A battler with a baton

I find him after rehearsal drinking cups of tea in his dressing room. The first thing that strikes most people on meeting Gergiev is his brooding intensity, the way that his presence fills an entire room – or, for that matter, an entire stage. He's a craggy kind of handsome, his voice a low Russian growl and his gaze unflinching. But he can be courteous too, even gentle. At 59 his sunken eyes and grey-specked grizzle vouch for the hours he usually keeps.

The LSO's concerts at this year's International Festival paired symphonies by Szymanowski and Brahms – a pairing that left a fair few audience members scratching their heads. Gergiev is happy to talk through the pitfalls of Szymanowski's overloaded scores: "It's very difficult for any conductor to make a strong case for the first symphony; we tried our best." He's also upfront about just how little experience of Szymanowski the LSO had at the beginning of the run. "They've grown through the week," he says. "They've come closer to the composer. It's how we did the cycles of Prokofiev and Shostakovich too – we worked and worked and worked them. Right now the LSO isn't the best Szymanowski orchestra in the world, but once we've taken these programmes to London and Paris they probably will be."

Gergiev seems to follow a theory of brute exposure: play enough notes enough times and they're bound to improve. "During [Szymanowksi's] Third Symphony last night I felt it coming together. Our playing made the public really listen and totally concentrate. I feel it in the air if the public is not concentrated. I hear if they're making noises or coughing too often. I think yesterday was a good experience." Few conductors would so readily risk waiting until the third in a series of four concerts for the playing to come together.

Gergiev returns to Edinburgh for the closing weekend of the International Festival, this time as artistic director of the Mariinsky Theatre. He's held the position since 1988 when it was still called the Kirov, and through sharp political nous and that ferocious work ethic he has rekindled the company's international clout. The bulk of his efforts have gone into operatic and symphonic programming, but as chief conductor he also tours with the ballet company.

Does his approach change when he's in the pit accompanying dancers? "I am aware of the presence of the dancers," he says, a little off-hand. "But the music is all-powerful. No dancer in the world can ignore Prokofiev when dancing Romeo And Juliet or Cinderella. We all have to follow the music: dancers, musicians, choreographer, conductor, designer, costume designer. In terms of theatre, a good choreographer tells the story clearly. But listen," – he seems ready to move the topic on – "the quality is always high at the Mariinsky. We have famous ballerinas. I don't think there's any reason to be worried." I hadn't expressed any worry, so perhaps the last comment was mainly for his own benefit.

Last year Gergiev was appointed Honorary President of the Edinburgh International Festival, a position only previously held by Charles Mackerras and Yehudi Menuhin. Certainly his relationship with the festival goes back long enough – back to the early 1990s, when he first started bringing huge works by Mussorgsky and Glinka, Prokofiev and Rimsky Korsakov. Some have questioned whether a conductor so busy and so volatile is the right man for the job; whether the festival might have chosen someone with more time and energy to devote.

Gergiev just shrugs and says he shares the festival's goals. "In its origins this festival was a message for peace and co-operation after the Second World War. You cannot burn in the flames of war the symphonies of Beethoven or Tchaikovsky." His voice quietens. "You cannot destroy the culture of Europe and Russia, the national traditions of opera singing, ballet dancing, instrumental playing. These are the collective survivors. We were very close to total disaster after that war and it should never happen again. I don't know how to help other than to bring very good music and theatre to as many people as possible.

"Last year we [Mariinsky Opera] performed Die Frau Ohne Schatten: music by Richard Strauss, who was famously involved in the Third Reich. But we represented St Petersburg, which was half destroyed by the Germans. Culture is more powerful than any political reality. Political leaders come, make mistakes, sometimes bring a lot of destruction, and then go. Great music lives on."

Gergiev reaches his conclusion and leaves a moment of silence hanging in the air. His PA indicates that my interview time is up. I pack away my recording equipment and put on my coat to leave, but Gergiev starts talking again. Not just talking – chatting.

He chats about food and whisky. He chats about the weather. He chats about driving up to the Highlands, about how hills make him feel at home. He grew up in the Caucasus and tells me that on a clear day he could see the tops of the mountains from his family house. He chats about politics, wants to know my opinions on Scottish independence and the future of Scottish Opera. He challenges the answers I give, and seems keen to be challenged back.

I ask whether he ever gets tired. "I started very, very young," he says, dodging the question. "I was 20 years old and thin like a ballet dancer. I was the winner of many competitions and it all just happened for me-" Later he moans that people only ask him to do vast projects. "It's always the complete symphonies of Mahler, the complete symphonies of Shostakovich or Prokofiev." Slight exaggeration, but telling. Gergiev has had doors open wide to him for decades. He seems bored with the ease of his success.

Finally he sighs and says he should probably look through his scores. Twenty minutes later he's on the podium conducting Szymanowski's Fourth Symphony, its monumental colours sounding light years more vivid than the LSO's ropey account of the First Symphony just a few nights before. After the interval Gergiev pulls apart Brahms's Fourth Symphony with such insistence that the piece almost disintegrates. Why does he do it? Why does he push at such limits? Maybe this is how he sets himself challenges that nobody else will. It's a dangerous game, sometimes played at the expense of the musicians he conducts. It's a game he sometimes loses – he lost, in my opinion, this performance of Brahms's Fourth. But it's a game whose danger is a thrill to witness.

Cinderella is at the Festival Theatre from tomorrow until Saturday.