In truth, Dan Snaith’s life and career haven’t ever really played out according to any script we might already be familiar with. Generally speaking, people with doctorates in mathematics aren’t really supposed to make a living playing to huge crowds in dance clubs. Musicians who have been plugging away for the best part of a decade making music for no other purpose than to entertain themselves aren’t meant to end up releasing a record that becomes a serious crossover success. That Snaith lived out these particular scenarios with his third album under the Caribou name, Swim, tells you a lot about how sharp his appetite seems to be - intentionally or otherwise - for confounding expectations.

It’d also possibly explain why the follow-up to Swim has seen Snaith move in a direction that’s hardly a typical next step for somebody just handed a commercial breakthrough. More of the same would have been the most obvious path to go down, but whilst Swim was dark and brooding, Our Love is considerably lighter in texture - emotionally open and sonically vivacious throughout. The Swim tour saw Snaith and his band turning the likes of "Odessa" into enormous, all-encompassing jams specifically designed for superclubs like The Warehouse Project or the cavernous arenas in which they opened for Radiohead; Our Love, though, is an intimate affair, one that you suspect will be reinvented for the stage, but that also - for the time being, at least - is as sharply geared towards the individual listener and their headphones as it is meant for communal consumption.

“I really noticed how the audience had grown, whilst we were out touring Swim,” says Snaith. “We were playing bigger shows, bigger festivals, and the thing that I kept picking up on was that, when people came to talk to me, they’d be young, old, from different musical scenes; it was a totally diverse audience. In the past, it’s always be people who looked like me, who were my age, who were into the same things - you know, coming over and saying, 'oh yeah, I can totally hear Silver Apples in that track.' This time, it was just this wonderful, and totally unexpected, mix of people, and that knowledge was pretty much the starting point for Our Love."

Everything about Snaith seems about as far as you could imagine from the typical profile of somebody who routinely packs out dance clubs with his music. He’s unfailingly polite, with every answer delivered in his clipped Canadian accent a thoughtful one. Recent press images of him - his hairline receding, NHS specs perched on his nose - make him an unlikely cover star. Our Love is the first record he’s made with the audience in mind, rather than for his own enjoyment, and the unselfishness of his approach to that transition, you feel, says an awful lot about him.

“I kind of thought, 'well, if I’m going to be thinking about who’s going to hear this album, I may as well just face it head-on.' I mean, I had two impulses; one, to make this record for the person listening to it at home, and then also to do something that would be interesting to the live audience, but it was the first one that really grabbed me. I wanted to be speaking to the listener one-on-one, to be engaging them in conversation. That’s probably why the album sounds a little bit more intimate. It wasn’t like I tried to be more populist because I knew I had a wider audience, and shaved off the idiosyncratic edges; I just wanted to be direct towards, and focus on, the listener, and communicate with them properly.”

Snaith’s wider plan for actually managing that, though, was thrown into some degree of disarray earlier this summer, when Our Love - albeit, at pretty rough audio quality - was leaked way ahead of time. “At the time, it felt so frustrating,” Snaith sighs. “I was just thinking that, you know, this was the first record I’d made with the intention of sharing it with people, with the intention for it to travel widely, and everybody getting to hear it at the same time felt like an important part of that, to me. When a friend of mine told me what had happened, I was pretty bummed out, especially because these days it feels like records don’t often leak that far in advance any more. In the long run, though, it won’t make any difference that a few people heard it earlier, and I did come around to thinking that it can’t be a bad thing that people were enjoying it. Nobody’s going to remember that happening in years to come - they’re just going to remember the record. It’s a shame that the quality was so poor, though.”

Even if the basic approach was different for Snaith, though, this remained a solo project in terms of practical application; the actual manner in which most of his collaborators influenced the album wasn’t necessarily a tangible one. “If anything, because there’s a lot of programming and sequencing on this record, there’s probably other albums I’ve done that have been more informed by the live band than this one. Those guys don’t play on this record, or anything, but they are amongst that group of people I go to for opinions; is this track any good? Is that track moving in the right direction? There’s those guys, my wife, Kieran Hebden from Four Tet; I have like a little advisory panel, I guess, but otherwise it’s all me. The only actual collaborations on the record are with Jessy Lanza, who contributed some vocals, and Owen Pallett, who plays violin on a few tracks."

The four-year lay-off, meanwhile, between Swim and Our Love was less to do with Snaith resting on his laurels or simply being pain-staking about the production process, and instead down to the fact that he put out his first album under the Daphni moniker, Jiaolong, in 2012. “The Daphni thing, for me, is a very strict kind of subset of the things that I love about music; it’s kind of about the excitement of going to a club and having new music to play, just by jamming it out and making it really quickly. The Caribou albums are the main focus because, as far as I’m concerned, they should contain everything I love about music; they should diarise, or document, my life in that respect. So, you know, one’s a part of the other. Caribou is always going to be reflective, because it comes from me putting things together, slowly, over the course of a year or so, and waiting for what feels right. Daphni, on the other hand, is like, 'right, I’ve got to get on an EasyJet flight in four hours, I’ve got two hours to make some music - go!' I can hear that energy when I listen back to it, and sometimes, I have to back and edit little things like the sound of me pushing the buttons being left on the track - it’s all done so quickly.”

