"I cannot be grasped in the here and now," the artist Paul Klee wrote as the first line of his own tombstone epitaph. Which in one sense is the opposite of the truth. There are few among the leading Modernist painters who are not more readily known than Klee, with his spindly lines and squared block colours, and even fewer who are more readily loved. Picasso and Matisse seize you with their vision, Klee has the knack of making you feel a partner in it.
And yet there is something elusive, unknowable indeed, about his compositions, hovering as they do between natural observation and imagination, neither the one nor the other: "Somewhat closer to the heart/Of creation than usual/But far from close enough," as he puts it in the same epitaph. It is a delicacy of feeling that has made him, however popular, seem less imposing than his great Modernist contemporaries and less obviously influential.
Tate Modern clearly intends to change his standing if not his popularity with its autumn blockbuster devoted to his work. The show may not have very much new to say. It is, after all, only 10 years since the last major show devoted to the artist. But this one aims to be the most comprehensive, taking Klee from his first entry on the art scene in 1910 to his death from a wasting disease aged just 60 in 1940. In between we are given, over 17 rooms, every phase of his enormously productive output (he completed nearly 10,000 works) and every twist of his endless innovation.
You can never have too much of Klee. The joy of his work, however small, and however persistent his themes, is that he seems so endlessly fresh in the result, whether you are looking at his first passionate embrace of colour on visiting Tunisia or in the bigger, freer oils of his final spurt in the last year of his life. Whether you come much closer to understanding the springs of this reticent man is more questionable.
For a man who taught with great effect in the Bauhaus schools as they moved from Weimar to Dusseldorf and finally Berlin in the 1930s, and had his lectures published and still read for their lucidity of thought, Klee found it remarkably difficult to express exactly what he sought in his pictures. Trained as a violinist and married to a pianist, he was remarkably precise in the way he organised his work. From very early on he catalogued his paintings and drawings in the way a composer does, giving each a number and date according to their sequence in time, keeping back some as special and taking great care over what he would send to show.
His working methods were equally deliberate. He would, according to visitors to his studio, work on several paintings at once, sucking for long periods on his pipe before going to one easel or another to develop his compositions. Like several of his colleagues in Germany, he believed and sought fundamental scientific principles behind his art.
If this gives the impression of an almost constipated effort, it was anything but. The process of reflection followed by application gave a sense of spontaneity to his work that made him popular with the Surrealists, seeing in him an artist of the unmediated subconscious. That certainly helps explain why the viewer feels so complicit in their working. "Take a line for a walk" was his injunction to students and that is what his lines seem to do.
And yet his paintings have too strong a sense of composition to be passed off as purely autonomic. Not for nothing did Klee earn his way as a professional musician until he sold his first pictures in his thirties. Nearly all his works, even the squared abstracts, have a sense of rhythm to them, a feel of a developing theme over the picture.
The Tate exhibition makes much of his continuous experimentation with technique, the "oil-transfer" method, which he developed in 1919, making drawings in paint by laying a sheet of painted paper on a blank one and then impressing the lines, the graded build-up of colour in the early 1920s and the later pointillist abstracts he produced with such glorious effect in 1932.
But technique was only part of his genius, and not the main one. One of the remarkable things about Klee is the consistency of his imagination. The watercolour of 1915, Brown Striving at Right-Angles, is not so very different in tone or mood from the squared abstracts of the early 1930s, nor the imaginary landscape of Above Mountain Summit of 1917 from the pointillist Plants in the Courtyard of 1932. He moves in and out of abstraction at all points of his careers and often in parallel, just as he skips from lighter to darker palette without any particular sense of sequence.
The exceptions seem to be when he becomes more emotionally involved with his work. There is a remarkable room in the show devoted to his work in 1921-2. It was the time when he first joined the faculty of the Bauhaus in Weimar and, free to spend time on his painting, he experimented with newer, more geometric compositions and gradations of colour.
What is so striking about these works is less their formal innovation than the way, for the first time, his compositions seem to be unstable. Even his Room Perspective Pictures of the period have an unnerving feeling of tension, while his oil-transfer drawing In The Wilderness and his later Ghost of a Genius from 1922 have the sense of a self-portrait of an artist out of focus.
He was not to express such unease again until the 1930s. With the rise of Hitler and the troubles with the Bauhaus (which was closed in 1933) events pressed in on him more personally. He was dismissed from his teaching post at the Dusseldorf Academy, his art was labelled "degenerate" and he was impelled to move to Switzerland at the end of 1933.
The titles of his works of the period – Fire at Full Moon, The Face of the Future, Bewitched-Petrified, Fear – give some indication of his state of mind and so do the paintings, with their compressed shapes and darker hues. A single image, Gaze of Silence from 1932, is deeply disquieting in its sense of menace and fear.
Klee was no Expressionist and it is a mistake to read too much of the personal in his works or to treat him as an ironic observer of the century in which he lived. He didn't really do angst. His reaction to the First World War, in which he lost some of his closest friends and was himself called up to do clerical work, was to make satirical drawings of warfare rather than confront it. A period of depression when diagnosed with illness in 1935 was followed by a surge of excited and open work in his final years. The colours brighten, the lines become more energetic and the show closes with a series of heart-warming flower pictures done with wax paint on burlap.
The Tate boldly declares this as the EY Exhibition, a title imposed, one assumes, as the price of a major three-year sponsorship deal with the company more commonly known as Ernst & Young. The question is whether the money is being used simply to reinforce what the museum expects to be popular or whether it will go to make possible shows that would be impractical without it.
Klee is a big name to start off a big exhibition series. Too big perhaps. The effect of so many rooms and so much wall space around the paintings is to monumentalise an artist who was anything but grand except in his ambitions. Klee's works are ones that draw you in to take part in his inspiration. Draw back too far and you view them as statements not processes of the imagination. You can sense him in as few works as many. Go, pick a few rooms and lap him up. He doesn't need a regimented march across his life story.
The EY Exhibition: Paul Klee – Making Visible, Tate Modern, London SE1 (020 7887 8888; tate.org.uk) Wednesday to 9 March