Singer-songwriter Molly Rankin serves her final cacciatore salami pizza of the day and makes her way across the street to a coffee shop for an interview with a music journalist. The Rankin Family scion has not only graduated from mopping washrooms at a Toronto pizzeria to serving customers, her music career is taking off as well. Her band, the indie outfit Alvvays – pronounced "always" – has just released a self-titled debut album of droning, jangling, beachy guitar-pop that's receiving attention from high-profile media outlets such as Time magazine and others that see in her It Girl potential.

At the coffee shop, Rankin leafs through a newspaper and sips from a small bottle of pineapple juice. The music writer arrives and congratulates her on getting the album streamed on the National Public Radio website in the United States. Rankin doesn't react. It's a big deal, she's told – to which she responds, "Is it?" To the surprise (but not to the offence) of the journalist, she then goes back to reading her paper, continuing to do so for the first few questions of the interview. When one grows up watching relatives accepting Juno Awards on television, it is hard to get worked up over a little media interest or radio play.

"It's been very gradual," the unperturbed-but-pleasant Rankin, 27, explains. "We've been slowly building something for a while." While the rest of the world sees the ascent of the Toronto-based Alvvays as an overnight arrival, the album is nothing monumental to Rankin. It was recorded more than a year ago, originally available on cassette as something to give to festival bookers as a demo and something for concert-goers as a souvenir. "It's been floating around for a while," the unexcitable singer says.

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Some of the Rankin/Alvvays buzz was created by the outspoken Stars singer Torquil Campbell, who, on the CBC Radio One show Q, proclaimed the song Archie, Marry Me (a fuzzy, droning nineties-styled tune that dryly comments on marriage as a goal) as his song of the summer. Rankin does not write autobiographically, and while her high-school friends are pursuing marriage, picket fences and jobs with pension potential, she craves a different kind of buzz. "It's an open assessment on the idea of marriage," she says, when asked about the tracks Archie, Marry Me and Adult Diversion. "Archie specifically is about this thing that you become an adult and now you should get married. Unfortunately, the love that gets no attention is the wild kind of attraction that leaves you in jail together. Nobody talks about the adventurous kind of love."

As it turns out, Rankin doesn't talk about her particular kind of love either. She has been linked romantically with guitarist Alec O'Hanley (her bandmate in Alvvays and formerly of the split-up Charlottetown power-pop troupe Two Hours Traffic), but declines comment when asked about the relationship. "We are in a band together," is all she says.

The band she gets asked about often, besides Alvvays, is, of course, the Rankin Family, the Celtic music-playing East Coast institution. She's a lapsed fiddle player and the daughter of the late John Morris Rankin, a fiddler and pianist with the Cape Breton act who died tragically in 2000 when the truck he was driving plunged into the St. Lawrence.

When she was quite young she played with her father at parties, and was the inspiration for one of her father's instrumentals, Molly Rankin's Reel. And while she participated in the Rankins' first reunion tour in 2007, this reel deal never had any intention of being in the family business. "I wanted to be a veterinarian when I was young," says Rankin, who studied theatre at Dalhousie University. Though she released a solo EP under her own name in 2010, she prefers the relative anonymity of being in a band today. "I'd rather not use my name. I don't want people leaning on any reference on what they think I should sound like."

With the interview complete, Rankin points out the restaurant she works in, across the street. Given her rising indie-pop status, has she given her two weeks' notice? She has not. "I'm on the schedule for the next two or three months at least," she says.

With that, Rankin and the writer part, with the former heading home and the latter beelining it to the pizzeria for dinner. There he chats with a bartender, who excitedly says that the staff all aspire to be something else and that they are all proud of Rankin's success. "Her album," he says, as fired up as the pizza oven itself, "is on NPR."

Well, you don't say.

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Alvvays plays the Wolfe Island Music Festival, Kingston, Aug. 9; the Horseshoe Tavern, Toronto, Aug. 14; La Vitrola, Montreal, Aug. 16.