It’s a story with so many incredible threads, one doesn’t know where to begin. Those with an endless appetite for all things hip-hop might be tempted to lock onto the Tyler the Creator angle. Modern R&B aficionados will be all over the Frank Ocean association. For jazz disciples, how about the way that BadBadNotGood is keeping the spirit of groundbreakers like Miles Davis alive but at the same time confusing those who profess to love groundbreakers like, well, Miles Davis.

And for anyone who’s ever formed a band that’s ripped up the rule book to little reward, there’s the fact that three kids from Toronto have, in four short years together, amassed a bunch of experiences that seem too amazing to be true. How amazing? Well, let’s just say that BadBadNotGood’s Chester Hansen is right at the top of those in awe of what he and his bandmates—keyboardist Matthew Tavares and drummer Alexander Sowinski—have managed to achieve since causing an overnight sensation on YouTube. Reached on his cellphone in Hogtown, the bassist suggests everything has been the kind of wonderful blur where coming up with one highlight is mission impossible.

“This has been the kind of stuff where you can’t really think about it too much, otherwise you start going crazy,” Hansen says. “You have to take it as it comes. When we started, we were literally, like, friends who were just jamming. We ended up recording a couple of things, and then putting them online. Since then we’ve been blown away by how into it that people have been, and really blown away by the fact that people really seem to like our weird instrumental music.”

For proof of that, consider how BadBadNotGood has spent the past few weeks. Hansen is a tad jet-lagged on the day he talks with the Straight, having just come off an extended European tour for his band’s third full-length, and first outing of all-original material, III. Despite being tired, he’s also completely jacked. Noting that BadBadNotGood has just done a swing that included Belgium, Russia, Scotland, England, and Ireland, Hansen says all the shows were amazing, but some were just a tiny bit more amazing than others.

“The one in Moscow stood out, maybe because it was on this really beautiful rooftop in the middle of the city,” he says. “And also because 1,400 people came, which is really crazy. The promoter wasn’t expecting half that many people.”

No one should have been surprised, considering the band’s trajectory. The beginnings of BadBadNotGood can be traced back to Ontario’s Humber College, where all three members of the group enrolled in the school’s jazz program. Pretty quickly, Hansen, Sowinski, and Tavares realized they were interested in something more than mastering the works of Charlie Parker and Charles Mingus.

“We ended up meeting through a bunch of random mutual friends, and then all bonded over being fans of hip-hop,” Hansen remembers. “A lot of guys in jazz school are really, really focused on jazz, in terms of playing but also in terms of listening. The three of us were among the few that were listening to what was new in hip-hop at the time. That was one of the main reasons we first got together and started playing. We jammed together randomly one night at school and found out that we had a lot of fun playing together. Then we jammed some new hip-hop stuff totally on a whim, and also ended up recording it on a whim, and everything stemmed from there.”

In the wake of that came two albums (2011’s BBNG and 2012’s BBNG2) that featured originals mixed in with deconstructed covers by the likes of Nas, Gang Starr, and Kanye West. With III, the group suggests that jazz and hip-hop aren’t the only obsessions in the BadBadNotGood tour van. The band certainly learned something at Humber, as evidenced by new album tracks such as the rainy-day melancholia of “Differently, Still” and the frenetic experimentalism of “Confessions”. But the group also makes a successful effort to ensure that no one is able to pigeonhole it as a jazz trio. The skittering “Hedron” nods heavily to late-’90s techno, while the cinematic soundscape “Eyes Closed” is built around desert-glow guitar and slow-burn synths.

Ultimately, III signals a group that’s coming into its own. That bodes well for the future, considering that the initial buzz around BadBadNotGood was about the band’s covers. The hype started with an early live clip called “The Odd Future Sessions Part 1”. Subtitled “A Jazz Tribute to Odd Future and Bangladesh Bastard—Orange Juice (Lemonade)—AssMilk”, the black-and-white video didn’t do much for the instructors at Humber College. That was perhaps partly because they weren’t used to seeing students performing jazz in pig masks, which Sowinski does when he gets behind the drums. (Memorably, Humber College brass responded to “The Odd Future Sessions Part 1” with such comments as “I didn’t find anything of musical value in this performance.”)

More enamoured with the band’s work was Odd Future enfant terrible Tyler the Creator, who tweeted “This Is Fucking Sick! Dave Brubeck Trio Swag.” Just like that, the members of BadBadNotGood were instantly the new cool kids on the block.

“The fact that Tyler the Creator posted it caused a 100,000 people to check it out in just a couple of days,” the bassist marvels. “Since then, everything has been pretty unexpected.”

Indeed, when the members of BadBadNotGood enrolled at Humber, they didn’t expect that they would one day be jamming with Tyler the Creator in Sowinski’s parents’ basement. They didn’t expect to be landing an artist-in-residence spot at Coachella, where they also served as the live backing band for shit-hot urban crooner Frank Ocean. And they didn’t expect to launch a thriving side career as hip-hop beatmakers, the band’s résumé including work on Danny Brown’s “Float On” and Earl Sweatshirt’s “Hoarse”.

Which one of those threads will impress you most depends entirely on your own musical predispositions. Hansen isn’t going to decide for you, mostly because he can’t believe any of it has happened.

“What I can say is that we’re really grateful,” he allows. “I don’t know if I can pick one specific moment that stands out above the rest—it’s more kind of like everything that we’ve done for the last three years has had so much response, and been received so well. We’ve reached so many more people than we ever thought we would.”

BadBadNotGood plays Venue on Tuesday (August 12).