Not so long ago, chamber music was among the most arcane of civilized pleasures for American audiences. "When we started," says Robert Mann of the Juilliard String Quartet, "we were lucky if there were 200 people in the audience. And most of them were émigrés from Central Europe who had moved to Buffalo or Cincinnati or wherever." But times and tastes have changed, and next week the Juilliard, America's reigning chamber ensemble, will celebrate its 35th anniversary before a packed house at New York's Lincoln Center. With more than 100 albums and three Grammy awards to its credit, the Juilliard stands at the pinnacle of its once-obscure discipline, unsurpassed in popularity or musical influence. Last spring the group performed at its first White House dinner and was stunned when President Reagan gave an off-the-cuff biography of each of the players. "From memory!" exclaims one dazzled musician. "We couldn't believe it!"

Neither, perhaps, would the classical music buffs who once regarded the Juilliard as explosively experimental. From the beginning, the group's style was more vigorous and intense than that of the illustrious Budapest String Quartet, and the Juilliard further flouted tradition by playing contemporary works as well as the classics. "We developed quite a reputation in the early days," says Mann, 61, the ensemble's founder and first violinist. "People used to say, 'If you want one of those awful new pieces, hire the Juilliard.' Then we'd play Bartók, Webern or Berg, and they'd say, 'Now don't hire them for another 10 years, because we don't want any more of those horrible pieces.' "

Only Mann remains of the original quartet. A native of Portland, Oreg., he studied at New York's prestigious Juilliard School of Music and started the chamber group with violist Raphael Hillyer and two Army buddies. During its initial year, the group, Juilliard's first resident quartet, played only nine paid concerts for $200 each. Today it makes more than 100 appearances a year, all over the world, for $6,500 a performance. Over the years a variety of celebrity musicians have performed with the group, including Artie Shaw and the late Albert Einstein.

But it is not only in concert that the Juilliard shines. Each of the four musicians—Mann, second violinist Earl Carlyss, violist Samuel Rhodes and cellist Joel Krosnick—teach at the Juilliard School, where the quartet's course is required for all string students. As a result, many of the top young chamber groups were trained by the Juilliard Quartet, among them the Tokyo String Quartet, the LaSalle, the Emerson and the Concord. "Ultimately, that is our legacy," says Carlyss. "We can help a lot of kids."

Although the Juilliard's members' styles are not uniform, the men try to run the quartet democratically, and most of the time they succeed. "Each of us plays very differently, and yet we are able to blend," says Rhodes, 40, a New Yorker who received his master's degree in composition from Princeton. A confirmed jogger, he is married to violinist Hiroko Yajima and has two children. His colleague Carlyss, 42, was born in Chicago and studied at the Paris Conservatoire as well as Juilliard. Married to pianist Ann Schein, he is the father of two girls and was briefly concertmaster of the New York City Ballet Orchestra before joining the quartet 16 years ago. Krosnick, 40, is a bachelor, born in New Haven, whose pediatrician father was once concertmaster of the Yale University Orchestra, and whose mother was a concert pianist. He is a Columbia graduate. Mann, married to Naumburg Foundation administrator Lucy Rowan and the father of two grown children, is the group's acknowledged anchorman. "Bobby is the Juilliard String Quartet," says former Juilliard School President William Schuman, "and one day, when he leaves, the quartet will go on in his image because his stamp is indelible."