Above all else, Miami trio the Axe and the Oak trade in scarcity and outsider-ness. The band, whose core is frontman Sander Willig and bassist Myles Kaplan, seem to almost deliberately cultivate mystery. Their live performances are rare, spooky, spookily polished, even. Their sound is tight but twangy, full of minor-key reverb and Willig’s emotive post-goth wail.

But instead of going all psychobilly-cheese on their twist on dark Americana, the Axe and the Oak refreshingly opt for tasteful restraint. The songs slowly creep through narratives, only exploding at the right moments, and the band members themselves appear onstage in a kind of high-fashion, natty-retro vision. So are they from Miami, or another planet entirely?

As much as they are aesthetic outliers, Willig and Kaplan are longtime veterans of the Miami scene, and first linked up around age 12. “We both rode skateboards and listened to punk and hardcore music which was really, really, really unpopular at the time — the other kids would beat you up for it,” Kaplan recalls. “So all the skaters hung out together. When we say all, we mean five, maybe 10 of us tops — safety in numbers I suppose.”

Their first band together came at age 14, a hardcore act named Avoid. Willig especially put in time in a varying series of both hardcore and more artsy, goth-inspired bands after that, before dropping music at age 22. The Axe and the Oak would not appear for another 10 years, and began with a few songs Willig had written over the years with zero intention of evolving into a proper band.

“Sander had played some old songs of his at an open mic night at Churchill’s just for kicks. After that he asked us to back him up for a few shows, we didn’t even have a name. So essentially what happened was people kept asking us where we were from and if we had a record they could buy and the venue kept booking us,” Kaplan says. “I guess we figured telling them we weren’t really a band wasn’t going to cut it and it all grew from there.”

In the ensuing handful of years, though, the group hasn’t exactly been prolific. The name of the band, in fact, reflects the full-time occupations of its founders: the axe for devoted firefighter Kaplan, and the oak for woodworker and carpenter Willig. Drummers have come and gone due to similar professional constraints, until the arrival of current member Alex Luna, who, fun fact, is also a doctor. (There’s been no resulting name addition to the band because of that, though.)

To date, too, the band has only released one 7-inch, a limited release for Record Store Day in 2010. All of this has led to a scarcity factor that’s gained the band a quietly growing, devoted following among tastemaker types. But all of that is window-dressing for the Axe and the Oak. “The writing aspect is totally solitary and egotistical. We are writing songs that we would want to listen to ourselves,” says Willig. “So when we get out of our cave and play this stuff for an audience it’s flattering, encouraging, and surprising to see the positive response people have.”

Still, this Saturday’s appearance at Sweatstock marks a gentle increase in activity, especially as the musical landscape in Miami changes for the better, according to Kaplan and Willig. More shows are in the works, and new material is due out later this year. “Having a sound that is different hasn’t proven as much of a barrier as we supposed it would have been at first, [and playing at suitable venues] is getting much easier,” says Kaplan. “As South Florida’s music scene keeps growing so do the options.”

The Axe and the Oak. Saturday at midnight on the Beached Miami stage inside Churchill’s, as part of Sweatstock.

Alex Luna, Axe and the Oak, Beached Miami, Churchill's Pub, Myles Kaplan, Sweat Records, Sweatstock