The Neva was still covered in a thick layer of ice in March. Skaters whirled, not safely close to the banks but out in the middle of the wide river on which St Petersburg stands. Cold, sombre sunlight filtered into the gallery high above. And that was when I saw them for the first time, the red bodies whirling and yet still, linking hands, looking downward at the green earth as they cycle in the blue: the dancers.
There in that wintry room in Russia's great museum, the State Hermitage, was the most beautiful painting of the modern world. One of the most beautiful of any age. There is an unreal and utopian quality to Henri Matisse's Dance. And its strange story has recently become stranger still. The painting, completed in 1910, was confiscated from its Russian owner after the 1917 revolution and vanished. It was nearly destroyed for being decadent during the Stalin era, eventually became visible to westerners after the downfall of the Soviet Union and, in the past few weeks, has been at the centre of a cold war-style diplomatic drama.
The painting was set to be the star of an exhibition of modern masterpieces, From Russia, at the Royal Academy later this month. Then, last month, the show was abruptly cancelled. Russia took fright that paintings taken into state ownership after the revolution, and now of contested provenance, had no protection from legal seizure in Britain. Only fast-tracked anti-seizure legislation has saved the show. The near disaster underlines how magical it is that Matisse's masterpiece is coming our way. It is the most glorious work of art we will see this year. Savage and classical, ancient and modern, civilised and barbaric: Dance is all these things. Its beauty comes from a time and a place when art was being remade. It is a blazing modernist banner of desire.
Matisse's five dancers are outlined in thick brown lines that notate simple anatomical details - breasts, buttocks, leg muscles - in a deliberately gauche and childish way. A foot at the far left of the picture is a tangled blur; and what's that bungled attempt to delineate the stomach of the dancer at top right? Yet no one can seriously mistake the roughness of Matisse's drawing for incompetence. The masculine-looking dancer at the left edge forms a perfect geometric curve - an abstract pattern that, if repeated, would turn this into a mathematical game. Instead, Matisse does everything he can to tie his ethereal imagination to basic physical facts. One dancer throws her head forward over a twisted, bulging stomach and muscular, wide-flung thighs. The rawness of her physique releases the painting's sheer visceral power. It's hard to miss the fact that she resembles some tribal carving or neolithic fertility idol. This painting reconnects jaded modern eyes with the primal origins of art. It takes civilisation for a spin on the wild side.
I saw Dance in St Petersburg four years ago, and it stayed in my imagination as a memory of pure colour, like an afterglow on a closed eyelid. It is a big, indeed massive canvas - nearly four metres wide and two and a half metres tall - in only three colours: blue, green and red. A full-scale 1909 oil sketch for the finished work survives in the Museum Of Modern Art, New York: here, the dancers are pink, the volcano of energy does not erupt. In the final, St Petersburg version, first exhibited in Paris in 1910, all is changed. Matisse is one of the greatest liberators of colour in the history of art - his only equals as a colourist might be Leonardo, Raphael, Titian, Turner or Goya - and here he puts colour at the service of a great revolt.
In the first decade of the 20th century, any adult - including Matisse, born in 1869 - would have grown up in the Victorian age. Even in Paris, where artists had long made a virtue of shocking moralists, sex was disreputable - it is what happened in brothels depicted by Toulouse-Lautrec. Now, suddenly, here is Matisse's Dance, a painting that declares there is no higher, no more human thing than to dance in naked ecstasy: to burn with passion.
And yet the image Matisse uses is one of the most ancient in art. From Etruscan frescoes to Botticelli's Primavera to Turner's landscapes, the story of art is full of dancers. They are inhabitants of Arcadia, the fabled land of the Greek golden age. For the Old Masters, whose works Matisse studied meticulously, the Arcadian dance can become loaded with philosophical and moral weight, as in Poussin's Dance To The Music Of Time. But first and foremost, as in Raphael's Three Graces, the image of the dance is an image of perfect beauty and poise. Dance pays homage to this image that reappears again and again in the sensual, physical art of the Mediterranean world.
To compare Matisse's Dance with, say, Botticelli's Primavera is to recognise Matisse's originality and modernity. From the Primavera, the goddess Venus looks out. That gaze has a basic way of connecting painting with the world we inhabit, and artists through the centuries have been scrupulous about using this trick when imagining realms of bliss that might otherwise seem too distant. If you're painting paradise, you don't want to make people feel like strangers.
In Matisse's Dance, only one of his dancers shows her face - and it is cast down. Why such secrecy? This is a private revel, a party we can watch but that will happen with or without us. The painting deliberately risks what every previous artist had gone to great lengths to avoid when painting the good life - an inhuman remoteness.
To grasp what it was that drove Matisse to such unprecedented creative fire, you have to look to a dramatic tension between two geniuses - those red bodies glow with the heat of rivalry. At the time no one seems to have noticed; only now, a century on, can we see this work as a stratagem in a competitive game with Pablo Picasso.
If earlier Bacchanalian paintings engage the beholder in sensual conversation, Picasso's Demoiselles d'Avignon stare you down. Five women stand in a compact, jagged curtained theatre. Even the slice of melon looks as if it could do you an injury. Two have grotesque masks for faces, yet all are looking at you. Eyes burst from the pink and blue canvas: staring black pupils madly penetrate the space between you and them. The young Picasso created his masterpiece, a big square picture on the ambitious scale of a history painting by Poussin, in 1907, in his studio in the ramshackle Bateau-Lavoir in Montmartre. Three years later, Matisse composed his answer to Les Demoiselles d'Avignon - for that is what Dance is.