Snaith isn’t kidding, either, when he talks about the reflective nature of his work as Caribou. In that regard, Swim was kind of abstract, characterised by constant sonic nods to the theme of water and often-claustrophobic sonic environments. Our Love, meanwhile, has him embracing clarity of thought and emotion in a way that’s still relevant to the clubs; he’s managed to make a record that’s relevant to the all-inclusive atmosphere of the rooms he plays, in a manner that’s in no way corny - there’s certainly no shades of Primal Scream’s "Come Together".

“The theme for this album is there in the title, totally explicitly,” Snaith explains. “I was thinking a lot about how my life has changed for the better with all the people I’d met because of Swim, about the fact that I’m in my mid-thirties now, with a daughter, and I was trying to just capture all of the love in my life right now. Not just romantically, with my wife, but with my family, my friends, and then my love of music, the love I feel when I talk to people who like my music, and the love that must exist in those crowds, between the people who are there for the same reason but seem to come from different demographics. It’s a lot broader, in that respect, than Swim, I think.”

The sheer intelligence that goes into Snaith’s work is difficult to overlook; Our Love, like everything that’s gone before it, is the sort of record that throws up another little nuance, another little subtlety, with every listen. The fact that he has a PhD - from Imperial College London, since you asked, on Overconvergent Siegel Modular Symbols - obviously goes a long way to reinforce the image of Snaith as a cerebral figure in the eyes of his audience, but it’s not a view he shares.

“That’s not my experience of it at all!” he laughs. “It couldn’t really be any further from it. The experience, for me, has always been about trying to generate some kind of emotional punch; I make music very intuitively, very emotionally, and not analytically at all. That’s just my perspective, obviously, and I’m aware that a big part of my personality is that I’m the same guy who did a math PhD, so that side of my brain has to be in there somewhere. I know it makes for a good press release, and it’s going to colour the way people perceive my music, and I’m cool with that, honestly.”

It’s also possible, of course, that the enduring nature of Snaith’s near-unanimous popularity with the critics has helped to cement the view of him as somebody who brings a serious level of thought to his work; after Swim, you wondered whether it might finally get to the point where the positivity surrounding Caribou began to turn into its own kind of pressure. “It’s funny, whenever I spoke to my record label in between Swim and Our Love, they’d say, 'Dan, are you OK? Are you feeling any pressure?'” laughs Snaith. “And, of course, I’d never even thought about it until they brought it up. I’ve been doing this so long now that I already have a level of confidence in the way I work. I mean, that’s something that might have been damaged if I put out a record that was slammed across the board, but I always just viewed it as affirmation; that it meant I should carry on doing what I’m doing, rather than worry about pleasing people.”

At several points during our conversation, Snaith makes reference to the contribution of his fellow Canadian renaissance man without being prompted; it’s obvious he had an impact on Our Love that went beyond the violin he lent to several tracks. “I’ve known him for maybe twelve years now,” recalls Snaith. “He was very much active in the Toronto music scene, like I was, when I first met him, and in some ways, it’s surprising to me that it took me so long to collaborate with him - we put out a Daphni/Owen Pallett split twelve-inch earlier this year, that we recorded back in 2011. We did that pretty much in a day; I asked him to come down because I knew that he was interested in dance music, but didn’t have much of a grounding in it. It was so much fun that he was like, 'why don’t I do something on the next Caribou record?', and I couldn’t believe I’d never asked him to do that in the past. He comes at things from such a different angle to me; he has a totally different set of abilities, and he gave me so much feedback on this record that I never would have thought about myself. That’s what I love about collaboration - that potential for it to upend what I’ve been doing, and send it in a different direction.”

Kieran Hebden of Four Tet, of course, has similarly close ties to Snaith, even if they’ve never worked on an out-and-out collaborative recording project to date. “It’s weird with Kieran; even though neither of us appears playing on each other’s records, I kind of feel as if we’re embedded in each other’s music pretty intensely. I was sending him early tracks for this record and getting so much help from him, about everything - arrangements, mixes, mastering - and I’ve done that with a lot of his stuff, too. He’s definitely my closest musical ally. We did the Caribou Vibration Ensemble together, too, where we got a bigger group of musicians and played the songs in a way that was much more open-ended; that’s one of my favourite things I’ve ever done, I think. We’ve got no plans to do something together properly, but you can always bet that we’ll have input into each other’s music, whatever we do.”

The pair will share a bill at The Warehouse Project in Manchester - not for the first time - on Halloween, and the live show is now Snaith’s primary focus, looking forwards; as usual, the version of Our Love that makes it to the stage is likely to be drastically different from its recorded counterpart. “We debuted a few songs earlier in the summer, and we’re in the process of working out the others, still. It’s a lot of fun, far less pressurised than when I’m making the album; that’s the kind of situation where, if I don’t do anything, nothing will happen, but there’s a real sense that everyone’s pitching in when we tear the songs apart and rebuild them for the live shows. The really cool thing is that, already, the songs we started playing at the beginning of the summer are sounding pretty different from how they did then; I’m really excited to see how these new songs are going to sound a year from now. It’s a constant state of evolution.”

Our Love is available via City Slang on October 6th. Caribou plays four UK dates in October.