When Radiohead recruited drummer Clive Deamer to their live line-up last year, he found himself being grilled by Rolling Stone about Britain's most innovative band, and had to swiftly put his interviewer straight. "He kept using this term – 'rock band', 'rock' this, 'rock' that, he was dropping that in all the time," Deamer recalls. "Halfway through the interview I had to say: 'Hang on mate, we're talking aboutRadiohead – what's rock about Radiohead?' Having spent nearly a year with them, they don't see themselves as a rock band, and they're not trying to maintain that."
Anyone who saw their meditative surprise Glastonbury set, or heard the intricate beauty of last album The King of Limbs, would agree. These days, they sound like what one might have imagined a Blue Note modern jazz quintet would morph into by the 21st century. The presence of Deamer, who with his band Get the Blessing won 2008's BBC Jazz Award for Best New Album, is a clue to a wider trend. He and Get the Blessing bassist Jim Barr are also Portishead's rhythm section, while both Adele albums and parts of Emeli Sandé's were built on piano parts by 2007 BBC Jazz award-winner Neil Cowley.
In the US, Wilco have been transformed by guitarist Nels Cline's mould-breaking improvisation skills. From Kasabian drummer Ian Matthews to Goldfrapp lynchpin Will Gregory, a music which gets little UK attention beyond the Mercury Prize's token jazz nominee has dug a role in the mainstream's heart.
It recalls the 1960s, when musicians from Ray Davies to Motown's studio band drew on early days as jazz players to lend rhythmic suppleness to artful pop hits. A bridge which seemed broken as rock bands who knew nothing else and house music's iron rhythms ruled, is tentatively being rebuilt.
When we meet as he prepares for Radiohead's tour, Deamer explains jazz's place in the band's world.
"There's a track on King of Limbs called "Bloom". That is the most complex, antagonistic, tangled rhythm. It's certainly not pop music, certainly not rock. It's got the same intensity as any Sun Ra, or [drummer] Elvin Jones burst with [John] Coltrane. And Thom [Yorke] and Jonny [Greenwood] have their jazz references that they like. Those things do get mentioned.
"I don't think they would call anything they do jazz. But they're keenly aware that there are obvious parallels – in the way that they deliberately try to avoid cliché and standard forms for the sake of the song, and of Radiohead as an idea. Rock bands don't do that. It's far more like a jazz mentality. Jonny certainly, there's a jazz player in there, along with all the other things."
When another of Deamer's famous employers, Portishead, played Alexandra Palace last year, the moment in "Glory Box" when the band massively crescendoed, then dropped out for Beth Gibbons to cry back in with its chorus, was epic jazz. "I first met Adrian Utley as a jazz player around Bristol," Barr says of one of the band's architects. "When Portishead curated All Tomorrow's Parties at Minehead Butlins, Adrian said the last time he'd played there he'd been doing a cheesy jazz thing with the redcoats."
Such fundamental jazz roots have been rare in recent pop, Get the Blessing saxophonist Jake McMurchie believes: "What pop bands look for now is someone who can turn up cheaply and go straight on Jools Holland."
Neil Cowley's contribution to Adele's albums developed into something more. "I popped in on the Monday," he recalls. "I was given the general chords, and improvised the intro to "Hometown Glory" on the piano, which is now endlessly used on American Idol. Since then, we always record all our takes just me and her live, and everything else is built around it. Jazz taught me less is always more. It can be one note, one chord. That works perfectly in pop."
Like Deamer, Cowley is far from a pure jazz player, turning his back on classical training aged 15 to play everything from funk to raves. He feels jazz has lost out equally in its sundering from pop, as its players these days train in conservatoires.
"The people that interest me tend to be the self-taught, unadulterated ones," he says. "The most fascinating guitarist I ever met escaped out of rehab for a gig and played one chord. I wait with anticipation to see whether people the conservatoires put out touch my soul like that. I think there might be an identity crisis within jazz. We are supposed to be the most avant-garde and forward-thinking music. But you could answer, 'no, it's dance now, or rock'. "
Just as Radiohead have adopted a philosophy of constant change that is more Miles Davis than Oasis, so some jazz bands are moving to meet them. The self-titled new album by experimental jazz's British standard-bearers, Portico Quartet, leaps from that music into electronic terrain neighbouring The King of Limbs, while the Neil Cowley Trio's The Face of Mount Molehill and Get the Blessing's OC DC use jazz technique to push moving pop melodies. The latter album includes a vocal cameo from Robert Wyatt, who has been here before. "He embodies all the things we've been talking about," says Barr. "He does jazz but makes it sound like pop, and vice versa."
The Neil Cowley Trio tour from 13 to 26 April (www.neilcowleytrio.com). Radiohead's UK tour starts 6 October
Independent.co.uk is partnering with Spotify to allow readers to listen to all the key music mentioned in the paper. Listen to the artists and songs mentioned in this piece at Ind.pn/radioheadjazz