The roofline was so beguiling, seen from my attic window, that I spent ten minutes each morning watching the frosty sky colouring and clearing behind the neoclassical cathedral tower, with the fantastical Borrominian helter-skelter of Vor Frelsers Kirke in the distance. As the colour changed from yellow to dusty rose to cold azure, small vermilion clouds sailed up into the space between that spire and the prodigious spire which crowns Christian IV’s Bourse, the tails of four lead dragons swirling up into the air making one spiral like a unicorn’s horn.

The Absolute Kings of Denmark were lords of the unimaginable territories of Finnmark and Greenland – the lands toward the pole, lying under the constellations of the Bears – and they reigned from a throne of narwhal ivory. Never in Copenhagen are you in any doubt of this element of Renaissance splendour, never in any doubt that this was once the capital of a great, strange, snowy, absolute monarchy.

But this is balanced by a delight in the pleasures and dignities of the quotidian, grounded in the early nineteenth century: the world of the Copenhagen pleasure gardens and of the modest manor houses of rural Denmark. The breakfast coffee was of that world, in that only cream was offered with it, never plain milk, nor the condensed ‘coffee-milk’ of Germany and the Netherlands. The lavishness of the breakfast itself went far beyond the abundance with which the Protestant north usually rewards those who get up early and do their eating in the morning. Five different sorts of bread, all excellent; great cubes of cheese; rolls; biscuits; almond pastries; fruit and jam. Lunch was the same – big pots of sweet butter with the black bread and shrimps and herring; a heaped plate of marzipan cakes automatically set down with the coffee.

I went round the Thorwaldsen Museum one afternoon. So great was the prestige of the sculptor that his museum and mausoleum are next door to the parliament building, on the same island as the parliament church and the royal library with its melancholy and austere formal garden. You enter the neo-Pompeiian villa from the side, opposite the sentry box for the bearskinned sentry who guards (anti-climactically) the builder’s hoarding around the ceremonial entrance of the parliament. On the ground floor, the length of the museum is lined with enfilades of small rooms. Their high, barrel-vaulted ceilings are painted in Roman style, sometimes with fictive architectural coffering, sometimes with loose vine-scrolls and songbirds. One ceiling has odd trophies of Gothic arms against a ground of deep cobalt, the figure of a Viking in feigned relief. The thermal windows are high, leaving all four walls clear for the display of the sculptures.

Once one has accepted the metaphor which clothes every notable (and many wealthy nonentities) of the early nineteenth century in flawless Roman fancy-dress, the collection is staggering. There is a kind of frankness and innocence giving life to the marmoreal correctitude. Russian princesses, all looking exactly like Paolina Borghese, share the luminous rooms with Danish national heroes, forgotten English milordi and panels of bas-relief strictly after the antique: Ganimed med Jupiters ørn, Amor på Løvensrig. Very occasionally, a trace of sentimentality comes to the surface: the taste of the frankly Biedermeier paintings in the upstairs recreations of Thorwaldsen’s apartments. There is an affecting bas-relief of a cupidon cuddling a fat and blunt-snouted puppy, surveyed from the other side of the room by the bust of the great English neoclassicist Thomas Hope of the Deepdene, his Romanitas just touched with emotion, the petrified hint of moistness in the deep-cut eyes. There are two different busts of Metternich, and one of Sir Walter Scott swathed in draperies which could be read as either the toga virilis or the plaid of a Border shepherd. All the eminent of post-Napoleonic Europe are here, life-size and in unflawed Carrara marble.

Because you enter the building at the side, the most stupendous room comes as a surprise at the end of the tour: a curtained doorway gives onto a double-height entrance hall with a pontiff, three times life-size, and a number of assorted Baltic national heroes on horseback, to a scale yet more tremendous. Their bronze realisations in the palace squares of northern and central Europe must have fallen, almost without exception, to the wars of the last century.

You see the quayside and the water through the long windows. Under snow – and all neoclassical architecture seems to call for snow to be seen in perfection – it would be one of the loveliest things in Europe, this view out between the pale statues over the inlet of the Baltic and across to the placid eighteenth-century houses.

I was taken for a city walk by a senior student of the university who had an effortless command of the most exalted register of English. We went searching in the rural-looking courtyards behind a street of antiquarian booksellers for the last (alas, vanished) toy theatre shop in Denmark. Its closure severs one link with the world of the 1840s which survives in the Royal Ballet’s repertory of Bournonville ballets, in the galops and polkas still played at the Tivoli pleasure gardens. We went to see the bits of British navy cannonball which had come through the roof of the university library (‘It has long been our policy to preserve national tranquillity at virtually any price’), and then we went to see the cathedral built on the site cleared by Nelson’s bombardment. It is all white and gold inside, with no colour at all, as though the whole ensemble – pillars, domes and Thorwaldsen’s heroic statues of the apostles – were moulded from snow.

