'I guess I'm just a natural multi-tasker," says Iain Burnside as he downs a coffee and finishes telling me about overlapping key structures in Rachmaninov's collected songs.

"I like using different parts of my brain," he shrugs. "If I do nothing but play concerts I go crazy."

Burnside is a Glaswegian pianist, teacher, radio presenter, writer, programmer and - well, what else, Iain? "Oh I don't know," he says with exaggerated coyness. "I suppose you'd have to add playwright to the list." Indeed: he has written several theatre pieces for students at the Guildhall School of Music and Drama - one about Britten, one about Brahms and Clara Schumann, one about Ivor Gurney. The Gurney script, A Solider and a Maker, has now been commissioned as a radio play for BBC Radio 3 and will be broadcast this summer.

But playing piano with singers is what Burnside does most. He's one of the UK's most in-demand vocal pianists (he doesn't like the term accompanist: "When a violinist plays with a string quartet he doesn't become an accompanist, right? I'm still a pianist when I play with singers"; it seems a fair point). We meet for lunch in Wandsworth, south-west London, around the corner from the National Opera School where he has spent the morning coaching trainee opera singers. He barrels into the small restaurant, flamboyant and animated, already talking at a rate of knots. The entertaining twists and turns of the proceeding conversation make us both very late for our next meetings.

Burnside has lived in London for more than three decades but still speaks with an unmitigated Bearsden lilt. He's a proud Scot: don't get him started on the lack of an expat vote in the referendum, or on why he isn't asked to play in his native country more often. He's more cheerful talking about his family roots. His grandfather conducted the Glasgow Gaelic Music Association for 50 years - "I remember singing in the Mòd, grandpa conducting, me looking out to try to find mum in the audience. Until I was about 16 I never performed in anything other than a kilt. That was all grandpa's influence."

He happened upon a very good piano teacher in Bearsden, a certain Mrs Campbell, who had been a pupil of Artur Schnabel. Despite little musical encouragement at Glasgow Academy ("I'm the only professional pianist I know who never played in a single school concert) he went on to study music at Oxford and a postgraduate at the Royal Academy of Music in London. But it was the two years he spent in Warsaw that really whipped him into shape. He arrived in 1979 just as the Solidarity movement was gaining force. "I've never worked so hard before or since. My teacher was in her 80s - an incredible woman with a terrifying work ethic. Playing music was a serious matter. It wasn't a luxury option."

He returned to London in 1981 and soon found himself immersed in vocal music thanks to a chance encounter with the tenor Peter Pears. They met at a concert and Pears asked him to play for his song classes in Aldeburgh. "I ended up playing for a whole generation of emerging singers - suddenly that's what I was doing." It wasn't planned but when he looks back it makes sense: he has always been interested in languages and poetry, "and let's face it," he says, "I have no aspirations to play piano concertos. Even if I practised every hour of every day I would never be Martha Argerich. That's fine with me."

The teaching, the writing, the broadcasting - all that "just sort-of happened" along the way. "The first time I went on radio my mum said, 'listen, you've been a bullshitter from day one; you might as well get paid to do it.' Anyway, it's a lot easier to talk about music than play it." Now it's the variety that keeps him on his toes. "I'm pleased that at my advanced stage I still don't know what I want to do when I grow up. If you start phoning it in - that's when you start to atrophy."

We've officially met to talk about Burnside's latest recording project: an extraordinary triple album that brings together seven different singers, all native Russian speakers, and covers all of Rachmaninov's songs. It's the first time the complete set has been recorded in 25 years. "Usually I'm a cherry picker, not a completist,but this project deserved the attention."

He's right: the set makes for a passionate, potently atmospheric and - for listeners like me who only knew the most famous songs - often revelatory collection. Rachmaninov didn't write his 73 songs as a cycle but they work beautifully in sequence, with moods and keys that grow and intertwine.

The vast majority are hardly ever sung, especially outside of Russia, "but there isn't a single duffer amongst them," says Burnside. "On paper a few look like monolithic slabs where the piano plays clunky chords and the voice declaims. But get a Russian singer who understands not just the language but the whole tradition of rhetoric and it makes total sense."

The singers for the project (sopranos Ekaterina Siurina and Evelina Dobraceva; mezzo Justina Gringyte; tenor Daniil Shtoda; baritones Andrei Bondarenko and Rodion Pogossov; bass Alexander Vinogradov) are all in their 40s or younger. "They belong to a generation who have strong links with the States and Europe but retain a Russian sound and heritage. For them, these songs mean everything."

This is the first time that Rachmaninov's complete songs have been recorded in 25 years.

Why has it taken a Scottish pianist and a Scottish record label to make it happen? "Because we were up for it!" says Iain Burnside. "A lot of record companies are too timorous these days, but Paul Baxter at Delphian took a punt on it. We recorded at the Queen's Hall, which was splendid. It worked out much cheaper to bring the singers to Edinburgh than to host them in London. After the sessions we took them for cocktails in the New Town. They loved it."

Rachmaninov wrote all of his songs between 1890 and 1916: he abandoned the form after he fled revolutionary Russia. For him, says Burnside, songs belonged to the world of the dacha, the forested estate. "Once the cherry orchard was cut down he couldn't go back." In his sleeve notes Burnside describes the last in the collection with real poignancy. "The piano postlude ends not with a bang but with a whimper, on a final, soft, ambivalent dissonance. A hundred years later it still waits to be resolved."

Rachmaninov's complete songs is out now on Delphian Records