For most of its dozen years, the Bad Plus has been a splendidly self-contained trio knitted together by its members' idiosyncratic sensibility.

While ostensibly a jazz group, drummer Dave King and bassist Reid Anderson (born and raised in the Twin Cities) and pianist Ethan Iverson (originally from nearby Menomonie, Wis.) have never shied from unconventional song choices, be it Nirvana's "Smells Like Teen Spirit" and Gloria Gaynor's "I Will Survive" early in their career, or their just completed European tour playing Igor Stravinsky's "Rite of Spring." Their original tunes likewise tend to tweak tradition through elements of humor, odd meters and a blend of sprawling rhythms and nerdy, polymath precision.

"We speak a tribal language that I think is based on coming from the same soil," said Anderson by phone from his New York home last week. "The Twin Cities is this culturally aware and vibrant, supportive community, but at the same time removed from New York. Growing up, we had to form our own ideas about how we were supposed to play jazz and what it can be, and we value that independence."

More recently, however, the trio has broadened its intrepid sense of adventure to include collaborations. When the three take the stage in St. Paul's Mears Park next Saturday as headliners of the Twin Cities Jazz Festival, they'll be joined by renowned tenor saxophonist Joshua Redman. It may seem an unlikely, if high-powered, pairing. Redman, who had a major-label contract at age 24, is a cavorting, bop-drenched stylist with a more mainstream approach.

"We like that surging thing," assured pianist Ethan Iverson in a separate call. "We'll take care of the math if Josh wants to play that beautiful, passionate saxophone."

Iverson first saw Redman play on a gig with his famous father and fellow tenor man Dewey Redman back in 1991. But a more potent memory occurred when Joshua performed at Dewey's memorial service in 2006. "He played a solo in this church that was one of the most powerful things I've ever heard. He should make a solo saxophone record."

In the spring of 2011, when New York's Blue Note jazz club asked the Bad Plus to include a special guest as part of its weeklong engagement celebrating the club's 30th anniversary, someone -- neither Anderson nor Iverson remembers who -- suggested Redman. "And everyone immediately said 'Great!'" Iverson recalls.

The shows went well enough that the trio and Redman reprised the collaboration at a jazz festival in Austria last August, and made plans for a three-week tour of Europe this summer. The St. Paul gig and another at the Toronto Jazz Festival this weekend will be the only North American dates before Europe.

"Josh is just a professional of the highest order," Anderson said. "He came in knowing our music to the point of memorizing it, with great positive energy."

Even so, some accommodation was necessary. The group picked songs in its repertoire with the most conventional chassis, including Iverson's blues-oriented "Guilty" and Anderson's "Prehensile Dream."

"We've done some things with guitarists, like Kurt Rosenwinkel and Bill Frisell, and we did a terrific full-scale project with vocalist Wendy Lewis [the all-covers album "For All I Care" in 2008], but Josh is our first saxophonist," said Iverson, who acknowledges that adding a horn to a piano trio is "a big shift." But as a student of jazz history, he said, he relishes not only engaging with Redman, but laying out and hearing the saxophone with just bass and drums, as Sonny Rollins and John Coltrane did in some of their most iconic recordings.

As for the Bad Plus, Iverson said it has a new record, "Made Possible," due Aug. 31. It's the first of their records to prominently feature electronic effects. "And after three solid weeks with Josh in Europe, I'd like to record, if we can work it out," he said. "We'll see what the future brings."