Bartok and Fibonacci

It is usually taught in music history courses and textbooks that the Hungarian composer Bela Bartok made generous use of the Fibonacci Series, or the Golden Mean, in his works, the most commonly cited example being his Music for Strings, Percussion and Celeste. When one looks a little deeper it starts to become unclear to what extent he consciously employed this formula as an organizing principle in his music and to what degree this idea is a creation of the musicologist and biographer of Bartok, Ernő Lendvai.

Today, as I was driving in my car, I heard a discussion on the radio about how it is that we can understand and appreciate music and what the evolutionary roots of music appreciation may be. At one point, while discussing the limits of our tolerance for what our expectations define as both acceptable and interesting, one of the panelists (who was a comedian and not an expert on music) said that he drew the line at Bartok. I often find myself annoyed at the idiocy of this kind of judgment of Bartok's music and not because I have a problem with other people's musical tastes. I find myself irritated because I am 99% sure that this person could not name a single piece of music by Bela Bartok and the fact that he is in the company of many "educated" and dilettante patrons of the arts. It is hard to believe that there are still so many people in the 21st century who listen to classical music who have such a great ignorance of Bartok (and of contemporary classical music in general ) when it is almost universally felt among musicians and music historians that he wrote some of the most exciting, innovative and profoundly beautiful music ever written, and that he is certainly one of the greatest composers of the 20th century.

No one is going to argue that Bartok is easy listening, his music does not generally serve well as musical wallpaper. Rather, it requires that we follow the narrative or musical argument as it unfolds in order to fully appreciate and enjoy the fruits of the composers labor. That being said, there are certainly some pieces that are more accessible to those unfamiliar with his work. The 14 Bagatelles are a good starting point for the new listener to begin to get an idea of Bartok's genus. In fact, Zoltán Kocsis's recording of works for piano solo, which includes the Bagatelles, contains works which are not too demanding on the listener. Of course, if the listener feels that music has no right to ask anything of them, to make any kind of investment in learning new things, of hearing in new ways, of challenging their prejudices. If the listener is content to let classical music serve as nothing more than a kind of auditory Prozac, then they are perfectly entitled to make that choice. But, they should refrain from pretending to make judgments about things they are completely ignorant about and simply do not understand.

Biographies of Bartok all recount his frustration with the fact that he was neither understood nor appreciated by many of his contemporary critics who mistakenly labeled his music as "atonal." Although it is true that he did take tonality to its outer limits there seems to be universal consensus today among people who have studied his music that he was not an atonal composer and that he was only continuing on the path that one could see opening as far back as J.S. Bach who, in spite of the fact that his music is truly a celebration of tonality in all its possibilities, began to experiment with prolonged and unresolved dissonance and tonal ambiguity in so many of his works. In Bartok's own words, "The time will come when it will be realized that despite the atonal inclination of modern music, the possibilities of building new structures on key systems have not been exhausted." One example of to what extent Bartok could be misunderstood is a story a friend of mine told me about a music teacher he had who said that Bartok composed his music by carrying little cut out pieces of paper with notes on them around and then just splicing them together. In other words, completely randomly and without any system or sense. Anyone who has studied Bartok's music knows how completely laughable and absurd this is as nothing could be further from the truth.

Two books that do an extensive analysis of Bartok's music, The Music of Bela Bartok by Eliott Antokoletz and The Workshop of Bartok and Kodaly by Erno Lendvai, should be enough to prove to the skeptical reader that Bartok's music has solid architectural underpinnings and that its content is rich in its musical references. It is interesting to note, though, that Bartok's use of the Golden Mean only occupies a footnote in Antokoletz's work and that it is only one of many different angles that Lendvai takes on his music. I was curious to see what Bartok's own opinions were regarding his use of the Fibonacci series, when he decided to incorporate it into his music and why. One of the things I discovered was a Website by a mathematician who stated "we know from his letters and diaries that Bartok consciously used the Golden Mean." This sentence aroused my curiosity and I was able to find copies of his letters at the SFSU library and after going through them I could not find a single reference to the Golden Mean. I was not able to find his diaries. This finding surprised me and I immediately wrote the Website author to get the references for this assertion. He wrote back to me rather timidly that he had actually not seen either his diaries or letters and he just assumed that this was the case and he apologized for the sloppy scholarship. I was rather stunned by this admission and it turned out that the more I pursued this subject the less certain I became that Bartok had consciously incorporated the Golden Mean into his work as a kind of blueprint. Instead of finding a wealth of information of how the Golden Mean was used in his music what I found were more and more references that raised doubt about the extent to which Lendvai's assertion of the importance of it was valid.

For example:

Laszlo Somfai-"Calculations of actual proportions of a composition, using Fibonacci, or other number systems, have not been found anywhere in the source materials of Bartok's music."

Malcolm Gilles-"Did Bartok know about the Fibonacci series of the Golden Section principle? Unfortunately, a good deal of doubtful arithmetic manipulation, practiced about all by Lendvai, deflecting attention from the essential truth that this fugue (and some other Bartok movements too) is an excellent example of natural growth principles in action, but that there is no evidence that Bartok was aware of these principles or designed any of his works according to them."

and from Bartok himself,

"I must state that all my music is determined by instinct and sensibility; no one need to ask me why I wrote this or that, or did something in this way rather than in that way. I could not give any explanation, other than that I felt this way...I never created new theories in advance."

Music for Strings, Percussion and Celeste 1.Fugue Graphic Analysis by Solomon

I am assuming that if you have stumbled upon this article that you are already familiar with the concept of the Golden Mean, also known as the "Divine Proportion,"  that is described by a series or sequence of numbers that was introduced to the West by the Medieval mathematician Leonardo Fibonacci in his Book of Calculation. In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers, starting with 0 and 1. This sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... If you divide two consecutive numbers in the series (starting with 34/55 or 55/34)  you get the ratio of .618 or 1.618 This proportion can be found in a number of things in nature including the spirals of Nautilus shells, Sunflower heads, Pine cones and Nebulae. This ratio was also used extensively in Renaissance art and architecture. While is seems reasonable to assume that this ratio is something one could appreciate visually, it is not quite clear how one would be able to discern Fibbonacci sequences on an auditory level and what impact they might have on our appreciation of music. In any case, where would we look to find the Golden Mean?  One approach could be to use the number of measures, as in the graphic analysis by Solomon, to mark out proportions. One could also count the number of notes played or the amount of time that has passed in a piece. What about counting the intervals used in a work? But the question is do we count 1,2,3,5,8, etc., as half steps or whole steps? Another place in which this series could be manifest is in the durational and rhythmic parameters. 



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