Uneasy is the head that wears a much bigger, papier-mâché head.

That’s the most noticeable oddity about Frank (Michael Fassbender), mad-genius leader of an experimental punk band and the focus of a quirky, utterly engaging film that manages to be amusing, dark, surprising and poignant without wasting a single one of its 95 minutes.

Promising up-and-comer Domhnall Gleeson (son of renowned Irish actor Brendan Gleeson) plays the film’s protagonist Jon Burroughs, an office drone and aspiring songwriter anxious to make his break. He lucks into a gig with Soronprfb, Frank’s unpronounceable band, after witnessing its current keyboard player try to drown himself in the ocean. Jon is immediately sucked into the odd, compelling aura of Frank, who sings with much more bravura than he projects offstage, and a gloomy clutch of bandmates including gruff Clara (a very scary Maggie Gyllenhaal), who has designs on Frank, or at least on controlling him, and sad-sack manager Don (Scoot McNairy, impressively stretching his range).

The band rents a house on a remote Irish island for a recording session, where they create their own raggedy mini-bohemia with Frank at its core, devising a new musical-notation system as he swans about in the retro cardigans and muscle tees he favors (if we can’t see Fassbender’s face, at least there’s his arms). They build new instruments out of old chairs, wire and drinking glasses as Clara barks out encouragements like “Play something!” and “Someone should punch you in the face.”

Frank, who Jon observes “finds inspiration in everything,” insists on a regimen of roughhousing exercise (the safe word when things get out of hand is “chinchilla”). Musical interludes range from the band’s usual atonal cacophony to Jon and Don’s wince-worthy attempts at soulfulness to a startlingly sincere and mellow ballad from Frank.

Frank’s charms are effective on strangers as well. When a German family shows up because they have rented the house after the band’s time has run out, Frank spirits the mother away for an impromptu dance across a field. She returns and instructs her husband to drive away, calling from the car window to thank Frank for showing her “my whole new truth.”

Meanwhile, Jon has built a small but steadily growing fan base via Twitter, and after posting a video of one of the band’s new songs, they get invited to play the ultimate alt-rockfest, South by Southwest in Austin, Texas. After a sudden tragedy devastates the group, casting a permanent pall on their idyll, they decide to press on and travel to the States, where Frank’s reception is mixed, leading to mishaps and revelations.

The character of Frank is partly a takeoff on the late Chris Sievey, a real-life English musician and television comic of the ’70s and ’80s. Sievey led a band called the Freshies before adopting the comic persona of Frank Sidebottom, who wore a very similar giant fake head. But he’s also a symbol of the blind adoration so easily inspired by people blessed with creative brilliance and charisma for whom an undercurrent of mental illness only serves to enhance their appeal.

So does Frank ever remove that ponderous noggin of his? To let on would be a spoiler in my book, and almost beside the point — either way is bound to satisfy on some level and disappoint on another.