The jihadists' rules were clear: No smoking. No drinking alcohol. No dancing. No singing. No music, period.

Singer Aliou Touré, bassist Oumar Touré and guitarist Garba Touré knew that violating the interpretation of shari'a law imposed by the brutal new rulers of northern Mali could mean public whippings, amputation of their hands—or worse. They'd seen it happening all around them since the radical Islamist movement Ansar Dine arrived in 2012.

So the young trio joined the tide of refugees fleeing to relative safety in Bamako, the capital city. They acquired a drummer, conservatory student Nat Dembele, and a name, Songhoy Blues—derived from the 14th-century Songhai empire that once ruled the Sahara, and the music that originated in Mali centuries ago.

"We talk about peace and war, the situation of Mali and politics there," says Aliou. (Despite sharing a surname, the three Tourés are not related.) "So many people don't have a microphone. We are lucky to have a microphone. The music is our way to communicate."

Songhoy Blues fought homesickness by playing its northern music and quickly found success in Bamako's lively club scene, even winning fans from rival ethnic groups like the Tuareg. In 2013, the band members learned that Blur frontman Damon Albarn's Africa Express project was recording in Bamako, auditioned, and so impressed Albarn that they not only scored a song on the compilation album but opened for Africa Express in London the following year. The quartet released its debut album last year, and recently became the first African band the august Atlantic Records label has signed in decades.

Growing up in Mali's musical melting pot of many North and West African cultural seasonings, the band members, still in their 20s, also had access to global influences their revered Malian predecessors—Salif Keita, Toumani Diabaté, Ali Farka Touré—lacked. While sizzling with the bubbling, trance-inducing electric guitar lines of famous Malian bands like Tinariwen, Songhoy Blues' Music in Exile, co-produced by the Yeah Yeah Yeahs' Nick Zinner, varies the traditional desert-blues slow grind with a potpourri of song forms and rhythms, including funk and reggae.

"We are lucky to discover all the music around the world," Aliou says. "With the Internet, now you can hear everything from the U.S. and Europe, and that has given us new ideas to [mix] Malian music with different kinds of music from around the world. Twenty years ago, when Ali Farka made music, they didn't have any Internet—they didn't have any idea about the rest of the world."

As likely to sport sequins, jeans and cowboy hats in concert as traditional keffiyehs, the members of Songhoy Blues have cited influences from B.B. King and John Lee Hooker to Jimi Hendrix and contemporary hip-hop, and are accumulating more with every touring mile. "Travel is the best school," Aliou says. "Hearing different kinds of musicians in different countries gives us new ideas for our music."

But while Songhoy Blues dabbles in the sounds and style of Western pop, with Mali still in turmoil, its lyrics are not likely to address pop music's usual subject matter.

"We can't talk about love yet because we are missing love in our country," Aliou says. "We hope to speak about it someday."

SEE IT: Songhoy Blues plays Doug Fir Lounge, 830 E Burnside St., with Little Star, on Monday, April 4. 9 pm. $13 advance, $14 day of show. 21+.