## Is there anything rational about math and music?

I’ve always been fascinated with the relationships between math and music, since they are both fields in which I take a great interest. This week I’ve been delving into some of the history that links the two. For instance, the Greek mathematician and philosopher Pythagoras (circa 569 - circa 475 BC) is probably best known for the Pythagorean Theorem. However, he also made significant contributions to music, the influences of which can still be seen today. He famously believed that everything in reality could be represented by numbers; that the universe was fundamentally based on mathematics. As the story goes, Pythagoras decided that the sound of blacksmiths hammering was especially intriguing, and decided to investigate, as he thought that the principle behind the appealing sounds could surely be applied to music. He discovered that the lengths of the blacksmiths’ anvils were simple ratios of each other, and decided to try to apply this pattern to musical tuning. Using this observation, he developed a system of tuning known as Pythagorean tuning.

You may have heard the phrase “music of the spheres”; this also originated from Pythagoras. Rather than referring to actual audible music, this term was coined to describe the movements of the heavenly bodies. It was thought that the Sun, Moon, and stars revolved around the earth in set celestial spheres, which themselves were believed to be related in the same whole-number ratios used to create common musical intervals, thus creating harmony.

Perhaps the most famous ratio out there is that of the golden ratio, or φ (phi). This numerical relationship (again, generally attributed to those ubiquitous Pythagoreans) has cropped up for centuries in mathematics, music, architecture and so on, as it is universally believed to be aesthetically pleasing. Wikipedia states that “two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.” This ratio is often expressed visually as a rectangle (the “golden rectangle”) with sides of length 1 and φ (approximately 1.618…) or used to create a “golden spiral” (see plot created with Maple). Successive terms in the Fibonacci sequence are also related by the golden ratio relationship.

Instances of the golden ratio show up over and over again in music; for instance, Antonio Stradivari apparently used it in the design of his renowned violins. Many people have tried to find it in the music of composers such as Mozart and Beethoven but some say any similarity to the ratio is mere coincidence rather than by design. It’s interesting, however, to speculate that great music owes something of its greatness to this seemingly irrational number. I’d like to issue a challenge for our readers. Can you find examples of the golden ratio in music, proving or disproving the claims? (Bonus points if you use Maple for your calculations!)

The relationships between music and mathematics have been studied for centuries, and much debate has ensued. Does playing Mozart to a baby really make them smarter at counting? What does it mean that the notes of our Western musical scale form an abelian group? I don’t know, but to help with these fundamental questions, we’ve linked a very cool video on the Maplesoft YouTube Channel that shows off the Fibonacci sequence with a much more contemporary twist.

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