Danish artist Vilhelm Hammershøi's paintings had a recurring figure: a mysterious woman with her back turned. Here, through letters and photos, her moving, sadness-tinged story is revealed.

In his 1901 painting Interior in Strandgade, Sunlight on the Floor, the Danish artist Vilhelm Hammershøi depicts a tall window, a lopsided white door and a woman at a table. We can't quite see her face, or what she is doing, but she isn't the painting's protagonist. That role is reserved for the light: the silvery Scandinavian sun that streams through the silent room to cast a pattern on the floor.

It's bewitching to see how Hammershøi (1864-1916) took these ordinary things and turned them into modern art. His paintings, which creak with an unforgettable, otherworldly atmosphere, prefigure the work of Edward Hopper and Andrew Wyeth in mid-century America, and the Minimalism movement of the 1960s.

Many paintings feature his home in the old mercantile quarter of Copenhagen. Hammershøi was besotted with its blue-grey walls, old-fashioned wainscoting (panelling), and the way its doors opened on to one light-filled room after another. So masterfully did he capture these that you imagine you can smell the polish that has made his table gleam, or hear the woman's soft breathing.

Her name was Ida Ilsted, and she was Hammershøi's wife – they married in 1891, when he was 27 and she 22. She features in about 100 of his paintings, sitting or standing in her lamp black dress, her thoughts elsewhere. More often than not, she has her back turned, dashing any hope of a psychological connection, or clue to her emotional state. Still, the urge to speculate is instinctive and overwhelming.

"The captivating scenes in the paintings of Ida give an initial impression that we are seeing into their intimate world," says Dr Felix Krämer, a leading expert on the artist, and the curator of Vilhelm Hammershøi. Silence at Hauser & Wirth's new Basel space, "yet on further inspection few personal details are shown, and for this reason the paintings remain powerfully unknowable and open to multiple readings".

The exhibition, which is accompanied by a new book, brings together 18 works from private collections and is the first ever solo showing of the artist's paintings in Switzerland. That might seem surprising given Hammershøi is now the most sought-after Danish artist of all time – his 1907 painting The Music Room, Strandgade 30 sold for $9.1 million last year. But for most of the 20th Century his work languished in obscurity, the modest reputation he had established in his lifetime (admirers included the poet Rainer Maria Rilke and the Russian impresario Sergei Diaghilev) not revived until the 1990s. The 2008 retrospective at the Royal Academy, London was his first in Britain, and even today there are only three of his works in UK collections.

Much comes down to the fact that the paintings do not slot easily into any of the "isms" and innovations that form the dominant narrative of Modernism. Indeed, though he was born the same year as Toulouse Lautrec and a year after Edvard Munch, Hammershøi always ploughed his own quiet furrow.

Though he and Ida might have travelled to all the artistic centres of Europe – Munich, Berlin, Paris, London and Rome – they shunned cafe society and the parties that lubricated that glitzy world. In Copenhagen, they lived as near recluses, turning their ancient apartment into a kind of laboratory for Hammershøi to study light’s dialogue with architecture to his heart’s content. He painted its interior 66 times in 10 years.

The motif of the lone woman in an interior probably came from Vermeer – whose paintings were rediscovered in the 19th Century and were very much in vogue – or from the Rückenfigur ("figure viewed from the back") of German Romantic painting – Caspar David Friedrich's Woman at the Window (1822) is a fine........

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