Fourteen years ago I went to a breathtaking show of Mondrian at the Tate and picked up the audio guide by Bridget Riley.
It was a revelation, her explanation of the artist's work clear and totally committed, her discussion of individual works enlightening and imaginative.
Of all modern artists – and she has long been regarded as something of the doyenne of British abstract art – Bridget Riley is the most analytical and the most concerned with explaining her art and its antecedents. Twenty years after appearing as one of the first of the National Gallery's Artist's Eye series of modern artists and their influences among the Gallery's pictures, this ever-evolving mistress of abstract form returns with a similar display of influences and new pictures, including two large works painted directly onto the Gallery's walls.
The pictures themselves are glorious. In a light-filled, carefully lit room, a full wall of circles on one side face a collage work of serpentine forms on the other. In contrast to some of her earlier works, which kept their distance from the viewer in their cool calculation, these works invite the viewer in. The wall of circles does it by the vitality of the interlocking lines (painted by her studio) and the way that they change slant the closer you get and the more sideways you look, and the Arcadia collage opposite by the rhythm and colour of its serpentine forms. Being applied to a wall, the flowing shapes of Arcadia can break the boundaries of the picture on each edge, a point emphasised by the display nearby of the artist's recent oil Blue (La Réserve) – the reserve being the base colour – where the lithe and leaning shapes are firmly framed not just by the edge of the linen frame, but by painted oblongs along the edges.
You don't need explanations or knowledge of influences to enjoy these pictures (the wall paintings are destined to be painted over). Riley's work in recent years has a colour, a sense of spiritedness – dare one even say joy – that is in marked contrast to the formalism of her past, as illustrated by the Black to White Discs of 1962 and the Saraband of 1985, which are also hung in the room. Colour has burst out, so has movement. You can see it most freshly in her smaller studies in gouache on paper, but even in the earlier wavy work Arrest 3 and the deep and embracing warmth of her Red on Red (2007).
For all the display and articulation of older masters in the initial room, Riley's new work here reminds one of nothing so much as Matisse, also an artist conscious of his classical tradition and always in homage to his major influences. Picasso was obsessed with the masters because he wished to compete with them; Riley, like Matisse and Cézanne, because she wished to learn from them, disassemble them and then take them further. Matisse is not here because the country and the National Gallery are so bereft of works from his earlier years, but the Arcadia collage was first exercised as part of the important Riley retrospective at the Musee d'Art Modern de la Ville de Paris in 2008, which does have one of Matisse's great dancer canvases. One would have loved to have seen the collage and the dancers in proximity.
Without Matisse or Cézanne (one of Riley's greatest influences) the show does have three wonderful oil studies by Georges Seurat, also a great hero of the artist, for his monumental Bathers at Asnières (also held by the National Gallery). The studies are workings of colour and composition, rough but confident, the work of a time when colour had become a scientific subject.
Raphael's Saint Catherine of Alexandria and Andrea Mantegna's The Introduction of the Cult of Cybele at Rome are in the show because they belong to the Renaissance, when perspective and form were similarly mastered. Raphael's work, in Riley's words on the wall caption beside, is an infinitely subtle, and harmonious canvas in which "we gradually realise that we have been directed by the artist through a slowly unfolding circular movement". In Mantegna's medallion-like frieze "the two-dimensional nature of picture making is held by the all-embracing rhythm with which he binds together the extraordinarily protracted horizontals and verticals of his format".
Now you can find this a little too cerebral or pretentious for your taste, but it does tell you where the modern artist is coming from and does make you look with renewed attention at the structure of these works. Anyone going to the exhibition should read her wall captions. It's also well worth looking at Riley's 20-minute film to accompany her 1986 Artist's Eye gathering of pictures by Poussin, El Greco and Titian which is being played continuously in an adjoining room. She's a wonderful lecturer.
Her devotion to artistic inheritance and the works of the great masters, says the National Gallery, has become somewhat old-fashioned, an enthusiasm no longer felt by today's conceptual artists. That may be true. But that doesn't mean that it's going out of style. Look at how many galleries are now trying to ginger up their collections by setting modern artists' work in amongst them and inviting contemporary artists to make their pick of the old. All art, from cinema to the novel, is self-referential to a degree.
And if you want to contrast, and complement, Riley's formalistic approach to art, then the National Gallery has an admirable single-room show of works by the "Hyperrealist" artist Clive Head, entitled Modern Perspectives. The exhibition has been set up to coincide with the Canaletto show in the Salisbury wing and to give a modern slant on the Venetian artist's precise and detailed views of his home town.
Clive Head's works are in one way completely opposite to Riley's. Not for him the use of colour and line to lock in his overall composition. Influenced by pop art and photography, his pictures here of a South Kensington café, Haymarket and Leaving the Underground are all about perspective and space. Colour is used to sharpen the field of vision and to give force, not to interact with other colours. There is nothing of Raphael's "unfolding circular movement", Mantegna's integration of background and foreground or Seurat's "contrasts of colour".
And yet Head shares with Riley a highly intellectualised approach to his work and an absolute belief in the paramountcy of paint. His slices of urban life may look like photographs, but in fact are a very considered assembly of viewpoints put together, as in Canaletto, by the artist's eye.
Installation hasn't made paint redundant, not judging by the work still being produced or the number of young people crowding into these exhibitions. Nor has a sense of forebears, a homage to past masters, become "old fashioned", let alone out of date.
Both these exhibitions are free, exactly the kind of things galleries should be doing to bring fresh interest to their collections and expand the horizons of their visitors. Unfortunately, with reduced funds they are exactly the kind of expenditure which museums may feel reluctant to indulge in. They need outside sponsorship. In Bridget Riley's case, that sponsorship has come from Bloomberg and all credit to it.
Bridget Riley: Paintings and Related Work, National Gallery, London WC2 (020 7747 2885) to 22 May