Agnes Martin didn’t decide to be a painter until she was 30: a good age to start something new. Martin (1912-2004) was a Canadian who lived most of her life in the United States. In New York she became part of the scene around Barnett Newman, Ellsworth Kelly, Robert Rauschenberg and Sol LeWitt. She always looked like an artist who was after something, and success came quickly. For the rest of her life, with multiple variations, she made abstract paintings in acrylic paint and graphite on canvases measuring 183cm square. Working in subdivisions of the frame, she created fields of minimal colour, banded in horizontal lines. The resulting expansions, contractions and experiments in this format are on show at Tate Modern in London and are astonishingly varied.

An Agnes Martin exhibition is an optical treat. People step forward, back, take glasses off and put them on again. They are wondering how the works draw their power; looking from near and far at basic surfaces, strokes and lines. Pencil marks in paintings are unusual. Martin made a feature of them and the ruled line is a mainstay of her work, a major player, but a slippery one. Often you don’t notice the lines until you get near. Then you start to look out for them. Pencil is generally seen as preparatory, intimate like drawing, yet here are rooms full of great expanses of painting that are clearly pencilled in, the marks not guides to be painted over. Pencil suggests provisionality, the idea that something will be firmed up later. With Martin there is no later: “My paintings have neither objects, nor space, nor time, nor anything—no forms.” No foreground, no background, no horizon, no particular emphasis on up or down. In front of the painting is where you are.

Drawing a straight line is hard on the weave and weft of canvas and her lines emphasise the stuff of the cloth. They register softness, minor meanders, shifts in pressure. The idea of the mark as expressive is tamped down. These works, for all their other-worldliness, are highly material.

For a while I stare at a grey painting. It’s like a flawed field, slightly watery, denser in the middle and lightening at right and left. The effect is vague, as if the fade is like some phenomenon on the edge of vision. But then I see it is created where the hundreds of pencil lines running across the width of the canvas stop abruptly. Nothing vague about it at all. Yet this precise step-change doesn’t read in any way as mechanical. The works are graphic in execution but not in effect. An intersecting chamber links the first and last rooms at the Tate and holds a display of small drawings. They are seismic: fields of tiny trig-point triangles, squashed eye shapes, dots and dashes, thin lines and thicker lines are all set in variations of the grid.

Lines delineate colour. In earlier work her colours are natural, like mud and dust. “Buds” (1959) is a series of pollen-coloured discs on a mustard surface. But thereafter the colours are increasingly prismatic: natural like the weather, optical rather than earthy. And even when things go grey they never shut down. In a series of austere grey paintings from 1977 the darks are shot through with light. There is a sense of impending precipitation, as if the deluge is somewhere imminent. To look at Martin’s paintings is to be very aware of water. But the overwhelming impression is of paleness: pale red, pale yellow, pale blue, pale white, as if bleached by the sun. These are too iridescent to be genuine fades, the pigment too pure. Apart from her years in New York, she worked most of her life in Taos, New Mexico, but hers is the world of the great indoors: the world inside the head.

It’s the sheer singularity of the artist that comes through. “On a Clear Day” (1973) is a striking set of screen-prints that marked her return to artwork after a five-year absence. At a pivotal point in her career she just took off: because she could, and because she needed to. This was in 1967. She said, “I must give independence a trial.”

Martin painted every day, and the last works, made when she was in her 90s, are extraordinary. “Gratitude” (2001) is like a song: palest green spanned by widths of red, yellow and white. In “Untitled” (2004) the horizontal bands are thicker, shaky, yet still utterly solid in their proportions. Many of her paintings are in numbered series, as befits a painter of grids. But here and there, particularly towards the end of her life, the titles indicate something else entirely: “Blessings”, “Gratitude”, “I Love the Whole World”, “Happy Holiday”, “Homage to Life”. Clearly, something was going immensely right. 

Agnes Martin Tate Modern, London, to October 11th