Though he has absolutely no reason to be, Burt Bacharach is exceedingly modest. "How many American composers can say they've had a song recorded by the Beatles? I'm very proud," he says, introducing Baby It's You (recorded not only by the Beatles, but by the Shirelles, too). Proud? With a catalogue that places him and his late writing partner Hal David in the very top tier of 20th-century pop songwriters, Bacharach might be said to have done the moptops a favour.

Yet he doesn't see it that way. This show, performed with an orchestra and three singers, is full of self-deprecating yarns about failed attempts to "maybe write rock'n'roll", and the drudgery of having to turn out novelty tunes such as 1958 track The Blob. He wrote the much-loved Casino Royale score, but feels compelled to explain it was the 1967 original, not the 2006 remake.

But if Bacharach seems a bit aw-shucks about his achievements, it's clear that the 86-year-old singer-pianist, dressed down in blazer and trainers, cares a great deal about the music. Frequently rising from the piano stool to conduct the orchestra with peremptory hand gestures, he comes across as the caretaker of his own legend.

For much of the gig, he's inconspicuous, letting the vocal trio take over most songs. Big mistake: though they're fine – no mortal could defile Anyone Who Had a Heart, or Walk on By – they're faceless, and long stretches of the show feel like Burt Bacharach Night on X Factor. When Bacharach himself takes over vocals, though, rasping like an exhausted playboy on Alfie and The Look of Love, the mood changes, and the songs' bittersweet magic is revealed. This isn't so much a gig as a primer in composing genius; see him while you can.

• Until 26 July. Box office: 020-7960 4200. Venue: Royal Festival Hall, London. Then touring.