Sam Fogarino, drummer with the band Interpol, has lost a woolly mammoth. “It was here,” he insists to the attendant at the Natural History Museum in New York. “ Here.” She shrugs, and suggests that the exhibit must have been removed. “I think she’s lying,” mutters the rugged-looking Italian, frowning from beneath his hat, as if Tony Soprano had been cast in an American remake of One of our Dinosaurs is Missing. The missing mammoth is crucial to his band, because it was beside it that Fogarino had a musical vision about their new single, Mammoth), which led them to use photographs of these and other stuffed animals on the artwork of their album Our Love to Admire.

Yet when they recently unveiled the new material at the Californian music festival Coachella, their music’s trademark moody darkness was ruined by paparazzi flashbulbs hounding a different kind of creature – the party animal Paris Hilton, who irked the band by standing on the side of the stage to show her devotion. Why Interpol are one of her favourite bands is unclear, but then everyone from Brad Pitt to Bono and Sadie Frost are fans of the New York quartet.

They formed while at college nearly a decade ago and became part of the cool rebirth of the New York guitar rock scene, along with the Strokes and the Yeah Yeah Yeahs. Yet while Interpol may have partied hard – the bass player Carlos Dengler, in particular, was known for his way with late nights and ladies – their music had a darker side, as did their overanalytical brains. They were the intellectual group, as interested in galleries as in groupies, and they have dealt with rock-star burnout by turning not to rehab but to the arts.

These days, the love of his life for Dengler, a philosophy graduate, is his greyhound (he admits it is a substitute baby, or perhaps wife), and his work on classical music sound-tracks for films. “That keeps me sane on the road, and believe me when I say that there is nothing in this lifestyle that is in any way encouraging that groundedness. Yes, I went insane before, several times. You’re constantly mobile and constantly being given attention, not just by fans but by the industry. The psychological adjustment that is necessary would make most people crumble. But listen, I’m not chagrined at all by my previous antics because I knew why I was doing it.”

Daniel Kessler, guitarist and band founder, is taking piano lessons, and rises at 7am to write songs in front of arthouse DVDs. “Tonight I’m gonna rest my chemistry,” sings the lead singer, Paul Banks, on a song about giving cocaine a rest. He tells me that he has had a literary breakthrough after finally finishing Moby Dick. Meanwhile, Fogarino has got married and is discovering museums, just like he did as a young child in Philadelphia, where he grew up in a family that were “bohemian but with nothing, just one step up from white trash. The art museum there had these recreations of rooms from far-away countries and what they did to my little brain – it was almost like a voice was inside my head, pushing outward – and it led me to music. You could just sit down and jump into the frozen moment.”

Frozen moments are what Interpol’s music is all about – the band share a love of atmospheric cinema, of songs that capture feelings rather than stories. Kessler put the band together after spending five years in New York “not finding a musician I wanted to play with – they were all too loose about their ambitions. Our personalities are very different, but when we played music we were reacting to each other in a way that conversation could not have done. In fact, conversation would have pulled us apart.”

It’s not that they object to bands who are more conversational, Banks explains. “I like the casual, off-the-cuff tone of Arctic Monkeys or the Streets – and the Strokes – it’s amazing and sort of relaxing. But I’m a jacked-up person psychologically and I don’t feel casual, ever, so I couldn’t do that anecdotal song-writing in the same way. I don’t tell somebody ‘this is how it is’ because I don’t know how it is – I just know that my perspective shifts constantly with my mood.”

Such lack of detail also saves the band from giving away too much about themselves. They claim to dress in designer suits every day, not just for performances, and they bemoan the casual nature of our times. As Fogarino explains with a grin: “We’re gonna have to learn these kids some sense of mystery.”

But when a “kid” approaches with a flyer for the Editors, the uptight Banks has to laugh – because Interpol are frequently compared to the British band. He claims that he liked one song of theirs that he saw on MTV but doesn’t know much more about them, even though they both played the T in the Park festival in Scotland. The very friendly Fogarino wanted to say hello, “but for the first time I felt like I couldn’t, because too much negative stuff has been put between us.

“But why I feel sorry for them is that we put out our first album Turn on the Bright Lights and we’re f****** Joy Division reincarnated, and then the Editors appear and they’re the British Interpol. Just leave them alone and leave us alone. If indeed they have influence from us I think that’s flattering .”

He says he can’t listen to Joy Division any more. “I remember when I’d just met my wife, Love Will Tear Us Apart came on in a club and she wanted us to go and dance, but I couldn’t. And once I was walking around the Soho record shops in London and saw [the producer] Martin Hannett’s mixes of Joy Division songs and I felt like I would have to send in some stranger off the street to buy it for me, because the minute the bloke sees me it’s like, found out! I knew it! It’s all the drummer’s fault!”

With four control freaks in one band, it’s a wonder they’ve released anything at all. But as Kessler says, they are actually better in a crisis, such as the day before, when they had played the David Letterman show with borrowed instruments after theirs had gone missing in transit. He says they find a strange sort of relief in such situations. Really? And what if they lost all their clothes and had to perform in pyjamas? There is a pause from Kessler. “Then there might be an issue,” he says, his Agnès B jacket glimmering in the Manhattan sunlight.

The album Our Love to Admire is out now on Capitol. Interpol’s UK tour starts on August 20. www.interpolnyc.com

More bands with brains

Dave Rowntree and Alex James, Blur
Both worked on the European Space Agency’s mission to Mars, Beagle 2.

Brian May, Queen
Recently awarded a PhD in astrophysics, regular contributor to the BBC’s The Sky at Night.

Chris Martin, Coldplay, and Tim Rice-Oxley, Keane
They first met in the classics department at UCL, where they both graduated in Greek and Latin – the other Coldplayers were there doing anthropology and mathematics.

Rob Coombes, Supergrass
A qualified astrophysicist and writes about it on his blog.

Julian Cope, Teardrop Explodes
Now writes hefty guides to Neolithic and Bronze Age monuments.

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