# Did Bartók use Fibonacci numbers in his music?

I love math, and I love music. I’ve played piano since I was 6 and composed music since middle school. So when I was planning my week at the JMM, I knew I had to attend the MAA Contributed Paper Session on Mathematics and Music.

Music inherently contains many levels of remarkable mathematical structure. And throughout history, some composers have devised clever ways to sneak even more math into their music (once, I represented exponential growth musically). Béla Bartók (1881-1945), a great Hungarian composer, might have been one of them—but historians are divided on the issue. In his talk “Bartók, Fibonacci, and the Golden Ratio: Fact or Fiction?,” Gareth Roberts of the College of the Holy Cross presented the evidence for and against the presence of the Fibonacci sequence in Bartók’s 1936 work Music for Strings, Percussion and Celesta

The Fibonacci sequence begins 1, 1, 2, 3, 5, … , with each number being the sum of the previous two. Bartók excelled in math and physics and collected plants like sunflowers and fir cones whose structures feature Fibonacci numbers. But he was notoriously tight-lipped about his compositional methods, leaving no written records that could confirm the use of Fibonacci numbers in his music.

The first page of the score of Bartok’s Music for Strings, Percussion and Celesta.

According to Roberts, the controversy was sparked by music theorist Ernő Lendvai, who in the 1950s published an analysis that seemed to demonstrate extensive use of Fibonacci numbers in the first movement of Music for Strings, Percussion and Celesta. The piece lasts 89 measures, with its climax (loudest moment) at the end of bar 55. The exposition in the opening lasts 21 bars, and the string mutes are removed in measure 34. Of course, 21, 34, 55, and 89 are all Fibonacci numbers.

But the case is not so clear-cut, Roberts explained. A later analysis by musicologist Roy Howat pointed out various flaws in Lendvai’s reasoning. The written score of the piece contains 88 measures (Lendvai argued that an extra bar of rest should be tacked on). Although the dynamic climax does occur in bar 55, the tonal climax is in bar 44. Some of the string mutes are removed in bar 33, some in 34, and the rest in 35. The exposition, Howat claims, ends in bar 20. And Lendvai neglected to mention the titular celesta, which enters after bar 77—nowhere near a Fibonacci number.

Top: Lendvai’s analysis of the structure of the piece. Bottom: Howat’s analysis.

The third movement of Music for Strings, Percussion and Celesta begins with a more unambiguous instance of the Fibonacci sequence, a xylophone playing a rhythmic pattern of 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1. (Apparently, this movement was used in the soundtrack of The Shining.)  But the bottom line of Roberts’ talk was that it’s a stretch to say that Bartók intentionally incorporated Fibonacci numbers into the structure of the first movement.

Roberts teaches this example in his undergraduate course on mathematics and music. The controversy draws students in, and the resolution offers lessons applicable beyond finding patterns in music. Lendvai’s analysis is a perfect example of cherry-picking data and confirmation bias.

It reminds me of the statistically insignificant “Bible code” that people have claimed predicts the future. Pick your favorite composer. Examine their oeuvre, looking for your favorite sequence of numbers. If you massage the data enough, considering all the possible ways your sequence could be hidden in the music, you’ll surely find it somewhere. But does that mean that the composer put it there intentionally? Unlikely. That being said, it seems plausible that Bartók played these sorts of mathematical games in other pieces.

Attending this talk gave me potential inspiration for my future composing. I think it would be fun to write a piece built around the Fibonacci sequence. And if I do so, I’ll make sure I explain it in writing to save future musicologists the trouble.

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