Last year, when the Richard Alston Dance Company (RADC) was at the Festival Theatre in Edinburgh, everyone - the company and venue heirarchies alike - agreed it would be great if RADC could do more than just parachute in, for one night, and then whizz off again.

This year, that notion has taken hold: for the past few days, the company has been in residence at the theatre's bright, airy studio annexe that looks out onto Potterow.

The full performance is tonight, but members of the public, and school parties, have already been in to watch open rehearsals of some pieces - including the new work that will premiere in Edinburgh: Burning, choreographed to Liszt's Dante Sonata by Martin Lawrance.

Tuesday's open rehearsal had a special buzz about it - the run-through was going to be the first time pianist Jason Ridgway had played the Liszt, live, for the dancers. Lawrance joked that there might be shouts of "slow down, slow down" - there are moments when the pace is so thrillingly turbulent, frenetic, you expect the piano keys to fly up in the air ... and the dancers to collapse in breathless heaps.

But Ridgway has been playing live, on-stage with the company, for over a decade and has a rare understanding of how to pitch his own performance to that of the choreography. His piano, if you like, is the 11th dancer onstage and the music he makes is the most supportive partner of all.

Costumes for Burning were still in transit, so this rehearsal was in the greys and blacks that most dancers favour for practice clothes. The subject matter of Burning, however, is colourful: it's Liszt himself, and his relationships.

Lawrance talks of wanting to give himself a different kind of challenge after his last RADC piece, Madcap, which was set to a soundscore by Bang on a Can. The challenge stretched beyond setting steps to the Liszt sonata. "I found myself being drawn into a kind of narrative," he says. "And that was definitely a first for me. But when I started looking at Liszt, at his career as a concert pianist, at the effect he had on women - I felt that had to be at the centre of the movement."

He goes on to introduce Liszt in terms that could just as easily apply to a modern celebrity-cum-babe-magnet - Justin Beiber springs to mind. "Women really did throw themselves at him, wherever he went," laughs Lawrance. "They even coined a word for it, Lisztomania. Even if he wasn't the one doing the pursuing, he didn't say no to these women either. The problem was that he already had a mistress, Marie, and he did, genuinely, seem to love her. But he just couldn't let go of the other women so she left him.

"She wrote to him 'My brain is spinning again and I cannot stand it, I cannot live in this state of perpetual agitation ... so goodbye.' And that's the story that I've used to make Burning."

In rehearsal, these passions translate into expressive duets and fierce onslaughts by smitten groupies, all driven by a piece of music that doesn't lend itself to the kind of rhythmic counting that choreographers often use to tie steps to sounds. "We've just had to let go of all that," says Lawrance.

"Obviously, there are landmarks. A change of mood - and Jason brings those out brilliantly - or a phrase that you can latch onto. But this is where you have to trust your dancers. Because they've been creating characters, not just learning steps, they have a real understanding of the music. Maybe a lift doesn't always happen on exactly the same note, but it happens in a flow of music and movement that tells you what those people in the narrative are feeling."

Also new to Scotland is Richard Alston's Rejoice in the Lamb, one of the works he created to mark the Britten Centenary in 2013. As ever, Alston has steeped himself not just in the music, but in its history.

"I can remember singing the cantata at school," he says, "but at the time I knew nothing, really, about Christopher Smart who wrote the words that Britten had set." When he looked into Smart's own life and work, a story emerged that teased at his imagination and choreographic instincts.

Smart's poem Rejoice in the Lamb was written sometime between 1757 and 1763, during a lengthy sojourn as a "curable patient" in a lunatic asylum. "It's not clear exactly why his family had him put there," says Alston. "But he seems to have suffered from a religious mania that made him rush up to people in the street, pull them to their knees and ask them to pray with him. Which was a problem in St James's Park, but inside the asylum they don't seem to have been particularly bothered. Against all the usual rules he was allowed to have a pet, his much-loved cat Jeoffrey, who is celebrated in what is probably the best-known section of the poem."

Jeoffrey is now a dancing presence in Alston's choreography, as is the Mouse whom Smart describes as "a creature of great personal valour." The thought occurs, however, that the figure of Smart himself chimes in with a recurring theme in Alston's work; one that harks back to, for instance, Movements from Petrushka (1994). It's of the "outsider" who doesn't dance to a different tune from the others - but who hears it differently, finds other textures and images in it, and moves accordingly. You could, in fact, think of Alston himself in those terms for his musicality is paramount in shaping his choreographic ideas. He listens in several dimensions, and this has again come to the fore with Rejoice in the Lamb.

He said: "I suppose, because I knew the work and actually knew Walter Hussey, the remarkable man who commissioned it - my grandparents were neighbours of his when he was Dean of Chichester - this music was always at the back of my mind. But when I went into the studio, the dance flowed in a way I hadn't expected. Kit Smart's words just seemed to dance. His awareness of nature as a spiritual manifestation of God - like Gerard Manley Hopkins, centuries later - had its own musicality. And when you're working with wonderful dancers..."

Alston's voice is full of the pleasure he takes in watching young dancers come into their own powers through the work he makes, and never more so than when he revived Overdrive - made in 2003 to a full-on, thrumming score by Terry Riley - for the forthcoming tour.

"You know Martin (Lawrance) danced in it originally" says Alston, "and they did it brilliantly at the time - but the current company are all new to it, and watching them has made the piece completely fresh, even to me."

The Edinburgh programme also contains Holderlin Fragments. "More lyrics by a mentally disturbed poet" jokes Alston, adding that: "the music is, however, full of Britten's own sense of innocence and beauty". Rehearsals over, all four pieces come onstage tonight.

Richard Alston Dance Company are at the Festival Theatre, Edinburgh tonight and at Theatre Royal, Glasgow on November 11