The music school has fallen quiet, for the time being. It's a sunny Friday morning in the labyrinthine warren of rooms and corridors of St Mary's Music School in Edinburgh. The only sound to be heard at first is Schubert. In the small chapel, five students - Richard, Sophie, Alexander, Hugh and Findlay - are rehearsing one of his quintets for a forthcoming evening concert at St Mary's Episcopal Cathedral, on the other side of Grosvenor Crescent.

By the time you read this, though, this former theological college will be alive with the sound of music: the new term will have begun.

St Mary's advertises itself as Scotland’s only independent specialist music school, and one which offers a "world class" standard of music and academic education. Its 83 pupils are aged between nine and 19. There are 22 choristers (St Mary's is the Choir School of the Cathedral), all of them day-pupils. The other 61 are instrumentalists, of whom 34 are boarders and 27 are day-pupils. Entry is by audition and assessment, based on musical ability and potential.

Many former pupils have gone on to make their name internationally, among them pianist Simon Smith (a "phenomenon", in the words of one critic), composers Helen Grime and David Horne, guitarist Paul Galbraith, conductor Garry Walker, and singers Susan Hamilton and Monica Brett-Crowther.

Cellist Philip Higham has just recorded the Bach cello suites. And Steven Osborne is one of the world's leading concert pianists, says the school's headteacher, Dr Kenneth Taylor.

Alexander Armstrong, the comedian and TV presenter, was a chorister between the ages of 11 and 13. Daisy Chute, part of the All Angels classical crossover group, also studied at St Mary's.

"We've been here for 20 years and have gradually upgraded the premises," Dr Taylor says as he takes The Herald on a tour. Coates Hall, to give the building its official name, is a splendid if cramped Victorian edifice, with tranquil gardens.

It's hard to think that Haymarket station is just two minutes' walk away. "The standard of accommodation is quite high," he adds, opening the door to a student bedroom. "We certainly compare well to other boarding music schools in Britain."

As we head for some of the classrooms, we hear John Cameron, the school's Head of Keyboard, at work. "We'll leave him to his Bach," smiles Dr Taylor, as he heads for his office.

The academic subjects at St Mary's range from English to art, mathematics, sciences and various languages. The musical curriculum offers tuition in all of the classical instruments you would expect, but more modern instruments - guitar, saxophone, jazz piano - are also catered for.

"The school," Dr Taylor says in his office, "started out rather modelled on the Yehudi Menuhin School [in Surrey] whereby it was strings and piano, but we offer high-quality tuition regardless of the instrument.

"Our focus remains on Western classical music but we do offer such things as jazz - every pupil does jazz for two days each month. We also have the occasional masterclass in traditional music."

Dr Taylor is 50, and has three children between the ages of 11 and 17. A chemist by training, he was educated at Dulwich College and studied at Edinburgh University. His academic background is in university science research and education.

For more than 17 years he held posts in various Scottish schools; and before arriving at Coates Hall he was depute head at Biggar High School, in South Lanarkshire.

"Biggar is a fantastic school but quite small and I was looking to move to a larger school. I've always had a passion for music [he's a keen singer, and plays the viola, piano and organ] and had set up choirs in some of my previous schools. I saw this job advertised.

"I thought it was unique, and did some research into the school and was delighted to find its provision is available to anyone with the talent, regardless of their financial circumstances.

"I suppose it was that combination of a love of teaching, a love of working with young pupils and my love of music that brought me here."

In terms of St Mary's, he continues: "We are ambitious. My ambition is to have a state-of-the-art music school where the facilities match the very high standards of academic achievement and music teaching. But one thing we don't have is a decent space to rehearse the orchestra and larger groups. That takes place at the moment in the dining hall."

Possibilities that have been explored include building on the current site, or moving elsewhere.

One intriguing possibility that has been raised concerns the city's former Royal High School. There is a proposal to turn the venerable building into a boutique gallery hotel, but the Royal High School Preservation Trust, waiting in the wings, has proposed St Mary's occupy the building instead.

The trust's website carries a quote from Alexander McCall Smith: the music school, he says, is a "wonderful institution that does so much for the musical life of Scotland. Nothing could be more suitable than to have it occupy a building that was once a school."

"The move [to the Royal High School] would fulfil many of our aspirations and would be hugely exciting, putting the school in a much more prominent position in Scotland's capital," Dr Taylor says now.

He believes a relocation to Calton Hill would establish St Mary's on an international platform; he speaks of the ability that would arise to host open masterclasses, workshops and concerts. No longer would the instrumental performance space be limited to the current chapel, which can only accommodate 60 people or so. But all of that, of course, lies in the unpredictable future.

Dr Taylor says that Scottish Government funding, up to 100 per cent, is available to help with the cost of tuition and boarding fees. "What I want to get across is what's on offer here, and the fact aided places are available. Any young musician with the appropriate talent and commitment can come here. We have some pupils whose parents pay nothing at all. If parents are wealthy, they pay the appropriate amount. It's obviously expensive, but we're proud of what we have achieved here."

Pupils who graduate from St Mary's go to music colleges, universities or conservatoires in the UK or abroad. The headteacher pulls up a map on his computer screen to show the geographical spread of instrumentalist pupils.

One is from Barcelona, others from the north of England. One, from Shetland, studies the trumpet; another, from Orkney, the piano. There are three Chaimbeul siblings from Skye. "We've had the other two sisters as well." Dr Taylor says. "They're all unbelievable traditional musicians.

"I think our reputation is an international one," he adds. "We have produced musicians of international calibre. Helen Grime and David Horne, for example, have had their works broadcast on the BBC and performed at the Proms.

"There are people in orchestras and teaching in conservatoires throughout the world and we hope to improve our network with them. That's how we'll get the good news about the school out there. I would love to have more international students here: it would raise standards that are already very high."

He talks about the school's outreach programme. A masterclass was delivered at Glasgow's Royal Conservatoire of Scotland earlier this year, with Professor Felix Andrievsky; last year, the world-renowned cellist Steven Isserlis took part in a masterclass in Edinburgh. Both helped to raise the school's profile further still.

Dr Taylor enthuses about one of the perks of his job: regular meetings with Sir James MacMillan, one of the school's distinguished Vice-Presidents (Isserlis and Dame Evelyn Glennie are others), and "the second most performed composer on the planet - to meet him is incredibly exciting".

As I leave he gives me a copy of a CD made by Causeway Trio, which was formed by three former pupils: Pàdruig Morrison, Peter Thornton and David Swan. Morrison is now studying music at Edinburgh University; Thornton and Swan are studying trombone and jazz piano respectively at London's Guildhall School of Music and Drama. Three more names to look out for in the years ahead? The omens are promising.