'His music is so utterly original, enjoyable, perfectly constructed but at the same time emotional...

For me it's the best of any composer living today." High praise indeed coming from one György Ligeti - but then it was Ligeti who first introduced the world to the weird and wonderful music of Conlon Nancarrow.

Nancarrow (1912-1997) was a one-off. He grew up in a leafy neighbourhood in Texarkana, Arkansas, a few blocks away from where Scott Joplin was born half a century earlier. As a kid he played trumpet in a local jazz band and started composing semi-formally around the age of 15. Eventually he studied music in Boston where he met Schoenberg (whose music he did not like) and joined the Communist Party. He fought against Franco in the Spanish civil war, returned to New York to compose full-time but soon fled to Mexico to avoid anti-communist harassment by the US authorities. He ended up staying in Mexico City until he died.

Nancarrow's self-imposed exile took two forms: political, but also musical. During his brief stint in New York he wrote music that was broadly considered unplayable, and before long decided to stop writing for conventional instruments more or less altogether. In 1947 he bought a player-piano - a pedal-pumped mechanical keyboard that was common in American parlours before the age of the synthesizer - and the 50-odd studies he wrote for this peculiar instrument eventually built his reputation as one of the most distinct and brilliant American composers of the 20th century.

What makes these player-piano pieces so special? A lot of it has to do with their outlandish rhythms, which bounce about like ping pong balls in a tornado. The effect is funny and mind-bogglingly complex all at once. Somehow, despite being written for a mechanical instrument, there's a deeply human frailty to this music.

"I first came across Nancarrow in the 1980s," says Irvine Arditti, leader of the Arditti Quartet, "when we played his First String Quartet. It's a very jazz-influenced and very, very difficult piece. There are incredibly complex rhythms that overlap all over the place. One member of the quartet will be playing in three, another in four, another in five, and we only synchronise on occasional downbeats. In fact, it was hearing bad performances of this quartet that made Nancarrow decide to stop writing for conventional instruments and go into isolation. I can't believe that no quartet could have played it, but perhaps no good quartet wanted to play Nancarrow at that time because his genius hadn't been recognised yet."

The Ardittis performed the First Quartet at the Almeida Theatre in London, with Nancarrow, then in his 70s, in the audience. "Afterwards he came up to us and said he'd been bowled over by the performance," Arditti recalls. "Then about a year later he got in touch and said 'I've got a present for you...' He had transcribed some of his player-piano music and turned it into his Third String Quartet." The Ardittis premiered the Third Quartet (a Second was never finished) in Cologne in 1988. At the concert Ligeti ushered Nancarrow onto the stage and led the applause for him.

I ask Arditti why he thinks Nancarrow was drawn to writing such extravagantly complex music. "I don't know why," he says, "but what's interesting is that he was doing it on his own. Ligeti kind of discovered him, and was amazed that they had both been working on similar techniques but in totally separate spheres.

"Of course, there were other composers, like Giacinto Scelsi, who were also recluses," says Arditti. "The latter part of the 20th century lent itself to such individuals. After the breakdown of total serialism's rules and regulations, what was required was individuals. Individuality is what gives richness and colour and credibility to contemporary music."

Irvine Arditti should know: his quartet has been championing contemporary music since the 1970s, and it's thanks largely to them - and their American contemporaries, the more crossover-orientated Kronos Quartet - that new string quartet repertoire continues to flourish. Several excellent younger groups have since cropped up (the UK-based Smith Quartet, New York's JACK Quartet, Montreal's Bozzini Quartet) but the Ardittis remain every bit the elder statesmen.

At this year's Edinburgh International Festival they play Nancarrow's two string quartets as well as transcriptions of studies for player-piano, which Arditti describes as "particularly virtuosic - probably the most difficult works in our repertoire". Coming from the man who can trot out Stockhausen's Helicopter Quartet or John Cage's Freeman Etudes, that's saying something.

Also on their EIF programme is Janácek's First Quartet, the so-called "Kreutzer Sonata". Why the combination?

"Janácek wrote a lot of motoric rhythms in his music, and often the rhythmic textures are as important as the melodies," Arditti explains. "That's how we see it, anyway. And when we played at the Janácek Festival in Brno" - the Czech composer's birthplace - "a few years ago, the director of the festival told us that he loves how we play this music: he said it's how Janácek would have liked to hear it." The point is that the Ardittis approach Janácek as modern music. His quartet writing is whimsical and earthy, but it's also driven by the clanging, clattering rhythms of the machine age. "And in that way there are many connections to Nancarrow," says Arditti. "These pieces are all about velocity of movement."

The Arditti Quartet is at the Queen's Hall on August 27