“ ‘OK Computer’ sent us on this bizarre trajectory,” Yorke told me. “I just wanted to get off.” Yorke found himself unable to write songs for a while; he retreated to Oxford, listened to a lot of experimental electronic music, particularly the paradigmatically blippy Warp Records catalog, and came back a year or so later determined to reinvent Radiohead. No more anthems. Rhythm over melody. Vocals supplying color and texture, not text. An overall aesthetic, borrowed from dance music, that would embrace the artifice of the recording studio instead of trying to conceal it. This kill-or-cure moment for the band stretched out into almost two years of hard labor in various studios — Paris, Copenhagen, Oxford. They were listening to Kraftwerk, Jamaican dub and ’70s Miles Davis; they were reading Naomi Klein’s “No Logo” (Yorke had become obsessed with the notion that by playing corporate-sponsored rock festivals in non-Western countries, the band had become hip advance men for globalization.)

The sessions were infamously fraught and doubt-ridden, which is interesting because the material they produced — enough for two albums, “Kid A”(2000) and “Amnesiac” (2001) — is fraught and doubt-ridden, too. It de-emphasizes the standard guitar-bass-drums format that has been rock’s default since the mid-20th century in favor of keyboards and weird time signatures and Charles Mingus-y brass and the hum and squelch of a not-O.K. computer murmuring to itself in the corner. It’s music that dramatizes the collapse of certainties at the dawn of the 21st century by forsaking the certainties of guitar-based rock. And yet, despite being executed with a toolbox of weird, theoretically hard-to-groove-to noises, these records were actually deeply traditional: a big, ambitious, where-we-are-and-where-we’re-going statement spread across double LPs, the kind we’ve always looked to “important” bands to make.

It’s also worth pointing out that it’s pretty difficult for a band like Radiohead to draw a real line in the sand — the oddness of “Kid A” may have pruned a few casual fans from the band’s audience, but chances are most people who followed them from the worrying man’s Britpop of “The Bends” (1995) to the sprawling, spiny neo-prog of “OK Computer” weren’t about to jump ship the minute a drum machine came into the picture. Every generation of rock-music fans since the Beatles has needed a Beatles of its own, a band whose every creative left turn they can roll with, and congratulate themselves for rolling with. To generalize hugely for a minute, people like pop music (or any mass cultural experience, really) because it lets them feel as if they belong to something bigger than themselves. And they like avant-garde music because it lets them feel as if they’re part of something everyone else isn’t hip to. Going to see Radiohead play willfully difficult, electronically warped anti-anthems about rabbit-borne disease and human cloning in a venue the size of the Theater at Madison Square Garden satisfies both those needs; you get a sense of community and a sense of adventure for the price of a single admission.

A lot of lead guitarists would have struggled with a new direction that so pointedly and thoroughly marginalized lead guitar. Greenwood isn’t that kind of guitarist. He came into his own as a multi-instrumentalist. He taught himself to program modular synthesizers — the ones with the big switchboard-operator patch bays. He mastered the ondes martenot, an electric keyboard instrument from the 1920s that can mimic a string section, or a flute, or a choir of ghosts imprisoned in vacuum tubes. (It’s featured prominently on the French composer Olivier Messaien’s “Turangalîla-Symphonie,” which Greenwood has said is his all-time favorite piece of music.)

Yorke had been sketching out ideas for the band’s Great Electronic Leap Forward at home on a cheap beatbox. “The sounds were all awful,” he says. “So it was good to have someone who was prepared to go out and spend money on all this wild, brand-new gear and come in and learn how to use it. That really helped us along, that he was willing to go straight into that.”

Image Jonny Greenwood, the Radiohead guitarist with a double musical life. Credit... Christaan Felber for The New York Times

That’s been Greenwood’s role ever since — he’s the guy who can take an abstract Thom Yorke notion and master the tools required to execute it in the real world. The most recent Radiohead album, “King of Limbs,” sounds as if it has less Greenwood on it than ever, until you learn that a lot of the music was pieced together using a bit of sound-looping software that Greenwood programmed. Post-“Kid A,” Greenwood says, Radiohead has evolved into a band of arrangers. They start with an idea — usually some chords, a melody and some kind of a speed — and figure out how to orchestrate it. Recording a song by playing it together in a room has become just one of several options they can pursue while recording: a setting on the machine that is Radiohead.