It is April in Berlin. Piles of old snow lie gathered on the streets, and walls of multi-coloured graffiti sing out against the still-wintery skies. At the Michelberger hotel, preparations are underway for tonight's concert: the National are set to play through songs from their new album, Trouble Will Find Me, before an audience of assembled journalists and exhilarated fans.

In the courtyard, a stage is being assembled and outdoor heaters fired up. There are blown-up prints of the album artwork on the walls, and a selection of themed cocktails, their names inspired by both the band and the new songs, are on the menu. The Dessner, I am told, is particularly lethal.

A little after nine o'clock the crowd has gathered, cold-faced and giddy, before the stage, and the night seems charged with all the thrill of a Christmas carol concert. When the band take to the stage soon afterwards – singer Matt Berninger clutching a bottle of wine and propping a lyrics sheet up on a lectern beside him – there is much clapping and whooping and fizzling. It seems not so much eager anticipation as a feverish delight that greets these songs tonight. They run through new tracks – Graceless, Don't Swallow the Cap, I Should Live in Salt – and weave in a couple of old favourites (Fake Empire and Bloodbuzz Ohio) for good measure. As he sings, Berninger's breath wreathes white in the cold night air.

The National are that rare type of band, regarded with reverence by audiences and critics alike. Though some might dismiss their work as miserablist dirge, an Eeyored monologue backed by guitars, this would be to overlook the National's keen sense of lyrical humour, their delight in language and musical experimentation. What they have perfected, over the course of six albums, is a kind of glistening melancholy, a strangely beautiful dourness.

The band have accumulated success since 2001 with their self-titled debut, but it was their last record, 2010's High Violet, that propelled them to a new level of acclaim – riding high on the critics' end-of-year lists and earning them a Brit award nomination for best international breakthrough act (on the night, they sadly lost out to Justin Bieber). Made up of Berninger and two sets of siblings – the Dessners (Aaron and Bryce, who are twins) and the Devendorfs (Bryan and Scott, who are not) – the band took shape in Brooklyn in the late 90s. All hail from Cincinnati, Ohio, however – the Dessners and the Devendorfs were childhood friends who played on the same high-school sports teams, while Berninger met Scott Devendorf at the University of Cincinnati.

They do not, it has to be said, make the most obvious of rock stars. At 42 and bespectacled, Berninger has the look of a hip college professor rather than a wild-living musician. But it is also a result of the band's unabashed intellect. Berninger, a graphic design graduate, is married to Carin Besser, one-time fiction editor of the New Yorker and sometime contributor to the National's lyrics. Meanwhile the Dessners – Bryce, with a masters in classical guitar from Yale, and his brother, with a modern European history and cultural anthropology degree from Columbia – can sometimes seem like the conservators of the modern alternative rock scene. Between them they have collaborated with such diverse musical talents as the Amsterdam Sinfonietta, Philip Glass, Jonny Greenwood and the visual artist Matthew Ritchie. They have written songs about planets, Abraham Lincoln and the Mayan creation story Popol Vuh. They have been invited to perform at contemporary music festivals across Europe, been commissioned by the Kronos Quartet, scored movie soundtracks, curated 2009's Red Hot compilation Dark Was the Night, founded a record label and a music festival and, more recently, oversaw a three-day festival in New York named Crossing Brooklyn Ferry. Such is the draw of the National that their albums often boast the input of many of their contemporaries – Sufjan Stevens, Sharon van Etten and Richard Reed Parry of Arcade Fire among them.

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A few days after the Berlin show, we meet on a warm Sunday morning in London to talk about the new record. Trouble Will Find Me is without question their best album to date – the songs are looser and more limber than those of its predecessor. They have a characteristic darkness, of course – the recurrent themes of death, and the passing of time, a still-lingering preoccupation with the metaphor of water – but they seem somehow poppier now, from the steady charge of the first single, Demons, to the late-night, bar-room waltz of Pink Rabbits, via the heartbreaking simplicity of I Need My Girl.

"I think part of it is that for at least half the record the original demos formed the basis of the recordings," says Aaron Dessner, his voice like thin twine. "And they have a spontaneous quality to them, because with some of them they were recorded literally just right after the idea was created." He pauses. "And with this record we played together in a way that we hadn't done in a while. So I think we captured more interaction in the music. It was less constructive, more casual."

It is Dessner who, with his brother, tends to spawn the National's songs. Working separately, each will weave the beginnings of a track – a "sketch", they call it – and send it on to their fellow band-members for additions, subtractions, embellishments, discussions and, ultimately, to Berninger for melody and lyrics. In Berlin, chatting to Aaron after the show, there was a levity to him – he talked readily of the pleasures and confusions of new fatherhood, of his home and his life in New York. But this morning, with his professional hat on, there is an intensity to his conversation – his thoughts specific, precise, his words carefully planted, that seems to echo the way he works.