Matisse, 12 years older than his Spanish rival, had come late to art. He was 19 when he asked for a paintbox; he was convalescing in hospital after a breakdown that came of trying to force himself into a law career. Soon after, he was studying under the symbolist painter Gustave Moreau and by 1905 was seen as positively dangerous. That was the year Matisse, André Derain and Maurice de Vlaminck disrupted the Salon d'Automne exhibition with paintings as daring and joyous as fireworks. Nineteenth-century art movements always justified their departures from convention with the - sincere - claim that they were painting the world as it really looked. For the first time, in 1905, Matisse and Derain started to splash on colour and made no such claim - they obeyed what was in them, not what was outside. A beach could be red, a face yellow. People called them fauves, "wild beasts".
Yet even fauvism came to seem tame in comparison with a new revolution that was brewing. It came not from art, or not from European art, but from history, trade and geography. In the late 19th century, European nations were competing to colonise the world. And with that a multitude of masks, drums, thrones, canoes and plaques, carved and cast by tribal cultures in Africa and the Pacific, started to reach museums and art dealers and markets. It was an alien and enigmatic art that seemed divorced from any attempt at a realistic portrayal of the world, and it rocked French artists.
It is exhausting to follow the rapid succession of events and ideas in avant garde Paris around 1906. Matisse was painting a primal scene of six naked dancers in Le Bonheur De Vivre. That same year, the collectors Leo and Gertrude Stein introduced him to Picasso, and Derain bought an African mask he showed to both artists. It still survives, a white, wooden oval with piercing eyes from Congo. The faces of Picasso's prostitutes in Les Demoiselles are blatantly modelled on it. But how does African art infect Matisse's classical arcadian Dance?
Les Demoiselles is so savage, so aggressive, it horrified even the avant garde artists who saw it in Picasso's studio. It was not exhibited until years later. Only since its acquisition by New York's Museum Of Modern Art in 1939 has it become an icon. One person, however, understood it - and that was Matisse. Dance, his response, infuses the ancient classical tradition of the Mediterranean with the "primitive" energy of masks and rough-hewn tribal sculptures. Most of the African and Oceanian masks these artists were collecting were, after all, made to be worn during ceremonial dances.
The Russian businessman Sergei Shchukin, a notable collector in tsarist times, fell for the art of Matisse after seeing Le Bonheur De Vivre. Their relationship became like that of Renaissance artist and patron, with Matisse accepting commissions from Shchukin. In 1908, they went to Picasso's studio to see Les Demoiselles d'Avignon - a visit of great significance for both.
Shchukin commissioned Matisse to create mural-scale decorative paintings for his Moscow mansion, and Dance is the result. Matisse's determination to outdo Picasso on the large, historic scale of Les Demoiselles could scarcely be more obvious. Both paintings present the same number of nudes, but where Picasso painted five women who challenge you the viewer, Matisse answers him with a picture whose five naked inhabitants don't care if you are there or not. Picasso is aggressive, Matisse self-contained.
Dance is modernist in a way that has not faded. When you look at it, you are unsettled as well as uplifted. It seems on the edge of emptiness. If Les Demoiselles contains within it the seeds of dada and surrealism and every impulse in modern art to unleash dark forces, Matisse, in the final, 1910 version of Dance, anticipates abstract art, minimalism and every artwork that invites the dismissal: is that all? This sumptuous masterpiece leads to the lights going on and off.
There's nothing in any other painting quite like the chromatic miracle of Dance. Terracotta flesh combines mysteriously in your mind with those saturated blues and greens, in a poem of absolutes. Absolute red, absolute blue, absolute green - a hymn of intensity. And here Matisse's understanding of the "primitive" becomes clear. For all its Hellenic grace, Dance is as much a wilfully barbaric painting as its rival by Picasso. The key to its barbarism is colour. There is something simple, childish, folkloric about rampant colour.
An unlikely source for the figures in Dance was an artist - a primitive in Matisse's eyes - who lived in the heart of a modern city: JMW Turner. Matisse made a special study of this overwhelming British colourist - who, like him, loved the Mediterranean - on his honeymoon in London in 1898, when he looked repeatedly at Turner's art in the National Gallery. You could hang Dance next to Turner's paintings and the emotional use of colour to blaze a path between the imaginations of artist and beholder would immediately strike you as similar. Turner's Mediterranean scenes are peopled, too, with Arcadian figures. The five figures in Dance look uncannily like a group of dancers in Turner's The Golden Bough and reminiscent of dancers in other Turner paintings such as Apuleia In Search Of Apuleius, which was on view at the National Gallery in 1898. What makes them so similar is the serpentine loose depiction of the bodies, which in Matisse is deliberate and in his model was an accident. Turner didn't paint people very well. His figures are ungainly, rough - in short, "primitive".
I think this is crucial to the meaning of Dance. The simplest, most humane discovery of modern artists at the beginning of the 20th century was that art is not a matter of drawing "correctly". There are no rules. Dance exemplifies the modern belief that to evoke the important things you don't need to pay attention to any protocols, any tradition. Turner had faults, but his vision makes these irrelevant, just as the imagination of unknown African sculptors made their distortions of the body wondrous.
Matisse's conception, a century after he painted Dance, is still lethally elusive. Those red dancers flit and flicker away from you like glinting fish in the water when you try to grasp them. They slide off, they dance away. Such bliss, such ecstasy - and it is yours, but not yours. It is a painting that in the end eludes the analyst, and it almost escaped us in not coming to London after all. Matisse's art suggests there is something, some experience, some truth, that we cannot touch with our hands, cannot capture with language, but must treasure in the moments it comes to us - if it comes to us at all.
· From Russia is at the Royal Academy of Arts, London W1, from January 26-April 18; 0870 848 8484 (royalacademy.org.uk) for ticket details.