Rock stars can be an arrogant bunch, famously full of bluster, swagger and off-the-scale egos. Particularly when it comes to guitarists. Jonny Greenwood, however, despite being the lead guitarist in a band who have sold some 30 million records worldwide, and revered as one of the most inventive guitarists of his generation, couldn’t be further from the tight-trousered, axe-wielding cliché.

“It could be all full of mistakes and nervous notes,” he tells me of his upcoming show at London’s Royal Festival Hall, where he’ll perform music by the pioneering American minimalist composer Steve Reich, alongside the renowned contemporary chamber orchestra London Sinfonietta.

Greenwood is no stranger to the world of classical music. Over the past 20 or so years, he has famously developed a parallel career to Radiohead as an acclaimed film composer, writing award-winning scores for features including director Paul Thomas Anderson’s 2007 , and the 2021 Princess Diana biopic .

As well as recording and touring with both Radiohead and The Smile – his successful offshoot trio with Radiohead’s Thom Yorke and drummer Tom Skinner – 54-year-old Greenwood has also been involved in myriad other projects, from setting up a classical music record label and recording Rajasthani musicians to selling olive oil harvested at the Italian country estate where he lives part of the year (alongside his Oxfordshire home), with his wife and their three children.

Greenwood has long had a particular interest in Steve Reich’s music, and the pair have a long-standing relationship. He first performed Reich’s “Electric Counterpoint” – one of the pieces he’ll be performing at the Festival Hall concert – back in 2011. “It’s a piece you play to a backing track of other pre-recorded guitars,” Greenwood tells me. “At one performance in Poland Steve was actually there. According to the sound guy, he punched the air when the bass guitar came in. Unless he was shaking his fist at me,” he adds, laughing. “Always possible.”

Greenwood has cited Reich as an influence on the phasing guitars in Radiohead’s songs “Let Down” and “Weird Fishes/Arpeggi”, and in 2012 Reich returned the compliment. In “Radio Rewrite”, which will also be performed at the Festival Hall concert, Reich reimagined the Radiohead numbers “Jigsaw Falling Into Place” and “Everything in its Right Place”, reinterpreting them in his own rhythmic style.

Having his own band’s work reinterpreted by a classical music luminary like Reich was, Greenwood tells me, “insanely complimentary”. “In fact, I can’t believe he did it.”

Classical music played an important role for Greenwood from an early age. As a teenager at school, he played the viola and recorder, and joined youth orchestras and recorder groups, both of which he says he “took very seriously”, despite being what he describes as a “terrible player”.

“School was important too,” he continues. “A teacher played us [French 20th-century composer] Messiaen, who was still composing then, and it felt as current as my New Order or Pixies records. So I just put them all on similar pedestals and learned what I could.”

Greenwood has talked in the past about the influence of Bach on Radiohead, and Bach was something else he learned at school. “I’m still kind of using that knowledge all the time today.”

Before Radiohead’s early success – they got their first record deal with EMI in 1991 when they were all only in their early twenties – Greenwood had begun studying for a music degree. Those early weeks similarly influence his work, he explains, both with Radiohead and beyond.

“I only managed six weeks before leaving to tour with the band,” he says, “which was fine because that’s where my head and heart already was. Still is. But luckily in those six weeks, they showed us a [Polish modernist composer] Penderecki score, and gave us access to a Portakabin which was full of old analogue synths. I’ve sort of run with those two things ever since.”

Combining all these influences, despite still being best known as a guitarist, he sees himself as much as an arranger as a musician or composer, particularly when it comes to Radiohead.

“It’s partly because,” he explains, “Thom will sit down and play a song like ‘Pyramid Song’ on the piano, and we’ll all look at it and just think, ‘Well, that sounds great already. How do we not make that worse?’ It’s not like I’m going in as a guitarist and going, ‘Well, when do I get to play my solo?’ That’s the last thing we think about. It’s about, ‘How do you not mess it up?’”

“Have you ever heard Leonard Cohen’s version of ‘Hallelujah’?” he asks me. “It’s kind of quite bad and the song is just glorious,” he says with a smile in his voice. “And I always find that kind of thing really interesting. That someone else can do a version of a song and show how good it is. That’s always a big fear that you’re not gonna do justice to what’s amazing material. Not to insult Leonard Cohen or anything.

“It’s about how we record I suppose,” he continues, “and about how music is arranged. Just thinking about what the song needs. I always get excited when a song is arranged really well. It’s very nerdy.”

Greenwood will soon be back in the studio, not with Radiohead, but to record the violin concerto that he has written for violinist Daniel Pioro with Manchester’s renowned Hallé Orchestra. He’s also due to write a new string quartet for leading young combo the Fibonacci Quartet.

Despite such a huge range of projects on the go, Greenwood is hungry to explore the world of classical music still further. “I don’t feel I’ve learnt enough,” he tells me. “I’m always sort of torn between wanting to greedily consume all this stuff and understand it. But also knowing that it’s a mistake to just have a kind of jack-of-all-trades approach to it all, because it just means you don’t really get a deep understanding of any of it. I just permanently feel like I’m in a kind of panic between the two things.”

Greenwood may be eager to learn more but it’s the guitar that has become his overriding passion. “I will sit and play guitar every day now,” he explains. “It’s funny. It’s something I kind of never wanted to do. I used to distrust any sort of fluidity and think that was somehow part of that thing of technical playing. But now I just feel different about it. And it feels lucky that my fingers do these things. I’ve got a certain facility with it, so I’ve decided to accept it, and have realised that I really love playing the guitar.

“In the early days of Radiohead I always felt like a clumsy guitar player and was always lazy but now, just through repetition, I’ve got hooked. I’ve just got addicted and want to play all the time.”

Steve Reich at 90: London Sinfonietta & Jonny Greenwood is at London’s Royal Festival Hall on Sunday (southbankcentre.co.uk)