Popular music has always had a place for eccentrics and visionaries, artists whose inventions sometimes take them to the borderlands of madness. “Frank,” a captivatingly low-key new film directed by Lenny Abrahamson, was inspired by the life and work of one such musician, a singer known as Frank Sidebottom, who flourished in Britain in the 1990s.

At one point, Sidebottom hired a keyboard player named Jon Ronson — though “hired” may suggest too conventional an arrangement — who later embellished, edited and updated his experience into the screenplay for this movie, which takes place in the present-day world of social media and South by Southwest.

Ronson’s alter ego is Jon (Domhnall Gleeson), a rock ’n’ roll dreamer living in middle-class tedium.One day, he finds himself recruited into a band led by a man in a cardigan and a large papier-mâché head.

That would be Frank, played by Michael Fassbender in a performance all the more extraordinary for his being masked and muffled. The others include the nearly silent drummer and bassist (Carla Azar and François Civil); the more overtly hostile theremin and synthesizer player (Maggie Gyllenhaal); and the manager (Scoot McNairy), who can hardly manage himself.

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Jon’s debut goes more or less smoothly and he is invited to join the others at a recording session in Ireland, which occupies much of the film’s running time.

Since the audience sees Frank primarily through Jon’s eyes, we share his image of a willfully peculiar, unorthodox genius. In Ireland, Frank is funny, charismatic and compassionate, but it is only later that we and Jon come to see how fragile and unstable he is.

Abrahamson’s main achievement, enabled by the sensitive and resourceful cast, is to find a tone that is funny without flippancy, sincere without turning to mush. “Frank” is an accumulation of memorably offbeat moments, like an album made up of B sides and lost demo tracks that you stumble across and can’t stop replaying.