Five years ago, the alto saxophonist Miguel Zenón was performing with his quartet at a club in Chicago when he was contacted by Julien Labro, a French accordionist based in Canada. Labro was in town making a record with Spektral, a Chicago-based string quartet that specialises in contemporary music. He had arranged a piece by Zenón, a racing tune called 'El Club de la Serpiente', for the session, and wanted to know if he would have any interest in recording it with them. Zenón went to the studio, and instantly clicked with the quartet. 'The guys from Spektral were really on top of the music, which made the session very fun and easy,' he told me. ('El Club de la Serpiente' appeared on Labro's 2014 album From This Point Forward.) When the Hyde Park Jazz Festival commissioned Zenón to write a work for local musicians, 'naturally I thought of Spektral.'

Zenón, who is 41, had written for string quartet on one of his early albums, but had never plunged so deeply into the literature. In preparation, he read scores by Bartók, Beethoven, Schoenberg, Alberto Ginastera and Philip Glass, soaking up 'the style and sound' of the quartets he loved. But his ultimate aim was to honour the music and folklore of Puerto Rico, where he grew up; the songs used in religious services, rituals and dances.

The album that grew out of the commission, Yo Soy la Tradición, is an hour-long suite for string quartet and alto saxophone, in eight movements. Before I say anything more about this breathtaking album, let me emphasise what it is not: jazz-with-strings, the archetypal middle-brow, mid-20th-century genre in which a jazz soloist improvises over an orchestral backdrop that would otherwise be of scant musical interest. No insult intended: I rather like jazz-with-strings, especially if the soloist is Charlie Parker. But the strings in jazz-with-strings seldom did more than create a lush ambience – an aspirational signifier of class and refinement at the time, now a quaint signifier of the time and its aspirations.

Nor is Yo Soy la Tradición a work of 'cross-over' or 'fusion', in which the music of different traditions either awkwardly collides or walks hand-in-hand in a we-are-the-world partnership, usually to the detriment of both. Instead, Yo Soy la Tradición is a brilliantly realised work of chamber music that interweaves and celebrates the languages of the string quartet, jazz and Puerto Rican vernacular music. For all the swagger of the title, I think Zenón might have sold himself short: the music of his homeland is only one of the traditions he embodies. In his fluent and imaginative adaptation of vernacular music for the string quartet, he builds on the tradition of Bartók, Janáček and Shostakovich, who forged a distinctive modernism in large part out of folk material. Like John King's recent quartet Free Palestine, which is based on Arabic modes, Yo Soy la Tradición demonstrates the string quartet's exceptional ability to channel – which is not to say 'elevate' – styles far afield of Western art music.

The writing on Yo Soy la Tradición is taut and disciplined, leaving little to chance. Still, both Zenón and the quartet improvise at specific moments. 'One of the main goals with the project,' he says, 'was to put something together that blurred the lines between what comes off as notated, and what comes off as improvised.' The cellist Russell Rolen's haunting introduction to the penultimate movement, 'Promesa', is entirely improvised. And in the sections where Zenón improvises, the parts written for Spektral are more 'interactive', because he wanted to achieve a sound 'closer to the type of energy you'd get from a jazz rhythm section'.

Zenón's tone is as precise as it is voluptuous, and he often uses it to soaring effect. He also has a balladeer's feeling for melody, notably in 'Viejo', which alludes to the Aguinaldo Viejo, a medieval genre of Jibaro. But it's his intricate use of rhythm that supplies the album with its deepest fascination. He often stacks rhythms into what he calls 'rhythmic layers'. Riveting in their patterns, these layers are at first reminiscent of minimalism, particularly of Steve Reich's work, which Zenón admires. But he emphasises that his ends are almost the opposite of Reich's: 'I don't mean for things to sound jagged or out of sync. I want these rhythmic counterpoints to come together and lock into a groove, something that can be felt right away.'

Yo So la Tradición is a buoyant, often romantic work, but it is also a complex, even demanding one, so full of incident that it's easy to lose the plot. Zenón thinks of composition 'in terms of events, and recycling material and using it again in the exact same way it was used before is not something I enjoy.' What this means, in practice, is that each of the eight pieces shifts busily between time signatures, keys and moods, and none ends where you expect it to. It's enthralling, if sometimes exhausting, and there's a kind of respite when, in the last minute of 'Cadenza', the sixth movement, all the musicians except Rolen put down their instruments and clap a rhythm. As Rolen plays a fierce staccato line on his cello, the other members of Spektral begin to play pizzicato – moving from their hands to their fingers, and only then returning to their bows, as if to underscore the music's corporeal roots. This is chamber music that never forgets the earthier pleasures of dance. In the words of Ornette Coleman, another alto saxophonist who wrote memorably for strings, it leaves you dancing in your head.