With Parker backed by his touring band, Tame Impala became a headliner at festivals worldwide; coming dates include Primavera Sound in Spain, Glastonbury in England and Lollapalooza in Chicago, along with Madison Square Garden shows on Aug. 21-22. To his own surprise, he found himself turning into something like an arena-rock frontman.

“I used to hunch over and cover my face with my hair and stare at my toes the whole time — to a room full of 200 people,” he said, chuckling. “But if I’m not going to perform, no one is. I’ve realized the value in me being an extrovert, or an exhibitionist, on stage. That it’s less about me and more about everyone in the audience.”

“And there’s something about 10,000 people all together, everyone screaming lyrics about, ‘Why am I a loner?’” he added. “A lot of other stadium shows are about having a great time and screaming ‘Let’s party!’ There’s a strange kind of togetherness about 10,000 people getting together and screaming about the opposite. Music for me has always been such a solitary, intense personal experience. I never expected to be able to have that experience in front of 10,000 people.”

Back in November, Parker played me 10 tracks in various stages of completion, most awaiting vocal tracks and lyrics. Some had frisky grooves hinting at Stevie Wonder and Michael Jackson; one reimagined the grandiose buildups of 1970s power ballads.

One became “Borderline,” the only other new song Tame Impala has released besides “Patience.” It is filled with questions — “How was I to know this dark emotion?” — and muses about being “caught between the tides of pain and rapture,” mirrored in music that layers bouncy house-music piano chords with the beat of a rock ballad and a wistful, minor-key melody. “I like the idea on this album that people will argue about what it sounds like, or when it sounds like it’s from,” Parker said.

Now, festival and touring season has arrived with the album still unfinished. In a phone interview last week from Guadalajara, Mexico, where Tame Impala was performing, Parker had no firm title or release date for the album. But its structure and themes had become clear to him. “It’s taken shape in my head,” he said. “When I start making songs for an album, I don’t know what each one’s role is. But by the time I’m finished, each one has a color, each one has an identity, each one has a purpose.”