which paint dribbles in rivulets are a metaphor not just for transience but embody too the sensuality of life. In his last decade Twombly said he worked "in waves because I am impatient — I take liberties I wouldn't have taken before" and the paintings are the proof. Exemplified by his extraordinary “Camino Real” (2010), they show a new interest in colour. They are pictures of supersaturated shades — inky reds, livid oranges, fizzing greens so unlike the tonal politeness of his earlier pale work.

Elsewhere the links between the three are more exact. Monet first encountered Turner's work when he came to London with Camille Pissarro in 1871 to escape the Paris Commune. The pictures he saw in the South Kensington Museum — now the VA — made a deep impact and engendered a sense of emulation. Some historians have suggested that the founding work of impressionism, “Impression, Sunrise” (1872-3), was painted as a direct result.

Turner was not an impressionist avant la lettre but Monet's Thames paintings and especially the series depicting the Houses of Parliament, painted between 1900 and 1905, were undoubtedly a response to the Englishman's own love of the river and his experiments with atmospheric effects and shifting light.

Paintings of Waterloo Bridge by both artists hang side by side in the exhibition. The idea of studying one motif under changing conditions was something Monet used again in his other series showing Rouen Cathedral and haystacks.

Turner and Monet also shared an immunity to physical danger while painting. Turner claimed to have been lashed to the mast of a ship called “Ariel” in order to witness the inside of a storm for a picture. Monet meanwhile nearly lost his life painting the cliffs near Etretat on the Normandy coast.

He had climbed down to be able to paint the Manneporte rock arch when he was taken unawares by a wave: "It threw me against the cliff and I was tossed about in its wake along with my materials! My immediate thought was I was done for, as the water dragged me down."

Twombly was less daring: "Mainly I sit and look", he said, "I can't get on a ladder all the time, it hurts." Where he most resembled Turner was in the frequency with which he dealt with myth and history. Turner's art is full of references to antiquity — from Dido to Ulysses — and also to contemporary events, whether it was the burning of the Houses of Parliament or the scandal of a slave ship captain throwing his dying cargo overboard.

Twombly used myth not as illustrative but allusive. By naming a canvas "Bacchus" or "Orpheus" he didn't so much imply a narrative but use the resonance of the name and its residual impact in the viewer's mind to give an extra depth. He invoked a sense of nostalgia for a played-out civilisation.

He too could nod to contemporary events though: his sculpture “Thermopylae”, referring to the battle between the Greeks and the invading Persians in 480BC, was made in 1991, the time of the first Gulf War. He gave the title “Lepanto”, the name of the last great sea battle in 1571 between Christians and Ottomans, to a series of pictures in 2001, the year of 9/11.

Twombly described himself as a "Romantic symbolist" and that could, at a stretch, be applied to Turner and Monet too. All of them used boats, for example, to express man's passage through life, whether it be Turner's wave-tossed sailing ships, Monet's rowing boat at rest on a still lily pond, or the one-way journey of Twombly's Egyptian funerary barques.

"Meaning", however, in all three artists is always elusive and mutable and this exhibition does not focus on what the symbols represent but rather on their painterly affiliation the shared poetry, the raging against the dying of the light, and the fact that the pictures invite a psychological reading.

All three were painters of immersive canvases, works without borders that draw the viewer in to a rich and often melancholic world. Above all, perhaps, their pictures give the tangible sense that Turner and Monet would have agreed with Twombly's definition of the painter's motivation being all about "the forming of the image; the compulsive action of becoming". These were artists determined to the very end to discover just what painting could do and who went about it, across the centuries, in remarkably similar ways.

Turner, Monet, Twombly: Later Paintings is on at Tate Liverpool until October 28.