Heather Mansfield of the Brunettes says that one reason she and Jonathan Bree share vocal duties in their New Zealand–based indie-pop band is because it allows for the idea that the two are arguing in song. As it turns out, Mansfield and Bree also argue on the phone. Speaking from Australia last week shortly before setting out on a North American tour that will bring the Brunettes to the Troubadour for a headlining show September 13, the pair can’t quite agree about the precise status of their relationship.

“We used to be romantically involved,” Mansfield says, “and now we’re having some time off.”

“Well, we’re still romantically involved,” adds Bree. “It’s just with other people now.”

“Wait, you said we were only taking a break,” Mansfield counters. A beat. Chuckles all around. “We’ve still got a pact between us that if we’re not married by the time we’re 30, we’ll marry each other,” says Mansfield. (Those deals always turn out well.)

Bree and Mansfield met in 1998 in Auckland at a show featuring each of their previous groups. Bree liked the sound of Mansfield’s voice (if not her band) and thought it would work well in the songs he was writing at the time. Mansfield figured, What the heck? “The Auckland music scene is very small,” she explains. “Finding someone who’s actually into half the kind of stuff that you’re into is a pretty good reason to form a band.”

Bree says the Brunettes began as more of a “recording project” than a proper band (whatever that means these days). The first fruit of their labor? A 7-inch EP that got played on student radio in New Zealand. “We sold all 35 copies,” Mansfield crows. “We thought we’d best get a live group together to get rid of the last five,” adds Bree.

The outfit’s early material adhered fairly firmly to the twee-pop tradition laid down by an international cabal of early-’80s acts.

Bree and Mansfield quickly made themselves a home at Auckland’s Lil’ Chief Records, which Bree founded in 2001 with Scott Mannion of the Tokey Tones. In 2003, the Brunettes opened for the Postal Service at a show in London and caught the ear of Sub Pop prez Jonathan Poneman, who has a “weird and cool infatuation with New Zealand,” according to Sub Pop A&R Tony Kiewel. Two years later, they landed a gig touring the U.S. with the Shins. “That changed our lives pretty much,” Mansfield admits. High-profile jaunts with Rilo Kiley and Clap Your Hands Say Yeah followed.

Last month, Sub Pop released the Brunettes’ first American full-length, Structure & Cosmetics, which is both their strongest and strangest disc yet. The album’s nine songs are still based on the familiar twee-pop model — strummy guitars, catchy melodies, rhythms guaranteed not to intimidate toddlers or grandparents (both of whom Bree says form a large chunk of the band’s audience) — but they’re also smeared with the sort of vaguely psychedelic studio gunk familiar to fans of the Flaming Lips and Mercury Rev.

Opener “Brunettes Against Bubblegum Youth” — how’s that for a mission statement? — layers Polyphonic Spree–like choral vocals over a driving garage-soul groove. Somebody should pitch “Wall Poster Star,” which features Mansfield commanding a dude to “stop acting like some modern Don Juan,” to Alicia Keys and/or Kim Gordon of Sonic Youth. Throughout the disc, Bree and Mansfield do their trademark duo-vocal thing, and they’ve honed it to an impressive degree of cutesiness-slash-perviness. Several tunes echo late-’60s/early-’70s stuff by Nancy Sinatra and the late Lee Hazlewood.

Principal songwriter Bree started work on the album here in L.A. while housesitting a friend’s Glendale pad. (In his spare time, he unsuccessfully hunted celebrity graves at Forest Lawn, which may or may not account for Structure’s slightly darker vibe.) He and Mansfield also recorded in Portland (using gear borrowed from the Shins), New York and New Zealand. Bree says his reasons for wanting to move away from the tidy three-minute pop songs of the Brunettes’ first two albums were no more elaborate than that he’d already demonstrated his ability to write tidy three-minute pop songs and fancied a new challenge. “I think it sounds — and I can say this because I didn’t write the songs — more confident,” says Mansfield. (The band’s paymasters at Sub Pop claim not to be distressed by this relatively noncommercial move. Says Kiewel, “I’m really happy that their evolution took them in a more experimental direction.”)

When it comes to talking about reimagining the complicated new material for the road, Bree and Mansfield are back to their arguing. “I know that some people will walk away from a night thinking, ‘That was a great show — it sounded just like the album!’ ” Bree says. “But I think that’s a little bit lame.”

“Plus, we can’t really do that,” Mansfield admits.

“Well, yeah — it’s always a little rougher around the edges,” Bree says, a touch defensively.

“Oh, come on,” says Mansfield. “The guitar solo is usually a one-off.”

“That’s true,” Bree acknowledges. “We’re big on charm, though.”

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