A HAWK AND A HACKSAW

With Dark Dark Dark and Pillars and Tongues at Brighton Music Hall, Sunday at 9 p.m. Tickets: $14. 617-876-4275. www.worldmusic.org

Jeremy Barnes won’t mind if you call him quixotic.

In fact, he will take it as a compliment. Like Don Quixote, whose adventures in the Cervantes epic birthed the adjective, Barnes set out long ago on an oddball yet high-minded adventure as the accordionist and co-leader of the New Mexico band A Hawk and a Hacksaw.

For Quixote, it meant tilting at windmills. For Barnes, a veteran of the ’90s indie band Neutral Milk Hotel, it meant starting anew in 2000 while living in a small town in France and recording an album based on obscure folk music from the Balkan region - for the sole and simple reason that it fascinated him.

A decade later, Barnes’s venture, whose quirky name (often shortened to AHAAH) picks up on quotes from Cervantes and Shakespeare, has shown sticking power that he never anticipated. Now a duo with violinist Heather Trost, plus a shifting cast of American and European musicians, the group has put out six albums, toured widely, and found its voice in a landscape of Balkan-influenced acts that includes Gogol Bordello, Balkan Beat Box, and Slavic Soul Party!

The band’s new album, behind which AHAAH plays at the Brighton Music Hall on Sunday, is called “Cervantine.’’ Another Cervantes reference, and, Barnes says recently from a tour stop in New York, for good reason.

“Reading the book and seeing the character inspired me,’’ Barnes says of Quixote. “He will never become the knight errant that he wants to be. But he creates his own noble character and goes on a completely different path. I felt this affinity to him.’’

His quest led Barnes first to visit remote villages in northern Romania. Later, he and Trost, whom he befriended over a shared appreciation for Bela Bartok, spent two years living and playing music in Hungary. Having immersed themselves in the music of Greece, Turkey, Bulgaria, and Serbia as well, they are now weaving in Mexican brass sounds on “Cervantine.’’

One song, “Española Kolo,’’ is imagined as a tribute to the gritty New Mexico town of Española, but it’s performed with the cadence and bass lines of Serbian folk dance.

Barnes makes no apologies for his fascination with a region to which he had no prior connection. And he isn’t shy about the effect his travels have had on his conception of a musician’s work.

“I love Eastern Europe,’’ he says. “It isn’t just the way they play music. It’s also the way they interact with each other. A lot of people think this music is all about a drunken party, and that’s not true. The musicians I was with were family men who also farmed, they had cows, they took care of everything, and their job was to play music.’’

“It’s a job that you do,’’ Barnes adds. “You work at your craft. You can be a goldsmith, or making baskets, or whatever. What you do is music, and you do your craft.’’

Craft is evident in all that AHAAH does. Styles like kolo or Greek rembetika are meticulously addressed. Players contribute on brass, strings, and percussions, including regional instruments like the bouzouki and riq. Some songs have vocals, sung in the original language by Trost or, on “Cervantine,’’ British singer Stephanie Hladowski.

Unlike some Balkan-influenced bands, AHAAH steers away from overt rock or electronica fusion. But there are moments - for instance, the eerie passage of tape manipulation that opens the song “At the Vulturul Negru’’- where Barnes’s indie roots show to great advantage.

Trumpeter Samuel Johnson, who appears on “Cervantine’’ and is also on the current tour, says Barnes and Trost have found ways to play within Eastern European traditions that somehow never turn into mimicry.

“This is not a repertory band,’’ Johnson says. “They’re writing original music in the idiom, highly influenced by their own experience. It seems genuine to me, because it’s honest.’’

It has certainly resonated enough to turn AHAAH into much more than the possibly one-off side project Barnes thought it would be.

“I never expected that I would be able to continue this long, or be able to tour, or all the things we’ve done,’’ Barnes says. “The ambition was and always has been minor. I don’t really want to be a popular band.’’

He’s happiest back home in New Mexico, in his home in the countryside with fruit trees and an old Spanish ditch system to irrigate his vegetable garden.

For that, too, he credits Eastern Europe.

“There’s a connection to the earth that I didn’t realize would affect me as much as it did,’’ Barnes says. “It made me go home and want to think about these things more consciously.’’

Siddhartha Mitter can be reached at siddharthamitter@gmail .com.