Bartók and the Golden Mean

I’ve been on an odyssey lately, time-traveling back to graduate school in Philadelphia, to Aesthetics course with Dr. Carrow, to my most profound “Aha!” moment reminded to me today by one of my best research papers ever. This "Aha!" moment occurred when I learned about the Golden Mean, a magical proportion that is eerily found all over in nature, but also applied to art, architecture, literature, and music in some cases before the ancient Greeks. Find the Golden Mean in the natural organization of sunflower seeds, the spiral of a conch shell, leaves, the Parthenon in Greece, playing cards, climaxes in drama, and the music of Béla Bartók and others. 

The parts of my Golden Mean research paper, including manually-typed text, hand-drawn charts showing forms of the music, hand-manuscript lists of scales Bartok used, and complete bibliography.

The Golden Mean can also be found in art. For example, a Golden Rectangle can be seen in George Seurat’s “Invitation to a Sideshow” or “La Parade de Cirque” (1807-8). We call it a Golden Rectangle because the relationship of the short side of the rectangle to the long side is the Golden Mean proportion. In this painting, there are Golden Rectangles within the painting (notice the background) that have Golden Mean relationships with the other rectangles. A reproduction of this painting hung in my band room in high school and I thought I knew it by heart. Then I learned about the Golden Mean and...Aha!

How many Golden Rectangles can you find?

 

Mozart, Haydn, Beethoven, and Brahms probably did not use the Golden Mean purposefully even though many music theorists have attempted to prove through complicated graphs and charts that they did. I suggest that the common first-movement form of their day, sonata-allegro form, easily relates to the Golden Mean because it is a pleasing proportion, so it’s there, but not on-purpose. J.S. Bach was very interested in numbers, so it’s possible that some of his preludes and fugues from the Well-Tempered Clavier were created with this proportion in mind, but we just don't know for sure. 

We do know twentieth-century composers tinkered with the Golden Mean, or sometimes based every miniscule aspect of their compositions on it, because it was discussed in their writings and ever-so-obvious in their modern compositions. Check out Luigi Nono’s and Karlheinz Stockhausen’s (above) music, for example, and then try to tell me you enjoyed listening without understanding what was going on. Béla Bartók’s music appealed to me—balanced, tasteful, and generally aesthetically pleasing—even before I understood the Golden Mean and its companion, the Fibonacci Series.

So what is this Golden Mean? The Golden Mean is the proportion where the larger part of something relates to the whole the same as the smaller part relates to the larger. (The small to large is the same proportion as large to whole. Got it?) The proportion was first proposed by Chaldeans in the third century BC, and then was used by the Greeks. Nobody talked about it much or used it until Pacioli used it at the end of the fifteenth century (in the Renaissance). He called it the “divine proportion.” 

The Fibonacci Series was discovered about 1202 by Leonardo of Pisa, son of Bonaccio (“Filius Bonacci” or “Fibonacci” for short). In this series, the numbers are defined in terms of previous numbers:

FIBO(n) = FIBO(n-1) + FIBO(n-2) for n>2

In other words, if you look at the series of numbers, each equals the sum of the two that precede it, after you pass ‘1’ (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…). Surprise! The Fibonacci Series is an approximate representation of the Golden Mean in natural numbers. 

As I mentioned before, the Golden Mean and Fibonacci Series have been used extensively by 20^th-century composers, including Karlheinz Stockhausen (especially in his time signatures and note durations, see video above), Luigi Nono ( with pitches and pitch relationships), and Béla Bartók. Hungarian music theorist Ernö Lendvai applied the Golden Mean concept to Bartók’s works and found that many of his forms correspond to this proportion. For example, the recapitulations of the first movements of the Sonata for Two Pianos and Percussion (see video below this paragraph), Contrasts (see the video beneath that one which features Benny Goodman, Joseph Szigeti and the composer!), and some of the Mikrokosmos occur exactly at the Golden Mean. (The recapitulation is where the listener hears the material from the beginning of the piece again, and is therefore a very important spot.) 

Bartók also uses the Golden Mean to come up with his basis of tonality, although this is harder to identify by listening without a score. He established his three basic scales  from the Golden Mean proportion applied to tonal music and used them to organize many of his compositions:

            Golden Section Scale: C Eb F Ab

            Acoustic Scale: C D E F# G A Bb

            Octatonic Scale: C D Eb F F# G# A B (alternating whole- and half-steps)

Bartók also used the Fibonacci numbers in compositions. The form of Music for Strings, Percussion and Celeste is a good example of this with sections of 8, 13, 21, 34, 55, and 89 measures. Listen to the following video when you've got some time to get lost in a half-hour's worth of music.

I could go on for pages (screens) on Bartók’s use of the Golden Mean and Fibonacci Series, but I won’t. If you are interested in learning more, get your hands on Lendvai’s classic book, Béla Bartók: An Analysis of his Music, which is a smooth read and likely to be found in most libraries with a decent music collection.

So here’s my point: back in Aesthetics class I spent hours analyzing music looking for Golden Mean and Fibonaccian proportions only to discover that most music I found was not made better by strict, academic adherence to it. It may provide a formal (as in ‘pertaining to form,’ not ‘wearing a tuxedo’) balance, a way to unify a composition, and a topic for scholarly discourse. Bartók’s music is the exception, for me, because it is sonically interesting, aesthetically pleasing, and unique compared to the work of his contemporaries. Learning that he used Golden Mean and Fibonaccian proportions still enhances the listening experience for me because I was drawn to this music from the moment I first heard it.