At the first performance of The Rite of Spring in May 1913, even well-informed members of the audience had no idea what they were experiencing. Afterwards, the critic of the French periodical Gil Blas recorded a conversation about the beginning of Stravinsky's prelude to the ballet: "'What instrument produces these sounds?' I say, 'It's an oboe.' My right-hand neighbour, who is a great composer, insists that it is a muted trumpet. My left-hand neighbour, who is no less of a musical scholar, declares, 'I should have thought it was a clarinet.' In the interval we ask the conductor himself, and we are told that it was a bassoon that gave us such heart-searchings."

We have it on Stravinsky's own authority that the only person in Paris who understood, completely and immediately, The Rite of Spring was the composer Maurice Ravel. The rest of the audience, apparently, were placed in a position where they could not even recognise a single note, played unaccompanied, by a standard instrument of the symphony orchestra. Their confusion proceeded from there.

Many of the accounts of that famous first night focus on the conditions of near-riot that met the performance. The audience had been well set up by the impresario Sergei Diaghilev, Diaghilev's lover, star dancer and choreographer of the Rite, Vaslav Nijinsky, and their company, the Ballets Russes. The new Theatre des Champs-Élysées was luxurious, and had been forced to pay through the nose for the company – 25,000 francs a performance, which shortly afterwards sent the manager, Gabriel Astruc, into bankruptcy. The audience was partly made up of advanced musical and intellectual society, but partly of the upper crust, including the immortal comtesse de Pourtales, who was recorded shouting "I am 60 years old and this is the first time anyone has dared to make fun of me." The programme was, by normal standards, disastrously ill-planned. The Rite was preceded by the soft-centred Les Sylphides, and followed by the new popular favourite Le Spectre de la Rose, a Nijinsky vehicle. The evening finished with the second act of Prince Igor, including the "Polovtsian Dances", previously considered a searing portrayal of the manners of primeval Russia. Diaghilev's motives were plain. He wanted to lure in the fashionable world, and confront it with his most advanced material.

The riot that followed is amusing, including the man who was so seized with excitement that he started to "beat rhythmically on the top of [Carl van Vechten's] head with his fists", and the argument between inhabitants of two adjacent boxes which turned, the next morning, into a duel. There had been other riots in Parisian theatres over the years, such as at the premiere of Victor Hugo's Hernani in 1830 and at the Paris premiere of Wagner's Tannhäuser in 1861. As with the Rite of Spring debacle, what interests us in these ancient battles are the limitations that gave birth to the outrage; in the case of Hugo, the violation of classical rules of unity of time and place, in Wagner's, the outrageous suggestion that the ballet which must take place in every opera might, on this occasion, be placed in the first act rather than the second.

The Rite of Spring's violations were more fundamental than that. Some of them were merely apparent, rooted in traditions that were relatively unfamiliar to a 1913 Paris audience. The designer and co-author, with Stravinsky, of the scenario, Nikolai Roerich, was a faithful student of archaeology. His long journey, from Diaghilev's sumptuous, nationalist magazine of art, Mir Iskusstva, to a deracinated mysticism can be traced in the beautiful museum devoted to his life on West 107th street in New York. Some aspects of the score, even, can be traced back to established Russian traditions. It has long been known that many of its themes, including the famous bassoon opening, are versions of Lithuanian folk tunes taken from a Krakow anthology published in 1900.

The folkloric aspects must have been difficult to follow or understand after the cultural journey to a Paris premiere. Few people, even now, would guess that in the successive dances "Rondes Printanières" and "Jeux des cités rivales" – or as they are sometimes garbled in English, "Spring Rounds" and "Games of the Rival Tribes" – the "rounds" and the "tribes" or "cités" are actually the same thing. They are renderings of the ancient Russian ritual display of khorovod, sometimes but not always round dances, that fascinated Roerich.

And what about the revolutionary irregular metres of the ballet? Richard Taruskin, in an astonishing and enormous study of Stravinsky's debt to the Russian traditions, points out that the irregular 7/4 metre of the end of The Firebird, Stravinsky's first major piece for Diaghilev in 1910, is foreshadowed by any number of finales in Rimsky-Korsakov, such as the 11/4 finale of his 1880 opera Snegurochka. Russian music had long enjoyed eccentric metres, such as the 5/4 waltz movement in Tchaikovsky's Pathétique, and shifting metres, as in the alternation of fives and sixes in the first movement of Moussorgsky's Pictures from an Exhibition. The terrifying speed of metre change in the Rite of Spring is extreme, but has small precedents.

There were other innovations in The Rite of Spring to startle the audience – Nijinsky's choreography offended even the dancers, with its insistence on turning the toes in, and landing flatly (and painfully) after leaps. But the core of the piece is, and has remained, Stravinsky's great score. At the Ballets Russes's first 1909 season, French critics had issued a warning; the first offerings had been visually and theatrically spectacular, but the music had been "insignificant", "woven out of any old remnants", "unspeakably vile". "The young Russian composers had better get to work seriously," the Mercure de France had said.

Diaghilev had responded, and had laboured to find a Russian composer to create a startling new score for the next season the following year. Diaghilev first tried and lost the composers Cherepnin, Lyadov, Glazunov and perhaps even the little-known Sokolov. Unsuccessful, Diaghilev unwillingly turned to an untried product of the Rimsky-Korsakov circle, a world he disliked, for the proposed Firebird ballet. Stravinsky was initially "one of the Korsakovtsi, is he? But they're all alike." After a huge success, Stravinsky cut his links with his former teacher's family and pupils, and was placed at the centre of Diaghilev's world. With successive commissions of Petrushka and the Rite, and commissions for French composers of the level of Ravel's Daphnis and Chloe and Debussy's Jeux, nobody ever complained about the trivial music at the Ballets Russes again.

