To many artists, he was one of the most inspiring figures of the 20th century. To some musicians, he is underrated: branded, unfairly, more important as a thinker than a composer. And to a large segment of the public, he’s a charlatan: a man who convinced some people that sitting onstage in silence for four minutes and 33 seconds could be construed as performing a work of music.

John Cage — composer, philosopher, visual artist, mushroom enthusiast — would have been 100 years old on Wednesday. This week Washington, usually somewhat conservative in its musical tastes, is challenging its own image with an eight-day celebration, opening Tuesday and spread throughout some of the city’s flagship arts institutions, that may be the largest Cage centennial in the country.

The festival honors Cage the polymath, touching on every aspect of his creativity: theater pieces, visual arts, dance, as well as music. Certainly it will include plenty of Cage’s music: more than 60 Cage works, many of them performed by heavy-hitting musicians who worked closely with him (Irvine Arditti, Margaret Leng Tan). But a look at the participating venues says a lot about Cage’s legacy — and about Washington. The National Gallery of Art, the Phillips Collection, the Corcoran, and the Hirshhorn are hosting shows, concerts, lectures. But the Kennedy Center, Washington’s leading performing arts institution, is not involved. Nor is the Washington Performing Arts Society.

In short: the art world has no problem accepting and celebrating the work of John Cage. But in the classical music establishment, Cage the composer remains suspect, too modern — and one of the most underperformed figures of any major composer.

Who was John Cage? The photographs and videos show an impish figure with a big laugh that creased his eyes shut; a tall, handsome, slightly sardonic man who looked not unlike a cross between Mickey Rourke (the ravaged skin, the mischievous glint in the eye) and David Byrne, with the air of slightly shy self-possession common in those who follow unusual visions and know that many people will receive their work with incomprehension, and laughter.

Laughter was fine with Cage. Music, he once said, is “a purposeless play,” its point to help us focus awareness of the world around us. Some of his scores are sets of instructions for what seems like play indeed. A grown man blows a goose call into a pitcher of water (in “Water Walk,” 1959); or makes a smoothie, and drinks it, on the stage of Carnegie Hall, as the conductor Michael Tilson Thomas did when he presented Cage’s 1970 “Song Books” with the San Francisco Symphony in March.

But the antic streak, the joy in creation that is so fundamental to Cage’s work, has helped foster a perception of him as a kind of musical clown, devoted to poking holes in tradition and countering the high seriousness of classical technique with deliberate randomness. In the 1950s, his student Christian Wolff gave him a copy of the first complete English translation of the I Ching or Book of Changes, the classic oracular Chinese text consulted by throwing sticks or coins, and it became the major tool in Cage’s compositional process, helping him to determine everything from duration to dynamics to the placement of notes on the page.