There is something oddly reassuring about the fact that Steve Reich lives in precisely the kind of house you might expect Steve Reich to live in. The cab journey to Pound Ridge, the tiny town on the New York-Connecticut border that the composer has called home since 2006, passes a lot of rather grand homes built in various classic styles, from colonial to arts and crafts. Reich, by contrast, lives in what appears to be one of the area's few examples of modernism. Inside, the rooms are huge and white. There is beautiful mid-20th-century furniture. It is clearly the home of someone of refined taste. You could even describe it as minimalist, if you wanted to use a term that the person generally regarded as America's greatest living composer is apparently not so fond of applying to his music.

On one of the walls, there is a framed score of Clapping Music, Reich's 1972 attempt to apply his phasing technique – in which two instruments playing the same part gradually shift out of unison – to music that "needed no instruments beyond the human body". On YouTube, there is a great video of Reich and Bang On A Can's Dave Cossin performing it not in the hushed and reverential environs of a concert hall, but in a tent at last year's ill-fated Bloc festival in east London, not long before the police closed the event down. The most enthusiastic audience members whoop and cheer throughout, greeting the moment when the two performers' clapping patterns finally lock into unison again like the chorus on a hit single. If nothing else, it is testament to Reich's willingness to take his music outside of the usual venues in which classical music is performed.

"Well, I take the Chuck Berry approach," he smiles. "Any old way you use it. In other words, music has to have legs. You could walk into a coffee shop and hear the Fifth Brandenburg Concerto. Well, it's perfect for just sitting down and having your coffee and making the atmosphere more pleasant. But you could take that same music home and play it on your headphones and take out your score and say: 'My God, this is the most unbelievable counterpoint I've ever seen in my life.' Anywhere you put it, any way you orchestrate it – Wendy Carlos, Glenn Gould, you name it – if the notes are right, the rhythms are right, it works. It may work better in certain situations, but it always works. Now, I ain't no Johann Sebastian Bach, that's for sure, but that's the model. So yeah, playing Clapping Music to a bunch of people screaming and walking in and out of a tent that looks like it could get blown away and too many people are coming in and where are they all coming from anyway – you know, it's distracting but, like, carry on, do what's expected of you and hopefully people get off on it." He laughs. "I mean, I'm sure they got off on other things that I wasn't even aware of, but, hey, you know …"

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Clapping Music, the Bloc Party festival, Bach's Fifth Brandenburg Concerto, Glenn Gould, Wendy Carlos: this, it turns out, is very much a characteristic Reich answer. Journalists have occasionally found Reich a slightly stern character – one writer who arrived at his house was greeted by the composer asking him when he was planning on leaving, and quizzing him on whether he had booked a cab in advance to ensure he didn't outstay his welcome – but today, he is friendly, engaged, utterly fascinating and extremely loquacious.

After completing his studies in composition at Julliard in his native New York and then at California's Mills College, Reich famously declined to continue in academia, preferring to support himself via a series of blue-collar jobs: at one point, he and Philip Glass started their own furniture removal business, which these days sounds less like something that might actually have happened than the basis of a particularly weird Vic Reeves sketch. On one level, that seems admirable, entirely in keeping with his longstanding desire to free himself from the rarefied world in which modern composition existed in the mid-60s, when composers schooled in serialism and 12-tone technique – his Mills tutor Luciano Berio included – tended to follow Arnold Schoenberg's line: "If it is art, it is not for all and if it is for all, it is not art." Reich says he always knew that serialism and its attendant snootiness wasn't for him: he chose to study under Berio, he jokes, because "as they say in The Godfather, you've got to stay close to your friends, you've got to stay closer to your enemies". It was "an artificial wall, kept up and replastered" by Schoenberg's followers, including Boulez and Stockhausen ("until Stockhausen found himself on the cover of Sgt Pepper," he notes, "at which point I think he started to whistle a slightly different tune").

"What my generation did wasn't a revolution, it was a restoration of harmony and rhythm in a whole new way, but it did bring back those essentials that people wanted, that people craved, but in a way they hadn't heard. Now, we're living back in a normal situation where the window is open between the street and the concert hall."

