Jon Burroughs (Domnhall Gleeson) goes around the beach trying to find inspiration for writing music. He recites lyrics in his head based on what he sees that catches his attention - a lady in a red coat, for instance. He then decides to just go home and tweet about his food.
On another trip to the beach, Jon sees a man trying to drown himself in the ocean. He is taken away by two paramedics. Nearby is a van for a band called The Soronprfbs. Their manager Don (Scoot McNairy) explains to Jon that the man, Lucas (Shane O'Brien), is a keyboard player for the band. Jon states that he plays the instrument, so Don invites him to join the group for a gig later that night.
Jon joins the band at a club, meeting the abrasive and foul-mouthed Clara (Maggie Gyllenhaal), Baraque (Francois Civil), Nana (Carla Azar), and their singer, a man wearing a papier-mache head named Frank (Michael Fassbender). Everybody starts playing, and while it sounds upbeat and fun, Clara plays too hard and breaks her instrument. She starts yelling at one of the other bandmates. Jon is left rather stunned after this.
Sometime later, Don calls Jon to invite him to join the band, since Frank said he brought something cherishable to the group. Though Don admits that he probably couldn't hear Frank under the giant head, so he may have said "perishable". Regardless, Jon seizes the opportunity to be part of an actual band.
The band brings Don with them to a private cabin in Ireland where they will begin recording their next album. Baraque and Nana clearly do not like or trust Jon, though Jon cannot tell since Baraque only speaks in French. At one point, Frank's fake head catches fire, which Clara puts out. Jon asks Don in private what the deal is with the head. Don says Frank never takes it off and nobody has ever seen him without it. He eats through a tube and Jon wonders what he must look like underneath it. Don also tells Jon that Frank was institutionalized earlier.
Before any recording is done, the band spends their time writing or hanging out. Frank puts the group through physical exercise, shown when he engages in a fighting match with Baraque, who yells "Chinchilla!" as the safety word. Jon documents the experiences on Twitter and Tumblr. Jon later tells Frank that he writes and composes his own music, so Frank calls everybody in to hear some of what Jon has written. Nervous, Jon starts playing but keeps stopping and making excuses.
At night, there is commotion outside. Frank is trying to comfort Don, who is attempting suicide. Don runs toward the lake, but Clara knocks him down with a stick. Jon asks Frank about this, mentioning Don's lack of interaction with real women as opposed to fake ones. One afternoon, Don comes in while Jon is playing the piano. Don has also tried to write music, but he thinks his stuff is no good. He wishes to be as good as Frank, though he feels he'll never get that far. He plays a piece for Jon, which Jon says is great.
One day, while the band is playing, a German family of three comes to the cabin. Don admits that they've spent all the rent money and were given 30 days to leave. Frank goes to talk to the family, being fluent in German, and Jon offers to use money from his inheritance to support them.
Almost a year passes, and the album has yet to be recorded, because according to Frank, they need to perfect every detail of the music. Jon has considered leaving, but he sticks around because he feels he's found his muse. He writes a full song and plays it for Frank, who says it's amazing. When they finally get around to recording the album, Jon tweets that the band will not use any of his compositions, which he is rather upset about. When the album is complete, the band celebrates, save for Don who is chugging a drink by himself.
The next morning, Jon steps outside to have a cigarette when he sees what looks like Frank's body hanging from a tree. The other bandmates run and bring the body inside, when they see the actual Frank walking in. Baraque removes the head from the body to reveal Don. The band holds a funeral for him and puts him on a pyre, sending him down the river after lighting it.
Sometime later, Jon gets responses from his social media activity. Since he's put a video of the band playing on YouTube, they have been invited to play at the South By Southwest Festival in Texas. Jon shows Frank the video and sees that over 23,000 people have seen the video, to Frank's astonishment. Clara later confronts Jon while he's in the hot tub about this, thinking he's playing at something for his own gain. He stands up, and she sees him nude. They have rough, angry sex in the hot tub before sitting there across from each other, having a smoke and talking about Frank. Jon realizes Clara is in love with Frank, and she threatens to stab Jon if he messes anything up in America.
On the road to the States, Frank stops and starts to spread Don's ashes across a desert from a container of Grownut. Clara grabs a handful and licks it, realizing it IS Grownut. Baraque admits he grabbed the wrong container, leaving Frank rather disappointed.
The band makes it to SXSW. They meet two girls at the registration booth, Simone (Hayley Derryberry) and Alice (Lauren Poole), who are both fans of their music. They invite them out to get pancakes. Frank mentions the amount of views they got, to which Alice says that's usually 1% in the room of people that would actually enjoy it. This crushes Frank's spirit, and he moans and crawls under the table.
Jon suggests that the band changes their sound to make more appealing and likable music. Frank is inspired to make his most likable music yet. He plays the keyboard and starts singing nonsensical lyrics. Clara dryly states that people will love it. By now, it's a little clearer that Frank is not feeling right. He leaves his room, and Jon goes searching for him at night, asking people if anybody has seen a man with a large fake head. He finds Frank and Clara sitting together in an alley. Clara gets angry with Jon and stabs him in the leg. As she walks away, she is grabbed by a pedestrian, and Jon says she was arrested.
Baraque and Nana sit by Frank's bed before telling Jon they never trusted him and have decided to leave the group. Frank decides to stay, leaving him with Jon.
Jon and Frank are getting ready for their performance at the festival. Frank is only more nervous, having put make-up on the fake head and putting on a dress. When they get up to perform, Jon starts to play his music, when Frank starts to scream and he collapses onstage. Jon asks him what's wrong, and Frank says the music is shit.
The two are left alone in a motel room. Jon is completely frustrated with Frank, while the latter is apologetic and passive. Jon reaches his breaking point and tries to pull Frank's head off. Frank runs out of the motel screaming, when he gets hit by a car. Jon runs outside to find Frank's broken head, but nobody else there. He asks the driver if he saw a man running, and the driver says Frank was limping. Jon then walks and is hit by a car himself.
Jon recovers at the hospital and tries to find Frank through the police, but since he can barely begin to describe Frank, he realizes it's pointless. He asks social media to help him find Frank. People send rude or misleading answers, until Jon finds Frank in Bluff, Kansas, Frank's hometown. He has gone to his parent's home, and Jon finally sees Frank without his mask. He has a scar across his forehead. Frank's parents tell Jon that Frank has a mental illness, and that the thing with the mask started when he was 14, and they regret playing along and encouraging him to wear it. Jon sees Frank trying to play, though Frank has already lost the will and spirit to keep doing it.
Jon brings Frank to a bar where he saw Clara, Baraque, and Nana playing. The three of them are quick to realize who Frank is, even without his mask, and they all start to play a song, "I Love You All", with Frank singing his heart out and tearing up. Jon watches the performance from the bar, until he decides to walk away before the band finishes.
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