The Weekend's Viewing: True Love, Sun, BBC1The Great British Story: a People's History, Fri, BBC2

"I should be paid as a writer," David Tennant told a Radio Times interviewer recently.

It was a joke, a reference to the fact that his leading role in True Love didn't come with a line-by-line script but a storyline only, supplied by the director Dominic Savage and then filled out in improvisation by Tennant and his fellow actors. Which still leaves open the question of how much you'd pay him as a writer, and what this way of working actually delivers to such an enterprise. And initially, anyway, it seemed to deliver a low-key truth. The first of five short dramas, to be broadcast on consecutive nights this week, True Love began at a family birthday party, with Nick toasting his "long-suffering wife", Ruth, and Tennant nicely capturing the tone of such an occasion. The bedroom scene that followed felt convincing too, warmly intimate and offhand.

But then Nick got an unexpected call at work from an old girlfriend, Serena (Vicky McClure), his face clouded and Roberta Flack piled in on the soundtrack to hint (if "hint" is really the word for such a thumping nudge) that there might be trouble coming. Back from Canada after 13 years, Serena wants to go back to a big fork in the road and take the other turning. Nick isn't as impervious to this suggestion as either Ruth or he might hope. And for the moment improvisation continued to deliver. After Ruth spots Serena walking down the prom in Margate (all five dramas are set in Savage's home town) she starts to worry, but can't yet name her dread. Savage constructs a scene that consists entirely of two people not saying what is on their mind, and the halting commonplaces of their exchanges only increases the tension.

What followed, though, was less subtle, and you'd have to lay at least part of the blame at Savage's door. Lovers might not care who sees them kissing in the street, but adulterers surely do, and Savage's desire to silhouette his two leads against a Turner sky takes them out of the realm of messy reality and into something more simplistic. The script followed. When Nick talks about his dilemma with a friend he expresses himself with a directness that doesn't seem to acknowledge the level of self-deception in a betrayal. There's a good moment in which he first tells a direct lie to Ruth, electric with false reassurance, but his scenes with Serena fall back on clichés of romance, again endorsed by Billie Holiday crooning "I'm a fool to want you". "Let's just go somewhere else and start again," says Nick, after an afternoon tryst in a local hotel, and you find yourself wondering whether he's forgotten that he even has children.

The last lines distilled the problem of this way of working. When he sees Serena off on the train, after staying with her all night, she says simply, "It'll be OK." Back home, cradling his wife, Nick says just one word, "sorry". And while the ambiguity of these moments is clearly intentional, you have no real way of knowing whether the inadequacy of those words, how far short they fall of the feelings they express, is a failure of invention or a triumph of art. Had a writer's name been on the script, though, I think there would have been some muttering about the quality of the final draft.

The title sequence for The Great British Story: a People's History is awful: Michael Wood's voice at its most rhapsodically soaring and sound and images that boost the idea of our proud island destiny. But for the trade union banners you could rerun it as a Ukip political broadcast. Happily, the programme that follows it is actually very good, full of local enthusiasm for history and full of the most exciting things a history programme can contain, ancient documents and ancient records. Come in a minute late and you'll be fine.

Twitter: @tds153