Bach's mentor was Buxtehude - so much so that he walked miles just to hear him play (no public transport!). Much of Bach's music owes its style to Buxtehude to some degree.

It's certainly true that Bach was a 'mathematically perfect' musician, but all good improvisers (or, in Bach's case 'extemporiser' would be more accurate) gradually evolve their compositions over a period of time, based upon buiilding blocks of what are now called 'riffs'.

An extemporiser takes a known melody, then takes off in another direction - his chorale preludes, for example. He was one of the last great polyphonic composers - where the melodies in the 'parts' could stand alone, they weren't just filling in the harmony.

It is well documented that the famous 'Toccata and Fugue' was essentially an improvisation (not an extemporisaton this time!) which evolved into the piece we now know. That is one of the reasons why the fugue doesn't follow his normally fairly rigid structure. He apparently used it as a test piece to 'try out the lungs' of newly built organs.

His fugues, in the main, were not improvised - they were carefully worked out; they only represent a very small portion of his work, however.

Sorry - I'm going off on a very wild thread creep here, but Bach has always been dear to me, as I am also a church organist... extemporisation is an essential part of our craft!