You never forget your first time. Sitting in the Concertgebouw in Amsterdam waiting for its resident orchestra to start a concert when you have never experienced the hall's astonishing acoustics before is a dream come true for any epicure of orchestral music. After all, it wasn't so long ago that the Concertgebouw Orchestra (the symbiosis of the orchestra and the hall is suggested by their shared name: both came into being 125 years ago today - 11 April 1888) was voted the world's best by Gramophone Magazine's panel of critics. Just being in the place where Gustav Mahler enjoyed some of his greatest successes as conductor and composer is a music-historical thrill. But none of that prepares you for the sonic shock of what the hall really sounds like. The first concert I heard there included Debussy's La Mer, conducted by Mariss Jansons (astonishingly, Jansons is only the sixth Chief Conductor in the Concertgebouw Orchestra's history; the second, Wilhlem Mengelberg, was in place for half a century, from 1895-1945). I've never heard a performance in which the colours of textures of Debussy's marine onomatopoeia sounded so thrilling, in which the mere sound of a gong or a celesta was transfigured into musical poetry simply by being played in that space by those musicians. Every sound the orchestra made, from the double-basses to the percussion, was perfectly mixed and blended by the acoustic of the hall before it reached my ears with a velvety sensuality.

Mariss Jansons, chief conductor of theRoyal Concertgebouw Orchestra. Photograph: Marco Borggreve /Free PR Handout

It's tempting to be jealous of the Concertgebouw players. After all, which orchestra wouldn't sound magnificent in their hall? And when you've a turnover of conductors and players that's a glacial as it is in the Concertgebouw (as one of the musicians said to me as we passed a panel with a long list of the names of the Concertgebouw players who had been there for more than 25 years, "once you get a job with an orchestra like this, you tend not to leave"), there is a guaranteed musical and cultural continuity. That's the reason the Concertgebouw Orchestra is capable of the minor miracle of making every hall they play in around the world sound just a little like their home turn: even in London's Barbican Centre, a world away from the Concertgebouw, they make their sound sing and soar.

But all is not as it seems. That much-vaunted acoustic in Amsterdam is actually a bit of a nightmare for the players. Seriously. When they're on stage, the musicians can't actually hear one another properly. At the geographical extremes of the orchestra, if you're trying to co-ordinate, say, the double-basses over at the right-hand side with the horns on the left, you can't rely on your ears. The players know that if they wait to hear their colleagues over at the other extreme of the ensemble, they will be too late, so they have to compensate by playing what they think is a fraction early. Something weird happens in this supposedly perfect place for orchestral music: to the players on stage, their playing doesn't sound nearly as perfect as it does to the audience. As Dominic Seldis, the Concertgebouw's principal bass, told me: "Even the greatest diamond needs polishing. And it's true that it's difficult to hear on stage here, but the hall corrects itself. If you sit in the audience, what you hear is a warm glow of sound. On the stage, the music might not sound together, but when it's projected out to the audience, it works."

That means the Concertgebouw players are really masters of an orchestral illusion, because they are continually trying to transcend the limitations of the hall's on-stage sound when they play. In fact, the world-conquering warmth of the orchestra's sound is a direct result of their sonic battle with the acoustics of the hall: that tension produces the uniquely Concertgebouw combination of musical alertness - which the players need to co-ordinate their ensemble - and the sensual halo that's made by a slight inexactitude between the sections of the orchestra.

Exterior shot of Amsterdam's Concertgebouw Photograph: Leander Lammertink

Dominic Seldis again: "when you have ten violinists, all with a slightly different tempo in their brain, but they're approaching the emotional and expressive qualities of a phrase with the same intensity – well, it's an amazing sound … If you're of the mindset that everything has to be right [or exactly in tempo, like a robotic click-track], this isn't the band for you."

So here's a quick celebratory revelation of the Mahler interpretations of Mengelberg, van Beinum, Haitinik, Chailly, and Jansons, performances that are some of the bedrocks of the Concertgebouw's success. As you'll hear, it's a story of sonic continuity but creative flux in the vastly different approaches to the composer from each of the orchestra's chief conductors. That's just tip of the iceberg, of course; to really get stuck into the Concertgebouw's story, you'll need these volumes of their recorded history. But by way of a Mahlerian amouse-bouche: here's Mengelberg's Dionysian Mahler 4, van Beinum's classicism in the same symphony, Haitink's combination of the visionary and the structural, also in the 4th, the clarity and wildness of Chailly's 7th, and the sheer epic intensity of Jansons's 2nd.

Happy Birthday Concertgebouw. Here's to the next 125 years.