On a hot day in May, the city of Bilbao was shaken by a velvety roar, not unlike the sound of a jet plane waiting on a tarmac. At one end of Calle de Iparraguirre, which cuts across the town, the silver shell of the Guggenheim Museum was glistening in the sun. For a moment, it seemed as if the noise were coming from there—as if the building were about to lift off and rocket elsewhere. In fact, the source of the disturbance was a local bullring, the Vista Alegre, where a German experimental d.j. named Christoph de Babalon was testing his gear. He was the opening act for Radiohead, a five-man rock band from Oxford, England. Radiohead fans were gathered at the entrance, staring up at the sound and asking what it meant. One had on a Kafka T-shirt, and others wore shirts with a Radiohead logo, which can only be described as a demon in tears.

Ed O'Brien, Colin Greenwood, Phil Selway, Thom Yorke, and Jonny Greenwood: combining boyish riddles with sophisticated musical games. Illustration by Jaime Hernandez

A side door led to a concrete corridor, where the bulls run on an ordinary day. From there, planks reached out to a temporary stage. Christoph was in the center, eying his mixer and his CD players. With ice-blond hair and black sunglasses, he looked like a young Bond villain, but he turned out to be a friendly, chatty sort of sonic terrorist. “I am familiar with dark, small clubs,” he said after the sound check. “Now it is like I am in a gladiator film.” He mentioned some of his musical influences, which included avant-garde figures like Merzbow and the composer Morton Feldman. “Sometimes I work with beats, sometimes with layers,” he said. “Tonight I do layers.”

Popular music in the year 2001 is in a state of suspense. No one can say where the mystery train is going. On the one hand, the Top Forty chart is overrun with dancers, models, actors, and the like; on the other hand, there are signs that pop music is once again becoming a safe place for creative musicians. The world fame of Radiohead is a case in point. Having established themselves with tuneful guitar rock in the nineties, the members of this band took the risk of doing as they liked, and they discovered things about the marketplace which others had missed. Last year, they released an album titled “Kid A,” an eerily comforting blend of rock riffs, jazz chords, classical textures, and electronic noise, which, in a demolition of conventional wisdom, went to No. 1 on the Billboard chart. “Amnesiac,” its like-minded successor, came out in June and is doing just as well. Radiohead’s selling point is not their identification with any one genre but their way of ranging over music as a whole. They have intensity, intelligence, a personality in sound. In Bilbao, they were present in spirit before they played. The idea of placing a German d.j. in a Spanish bullring was Radiohead to the core—a tricky, eclectic song waiting to happen.

Christoph went to his dressing room, and the stage was empty for a while. Two shirtless old men sat on stone steps, looking as though they had not moved since Franco died. Radiohead’s gear basked in the sun. On the left-hand side was a rack of guitars—twenty-three in all. Up front, in an area set aside for Ed O’Brien, one of three guitarists in the band, was a tangle of pedals, samplers, and inch-thick cords. In the center of the stage, to be shared by Colin Greenwood, the bass player, and Thom Yorke, the singer, were various keyboards, a piano, and an upright bass. Equipment for Colin’s younger brother, Jonny Greenwood, stood on the right: more guitars, more keyboards, a xylophone, a transistor radio, a sci-fi stack of analog synthesizers, and a modified ondes martenot, one of the earliest electronic instruments. The ondes martenot is controlled by a ring that slides along a wire; fewer than a hundred people have mastered it, and Jonny is one. The only really conventional-looking apparatus was the drum set, although you could tell by some distress on the edges of the drums that the player, Phil Selway, had attacked the sides as well as the skins.

