The great Russian pianist Mikhail Pletnev recently caused feathers to fly on Radio 3 when he asserted that the history of classical music was completed and therefore "over". In his view there would be no more great works - the well had run dry. But that hasn't stopped him dedicating his life to playing Beethoven and Schumann with fierce conviction.
So what is "classical music"? When we talk about it, what do we mean? We should more accurately add the word "Western", because there are several other classical traditions which are just as venerable and distinguished.
But no other music could rival the Western tradition in intellectual sophistication, and no other has thrown up anything so exalted as Bach's Mass in B Minor, Mozart's Requiem, or Beethoven's Ninth Symphony.
What was it that allowed the music of the West to surpass all others in its transcendental flight? The Christian religion, which inspired the first two of these works, may have had something to do with it, but it could also be argued that the key lay in revolutionary political ideas, since those were what animated the third. The answers lie embedded in the nature of Western music: in how it evolved and in the fact that - as Pletnev believes - it arguably has had a beginning, a middle and an end. Whether or not we agree, we have to accept that classical music's language is bounded by strict time limits. Before the mid-16th century, that language did not exist, and without it Western classical music could never have taken off as it did.
Like all languages, it had a variety of roots and influences. The major and minor scales on which it is based emerged from the "modes" which dominated European music from ce400 onwards, and those modes had evolved from the scales used by the Ancient Greeks. Serious music in medieval Europe was based in the Church, where strict rules were handed down from abbots to junior monks as to how the liturgy should be sung.
But in the 11th century an Italian monk named Guido d'Arezzo invented notation, and that proved revolutionary. Earlier, the shape and pitch of a piece of music was never written down on paper. As composer Howard Goodall puts it: until that point, music tuition had been like "teaching maths without numbers and equations, or geology without relief maps. Now sounds had become ink". This meant that music didn't have to rely on one person's memory for passing it on.
Music, which had hitherto been spontaneous and simple, could now encompass systems of ideas and could become complex architecture in sound. Guido thus paved the way for a new breed: composers, who could map uncharted territories for other musicians to explore. Before this point, music had been a shared common currency. Now it could be a one-off work of art, and it would henceforth belong to its creators, not to the musicians who merely recreated it.
But the language of classical music as we know it was not yet born - notation had merely made its birth possible. One of the first things which Guido's invention encouraged was counterpoint - the fitting of two or more tunes together to make a pleasing whole. Without notation, as Goodall neatly puts it, putting tunes together "is like trying to play Scrabble without the board or the plastic letters". But fitting tunes together led to that other great device in the classical toolbox - harmony - and once they started playing with harmony, the composers of the 16th century were able to generate an endlessly bewitching range of new creations. Now classical music had its fully-fledged language, and - aided by the printing press - could enter its golden age.
Much of it was vocal, and educated families in France and Italy would sing counterpoint in their homes. It was as much a disgrace not to be able to sight-read music, as it was not to know the steps of fashionable dances.
But composers were also experimenting with combinations of instruments and fitting instruments to voices, and they were creating magical effects by strategically positioning their players and singers in the halls and cathedrals where their works were performed.
The birth of the orchestra dates from when the Venetian composer Giovanni Gabrieli indicated in his printed scores the exact instruments he wanted for each part.
Initially this music existed for ceremonies in royal courts, or to glorify God. But at the beginning of the 17th century opera was born, with Monteverdi's Orfeo - premiered for an exclusive group of intellectuals in Mantua - being its first great achievement. The Florentine composer Lully installed opera at the court of the French "Sun King" Louis XIV; opera took off in England, meanwhile, thanks to Henry Purcell followed by George Frideric Handel. But Italy was central to Europe's musical life, as reflected in the fact that the words describing the requisite speeds were everywhere written in Italian, as they still are: adagio (slow), allegro (quick), crescendo (getting louder), legato (smoothly), piano (softly).
Even the great German composer JS Bach knelt at the shrine of Italian music: as a young man he studiously rewrote Vivaldi's string-orchestra music, so that he could play it on the harpsichord.
In their different ways, Vivaldi, Handel and Bach composed in what's called the Baroque manner, which for many people is still the quintessence of classical music. This was powered by the belief that music no longer needed to justify its existence by royal or religious usefulness. The beauty of an aria or the brilliance of an instrumental solo became the goal. The listener was drawn into a world of beguiling harmonies and forms - a world with which Classic FM radio, pumping out Baroque favourites by Albinoni and Pachelbel, holds its six million listeners spellbound.
The Baroque style set music free from its previous constraints, but the other great emancipation was the revolution in tuning known as "equal temperament". Of all the composers who profited from this liberation, Haydn and Mozart were pre-eminent. Commissioned more by princes than by the Church, they wrote for many combinations of instruments and voices and experimented with many forms, but their work in one particular form laid down ground rules for generations of composers who came after.
This was sonata form - from the Italian suonare, to sound - and though just about anything written for instruments in the 17th century could be called a sonata, the "classical" sonata form which Haydn and Mozart used as the basis for symphonies and chamber works followed a fairly set pattern. First came an agenda-setting first movement, then a slow, lyrical movement, then perhaps a quirky one, but it almost always wound up with a liberatingly energetic finale.
The pattern has proved enduring: even today, symphonies and chamber pieces still largely stick to it.
The third founding father of sonata form was Beethoven. His personal bid for creative freedom embodied two strikingly new ideas. He was the first composer to make himself the hero of his own works with their glorification of his passions and his aggressive originality. And he was the first to write deliberately for posterity, disdaining anyone who failed to catch his drift. One of those who did catch it was his ardent young admirer Franz Schubert, whose attitude to worldly success was even more take-it-or-leave-it, but he died young and impecunious, before some of his greatest works were publicly performed.