But on another level, his abandonment of academia seems almost a shame: Reich would clearly have been the best kind of teacher. He talks about music with an intense, communicable enthusiasm. At one point, he tells me about discovering John Coltrane's Africa/Brass. I know the anecdote – he has trotted it out almost word-for-word in any number of other interviews – but even when he is repeating himself for the umpteenth time, he somehow makes hearing the saxophonist's 1961 album seem like a matter of the utmost urgency. Listening back to the interview, I realise his answer to my first question alone went on for the best part of 15 minutes, variously taking in his love of Coltrane, a visit to Birdland to hear Miles Davis, the appearance of Brian Eno backstage at a 1974 Reich concert at London's Queen Elizabeth Hall, the disappearance of progressive rock radio in America, David Bowie, the 1999 compilation Reich Remixed on which various DJs and producers were unleashed on his back catalogue ("I got all the royalties for their music," he notes), the wrangling over which label was going to release his 1976 breakthrough composition Music For 18 Musicians, the artwork used on the cover of Music For 18 Musicians, the Orb's sampling of Electric Counterpoint on Little Fluffy Clouds and why it's easier to "write complex interlocking patterns" for the bass guitar than the acoustic standup bass, as he did on 2008's 2x5, scored for two quintets of bass, electric guitar, drums and piano: "One of them goes 'bong' and the other one goes 'bop'."

It is perhaps worth noting that my first question was a fairly routine inquiry about Radiohead, on whose music Reich's latest piece, Radio Rewrite, is based. The piece, commissioned by the London Sinfonietta, wasn't initially intended to have anything to do with Radiohead, he says: he was trying to write "a giant counterpoint piece" in which 15 musicians played against a recording of 15 other musicians, but that "was like an elephant, it was going absolutely nowhere". Then he met Radiohead's Jonny Greenwood at a festival in Krakow in 2011, where the guitarist had assembled a new version of Reich's Electric Counterpoint: "He'd done his own backing tracks, which means he'd put a lot of time and effort in and it was kind of like: 'Oh, this guy's serious about this thing.'" Impressed both by Greenwood's ability to read music and his soundtrack for There Will Be Blood ("I didn't know this guy was a rocker, I thought he was just a composer who likes Messiaen"), he found himself looking up Radiohead's music on the internet. "And these two songs just said: 'I'm your ticket.' I let the people at the Sinfonietta know, this is going to be the source of where I'm getting my juices flowing, I want to do this, I don't exactly know how I'm going to deal with these songs, but I'm ready to go, I want to get going."

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The two songs in question – which Reich quickly points out his finished pieces don't much resemble – are Jigsaw Falling Into Place, "just a beautiful song", that snared Reich's ear thanks to its "elaborate harmonic movement, it's got an interesting move from B minor to A sharp" and Everything In Its Right Place. Reich describes the latter, winningly, as "three-chord rock", which does make you feel for Radiohead, nearly killing themselves to come up with something strange and unprecedented on Kid A, fuelled by Thom Yorke's belief that rock music had run its course. Then again, if you spent your 20s studying serialism under Berio, maybe it does sound like Louie Louie. "Well, it's three-chord rock but it's not, it's very unusual," says Reich. "It was originally in F minor, and it never comes down to the one chord, the F minor chord is never stated. So there's never a tonic, there's never a cadence in the normal sense, whereas in most pop tunes it will appear, even if it's only in passing. The other thing that really struck me about it is the word 'everything', sung to one-five-one, the tonic, the dominant and the tonic. The tonic and the dominant are the end of every Beethoven symphony, the end of everything in classical music, that's the way it goes. In the tune, those notes actually sound kind of distant because of the harmonies, they don't sound like the tonic and dominant. And the word: 'everything.' I'm sure Thom did it intuitively, I'm sure he wasn't thinking about it. I've seen him playing the piano and he's completely lost in it, the way he should be, but it's perfect, it is> everything."

For all the influence his own work has had on everyone from Bowie to Tortoise – whose music 2x5 bears a passing resemblance to – Reich was never much of a fan of rock music. "I was a kid who grew up with jazz. I was born in 1936, so that was my quote unquote popular music. In 1950, I heard bebop right after I heard Bach and then The Rite Of Spring and those three musics basically form who I am. To tell you the truth, when I was a kid and I heard Bill Haley and Elvis and Fats Domino, I couldn't care less. I was just like who would listen to this? And I just went back to listening to Miles Davis and I really didn't pay attention to rock and roll."

Over the years people tried to interest him, not least his Mills College contemporary Phil Lesh, who ended up as bassist in the Grateful Dead after having an epiphany while listening to Mahler's Sixth Symphony under the influence of LSD. "Well, at the time, he was Phil Lesh the composer and ex-trumpet player who was writing this Stockhausenesque orchestral piece. He told me I had to hear Revolver by the Beatles. So I listened to it and I like it but I couldn't name you a tune that's on it. Sgt Pepper came out, and I heard A Day in the Life and I thought, wow, these guys are really remarkable, but it didn't have any influence on me. They were really bright, intelligent musicians with a vast capability and enormous talent and a lot of charm. But Coltrane, I listened and said: 'Wow, thank you very much, I've learned something.' That never happened with rock and roll."