Backstage, the members of Radiohead were eating dinner. Colin Greenwood hovered over the catering table, inspecting an array of Basque dishes. He is typical of the group in that he looks nothing like a celebrity who has sold fifteen million records. He is thirty-two years old, with jet-black hair and large, kindly eyes. He is easily distracted and delighted by the world around him, favoring the words “mad,” “brilliant,” and “amazing”—the last spoken with a long, liquid stress on the second syllable. He has a habit of suddenly burying his face in his hands, as if he were sinking into despair, or falling asleep; after a moment, his face lights up again. Lavishly well-read, he can talk at length about almost any topic under the sun—Belgian fashion; the stories of John Cheever; the effect of different types of charcoal on barbecued meat—but he gets embarrassed by his erudition and cuts himself off by saying, “I’m rambling.” He is not above wearing a T-shirt that says “Life’s a beach and then you shag.” You might peg him as a cultish young neo-Marxist professor, or as the editor of a hip quarterly. But he is a rock star, with several Web pages devoted to him.

“It’s full on out there, isn’t it?” Greenwood said, looking toward the bullring, which was filling up with fans. “I’m scared.” He occupied himself by talking about “Faust’s Metropolis,” Alexandra Richie’s thousand-page history of Berlin. Outside, Christoph began to play for real, and was received in bemused silence.

An hour later, Radiohead hit the stage with a confidence that had been invisible before the show. The sound was huge, but it was awash in colors, contrasts, and detail. It was grand in effect, cool in tone, dark in mood. The set was a mixture of older tunes, from the band’s breakthrough albums of the mid-nineties—”The Bends” and “OK Computer”—and newer ones, from “Kid A” and “Amnesiac.” The old songs had choruses that the fans knew by heart, but the new ones, which have been described as “anti-commercial” in the rock press, made the crowd dance harder. “Idioteque” set off fierce rhythmic clapping, even though it was dominated by jagged beats, computer-music samples, and squawks from the old-school synths. It must have helped that the singer, when he was done singing, launched into a demonically silly dance, kicking his legs as if someone were firing a gun at his feet. It may also have helped, on an unconscious level, that love-drunk chords from “Tristan und Isolde” lurked at the heart of the song, courtesy of a Paul Lansky composition called “Mild und Leise.”

In the middle of the set, Radiohead played a song called “Airbag,” which showed why this band is taken as seriously as any since the Beatles. It was a rugged ritual, full of cabalistic exchanges, with each player taking a decisive role. Jonny started off with a melody that snaked along in uneven time—one-two-three-one-two-three-one-two—and swayed between A major and F major. O’Brien added leaner, brighter curlicues on guitar. Selway came in with a precise but heavily syncopated beat. Then Yorke began to sing, in a well-schooled, plaintive voice, an oblique account of a near-fatal collision: “In the next world war / In a jackknifed juggernaut / I am born again.” At the mention of war, Colin let loose a jumpy bass line, giving a funky spin to the hymns in the treble. The music cut through a jumble of verses and choruses, then held fast to a single chord, as Yorke fell into synch with O’Brien’s chiming lines. Just before the end, Colin grinned, leaped in the air a couple of times, and seized hold of his brother’s tune, the one that had set the song in motion. The doubling of the theme had a kind of thunderous logic, as if an equation had been solved. The interplay was as engaging to the mind as anything that has been done in classical music recently, but you could jump up and down to it.

In the old days, rock bands had a haircut, a lingo, a house style. The disconcerting thing about Radiohead is that its members do not really look or act alike. They are basically a group of smart English guys in their late twenties and early thirties. They read books, but they also check the football scores. Three are married; two have kids; all live in Oxford, where they grew up. Thom Yorke, who writes most of the songs, is compact, boyish, and impish; he has a lethally quick mind and a subtly powerful charisma. Ed O’Brien, almost a foot taller than Yorke, has the jutting jaw and floppy bangs of an actor in a period war movie; he is suave and direct and seems to have rolled in from a different posse. Jonny Greenwood, a lanky figure with unruly black hair, is more cautious than his brother Colin, but when he starts talking he excitedly involves himself in dense, Victorian sentences, biting clauses out of the air. Phil Selway is bald and sweet-faced, and talks in a gentle voice. He looks like the nice, ordinary one, but he often has a trace of a wicked smile.