But Schubert's wonderful songs marked a turning-point for music. From form and structure, the emphasis now moved to raw emotion and expressiveness.
This was the dawn of Romanticism. It was entirely in keeping with the mood of the moment that when Schubert died, his creative successor Robert Schumann - who only knew him through his works - wept all night. Schumann's piano pieces were intensely emotional, but also highly inventive. Indeed, this point in the 1830s was one of history's magic moments, because Schumann, Chopin, Mendelssohn, and Liszt were contemporaries, friends and rivals.
Each member of this flamboyant circle had immense influence on the course of classical music. Chopin showed what singular beauty could be achieved in miniature forms like the prelude and mazurka; Mendelssohn campaigned to put the music of JS Bach - which had been largely forgotten - centre-stage, where it has been ever since; Liszt - who invented the piano recital, and whose path was strewn with roses by adoring fans - blazed the trail for the harmonic experiments which would plunge classical music into uncharted seas 50 years later.
Orchestral Romanticism reached its twin peaks with Brahms and Tchaikovsky, but its naive force was then sapped by that same combination of social, political and psychological uncertainties which permeated literature, theatre and visual art by the end of the 19th century. But if Romanticism had run out of steam, the Romantic notion that each new composer had to "make it new" still impelled people to go on trying to do that in some way.
Attention therefore shifted to the musical language itself, which had scarcely changed in three centuries. But how could that be "made new"?
Richard Wagner, Claude Debussy, and Richard Strauss had already been pushing harmonic experiments in new directions, and Stravinsky was railing against what he called the whole "superannuated system of classical tonality". Was the language which had so nobly served Monteverdi, Bach and Beethoven about to be destroyed by its leading contemporary lights? The answer from Arnold Schoenberg - the most influential composer of the early 20th century - was "yes, if possible".
Schoenberg and his followers flouted the classical rule which held that every awkward-sounding dissonance had to be resolved into consonant harmony. They replaced the diatonic (do-re-mi) scale with the "chromatic" semitone scale, in which all 12 notes were to have equal importance. The result was a celebration of dissonance and not comfortably hummable.
This was the direction in which composers like Debussy and Strauss had been heading, but something had died as a result. Gut instinct was now being ousted by intellect. Champions of post-Schoenberg "atonal" compositions insist that if you listen with an open mind, you'll find a new kind of beauty, and atonal music has indeed contributed interestingly to the contemporary classical mix, as well as being harnessed very effectively in the service of dance. But by and large the work of his followers has made ferocious intellectual demands on the listener and given little pleasure in return.
Meanwhile, Schoenberg's deconstruction of music's language has been taken further by "experimental" composers who decree that any sound - even two pieces of sandpaper being rubbed together - can be classed as music. While a few composers like Leonard Bernstein and Benjamin Britten created works which were both "tonal" and triumphantly original, between 1950 and 1990 contemporary classical music in Europe and America retreated into an avant-garde ghetto, and traditional classical audiences were forced to seek satisfaction elsewhere.
Having turned their backs on their natural constituency, composers began to reap the hostility they had sown. The recent "return to tonality" of Americans like John Adams and Philip Glass and Europeans like Arvo PArt and Henryk Gorecki, reflects the pained awareness of the classical composing community that it had vanished down a cul-de-sac. Composers are desperately employing tunes to win back their traditional audience.
Audiences have always been crucial. You could tell the whole history of classical music in terms of how it's been listened to, and by whom, and in a recent book, Music: Healing the Rift, the critic Ivan Hewett has done just that. As he sees it, the story begins with music firmly embedded in its social function - to please God in church, or to please the king at court. Then came the notion that music could be transportable: a Mass could be taken out of church and staged in a concert hall. Next, music started its retreat from the public domain. It began to be increasingly enjoyed en famille, then to be listened to in one's room via a record player, until, finally, the Walkman reduced its operative space to six inches between the ears.
Hewett thinks this "privatisation" of music led to the fragmented situation we have today, where each composer speaks in his or her own newly-invented language to a small band of disciples. It's hard to imagine any "classical" music galvanising the crowd now, the way Verdi's Va Pensiero did when Italians made it their rallying-chorus in the fight for political unity.
It's almost impossible to imagine a new opera now being whistled by tradesmen in the street, as Mozart's Figaro was in 18th century Prague. But football crowds do sing along with "Nessun Dorma" - Puccini's music still casts the spell. And Classic FM's Hall of Fame is topped by Rachmaninov and Mozart, with Beethoven, Tchaikovsky, and Bach close behind.
More than half of the top 50 works chosen by a sample of Classic FM's six million listeners were composed more than a century ago. This tells us something significant about classical music before Schoenberg got at it.
Music in its most exalted forms is perceived by its listeners as genuinely timeless. Composers and audiences can mix'n'match musics from 1,000 years of history and from cultures all round the globe: the idea of music following a simple linear development just won't wash. Great composers will go on finding fresh ways of turning old elements into something bright and new. But people will always love Beethoven's Emperor, Mozart's clarinet concerto, and Vivaldi's Four Seasons. In that sense, Western classical music is indeed complete.
Howard Goodall's book Big Bangs is published in paperback by Vantage (Pounds 7.99); the 2000 TV series is available on videoDVD from Channel 4 Learning http:web.channel4.comlearningmainnetnotesseriesid271.htm
Music: Healing the Rift. By Ivan Hewett is published by Continuum (Pounds 14.99)