Of all the jazz artists and jam-banders who might spring to mind when it comes to improvisation, the superstar musician who may be the greatest improvisationalist of all time is well-known to the public for everything except his improvisation.  Yet so intense was this performer's dedication to jumping beyond the music as written that otherwise adoring audiences and back-up musicians sometimes became annoyed at the length and off-the-wall intricacy of his extemporaneous musical wanderings, and he lost gigs over it.  Even when sitting in with other musicians he couldn't resist changing their compositions on the fly.
     The irrepressible improviser was Johann Sebastian Bach.  If you're wondering how Bach managed to reconcile improvisation with the highly ordered, intricate, precise domain of Baroque music, consider that in the 18th century improvisation was regarded as an integral component of serious music.  It would have been a second-rate performer indeed who thought to limit recitals to prescribed notes.  The opportunity--the imperative, really--for improvisation was explicitly written into Baroque compositions, and in more than one way.  Bach and other composers of the time rarely spelled out parts for cello, bassoon, harpsichord and organ note for note, instead providing the players of these and other low-range instruments suggested chords on which they were expected to riff.  Concertos contained cadenzas that challenged the soloist to cut loose from the confines of the sheet music, and the resulting long, furious improvisations were often the highlights of performances.  Even when notes were specified on the sheet, musicians routinely threw in improvised flourishes, like a chef sprinkling an extra dash of this and sprig of that, and the ability to spontaneously insert such ornamentation was considered a basic element of musicianship.
     Bach was thus hardly a renegade in his time for being fond of improvisation, but he was infamous for the extent and boldness of it.  He would embellish at length on the organ even in the middle of church services, apparently sometimes dismaying the officiators, choirs, congregations and others who were simply trying to get through the liturgy.  In other performances he would take musical themes tossed at him from the audience and immediately improvise around them, much in the style of a contemporary nightclub comedian.  Bach also enjoyed the then-popular practice of challenging other composers to musical duels of improvisation, as he did with the renowned French organist and composer Louis Marchand.  So prodigious was Bach's improvisational bent that today experts believe we only have a portion of his compositional oeuvre, much of it having been whipped up in performance and left untranscribed. 
     Bach and his colleagues could not have predicted that by the middle of the 20th century the improvisational elements of their compositions, and of all classical music, would have been gradually and thoroughly excised.  Composers had later filled in the bass lines and cadenzas with note-for-note versions, so that today musicians play only what's on the page, and every performance is melodically identical to every other.  Taking it on oneself to add off-the-cuff flourishes to the music in the middle of a performance would be seen not as an exhibition of musicianship and creativity but as a career-ending stunt that violated the music--even though to do so would be far more in keeping with the composer's intentions than the frozen-in-place versions we hear today.  It has been a centuries-long organizing project that almost certainly would have appalled some of the very composers we most ardently lionize.