Reich did briefly operate on the fringes of what came to be known as the counterculture:performing with Leshaccompanied by a light show and working with a street theatre troupe: "it was kind of a release after Berio". He chose to excuse himself when Ken Kesey and The Merry Pranksters rolled into the Bay Area, preaching the benefits of mind-expansion. "In folk terms I'm too young to be a beatnik and too old to be a hippy. I think when they did The Trips Festival, which was kind of like, 'Ta-da, we're here' and the whole Jefferson Airplane thing hit, I really felt this was not my scene and I was ready to go back to New York. I don't think I ever heard a note of rock and roll in all those San Francisco bands that I can ever remember at all – I mean, the bands meant absolutely nothing to me whatsoever. There was no reason to stay." He talks about being unable to put together a group in San Francisco, because of the lack of musicians, but there's also the sense that, as with the Velvet Underground, hippydom just didn't sit right with his New York sensibility. "There are different mentalities between people who are born and raised out on the west Coast and people who are born and raised back east."

Reich performing live in London, 2006. Photograph: Getty Images

So he went back to New York, formed Steve Reich and Musicians – who, in an early echo of events at the Bloc Festival, performed everywhere from art galleries to discos – humped furniture with Philip Glass and developed his phasing technique. He went to Ghana to study African drumming. "You could get a cheap ticket if you went to some place in east London and got some ticket that said you were a Nigerian flight engineer," he frowns. "Do I look like a Nigerian flight engineer?" He came back with a case of malaria, and a renewed conviction that he was following the right path. "It was like getting a pat on the back saying, yes, people have been writing repeating patterns for hundreds, maybe thousands of years, percussion is more interesting and magnetic to the ear than electronically generated sound, go home and continue. It was very encouraging because people were looking at me like I was crazy because I wasn't writing 12-tone serial music."

His name first really came to prominence in 1973, as a result of what Alex Ross's history of modern classical music The Rest Is Noise calls "the last great scandal concert of the 20th century": a performance of his piece Four Organs at Carnegie Hall in 1973, which provoked such uproar in the audience that the musicians onstage couldn't hear themselves. Today, Reich suggests the scandal was perhaps a little more stage-managed than history likes to paint it. "I'd played that piece at my own concert at the Guggenheim museum and 300 people came and loved it. This wasn't my concert, this was the Boston Symphony Orchestra giving a Sunday afternoon programme to little old blue-haired ladies. The conductor, Michael Tilson-Thomas, had really kind of loaded the dice. He's not a naive guy, he knew the orchestra he was dealing with and he knew the audience. Everything kind of exploded. Even the most conservative critic in the New York Times said the audience reacted as if red hot pokers had been placed beneath their fingernails, but at least there was some life in the audience, which there usually isn't."

It feels a little strange to think of Reich as a riot-provoking iconoclast. At 76, he shows very few signs of slowing down to bask in his reputation. Radiohead Rewrite is already "ancient history", he says: he is working on a new piece called Quartet, scored for two pianos and two vibraphones, that is due to be premiered in London next year. Or at least he is trying to. "Well, I try to work as regularly as I possibly can. The problems of being successful can be greater than the problems of being unsuccessful. You've got all these emails to answer and requests that people want you to do, all kinds of imaginable and unimaginable stuff that has to be dealt with. You know, tomorrow there's going to be a concert of music at the Museum Of Modern Art, then I'm going to discuss Debussy with [the composer] David Lang, then go out to dinner, so I doubt if I'm going to get anything done then. But as soon as you're gone today, I'll go back to the piece I'm working on and write. I'm happy to say I'm well ahead of schedule," he adds, lest I think he's throwing me out.

I have to go anyway: mindful of some previous journalists' encounters with Reich, I took the precaution of ordering a cab in advance. "I think you've got enough stuff to write," he nods. "I talked so much, I expect to have the entire edition of the Guardian devoted to me. If it isn't," he frowns, offering a handshake and showing me out of the door, "I'll be calling my lawyer."

The London Sinfonietta gives the world premiere of Radio Rewrite at the Royal Festival Hall in London on 5 March, followed by Birmingham Town Hall, 6 March; Brighton Dome, 7 March, Glasgow Royal Concert Hall, 9 March.