How these five mildly eccentric young Englishmen became the Knights Templar of rock and roll is anyone’s guess. They are not too sure themselves. “Everyone comes to us with their heads bowed, expecting to be inducted into the mystery of Radiohead,” Selway said. He made a King Tut gesture with his arms. “We were hoist on our own petard with that. At a certain point, around 1997, we were simply overwhelmed and had to vanish for a bit. This was our honest reaction to the situation we were in. But some people thought we were playing a game, or had started taking ourselves too seriously. Really, we don’t want people twiddling their goatees over our stuff. What we do is pure escapism.”

What happened to Radiohead in 1997 was that they caught a wave of generational anxiety. The album “OK Computer,” with titles like “Paranoid Android,” “Karma Police,” and “Climbing Up the Walls,” pictured the onslaught of the information age and a young person’s panicky embrace of it. Yorke’s lyrics seem a mixture of overheard conversations, techno-speak, and fragments of a harsh diary: “I trust I can rely on your vote”; “An airbag saved my life”; “Ambition makes you look pretty ugly.” The songs offered images of riot police at political rallies, anguished lives in pretty suburbs, yuppies freaking out, sympathetic aliens hovering overhead. Yorke even dared to describe the feeling of letdown that follows a blast of hype, such as the one his band was producing. “When it comes, it’s so-so,” he sang. “It always ends up drivel.” The album sold more than four million copies worldwide, enabling the group to become, by 1999, an independent operation. Radiohead were the poster boys for a certain kind of knowing alienation—as the Talking Heads and R.E.M. had been before.

Radiohead remain a magnet for misfits everywhere, but their outsider status is only a part of their appeal. What fans seem to like, even more than the content of the songs, is the sense that the band members have labored over every aspect of the product. They are skilled, first of all, at inventing the kinds of riddles that teen-agers enjoy unravelling. The records, the videos, the official Web site, even the T-shirts all cry out for interpretation. Why are words spelled funny? What are all these charts and diagrams? What about the grinning bears and crying Minotaurs? “We liked worrying over that kind of thing when we were kids, and we’re still in the same mind-set a lot of the time,” Selway said. “But it’s a bit incidental. We’re dead set on the music. That’s the thread running through this whole thing. We met at school playing music together, and we still get together over music now. We like solving musical puzzles. That’s what Thom gives us.”

A Radiohead song is usually written in three stages. First, Yorke comes up with a rough sketch; then Jonny, who studied classical composition in school, fleshes out the harmony; finally, the others digest it for a while, working out their parts on their own. It can be months, even years, before a song comes together in a way that satisfies all of them. Take away any one element—Selway’s flickering rhythmic grid, for example, fierce in execution and trippy in effect—and Radiohead are a different band. The five together form a single mind, with its own habits and tics—the Radiohead Composer. This personality can be glimpsed in the daily bustle of the group, but you can never meet it face to face, because it lives in the music. A lot of what has been written about Radiohead—there are six books, hundreds of magazine articles, and millions of words on the Internet—circles around an absent center.

During Radiohead’s recent tours, the best picture of the band at work came during the sound checks. Yorke led these pre-show rituals with the assurance of a seasoned conductor with limited rehearsal time. He’d shout “Next!” just as a song was building to its climax. All problems had to be addressed on the spot. At one point, Colin said, “That’s something we can work on later,” whereupon Yorke muttered to himself, and Colin added, a little wearily, “It is also something we can work on now.” There were moments, though, when Yorke was at a loss. Once, during a run-through of “The Tourist,” he forgot the final chord, and so did everyone else—including Jonny, who had written the song. “Does anyone have a copy of ‘OK Computer’?” Yorke called out. No one did. The Composer was taking a nap. A couple of weeks later, the problem still hadn’t been solved. “Shall we nominate an ending and play through it to see what it sounds like?” O’Brien said, impatiently. “What do you have in mind?” Yorke said. “D7, perhaps?” “O.K., let’s try it.”