If classical music of the 20th century had enclosed itself in Jericho walls to separate it from the greater musical world, then the Turtle Island Quartet is the equivalent of the horns that knocked them down.

From its founding in 1985, its members have striven to take the string quartet beyond its traditional role as an exclusively art-music entity while at the same time reasserting aspects of older practices of classical music performance.

"It is kind of a new classical music," says Mark Summer, cellist and co-founder.

The group of conservatory-trained musicians has touched many bases in is multi-decade career: folk, bluegrass, swing, be-bop, funk, R&B, new age, rock, hip-hop and Indian. Along the way, the members have collaborated with the likes of guitarist Leo Kottke, pianists Billy Taylor and Kenny Barron, The Manhattan Transfer, clarinetist Paquito D'Rivera and the Parsons Dance Company. The latest to sit in with the group is mandolin player Mike Marshall, who performs with the quartet Tuesday in a concert presented by the Pittsburgh Chamber Music Society.

In a sense, Turtle Island Quartet reflected obvious contemporary trends in music consumption simply by not excluding them en mass. It wasn't a stretch because the members already approached music with the same openness that most people do today.

"I grew up playing Bach suites and being in a rock band and then jazz, and people have grown up with The Beatles and John Coltrane," Mr. Summer says.

Combining and mixing genres didn't mean they were going for what sells. "We want people to take us more seriously than classical crossover."

The irony is that the string quartet has won Grammys in Best Classical Crossover on two separate occasions.

The other major element of the group is its reaching back beyond the traditions of the 20th century to those of earlier times, when improvisation and arrangement was common in classical music.

"Turtle Island is the first group to be a string quartet in that tradition in which everyone in the group were jazz improvisers," says Mr. Summer. "We have always been about improvisation."

The quartet dropped the word "string" from its name several years ago to emphasize that it plays music far off the standard repertoire map for string quartets.

The Turtle Island Quartet's program at Carnegie Music Hall looks at different celebrations in December, from Christmas to Hanukkah, from Diwali (Hindu Festival of Lights) to Solstice. The selections come from each member's personal traditions, but the program isn't set in stone. Improvisation and diversity clearly flow through this group on many levels.

"We don't decide what we will play until we get together," Mr. Summer says. "Who knows!"