The rest of the site cleared by Lord Nelson is now a formal square, the aula of the university facing the flank of the cathedral. Along the side of the cathedral are herms of distinguished bishops; along the façade of the aula are distinguished professors. Both buildings are equally correct, equally classical. What gives the whole ensemble an air of distinction and fantasy is a simple matter of scale, because the herms are enormous, twice life-size at least, their granite plinths higher than your head. In the entrance-hall of the university are more frescoes derived from colour-plate books of excavations in Campania, cameos against grounds the colour of summer twilight. A series of big historical canvases around the walls of the aula magna itself ends with a delightful scene of a June evening party, a professor-poet reading his verses in a garden to the white-capped students. There is a musicians’ gallery high away in the cove of the ceiling. ‘It is from that gallery,’ Caspar says, ‘that we sing, every year, the song called “The Living Flame”.’

So we wander on to see the garden in the painting, the garden of the student house where he lives. Not in the ancient universities of England, nor in Coimbra nor Harvard are students housed with such magnificence: great carved staircases and vast doors all painted in the beautiful drab grey-green of the late Baroque, and an enormous garden, right in the middle of the city, its trees overhanging the brick walls and shading the cobbled streets. It even has garden pavilions. Surely this is the only community of students in the world to have at their command a formal garden with ornamental temples, and still to run their house and garden democratically, like the nationes of the mediaeval universities.

In the Hirschsprung collection, housed in another recreated Roman villa, in a landscaped park on the other side of the Rosenborg Palace, there is a painting of a scene in one of these student houses as it was in the year 1830. It is immensely civilised: the sparse Biedermeier room with its bare boards, its Empire sofa, its pots of flowers at the windows, a couple of family portraits hanging unframed from nails in the panelling. The two students depicted were the brothers of the painter William Bendz, who died in his twenties, scarcely out of his own student years. They wear the gallant, becoming clothes of the era, and their braided and tasselled student caps hang on their hooks. When I recalled this painting to my guide he thought for a moment and concluded that, on the evidence of the panelling, the room must be in the student house on the other side of the street, of whose inhabitants, he pronounced in his astral English, ‘We think they lack polish on the whole.’

Also in the Hirschsprung is a delicate painting from the history of Danish neoclassicism, by Christen Købke, another short-lived master of the early nineteenth century. It is a wry picture of a young curator, seen from behind so that the full glory of his beautiful frock coat is much in evidence, solemnly dusting the cyclopean plaster casts of the royal collection.

Købke’s painting of his young contemporary Frederik Sødring, in the National Gallery next door, is extremely moving. So much affectionate attention is given to recording the quotidian details of a shared studio in the year 1832, that this is one of the paintings which offers for a moment the illusion that the past can be known and recovered. The big double doors between rooms have been locked fast, and a table with a straggling pot of ivy pushed up against them; prints of classical landscapes are tacked to the panels; a mirror hangs across the doors. The sitter leans back for a moment, his palette still in his left hand, his strong-fingered right hand resting on his knee. The folding stool for landscape painting excursions leans on the doorframe. Across the grey-painted panels and the folds of his loose-sleeved grey shirt, the early summer light floods the picture.

Walking round the gallery, you can only be struck by the gentleness of the subjects of the paintings. They have the modesty of Dutch interiors and streetscapes, but are generally rural and squirearchic, rather than urban and mercantile: the young baron and baroness pointing with quiet pride to the little manor house in the valley below them. Even the military subjects seem as innocent as on the stage of the comic opera or the ballet: soldiers at ease in the sun on the grassy ramparts of a coastal fort – a painting wonderful for its precise capturing of the first really warm day in spring, racing clouds high in the pale sky, and the water tinselled with sunlight.

But for me, the real discovery was the works of Vilhelm Hammershøi (1864–1916). Once seen, these haunted and coloured my view of the city for the rest of my stay. His range of subjects was extremely restricted: a few portraits, one or two views of Copenhagen and the surrounding countryside, and a vast number of views of the interiors of the apartments where he lived with his family. It is hard to describe the hypnotic and nostalgic quality of his work – think of the most twilit work of Whistler, or perhaps of the nocturnes of Atkinson Grimshaw and you begin to have some sense of this long series of dim interior paintings. Almost all of these rooms to which he returns again and again are painted by evening light, winter light usually, and frequently by the very last of the twilight. Often the figure of his wife is present, reading or sewing, turning away from the painter.

He made at least one foreign excursion in the last few years of his life, a visit to London, and painted there two of the most atmospheric views of the Edwardian city that it is possible to imagine. One shows Montague Street at evening with the railings of the British Museum massive in the foreground and the lights of the houses glimmering in reflection on the wet pavements. His other London painting is a view out from a first-floor drawing room in Brunswick Square, with the iron railings of the balconies standing out against the fog. The painting conveys the essence of the cold in the eighteenth-century rooms – the chill, the damp, the inadequate heat from the little coal fire in the cast-iron grate.

Yet Copenhagen was his subject and his obsession; he was the poet of the failing light and the snowy city under his windows. His views of the waterfronts under snow offer a talisman of the city which governed once almost all of the snowfields of the north, of which era the narwhal-horn spire and spiralling ivory throne are reminders; a sense that this has been a place of departure for the farthest and remotest regions, of great ships setting out for the blizzard and the ice.