The creators, who seem to have strict formal principles, in fact often depend on improvisation.
So was the case with Andrei Tarkovsky. He reached the classical forms of "Andrei Rublev", "Mirror" and "Stalker" only after adding numerous corrections to the initial screenplays and film materials(See my book "Tarkovsky and His Time"). Robert Bresson's attitude to formal principles was more rigid, but he also depended on his intuition, which allowed him improvise in shooting.
It seems natural that Tarkovsky estimated Bresson very highly but didn't find any aesthetic value in a later film of Yasujiro Ozu that he had watched on television. For Ozu, his individual canons were above all chances. For him there was no room for improvisation.

When I made documentary films about Japanese and Russian musicians, I always felt that improvisational elements are inevitable to make them vivid and true. Some of their albums or performances seemed to me more sublime or more inspired than canonical (or mathematical?) performance of J.S.Bach by more skillful players(for example, "The Art of Fugue"played by Yuji Takahashi).
For me "KENKA"by Jack Or Jive, "SATOR"by Moon Far Away seem to be more sincere, or religious than technically excellent performance of Bach's masterpieces.
Below is some albums of those independent groups, whose music sounds sometimes very profound.
  

A year of Eclipse

KISMET(documentary, subtitled in English ) part1

At last I published the English edition( Kindle e-Book ) on Amazon.
(The image below is a direct link to Amazon.com Kindle store)

While writing and translating this book, I could understand his works and thought deeper than before.

Though there are contradictions between Tarkovsky's film aesthetic and practice, his ideal in creation and his private life, as a whole his life was consistent from higher point of view.

I can't help but feel that his life was constantly under some other worldly power. It may be called Destiny, Fate or Providence, if you like...

And this book seems also to be under such power, though not so clearly as Tarkovsky's case. Below is part of its "The preface to English edition.

"The book you hold now is a translation of the Japanese first edition, published on April 4 of 2011(Andrei Tarkovsky’s 79-th birthday). At first author intended to publish the book as an newest introduction for the great Russian filmmaker for those audiences, who know his films well but not so concerning his biography, especially in connection with Russian culture and late Soviet cinema. In addition, he intended it to be a preparation to some unpublished documents and visual materials, which might be released in Japanese towards the end of the same year.
However, as everybody knows, because of the unprecedented disasters on March 11 we must have resigned most of the plan.Especially we are so sorry for quitting release on DVD "Rerberg and Tarkovsky: Reverse side of "Stalker", a feature documentary film by Igor Maiborod, who helped the author by encouragements from Russia. I hope Mr.Maiborod’s film will be released someday in Japan and in English speaking countries, where this book must have enough readers to attract distributors’ attention to the film.""

Yes, I made a decision to publish the English edition instead of those unrealized plans. As a results, though, the book is now available both in Japanese and in English, just before the 25th anniversary of Tarkovsky's death.
I remember also 1986 was the year of Chernobyl' nuclear power plant accident. In 1986 I was yet a teenager and just began to be influenced by Tarkovsky's cinema(the first of his films I saw was "Nostalgia", in December of 1985.). And this year was, sadly enough, marked by Fukushima accident, as catastrophic as that of Chernobyl'.

I don't intend to mystify the course of things, but the fact is fact.

"Life of certain artistic geniuses, in the eyes of the ordinary man, is so tragic that it looks even “artificially”. Pushkin’s life is such, and Vrubel’s life, too. Andrei Tarkovsky is one of such geniuses. Their life is very similar in this sense."(from "Afterword" of "Tarkovsky and His Time")

Yes, in their life we can't find any "accidental" event. Even their mistakes and worldly pressure...
But they were also human, not angels, or saints. Life of an artistic genius is usually a tragedy, more or less.
So we should not idealize them and make a fetish from their works.

"Only by understanding artistic genius as flesh and blood and we can release their works from posterity's aesthetic consumption and making fetish of them, the author is sure of that."(Ibid.)