Longstreth has not yet made an LP of cricket duets, but you could almost imagine it. He is a playful conceptualist whose music moves in a handful of directions all at once. Dirty Projectors’ critical breakthrough was the album “Rise Above,” from 2007, in which Longstreth tried to recreate Black Flag’s 1981 punk landmark, “Damaged,” wholly from memory. The result — a covers record riddled with the flubs and inventions of forgetfulness — was a jumble of intricate syncopations, vertiginous time changes and splintery guitar work. “Rise Above” was Longstreth’s fourth album, and his first to feature the singer and guitarist Amber Coffman, who joined the cast of musicians Longstreth relied upon to tour and record. Coffman became his girlfriend, and she proved crucial to the success of the band’s next album, “Bitte Orca,” which came out in 2009 and won Dirty Projectors a wave of new fans. Longstreth’s ideas about harmony, rhythm and arranging remained unconventional, but his songwriting grew brighter and more direct. Coffman’s voice, with its strong, clean phrasing, helped this music to pass, after a fashion, as a kind of alien pop.

At the couple’s apartment in Greenpoint, Brooklyn, domestic rituals bled into the creative process, and vice versa. When they watched Wim Wenders’s “Wings of Desire” together, Longstreth asked Coffman to write down dialogue that resonated with her. These jottings became lyrics for the band’s biggest single, “Stillness Is the Move,” in 2009, on which Coffman emulated the octave-somersaulting feats of R.&B. stars like Mariah Carey and Destiny’s Child. Accompanying her, Longstreth played a West African-tinged guitar riff that sounded like something colorful shattering. The song put Dirty Projectors at the forefront of the booming Brooklyn indie-rock scene, alongside simpatico acts like Vampire Weekend, Animal Collective and Grizzly Bear. These bands booked worldwide tours, landed prominent festival slots, licensed songs to ad campaigns and crossed over to mainstream audiences. Among Dirty Projectors’ converts was Jay Z, who sent a handwritten note asking them to join a festival he was organizing; the electronic-music star Diplo, who flew Coffman and Longstreth to Jamaica to work with him; and the French rockers Phoenix, who invited the band to open for them at Madison Square Garden.

In 2013, after a year of intense touring to support Dirty Projectors’ assured sixth album, “Swing Lo Magellan,” Longstreth and Coffman split. The album was not the commercial success Longstreth had hoped for. “You think an album’s gonna propel you forward, and then it doesn’t, at least not immediately,” says Brett Williams, the music manager who represents both Longstreth and Coffman. This shortfall, paired with the breakup, sent Longstreth into a depression. “I was super bummed,” he said. “The band and my relationship with Amber had become so intertwined that, when we broke up, it felt like everything that had defined my life for a decade was suddenly gone.”

Gradually he began making music again “to try to work through what I was feeling.” The first song he released was the sparse, glitchy “Keep Your Name.” In writing the lyrics, he drew on a time-honored country-music convention: “It’s a divorce song,” Longstreth said. The opening line is “I don’t know why you abandoned me,” and he soon drops a knotty clue about fissures between him and Coffman: “What I want from art is truth — what you want is fame.” In building the vocals, meanwhile, Longstreth digitally lowered the pitch of his singing in a nod to DJ Screw, a Houston hip-hop innovator who wrung from this effect a narcotic quality that Longstreth was curious to explore in his song about heartbreak. The single, smoldering with recrimination, offered the first outward sign of turbulence within the group. Two weeks after its release, last fall, Coffman put out her own single, “All to Myself,” about finding solace in solitude. It emerged that she had an album of solo material due in 2017. Conjecture filled blog posts and comment sections. What was going on with Dirty Projectors? What did these dueling releases mean?