Vesuvius in Eruption, with a View Over the Islands in the Bay of Naples, c.1776-80 (detail) by Joseph Wright, which ‘effectively paid his rent for a time’. Photograph: Tate, London 2013  Joseph Wright arrived in Bath from his native Derby in November 1775 convinced that the spa town would provide him with a lucrative living as a portrait painter, just as it had done for Gainsborough before him. After all, what else had its idle pleasure-seekers to do with their long afternoons? But it was not to be. Wright's immense meticulousness and the bracing demands of his new clients – mostly visitors, they required near instant gratification – were never going to be a match made in heaven. By January of the following year he was complaining to his sister Nancy that he had "not had one portrait bespoke". Even when Bath society did finally yield a commission, he found himself exasperated. "The Duchess of Cumberland is the only Sitter I have had, & her order of a full length dwindled to a head only," he wrote to his brother, Richard. "The great people are so fantastical and whimmy, they create a world of trouble."  Wright's 18-month stay in Bath was, then, neither a particularly productive time nor a particularly happy one, and it's for this reason that the Holburne Museum's dinky survey of it is carefully subtitled Bath and Beyond (several of the works on display were completed during the spa's quiet season when the artist, worried about his expenses, headed home to Derby, and still others after he'd left it altogether). Nevertheless, it had its moments. For one thing, Wright broadened his love and understanding of landscape painting in these months; with time on his hands he was able to devote himself to painting subjects discovered during his recent travels in Italy, such as Vesuvius and the annual Girandola in Rome (one of these vast oils, Vesuvius in Eruption, with a View Over the Islands in the Bay of Naples, effectively paid his rent for a time, huge numbers of visitors having coughed up to enter Wright's studio in Brock Street in order to gaze on it). For another, those portraits he did manage to produce remind the visitor that while Wright was always kind to his sitters – even those flibbertigibbets temporarily resident in Bath – he was not in the business of turning out visual blandishments. Ignore their garb and stylised poses and his subjects' faces strike you as wonderfully modern. Each tells a story, and not always a happy one.  The show begins, though, with the combination of candlelight and science for which Wright is now most famous. The Alchymist in Search of the Philosopher's Stone was painted in 1771 and is included on the grounds that it may have been in his Bath showroom (it had unaccountably remained unsold) – though it's hard to imagine society swooning over this canvas as they did over Vesuvius. Thanks to the fact that our bearded alchemist has fallen to his knees in amazed reverence (he is staring at a white plume of phosphorous), it has the feeling of a religious painting – the influence of Caravaggio is clear – and I could only picture his fashionable visitors moving swiftly on. At the Holburne, however, it's a splendid scene-setter: like a curtain rising, it both demands hush and stokes expectation.  After this, several portraits. The Rev Dr Thomas Wilson and His Adopted Daughter Miss Catherine Sophia Macaulay (1776) was painted "for reputation only" – in other words, as a kind of business card – though it doesn't show; Catherine Sophia, the daughter of the controversial historian Catharine Macaulay, has a piercingly intense and searching expression on her face, as if she's hoping to discover her place in the world in the doughy chin of her new protector (Wilson was sharing his home with her). Agnes Witts, nee Travell (1776), the wife of a landed Cotswold wool-stapler, may have come to Bath for help with fertility problems, and the set of her mouth – her lips pressed together but hardly smiling – speaks of a certain stoicism, in spite of her fashionable dress. Most engaging of all is Anna Romana Wright (1776-7), a portrait of the artist's daughter aged two. Ignore the ball of fluff at her side – it's a dog – and concentrate on her face. She looks comically worried, as toddlers are wont to do, and it's lovely to see.  The Italian paintings that follow are notable not only for their sensational, even devilish, effects – fire, smoke, coal-black lava – but for their scale, any humans appearing only as ants. Later, Wright would paint Matlock Tor by Daylight (1778), a pseudo-Italian version of a beauty spot near Derby featuring freckle-sized travellers and a high limestone crag on which the late afternoon sun lingers in determined southern European fashion, even as the gloaming sets in down below. Next to Matlock Tor is Wright's portrait of his neighbour John Whitehurst (1782-3), a geologist who had shown that the origins of Matlock Tor lay in the same "subterranean fires" as the volcanoes of Naples. By now you may feel that the connection to Bath of the work is somewhat tenuous. But perhaps there is a link, for this is a marvellously plain-speaking portrait, one that reminds you why Derby, with its industrialists and its professionals, suited Wright's genius so well, and why Bath, so affected and so silly, did not.  And so to what was, for me, the exhibition's highlight: a pair of portraits painted in 1776, when Wright had swapped deserted summertime Bath for Derby. These oval oils, styled to ape sculpted cameos, are painted using grisaille – and so effectively that, from a distance, you wonder for a moment whether they aren't photographs. One is of Wright's friend, the poet William Hayley; the other of his wife, Eliza.  The story behind these two exquisite jewels is sad, for Eliza suffered from mental illness and was eventually sent away; William could not cope with her. Wright, consciously or otherwise, tells this tale for all that it's still in the future (the couple separated in 1789): Hayley looks fond but wary, while Eliza seems to be gazing at something no eye but hers can see. Oh, you can feast on this pair for minutes at a time. His wig, her pearls; his cravat, her silk. How on earth did Wright do it? The record states that these likenesses were "hasty", but the visitor sees only boundless skill and delicacy, every brushstroke a testament not only to the play of light and shade on cloth and skin, but to the artist's calmly uningratiating verisimilitude.
Joseph Wright's time in Bath was grim but, as an exhibition in the city illustrates, it was also the making of him, writes Rachel Cooke