Mr. Shankar, a soft-spoken, eloquent man whose own virtuosity transcended musical languages, was trained in both Eastern and Western musical traditions. Although Western audiences were often mystified by the odd sounds and shapes of the instruments when he began touring in Europe and the United States in the early 1950s, Mr. Shankar and his ensemble gradually built a large following for Indian music.

A Beatle Was Intrigued

Western interest in his instrument, the sitar, exploded in 1965 when Harrison encountered one on the set of “Help!,” the Beatles’ second film. Harrison was intrigued by the instrument, with its small rounded body, long neck and resonating gourd at the top, and its complexity: it has 6 or 7 melody strings and about twice as many sympathetic strings, which are not played but which resonate freely as the other strings are plucked. He soon learned its rudiments and used it that year on a Beatles recording, “Norwegian Wood.”

The Rolling Stones, the Animals, the Byrds and other rock groups followed suit, although few went as far as Harrison, who recorded several songs on Beatles albums with Indian musicians rather than with his band mates. By the summer of 1967 the sitar was in vogue.

At first Mr. Shankar reveled in the attention his connection with popular culture had brought him, and he performed for huge audiences at the Monterey International Pop Festival in 1967 and at Woodstock in 1969. He also performed, with the tabla virtuoso Alla Rakha and the sarod player Ali Akbar Khan, at an all-star concert at Madison Square Garden in 1971 that Harrison had organized to help Mr. Shankar raise money for victims of political upheaval in Bangladesh.

But his reach went much further. He composed for films (including the score for Richard Attenborough’s “Gandhi” in 1982), ballets, electronic works and concertos for sitar and Western orchestras. As his popularity spread, societies for the presentation of Indian and other traditional music began springing up — the largest one in New York is the World Music Institute — and a thriving world music industry was soon born.