If there's one band that treads the shared ground of adventurous instrumental rock, sweltering post-rock, bumping jazz fusion, and percussive minimalist classical, it's Tortoise. For the last 20 years the Chicago-based five-piece has been exploring those crossroads, and as to which camp the band officially resides in, it's best to just go along for the ride.

Tortoise's songs are both coolly calculated and explosive. Some seem to be circular reflections on rhythm, while others descend quickly into searing solos. A smooth bass groove slides into glassy keyboards and guitar arpeggios that hang in the air like digital haze, and then there's the percussion - shakers, marimba pulses, two full drum kits going in full-steam tandem. Band members are quick to trade instruments as the songs demand, or even drop out for a few bars to open up the sound.

"Our approach to music is unconventional," says guitarist Jeff Parker, "but we all consider Tortoise a rock band."

Tortoise's unique rock stew touches on a slew of creative influences, but the touchstones all fall short. The band's latest LP, 2009's "Beacons of Ancestorship," is a wide-ranging work that only scholars and obsessives could conjure, so it's not a stretch to imagine Tortoise's free time spent crouched over an obscure LP or a recording console. Keeping tabs on their constant side-projects and collaborations can be a dizzying task.

In addition to Tortoise duties, bassist Doug McCombs fronts his own sonic experiments and plays with his long-running alt-rock band Eleventh Dream Day. On one side-project, percussionists Dan Bitney, John Herndon, and John McEntire collaborate as a hip-hop beat-making team simply called Bumps. McEntire, in particular, is known for his prolific work in the indie sphere, recently recording Canadian collective Broken Social Scene's newest album, which managed to pick up a touch of Tortoise's trademark jazzy sound.

As a group, Tortoise has been working on a piece commissioned by the city of Chicago, set to be debuted in late July. Only a month away from the premiere, it's still in the brainstorming phase, waiting for the band after its current tour wraps. But Parker doesn't seem frazzled by the hectic schedule.

"All of us individually have been coming up with ideas for that, and we start rehearsals in a couple weeks," says Parker. For that piece, the ensemble will add a number of Chicago jazz musicians to the mix: reed player Ed Wilkerson, alto saxophonist Greg Ward, flutist Nicole Mitchell, pianist and synth guy Jim Baker, and cellist Fred Lonberg-Holm. What does writing for those new hands mean for the band?

"Honestly, I have no idea what it's going to sound like. People keep asking me, and nobody knows," says Parker. "The city is basically paying Tortoise to make a new album, which is really, really cool - we're going to record that after we do the concerts."

Even if not for the commission, Chicago has proven to be a fertile home for the band. "We've all been in Chicago for such a long time. I think it's pretty hard for us to separate ourselves from our experiences in the local community - it's such a big part of who we are," says Parker, who moved to the Windy City after college in Boston. "It was 180 degrees different from [Boston], in terms of being grassroots and trying to make a living playing music, just simple stuff, being able to walk into a club and ask the person behind the bar for a gig."

When the group plays this year's All Tomorrow's Parties New York - the United States edition of a UK-based fest for indie-rock aficionados, hosted in a cozy Catskill resort - it will see quite a few familiar faces from the Chicago scene its members has helped to nurture. One act, Papa M, is the stage name of former Tortoise member David Pajo (who coincidentally just became a touring member of the band Interpol, also in Rochester this week). Legendary Chicago producer Steve Albini's post-punk group Shellac serves as the house band, and Albini also runs a card table on the side. It'll be Tortoise's third time at the fest.

"Playing at ATP, it's awesome. It's just like a big party. Everyone wants to do well. No one wants to play a shitty show at ATP, especially if all your friends are watching," says Parker. "It's a friendly, competitive vibe. It's a nice feeling, playing with super die-hard music fans."

Tortoise is clearly among Chicago's indie-rock royalty, but their connection to the city's musical vanguard goes back even farther. Since the 60's, Chicago's AACM (Association for the Advancement of Creative Musicians), a forward-thinking jazz non-profit, has been a driving force in the art form's continued development. Parker, a card-carrying AACM member, gigs frequently as a sideman on his off-time, playing with the likes of drummer Brian Blade, Chicago Underground Duo cornetist Rob Mazurek, Chi-town sax pillar Fred Anderson, and more. It was his workaholic, jazzing ways that actually got him the gig in Tortoise (he joined the group before their third album, "TNT," in the late 90's).

Parker recalls, "I used to play with this group out of the AACM called the New Horizons Ensemble, and Johnny Herndon was a fan. I went into this place where Johnny used to tend bar and he said [that] we should get together and play sometime...and we got together and played and became friends, and I heard Tortoise and they started asking me to sit in on some of their shows whenever they would play in town. I just kind of worked my way into full membership in the band from there."

Even with those entrenched jazz and rock roots - and frequent booking at jazz fests around the world - the group maintains its ambitiously omnivorous approach and working band attitude. "We prefer to play in rock clubs, you get better energy from the crowd," Parker says. "But jazz festival gigs pay better."

Tortoise

w/Sulaco

Tuesday, June 22

Bug Jar, 219 Monroe Ave

9 p.m. | $18 | 454-2966, bugjar.com