Bill Linderman muses on Math & Music
By Leslie Grace | A! Magazine for the Arts | February 27, 2013Bill Linderman is a mathematics professor at King University in Bristol, Tenn. He is also an accomplished pianist, sings with the First Presbyterian Church Choir and composes.
"When I think about the relationship between mathematics and music, I think about the study of patterns. Both disciplines involve elements (numbers, shapes, sounds) that create structures with interesting and complex patterns. The way the structures interact and fit together, creating harmony, is what makes both subjects so rich and appealing. The patterns might be simple and elegant, or they might provide striking revelations. Although there is an obvious relationship between mathematics and music when it comes to counting and rhythm, I think the more distinctive connection involves the understanding and study of patterns.
"Mathematics is a highly creative field of study, although I know it is not generally regarded this way. Good problem solving requires ingenuity. There is a scene in "Apollo XIII' where a team of engineers is given the assignment of fitting a square peg into a round hole using the limited items the astronauts have onboard. The engineers needed some creativity to complete the task given the tools they had to work with. Higher mathematics involves proving new results, where the tools at your disposal are known results or theorems. Of course, having a good understanding of how other results have been proven is helpful, but sometimes it just takes a new way of looking at things to construct a proof that has eluded others.
"Non-mathematicians might be surprised at how often mathematicians use the words "beautiful' and "elegant' in describing mathematics. About 2,000 years ago, Euclid proved that there are infinitely many prime numbers. His proof is regarded as elegant because it is hard to imagine any proof of this result being more straightforward and clear. It reminds me of the music of Mozart, which is known for its purity and clarity. Mozart's music has a natural quality to it, as if it was revealed to him fully formed. In "Amadeus' Salieri describes Mozart's music by saying "displace one note and there would be diminishment, displace one phrase and the structure would fall.' Euclid's proof has a similar perfection.
"In the Fibonacci sequence each term in the sequence, after the second, is the sum of the previous two terms. The ratio of consecutive terms in the sequence approaches the golden ratio, called phi. This number is approximately 1.618. (A ratio is arrived at by dividing two numbers).
"A rectangle with proportions 1 to phi is called the golden rectangle and its dimensions are said to be pleasing to the eye. Much has been written about the use of the golden rectangle, from the measurements of the Parthenon to dimensions used in the "Mona Lisa.' Much has also been written debunking these claims. In terms of music, there are claims that some composers either used the Fibonacci numbers or used the ratio 1 to phi as a point of division in their works. Did Mozart use the golden ratio in some of his sonatas for determining the length of the exposition compared to the length of the development and recapitulation? Does the climatic moment in some pieces of Debussy create a division that satisfies the golden mean? Did Bartok use Fibonacci numbers in some of his works? Assertions have been made that this is true. Some scholars claim that these composers were not using the golden ratio, and others say that an unconscious use only gives more credence to the beauty of this division. The debate continues. The Fibonacci numbers, however, do occur in nature, in starfish and sea shells, flower petals, pinecones and pineapples.
"I've done a little bit of composing through the years. I had the wonderful opportunity to spend a sabbatical year as a visiting associate professor at Cornell University. I taught two courses and was able to take composition courses and study piano performance. An assignment in one of the composition courses was to compose a song using the set 1,3,4. This meant that any cluster of notes would be separated by one, three or four half-notes, possibly in different octaves. It was a restrictive assignment, and I worked hard to try to create a piece I was pleased with. The restriction helped to give the piece a unifying structure.
"An undergraduate professor once told me that it is a common mistake for young artists to try to use too many ideas in their compositions. Just using a few ideas or themes or patterns could give a piece more consistency. The themes could certainly originate from something mathematical. Part of the genius of Beethoven was his ability to develop simple themes into something extraordinary. The "Ode to Joy' theme is really quite simple, almost childlike, but Beethoven used it to create what is arguably one of the most beautiful pieces of music ever written."
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