Before she made the first of her three albums, Lykke Li was a scrimping 21-year-old Swedish transplant to Brooklyn with that number three floating in her mind.

“I woke up on a couch in New York, stoned and depressed, and I was like, ‘I need to go out and make a tattoo that has three lines,” says Li, now 28. “I didn’t know what it meant at the time, but to myself, I said, ‘I have to follow this to the end.’ ”

She got parallel lines tattooed on her wrist, then realized that they matched trigrams of the I Ching, the ancient manual of Chinese wisdom. “I really do believe in trinity, and I believe in closing a door because only then can you walk through the next.”

Li closed the door on New York, returned to Sweden and met up with Bjorn Yttling of indie-rockers Peter, Bjorn and John. Yttling produced her first two albums, Youth Novels and Wounded Rhymes, which often cushioned her coy vocals in synth and guitar hooks atop an electro-pop heartbeat. The ephemeral nature of some tracks only made the tribal empowerment tune “Get Some,” with its playful line “I’m your prostitute, you gonna get some,” seem more provocative.

But Li takes a more sweeping turn on her May release I Never Learn, which she co-produced with Yttling and Greg Kurstin (Kelly Clarkson, Lily Allen). The new album shifts focus to her voice in power ballads of wrenching heartache inspired by a self-instigated breakup. And that voice is writ large in Phil Spector-esque reverb that serves to broaden both her vocal breadth and emotional urgency.

“I wanted to step away from what I’d done before,” the singer says from Seattle on the eve of a U.S. tour that brings Li and her five-piece band to House of Blues on Oct. 3. “I didn’t want to entertain anymore, or provoke. I just wanted to be calm—and a songwriter in its purest form.”

I Never Learn may not seem savvy for the youthful pop market, but Li says it’s not about the public. “I only make it for myself, and then you have to decide whether to release it or not,”  she says, citing artists like Neil Young and Bob Dylan. “The people that I admire and respect  and who have changed me seem to do the same thing. They’re so honest that, in the process, they’re able to reach other people, too, because human emotion, heartbreak and confusion are universal.”

Li suggests a bit of an anti-pop star, given her penchant for black fashions (though she admits, “Because I’ve moved to LA, I’ll put on a white shirt when no one sees me”) and her introverted personality. “At the same time,” Li adds, “I’m adventurous and I want to try new things, if it’s eating grasshoppers in Mexico or singing a U2 song. You can’t just stay in your comfort zone.”

Yes, that’s her singing with Bono on “The Troubles,” the closing track on U2’s surprise iTunes release Songs of Innocence. “I didn’t even know if it would make the record or if I was still on it,” she says. “People change their minds a lot.”

As for the adventurous globe-trotting, it comes easily after living in places like Portugal and India with artistic parents. “It always annoyed me when people were like ‘Oh, are you a musician ’cause your daddy’s a musician?’ No, it doesn’t have anything to do with it. But I realize now that what I didn’t have to fight against was parents telling me, ‘Are you crazy? You can’t do that. You have to get a real job.’ ”

Li favored dancing until age 16, when singing and writing songs took over. “I wanted to sing because, for me, it’s the most spiritual thing I could be doing,” she says. And with the trilogy now complete, it’s her job to share. “What I did purposely on this record was I put closure to my f—ing crazy youth, my past, and I opened the door to my womanhood,” she says of I Never Learn. “We’ll see where it takes me.”

Lykke Li plays House of Blues on Oct. 3.