Background: Baroque Music

The term “Baroque” is usually used to describe an artistic movement that spread around Europe in the 1600-1700s. This artistic movement revolutionized music as much as it did visual art. The Baroque Era was characterized by motion, detail, and grandness. However Baroque music focused much more on ornamentation.

Some of the great composers of the era include Johann Sebastian Bach, George Frideric Handel, and more. The composers of the era developed many musical forms as well, such as the concerto and sinfonia.

 

One of the major patterns in music that was used during this era was
the concept of contrapuntalism, or counterpoint. In this device, two or more independent melodies are played together to create a harmony. Often, a melody would be played on one layer of a piece and continue on as the melody is restated in another layer of the piece a few beats later. This musical device was very prominent in major works of the era and we will delve deeper into its effects on the music and listeners next time.

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Background: The Golden Ratio and the Fibonacci Sequence

The Golden Ratio

The Golden Ratio is a value that shows up in many areas of mathematics as well as nature. While usually represented by the Greek letter phi (\varphi), the numerical value of this ratio is about 1.6180339887…

This number is derived from the ratio as shown below.

The Golden Ratio

The Golden Ratio

In the diagram, a/(a+b) = b/a = \varphi. Line segment b can be further divided in this way to create two line segments that add up to b, but have the same golden ratio as a and b.

These patterns can be continued on infinitely, as can be visualized with this picture using the golden ratio for sides of rectangles.

The Golden Ratio with rectangles. from http://www.riggspartners.com/wp-content/uploads/2012/04/golden-mean-spiral.png

Now, one might wonder: how can we find such a pattern in music? From the intervals between notes to the organization of sections or sequences of keys, there are many ways to see and interpret this ratio in music.

The Fibonacci Sequence

The Fibonacci Sequence is a sequence of numbers beginning with 1,1,2,3,5… where each proceeding number in the sequence is the sum of the two numbers before it. Thus an extended version of the sequence shows

1,1,2,3,5,8,13,21,34,55,89,144,233,377… and so on.

The Fibonacci Sequence is related to the Golden Ratio such that when you divide a term by the term before it, the result gets closer to \varphi as the numbers get larger and larger. Therefore we get the expression \lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi. Where F(n) is the nth term of the Fibonacci sequence.

Music? Math? What?

Welcome to Musematics. Here, in this very blog, I will detail closely, my progress in researching, discovering, and observing mathematical patterns and principles in various types of music.

Why math and music? Both these areas of study greatly interest me, and I have been involved in both since a very young age. I began playing the piano at the age of 4, clarinet at 11, and have in the years since have attempted to take up numerous other instruments such as guitar, ukulele, and accordion, and activities such as composing and arranging. Meanwhile, on the math side of things, I had always had a focus on math throughout elementary school, which led to my participation in numerous math competitions and contests throughout middle and high school. These two topics have arguably had the biggest impact on me during my life.

For many, many years, music has been associated with and related to nature. Many mathematical patterns associated with nature, such as the Golden Ratio and the Fibonacci Sequence, can also be found in pieces of music. How can we find and identify these patterns? What effect do they have on the piece? These questions will guide me throughout my research in this project.