Later Bryce, his manner more leisurely than that of his twin, will tell me: "I know that basically my brother goes crazy every time we make a record. So, now I know that, I just work with him and respect the process. He becomes despondent over the slightest thing … if some detail has been added [to a song] that derails anything, it becomes a massive emergency. That's kind of how my brother works."

Berninger provides the counterpoint to Aaron. He is a tall, imposing presence with his own more assured breed of intensity. He is something of a raconteur, and enjoys the fact that during the recording of an album the rest of the band will remain sober while he merrily boozes.

"This is the first record since our first record that I had fun making," he says of Trouble Will Find Me. "Because I was relaxed. I enjoyed it. It's not that I didn't work really hard. It wasn't that it was 'easy'; there was a lot of work. But the time I spent doing it I was happy. Even though the songs are dark … a lot of them are sad and about death, but I was finding myself totally being sucked in and finding such a soothing enjoyable process of looking for melodies and trying different things, finding different words."

It was not always this way. "For me, I used to be the type of writer who would describe writing every word as bleeding drop by drop from the forehead," he explains. "This time I was complete blood-letting into the thing. And that's the first time it's happened for us in 10 years."

The plan had been to take time off after High Violet, to enjoy the high of their success a little, recharge and savour time with their partners and families after a long tour. "But then," recalls Aaron, "towards the end of High Violet I had some ideas and I was home and I started writing them and Matt heard them and he started writing songs immediately." One of them was the album's opener, I Should Live in Salt. "And there was something about that song that made me feel like maybe we were making a record already or something. And because that song has a funny extra beat and has a different feeling to High Violet – and I think to anything else we'd done – it got me kind of excited." Before they knew it, there were 25 or 30 ideas, sketches of songs – enough, Berninger reveals, that they are considering putting out another record in a year's time.

Recording sessions have, in the past, been tense, but the band say there were fewer arguments this time around. "On High Violet," Aaron says, "Matt was pushing us towards a particular sound. So there was a little bit of a struggle, in a sense, and there were a lot of battles in that recording process. This time, there was more of a self-confidence, there wasn't as much anxiety about it, and everybody knew that each idea was strong. Generally, there wasn't a big debate."He hesitates. "Though there were a couple of isolated debates. And in the vocal recording process there were a few battles. It was the last six weeks of the two months that Matt finally was singing, and at some point it seemed like he was headed in one direction in the vocals, and then he was headed in another direction, and we needed to pull it back together somehow." He gives a small, tight smile. "And there was a moment of tension."

"It's mainly Matt and I that get into strong arguments," Aaron admits. "If he's taking issue with some core idea. And, actually, it's usually that he likes the orchestration but he's extending something or he's removed a part." He looks a little testy. "Now he has Garage Band he'll go in and edit it out …" Bryan shakes his head: "It's dangerous," he says. "And, all of a sudden, it gets tricky," Aaron explains, "because the music doesn't have this one transition or other part that made the whole thing work, in our mind. But, actually, what I think happened with this record is we've moved beyond a lot of the creative friction and embraced the chemistry we all have."

It was the song Slipped that really swung it for him. It was, he felt, the song he had always wanted the band to write. "This lugubrious, swampy ballad that Matt's voice would dominate, and just feel timeless," he explains. He rearranged the chords from one of his favourite Dylan songs, Not Dark Yet, and was pleased by the way that Berninger was able to trace the song structure's complexities and turnarounds in a natural way. "It's not that the song is God's gift to songwriting or anything," he says apologetically. "But it felt to me something that I'd always hoped we could do."

For Berninger, harbouring a new-found love of Roy Orbison melodies, the crucial song was Pink Rabbits. "I think Aaron actually regretted sending me that song," he says, "because when he sent me that one I'd been working on a bunch of other tracks that he'd been excited about. And suddenly I just stopped in my tracks, dropped everything else and I spent three weeks only working on it because I just couldn't help it – it was a total, total obsession of mine, because I loved it."

Bryce starts explaining the technicalities of how the song works, while Berninger stares at the table. "It's a harmonic thing mostly," Dessner says, "that song modulates in ways and it resolves in ways that have a slightly more classic American songwriting feeling about them. It has that piano melody, that lazy, swung rhythm that is comforting …"

"I have eight different versions of that song," Berninger continues. He lifts his gaze and frowns. "In fact, three of the other versions are actually about death, I just remembered. But in really fun ways. And I want to put out a whole box set of them because I love them all." Bryce smiles and slides his eyes towards Berninger. "A box set of death?" he asks. Berninger shrugs. After all, isn't that what people expect of the National?

Trouble Will Find Me is released on 20 May.