"I FEEL that people are searching for something that is a bit more open. They want something else," says Kamasi Washington when asked why his music has resonated so widely. The Los Angeles saxophonist and composer is jazz's biggest success story in years, winning rapturous acclaim for his 2015 debut The Epic, a vastly ambitious three-disc set that puts a contemporary spin on the jazz, funk and fusion of the late '60s and early '70s.

Steeped in the West Coast jazz tradition – his father Ricky is a saxophonist and music teacher – the 34-year-old is equally well-versed in hip-hop and r 'n b, having played as a sideman with Snoop Dogg and Erykah Badu, and contributed horn parts and string arrangements to Kendrick Lamar's 2015 jazz-rap masterpiece To Pimp A Butterfly. Another contributor to that album, experimental beat-maker Flying Lotus, released The Epic on his Brainfeeder label, expanding its reach beyond the jazz world.

While jazz peers such as Robert Glasper and Soweto Kinch (who plays Glasgow Jazz Festival on Wednesday June 22) have explicitly worked hip-hop elements into their music, Washington's engagement with the genre is more subtle. "The first tour I ever did was with Snoop and so for years I was playing more hip-hop than I was playing jazz really. There was an emphasis on phrasing and timing and where you place what you play. That sensibility definitely influenced the way I play jazz."

A grounding in jazz, gospel and hip-hop is what gives Washington and his fellow members of the West Coast Get Down collective their distinctive sound.

"I think that our music is very personal and that maybe comes from the fact that we grew up together," he says. "It feels very open and I think people attach to that. At the same time I think that we've all played so many different styles of music, part of that is in our music, so it feels familiar to them as well. It's like, 'Oh, it kind of feels like something that Snoop or Chaka Khan or Lauryn Hill or all these other people that I listen to would make, but it also has this other thing that I haven't heard before', like a mixture of familiarity and exploration."

The Epic may be Washington's debut as a leader, but it was several years in the making, having emerged from marathon recording sessions in which he and his band The Next Step laid down over 190 tunes. Forty of those were Washington's compositions, 14 of which appear on The Epic. The stylistic breadth of the album is remarkable, taking in spiritual jazz, hard bop, gospel, soul, funk, fusion and classical music. Washington and his bandmates contribute impassioned, wildly virtuosic solos, while celestial choirs and strings elevate the music to a cosmic realm. There's also a trio of uplifting r 'n b songs featuring vocalist Patrice Quinn, and a gorgeously languid take on Debussy's "Clair de Lune".

"We could do what we wanted to do, there were no restrictions," recalls Washington. "It was one of the first times we'd all been able to commit ourselves to making music with each other like that, so it felt like the possibilities were endless. I felt that we had a sound and a style and an approach that was unique and I wanted to document that. And there were so many facets to it, but at the same time, it's united, it's one thing."

Although The Epic is largely instrumental, it communicates a strong sense of pride in African-American cultural traditions. As a result, it has been seen as one of the foremost expressions of the Black Lives Matter movement. Washington is proud to represent Black Lives Matter, but he is keen to note that while the movement may be new, the struggle against America's structural racism is not.

"It's something that's been an issue for my whole life. So it's deeply embedded into the music and the feeling," he says. "Jazz, any instrumental music, communicates in a different way to music that has lyrics. It's an energetic level, it's almost like when you listen to a person playing an instrument, you're feeling their experience, so it's like, I can talk about police brutality, I can also share with you the feeling that you get from it. What does it feel like to have the police be the people that you fear the most, even though you're not a criminal? For no reason other than the way you look. And it's like okay, I want you to know that, and this is how it feels."

Since the album's release in May last year, Washington has been touring extensively with an eight-piece version of the Next Step, and now he is bringing his band to Scotland for a Glasgow Jazz Festival date. Their UK debut at the cavernous Barbican theatre was the hot ticket at last year's EFG London Jazz Festival, and they didn't disappoint. Powered by Tony Austin and Ronald Bruner Jr's double drum athletics, the Next Step brought a fresh energy to tunes from The Epic, contrasting Washington's beatific poise with Cameron Graves' outrageous funk synths and Miles Mosley's growling wah-wah bass. So potent was their sound that people were dancing in the aisles; not a sight you see at the average jazz gig.

"Yeah, a lot of people dancing," grins Washington. "They really respond to the music. It's reaching them in the way we hoped it would, it's really cool."

Kamasi Washington & The Next Step play Queen Margaret Union, Glasgow on Monday June 27. www.jazzfest.co.uk/