One of the more unexpected side effects of the digital revolution, it seems, has been to turn us all into remixers. Unthinkable even a decade ago, we all now routinely borrow and collage, recombining existing text, music or images and mashing them into something new. We crop and manipulate our photos; take a tune and refashion it into our ringtone. DJs chop, layer and loop tracks; visual artists and writers are busy referencing each others' work. "Remix culture", as the American academic Lawrence Lessig calls it, has in a short time become part of our everyday language.

In classical music (often perceived as the most staid and stuffy of all the arts), composers have been at this for centuries. And one man in particular has stimulated other musicians to borrow, refashion and remix: JS Bach.

Bach might not seem the obvious choice for a bit of iconoclastic smash-and-grab. After all, there are composers whose music, and indeed whose lives, seem less privately contemplative, more obviously playful, dramatic or public. Bach's contemporary Handel, for instance, travelled merrily round the great centres of Europe; Mozart practical-joked his way into women's beds and died writing a Requiem; Beethoven wrote music of searing intensity and was the ultimate Romantic hero. He grappled with legal battles, failed love affairs and profound deafness. Bach, it seems, either stayed at home or went to church.

The well-known portrait of him by EG Haussmann hasn't helped Bach's famously stuffy image. Painted in 1746, it depicts a paunchy, middle-aged German in a white wig: stern, serious and brandishing a piece of music manuscript paper. In reality, though, the composer was far more flawed and infinitely less reverential a figure than we might think, not so much a reclusive genius as a pipe-smoking, caffeine-addicted human dynamo. He drank too much, fathered 20 children (10 of whom died in infancy) and had two marriages. He was also a ruthless workaholic, writing music at a pace that outstripped pretty much any composer before or since. If he wasn't writing music, Bach was obsessively studying it, playing it, revising it, looking at what other musicians were doing. He once walked 260 miles just to meet the organist Dietrich Buxtehude and learn more about his compositional secrets. Music consumed Bach: even relaxing in Zimmermann's coffee house in Leipzig, he couldn't resist dashing off a quick cantata about the addictive properties of the black brew.

It's this multi-dimensional Bach that the Proms is hoping to explore this year, devoting a whole day of the festival exclusively to his music. First, John Eliot Gardiner and the English Baroque Soloists are bringing us Bach in its purest form – stripped of all ornament, played on the period instruments that the composer would have heard and in the style he would have been familiar with. But the day closes with a rather less authentic nod to the master: a series of transcriptions, or reworkings, of Bach from conductor Andrew Litton and the Royal Philharmonic Orchestra. Their concert features everything from the full-blown blowsiness of a symphony orchestra – Leopold Stokowski's brilliantly garish arrangements for Walt Disney – to stripped-back, contemporary rethinkings of Bach's music from white-hot young writers Tarik O'Regan and Alissa Firsova. What the transcriptions all share is the essential DNA of the originals (whether melody, harmony, mood or character), using them as a starting point for a brand-new piece.

But just what is it about Bach in particular that makes his music so endlessly malleable? Simply, its directness. A whole Bach piece can be boiled down and reduced to a single short idea; whereas in Mozart or Beethoven, the original raw material is more worked out, the ideas are longer and more intentionally complex. As O'Regan explains, "With Bach, I feel I'm allowed to take his snippet of an idea and play my own games with it, without damaging the original. With Mozart and Beethoven, I feel like I'd be tinkering with something that is already much more revised and refined in its exposition. One feels that – in some way – Bach's music is designed for invention and reinvention."

It's this economy, combined with expressive power, the ability to conjure immense breadth and depth through the leanest of means, that has repeatedly drawn other composers back to Bach. And it's not just a recent fad: since his death in 1750, musicians have turned to this great repository of music for inspiration. Within the staggering volume of music he produced, in almost every conceivable genre, composers have found a kind of ready-made musical encyclopaedia, and the archive of Bach pieces just keeps on growing as new musical minds present us with their perspective. There's Bach for marimbas, for ukulele and harmonium. There's Keith Jarrett's classical interpretations and Jacques Loussier's breezy Play Bach Trio, which brings cool jazz to classical music. Bud Powell often included music by Bach in his sets, such as the 1957 recording Bud on Bach. Tiempo Libre, Jethro Tull and the Grimethorpe Colliery Brass Band have all had a go. Bach's shadow extends across jazz, Afro-Cuban dance, folk and rock as much as it does across the classical world.

All this remixing would be sure to delight old Johann Sebastian, himself a master arranger and musical magpie. He reworked Vivaldi's violin concertos for keyboard and orchestra; he also recast his own music to suit pragmatic ends, so what was once a lute suite ended up the next week rewritten for solo cello. Rather than seeing himself as a genius master-composer, you suspect he would have viewed his job more as that of a workaday, gigging musician. What the Proms celebration should prove, if nothing else, is that pure, authentic Bach and idiosyncratic, remixed Bach can happily co-exist – and the man himself would almost certainly have approved.