Beauty and Bartok’s Shopping List

In Part 1, we began to consider the questions below, and concluded that there is an evolutionary basis for the existence of music.  In Part 2, we will focus on the second question: what makes music beautiful?

  1. Why does music even exist?  Is it an evolutionary adaptation, or an accident — an evolutionary parasite?
  2. What makes music “beautiful”?
  3. Why do we derive pleasure from music?  What is it about music that “moves” us?
  4. Why do individuals prefer one type of music over another?

At the end of Part 1, we introduced the Golden Mean, an irrational mathematical constant with an approximate value of 1.6180339887…  What does this have to do with music?  Well, according to Pearl, it makes drums sound better:

The Golden Ratio, also known as the Golden Section, Golden Mean, Golden Rectangle, and the Divine Proportion, has fascinated mathematicians since the time of the Pharaohs. It is a mathematical constant (1.61803) that is found repeatedly in nature and has been used by artisans for generations to create art and structure with pleasing proportions.

Stradivarius applied the Golden Ratio to define the location of the  “f-holes” and proportions of his masterwork violins. The exterior dimensions of the Parthenon are said to form a perfect Golden Rectangle. The proportions found in Leonardo da Vinci’s “Last Supper” follow the Golden Ratio.

Intrigued by this, our engineers asked if the Golden Ratio was applicable to drums and more specifically to the location of the air vents. For every drum depth there is an upper and lower Golden Ratio location, and testing revealed that the upper location noticeably improves attack, which is perfect for snare drums, while the lower position improves low frequency response, ideal for toms and bass drums. Pearl’s revolutionary new Golden Ratio air vents are so unique they’re patent pending. Experience the sound of Golden Ratio air vents, centuries in the making, exclusively on Masterworks and Master’s Premium series drums.

~ Pearl Drums – Masterworks

Let’s say we’re convinced (after all the drums and the Stradivarius sound pretty darn good.)  Can the Golden Mean explain why we find music beautiful? And what does all of this have to do with the Fibonacci sequence anyway?

Let’s take up the latter question first.  If you did the extra credit work from Part 1 and watched the YouTube Short from Jane Weavis, or read the Theory Behind The Numbers from Cristóbal Vila, you already know the answer.

If we divide each value in the Fibonacci Series by the previous, the result tends to Phi. The higher the value, the greater the approximation (consider that Phi, like any irrational number, has infinite decimals).

~ Cristóbal Vila

Now that we’ve got that settled, what does it all mean?  Well it’s easy to see and measure these relationships when you’re dealing with physical objects such as bodies, buildings, and the human face.  And a fairly reasonable case can be made for the relationship between physical beauty and the golden mean.  (Although some argue that powers ascribed to the golden mean are often better explained by symmetry or other factors.)

Demonstrating the relationship between the golden mean and beauty in music is much more difficult.  First, despite the adage that “beauty is in the eye of the beholder,” there is general agreement, especially within cultures, on physical beauty.

Research shows that, despite the elusiveness of physical attractiveness, people overwhelmingly agree in their judgments about the physical attractiveness level of other persons, and can distinguish points along a physical attractiveness continuum that are consistent from one time period to another.

~ Gordon L. Patzer, The Power and Paradox of Physical Attractiveness

Conversely, there is much less agreement about what constitutes beautiful music.  And these differences exist not just across cultures, but among and within groups from the same culture.

Even if we ignore musical tastes, how do we measure the Fibonacci influence or “Golden Meanness” of a piece of music?  Rather tortuously, as it turns out.

Below is a brief survey of attempts to find the Fibonacci sequence and the Golden Mean in musical compositions, followed by some of the difficulties with each approach.

  • Some find Fibonacci numbers in the most commonly used used scales: Pentatonic (5), Major and Minor (8) and Chromatic (13).  The issue here is that major and minor scales actually contain seven tones, and the chromatic scale contains 12.  They only reach 8 and 13 if you include the octave, in which case, you need to drop the pentatonic scale.
  • Others claim that the interval relationships in various pieces of music are based on the Fibonacci numbers. If we include the octave, then the numbers 1, 2, 3, 5, 8 represent more than 62% of the available notes in a major scale.  These notes can also form all intervals within the major scale except for the most dissonant (minor 2nd, tritone, and major 7th.)  So it would be more astonishing to find pieces of music that did not use the Fibonacci numbers.  Also, some musicologists use scale tones to demonstrate the relationships, while others use semitones.
  • Both the Golden Mean and Fibonacci numbers are found in examining the structure of compositions, with the climax coming at the .618 mark, or the sections starting at bars with Fibonacci numbers.  Many of these calculations need some nudges to hit their marks, e.g., you need to leave out introductory measures, or ignore repeated sections. And in some cases, changes in time signatures or tempo move the temporal location of a section to a place that does not correspond to its bar count.

