Guest post: Mathematics in the Music of Bach

This is a guest post by Jeremy Fordham who is a contributing writer for onlinephdprograms.net. Jeremy is an engineer who enjoys addressing topics at the crossroads of many different disciplines, with the hope of inspiring conversation in unique niches.

Johann Sebastian Bach was well-known for his skills as an organist and general keyboard musician during his lifetime, but his gifts as a composer were not recognized fully until the advent of giants like Frederic Chopin (who was a huge Bach proponent). Bach passed away in 1750 and only many years later did scholars begin to critically examine and appreciate the compositional beauty within his music. Since then, his work has been featured countless times on television and in movies, and his development of harmonic theory and counterpoint are still directly applicable to all types of music today. Scholars of renowned Ph.D. Programs and musicians alike have long argued various points of significance in Bach’s work, but by far one of the most contested (and interesting) is Bach’s use of mathematics.

Theories based on Bach’s reliance on mathematical principals as a composer range from the reasonable to the absurd. Some of the more outlandish theories suggest that Bach did everything from using a number alphabet to mathematically predicting the date of his own death in his music. However, other theories concerning Bach’s usage of mathematics in his musical compositions are more readily apparent and were clearly deliberately used by the composer. Understanding something of Bach’s life, and the times in which he lived, leads to a greater appreciation of how and why he applied mathematics to his musical creations.

Bach was born into a musical family in 1685 in northern Germany. By 1695, both of Johann’s parents were dead and he was being raised by an older brother who likely gave Bach his first formal keyboard lessons. Bach’s vocal talents also gave him a place in a select church boys’ choir by 1700. Bach was good at his academic studies and his experience in the church choir turned his musical attention toward sacred subjects. As such, many of his future compositions would be dedicated to God and his upbringing as a Lutheran made him a devoted Christian for the entirety of his life.

Bach’s lifetime encompasses what is now called the Baroque Era, a time of great religious conflict and upheaval. Yet the Baroque Era was also an age of reason as Isaac Newton was performing experiments and devising new laws about gravity, light, and motion. Thus, it is conceivable that Bach was influenced by both religion and reason as he composed the many works for which he would one day be recognized.

Bach specialized in fugues and canons of a mathematical nature. He also appears to have enjoyed games of logic as, for example, the written form of his “Crab Canon” piece can be turned upside down and played without disturbing the melody. Therefore when played correctly, this piece should sound the same whether it is played backwards or forwards. This characteristic evokes the communicative property, a basic arithmetic principal known to mathematics students everywhere.

Similarly, Bach liked to utilize numbers that had special Biblical symbolism throughout his compositions. Many of his works included multiples of three, which many scholars believe symbolizes the Christian trinity. In his work “St. John Passion,” he inserts five notes to signal Pilate’s order to execute Jesus. Some scholars believe that these five notes translate to the Fifth Commandment: “thou shall not kill.”

Bach also displays a special affinity for the number six. For example, in his “O Gott, du frommer Gott,” Bach composed a six line chorale. The work harmonizes on the sixth triad and Bach also includes six notes not usually found in the key of G major. Scholars think that Bach’s fascination with the number six is again related to the Bible. God is said to have created the Earth in six days, thus reserving the seventh day as a day of rest. Throughout the Bible scholars find references to six as a number symbolizing work and it may be that Bach was highlighting this significance.

Perhaps Bach’s greatest achievement at merging the mathematical and the musical came with his “Well-Tempered Clavier.” This work was a collection of 48 pieces of music for the solo keyboard. The preludes and fugues in “The Clavier” were composed in all 24 major and minor keys. Thus, by creating this work, Bach essentially devised a new tuning system in which music could sound pleasing in any key. To do this, he relied on the teachings of Pythagoras who, in ancient Greece, discovered that better sounds could be created by relying on simple ratios from frequencies. Bach took these theories further, embellishing on them until he had come up with a system that was more universally applicable in a range of keys. Later composers would also build on Bach’s theories, making him a precursor to many other musically inclined pioneers.

Today it is not unusual to find musicians in the math departments of colleges across the nation. The reverse can also be said to be true. Most scholars now acknowledge the connections that exist between math and music and those with an interest in both subjects often utilize their knowledge and experience with one to inform the other. Students who study music at a young age tend to perform better in school than those who do not, so clearly a relationship exists between music and intellectual development. It is a connection that Bach certainly would have acknowledged and, in fact did acknowledge, time and again with his music. His appreciation of certain numbers as symbols demonstrates a highly developed intellectual curiosity and a love of learning that manifested itself in some of the greatest musical pieces ever composed.

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