One room attempts a partial reconstruction of a 1915 exhibition, The Last Exhibition of Futurist Painting 0.10, in which Malevich's suprematist paintings hang on a tight array around the walls, a Black Square being their nodal point. This is where the real thrust of the exhibition, and Malevich's art, lies. With its discordant music and nonsensical libretto (based on Kruchenykh's idea for a language 'beyond reason' called zaum), the opera was created at a time when Malevich was developing what he called 'alogical painting', wanting to wrest painting away from its duty to render a world of myths, stories and representations. In 1913, Malevich collaborated with musician Mikhail Matyushin and poet Aleksei Kruchenykh on the production in St Petersburg of an opera, Victory Over the Sun, a mad (and maddening) futurist-inspired show, restaged at UCLA in 1984 and screened alongside a number of austere geometric suprematist paintings. They became filled with nonsense words there was a meeting between a cow, a violin, a fish and a wooden spoon (like the spoon that poked out of his jacket pocket as though to announce his outre modernity). His variants on cubism contained touches of realism: a collaged image of the Mona Lisa, and the head of a highly representational bearded middle-class man. Malevich's echoes of Matisse, Picasso and the rest were combined with imagery depicting the lives of the Russian peasantry. And here, men in smocked shirts doing a lumbering dance as they polish a floor with lumps of pumice under their bare feet. Here is a church, seen through bare trees, the whole painting in bleached snow-light and thin yellow sun, the paint heavily flecked and crusted (I imagine the crunchy sound of boots on frozen snow) then a Gauguinesque self-portrait, the artist full-faced with an almost glowering concentration. But he was never playing catch-up, and he managed early on to effect a synthesis of these influences.

Symbolism and impressionism, cubism and futurism – all found their way into Malevich's painting. All of whose work Malevich had seen as early as 1905, when he encountered two of the greatest collections of western avant-garde art in Russia, or indeed anywhere, belonging to Sergei Shchukin and Ivan Morozov. The opening rooms of this extensive retrospective show a bewildering set of influences: Russian folk art and dutiful 19th-century portraiture, Gauguin and Van Gogh, Monet and Cézanne, Matisse and Picasso. Born in Kiev to Polish parents in 1879, Malevich lived through artistic and political revolution. Malevich's art advances and retreats throughout his turbulent career, and now fills a suite of galleries at Tate Modern, complementing the exhibition of Matisse cut-outs that continues until September. Red squares met black rhomboids, circles and wedges. As well as black squares, there were black quadrilaterals and crosses, white-on-white squares, and an off-square red shape titled Red Square (Painterly Realism of a Peasant Woman in Two Dimensions). It appears again in a little drawing, seen in perspective, being borne flat on a stretcher like someone wounded. A black square was wedged between the headlamps of the truck that carried his coffin on a grey May morning in Leningrad in 1935. It took the place of a signature in Malevich's last, figurative paintings. The Black Square was hung across the spot where walls and ceiling meet, like a Russian religious icon. Photograph: State Tretyakov Gallery, Moscow A statement so final that everything else orbits it … Black Square, 1929.

Malevich metaimage