Enough.  There’s more, but you get the idea.  Given a sufficiently complex piece of music, you can conjure almost any mathematical relationship between its intervals, rhythms, and structure.

Even the simplest pieces of music can yield results.  Think of “Happy Birthday” in F Major.  Eight measures long, and the climax comes with the high C at measure five (the third “Birth…”.)  Eight divided by five equals 1.6.  Hmmm.  (Of course if we count the pickup as bar 1, we get 9/6, or 1.5.  Or if we count beats, we get 25/14, or 1.78.  But close enough, right?)

The composers most often trotted out to support the Golden/Fibonacci theories are Bartok and Debussy.  And the piece of music mentioned most frequently is Bartok’s “Music for Strings, Percussion and Celesta.”  Erno Lendvai write extensively on the topic in his book, Bela Bartok: An Analysis of His Music.  But not everyone agrees with his conclusions.

Some musicologists do not accept Lendvai’s analyses.  Lendvai himself admits that that Bartok said nothing or very little about his own compositions, stating: “Let my music speak for itself; I lay no claim to any explanation of my works.”  The fact that Bartok did not leave any sketches to indicate that he derived rhythms or scales numerically makes any analysis suggestive at best.  Also, Lendvai actually dodges the question of whether Bartok used the Golden Ratio consciously.  Hungarian musicologist Laszlo Somfai totally discounts the notion Bartok used the Golden Ratio, in his 1996 book Béla Bartók: Composition, Concepts, and Autograph Sources.  On the basis of a thorough analysis (which took three decades) of some 3,600 pages, Somfai concludes that Bartok composed without any preconceived musical theories.  Other musicologists, including Ruth Tarlow and Paul Griffiths, also refer to Lendvai’s study as “dubious.”

~ Mario Livio: The Golden Ratio: The Story of PHI, the World’s Most Astonishing Number

Having introduced dubiousness into the conversation, consider this unsubstantiated comment from a forum discussing Bartok and the Golden Ratio.

For a long time, there was no evidence at all to suggest that Bartok actually did all this on purpose, that he ever calculated anything. No papers, nothing.

Then, somebody found something. It was a paper in Bartok’s documents that was just covered with numbers from the Fibonacci series. Different calculations and so forth.

It raised a huge controversy; kind of like the rosetta stone, as one of the professors here described it, of Bartok study.

Well, you know what it was? A shopping list. Bartok had just been adding and subtracting prices, and figuring out taxes, and it’s just a coincidence that these prices happened to be fibbonaci (sic) numbers!

~ Sam, #353863

Suppose evidence of Bartok’s use of the Golden Mean was found, would his compositions be heard or appreciated any differently because of it?  Even though “Music for Strings, Percussion and Celesta” is one of my favorite pieces of music, not everyone would describe it as “beautiful.”  My daughter actually seemed a bit frightened by it during her first listen.

We may never find a completely satisfactory answer to the question of what makes music beautiful.  Victor Hugo said, “Music expresses that which cannot be put into words and cannot remain silent.”  Robert Jordain concludes his book Music, the Brain, and Ecstasy, with some thoughts on beauty and music.

The experience of unsullied order persisting simultaneously at every perceptual level may be taken as a working definition of the word “beauty.”  When events in everyday experience come together perfectly, we’re apt to exclaim “Beautiful!” and to register the pleasure of anticipations perfectly met.

Many people say that it is beauty alone that draws them to music.  But great music brings us even more.  By providing the brain with an artificial environment, and forcing it through that environment in controlled ways, music imparts the means of experiencing relations far deeper than those we encounter in our everyday lives.  When music is written with genius, every event is carefully selected to to build the substructure for for exceptionally deep relations.  No resource is wasted, no distractions are allowed.  Thus, however briefly, we attain a greater grasp of the world (or at least a small part of it), as if rising from the ground to look down upon the confining maze of ordinary existence.

~ Robert Jordain, Music, The Brain, And Ecstasy: How Music Captures Our Imagination

Yeah, what he said.  Come back for Part Three, where we look more closely at what exactly it is about music and its ordered relations that so move us.  None of this means you can’t consciously use the Golden Mean or Fibonacci numbers as a compositional tool (see the video for Lateralus below) — just that they aren’t necessary or sufficient to create beauty in music.