The great Dutch conductor Bernard Haitink once described Amsterdam's Concertgebouw as the best instrument of the orchestra that it houses.

There are many iconic concert halls in the world: Berlin's Philharmonie; Vienna's Musiekverein; London's Royal Festival Hall; Reykjavik's new Harpa - the list could go on. Many have settings more dramatic or architecture more bling, but none can top the Concertgebouw's trump card: the best acoustics in the world.

What's more, those acoustics have helped to craft what is arguably the best orchestra in the world. The Royal Concertgebouw Orchestra returns to Edinburgh for two concerts during the closing week of the International Festival.

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Their visit last year brought a towering performance of Mahler's Ninth Symphony under Mariss Jansons. This year the orchestra and Jansons offer two unreservedly sumptuous programmes: first the shimmering colours of Ravel's Daphnis et Chloe, then the heft and charisma of Strauss's Ein Heldenleben. The concerts will surely be a highlight of the 2014 Usher Hall programme.

But back to Amsterdam, to that building and to its unique symbiosis with its house band. Would the orchestra's famously plush sound have been incubated in a lesser hall? Who can say. Would the acoustic have gained its global reputation were it not for the orchestra's century of top-class touring and recording? Probably not.

The handsome neoclassical Concertgebouw (literally concert building - typically understated) was constructed in the 1880s on the soggy fields of Nieuwer-Amstel, then the outskirts of Amsterdam. A competition was put out to find the best architect for the job, and although Adolf Leonard van Gendt had never built a concert hall before - and admitted he knew nothing about music and had no ear for acoustics - he won the commission chiefly because of one idea. He suggested the Concertgebouw should have a garden and a bandstand where there could be outdoor concerts and a bar to provide a lucrative side income for the concert hall.

In the early days, before the noise of 20th century street traffic closed it down, such a garden did indeed exist out the back of the Concertgebouw. It was lined with poplars, rose trellises and gas lanterns. The bandstand provided entertainment at cheap ticket prices. There were even so-called "fence subscribers" - locals who hung on the fence to hear the music free. After the arrival of trams in the 1920s the outdoor concerts were deemed too noisy, and in the 1940s the garden was sold to the city for housing.

When it came to designing the building itself, Van Gendt essentially hit on a major stroke of luck. He decided to model his classic shoe-box design on the main concert hall of the Neue Gewandhaus in Leipzig, although he had never actually been there. Granted, the Concertgebouw's acoustics took some time to perfect: at first there was too much echo and the brass were too dominant. But after some trial-and-error experiments, he somehow stumbled on the perfect formula. (The art of acoustic design is exactly that: an art, not a pure science. The wonderful thing is that even in this day and age of fearsome technical precision, there is still a mystique around what makes for perfect acoustics. And as so many vastly expensive and duff-sounding new concert halls prove, it is still easy to get it wrong.)

The Concertgebouw was officially opened in 1888 with a concert of orchestral music by Bach, Beethoven, Wagner and Handel. Above the main entrance stood the busts that are still there today: Bach, Beethoven and Sweelinck, sternly surveying the Amsterdam audience from between stately columns. Presiding over the whole building is the iconic gold-painted lyre.

Visiting the Concertgebouw today is a thrill, partly because nothing much has changed. There is some wildly 1980s interior decors and a controversial bar extension that was added during major renovations 25 years ago.

The surrounding streets have been developed from marshlands into one of Amsterdam's most exclusive neighbourhoods, lined with jewellery shops and expensive restaurants. But above all, the building - like its orchestra - feels rooted in a very rich, very slow-changing tradition.

In its 125-year history, the orchestra has only had six chief conductors. The first four were Dutchmen: Willem Kes, Willem Mengelberg, Eduard van Beinum and Bernard Haitink. Then came the Italian Ricardo Chailly and Jansons, a Latvian. Jansons took over in 2004 and recently announced he will be standing down for health reasons. The long, careful search for the seventh chief conductor is under way.

The players, too, tend to stay put. There is a plaque in the building that lists the names of players who have been in the orchestra for 25 years or more - and it is a long list.

One of the most recent additions is the American horn player Sharon St Onge, who joined in January 1982. She had come to Amsterdam as a student ("in the 1970s there was nowhere better to go as a horn player," she says) and she never left. Part of the allure to stay, she tells me, was quality of life in Amsterdam: the bikes, the social welfare system. But mostly she stayed because of the orchestra.

It certainly is not a money thing. Salaries for Concertgebouw musician are not particularly high (they earn roughly half what the Berlin Philharmonic players do). "Yet we have some of the best orchestral players in the world" says St Onge. "They could go elsewhere, but they choose to stay. Maybe it has got something to do with the healthy and egalitarian life here."

She adds: !Amsterdam is like any European city in that it is getting expensive to live here and ordinary people are being pushed out of the centre. But that means a 20-minute bike ride to work rather than a five-minute bike ride. Compared to London we have got it pretty good. I guess that plaque on the wall speaks for itself."

As for the building? Funnily enough, although it is the ideal venue in which to listen to orchestral music, it is not always the ideal venue in which to play it. "It is almost too forgiving," says St Onge. "It beautifies the sound almost too much. We are not talking crystal precision - more an overall glow. That is great for the audience, but for the players it is tricky. We can't hear the other side of the stage. But we learn to adapt."

St Onge laughs about her typically undemonstrative fellow Amsterdamers, who tend to be matter-of-fact about their top-flight orchestra. "People are not inclined to look up to us," she says. "Some musicians find that hard to stomach - they want a little bit of status. But most of us like how normal it feels to work in this orchestra. We take our music making incredibly seriously, then we pedal home on our bicycles at the end of the concert. I like that."

The Royal Concertgebouw Orchestra is at the Usher Hall, Edinburgh, on August 27 & 28