The score remains astounding, despite its familiarity. It draws analysts like moths to a flame for a simple reason: it is, clearly, harmonically coherent from beginning to end, but tantalisingly difficult to explain. There were more extreme harmonic experiments in the air – Schoenberg's first atonal works were five years old, and there are chords in Debussy and Richard Strauss as extreme as anything in the Rite. The Rite is clearly not atonal – there are passages in B flat minor, though it is often impossible to say what key the music is in. Careful analysis seems to show that it relies on combinations of evolved modes, like Debussy's whole-tone scale. Prominent, too, is the octatonic, or eight-toned scale, regularly alternating semitones and whole tones that many of Rimsky-Korsakov's pupils so loved, and which gives Firebird such a sumptuous, static, somewhat orientalising flavour. Whatever the technical means, the score has evolved a language which makes perfect, coherent sense on its own terms and defies easy analysis through recurrent themes, chords or even instrumental sounds.

Its excitingly new quality rests in its rhythmic invention, which at first Stravinsky could hear but had no idea how to notate – often, the notation of the Rite had to be revised in coming decades. As late as 1944 for some concerts in Boston, Stravinsky rewrote the whole of the "Danse Sacrale", doubling the note values to make it easier for an orchestra to understand. There is, too, the extraordinary instrumental invention that so bemused the first audience. Stravinsky was a great orchestrator from the start – the fortissimo chords punctuating the "Danse Infernale" in Firebird always seem to have more resonance than anyone else's, thanks to the skill of the instrumentation. The Rite of Spring discovers joyous new noises from familiar and unfamiliar instruments – the high pizzicato solo double bass in the "Adoration de la Terre", the low and sinuous alto flute, curious percussion instruments, such as the guero rattling through the "Cortège du Sage", and the pair of tenor tubas that crash with such dull, moronic insistence into the "Jeux de cités rivales". Stravinsky went on inventing orchestral sounds and textures all his life, from the unique ensembles of Les Noces and the Symphony of Psalms until the "Gaillarde" from Agon, scored for flute, mandolin, harp and lower strings, with double bass harmonics teasing the bemused ear at the very top of the texture. It was with the Rite of Spring, with a larger orchestra than he ever used again, that he first laid all his cards upon the table.

Like many of Diaghilev's commissions, The Rite of Spring made its way easily in the concert hall – it is a technical challenge that few professionally virtuoso orchestras could ever resist. The Ballets Russes themselves were never the same after the extreme experience of 29 May 1913. The Rite was performed only six more times, in Paris and London. Nijinsky, and the rest of the company, minus Diaghilev, left in August for a tour of South America. On the boat was also an aspiring dancer only loosely attached to the troupe, Romola de Pulszky. The boat departed on 15 August. On 25 August, Nijinsky and de Pulszky first spoke to each other, inadequately – they hardly had a language in common. On 30 August, Nijinsky sent an emissary who actually spoke French to ask de Pulszky to marry him. Once she established that it was a genuine offer, she accepted, and they married on 10 September in Buenos Aires.

When Diaghilev heard of the marriage, in remote Venice, he took the first opportunity to sack Nijinsky. Nijinsky's career was more or less over: his biographer Richard Buckle points out that "by the time of his marriage Nijinsky had created all his famous roles, bar one". Diaghilev tried to replace him in the Ballets Russes with a new star/lover, Léonide Massine, and to lure back the creator of the Spectre de la Rose and the "Polovtsian Dances", Michel Fokine. Fokine's condition, after a four-hour telephone call, was that none of Nijinsky's ballets should be staged under him. "When Diaghilev wanted to revive [The Rite of Spring] seven years later," Richard Buckle, writing in 1971, said, "no one could remember the choreography and Massine had to start again from scratch."

In the decades that followed, many choreographers created their own versions, including Pina Bausch, Kenneth MacMillan, Maurice Béjart and Michael Clark. This year, for the centenary, Sadler's Wells is hosting a two-month season of Rite-inspired works, including contributions by Akram Khan, Michael Keegan-Dolan and Mafalda Deville. The original choreography is no longer, as Buckle thought, lost. The scholar Millicent Hodson pieced it together over years of analysis of prompt books, sketches and photographs and detailed recollection, and in 1987, the Los Angeles Joffrey ballet gave a performance of the reconstruction – it has since been taken up by the Kirov. Speaking in the 1950s, Stravinsky said he would not conduct the Rite in the Soviet Union, since years of cultural restriction had left orchestras technically incapable of mastering its demands. In recent decades, the score and the ballet has returned in grand style to the place of origin.

Is it losing its force? Certainly, it no longer presents the terror to orchestras it once did – I was startled, at the Proms a few years ago, to see an American orchestra who hardly seemed to be breaking sweat, even in the "Danse de la terre". Sound recordings and films have turned the Rite from the object of pilgrimage in concert hall and theatre to something you can drop in on at any time. I once even heard it played as background music in a Pret a Manger. It is not his most perfect ballet, which must be Les Noces, or his most musically exciting, which is Agon; it has strong sections and weak ones, such as (as Stravinsky himself said) the passage before the "Glorification de l'Elue" with the two solo trumpets. It hardly matters. Pay attention to it, even 100 years on, and it continues to seize audiences by the ears, and by the jugular.