Music and Mathematics - Mathematician & Concert Pianist Eugenia Cheng

Summary

This video explores the fascinating connection between mathematics and music, focusing on the evolution of the 12-tone equal temperament system. It explains how mathematicians, by calculating the 12th root of 2, enabled the division of the octave into equal intervals, which in turn allowed composers like Bach to write music in every key. The video highlights the harmonic series, dissonance in music, and how abstract mathematical theory influenced musical composition, particularly in Bach's '48 Preludes and Fugues.' The discussion blends mathematical precision with the art of music in an insightful and engaging manner.

Takeaways

• 😀 Mathematics can solve problems in unexpected ways, sometimes providing solutions to issues it wasn't initially intended to address.
• 🎶 Before dividing the octave into 12 equal notes, some musical intervals were favored over others based on harmony and dissonance.
• 🎵 Certain notes in music were historically considered less important, with some deliberately left sounding unpleasant to maintain the harmony of the scale.
• 🎼 The interval F# was once known as 'Diabolus in Musica' or 'the devil in music' because it was the most dissonant interval in the musical scale.
• 📐 The harmonic series in music consists of a fundamental tone and a series of higher-frequency tones (overtones), which are integer multiples of the fundamental frequency.
• 🎹 In music theory, an octave is a doubling of frequency, and intervals like the perfect fifth are based on simple integer ratios, but these don't exactly match with modern tuning systems.
• 🔢 The equal temperament tuning system divides the octave into 12 equal intervals, achieved by calculating the 12th root of 2, which was a difficult mathematical problem.
• 📚 The ability to divide the octave into 12 equal parts allowed for the creation of music in every key, facilitating greater musical flexibility and complexity.
• 🎶 Johann Sebastian Bach was excited by the possibility of writing music in all keys, which led him to compose the 48 Preludes and Fugues, a monumental set of works.
• 🧑‍🔬 Higher-dimensional category theory, a branch of abstract mathematics, plays a role in understanding structures and relationships within mathematics itself, often beyond practical application.
• 🎶 The intersection of mathematics and music is exemplified in how mathematical breakthroughs, such as equal temperament, influenced musical composition, as seen in Bach's works.

Q & A

What was the original reason for the uneven treatment of notes before the octave was divided into 12 equal notes?

Before the octave was divided into 12 equal notes, certain chords sounded better than others, and some notes were considered less important. The most important notes sounded fine, but others were intentionally left to sound unpleasant or 'awful'.

What does the term 'Diabolus in Musica' refer to in the context of music theory?

'Diabolus in Musica', meaning 'the devil in music', refers to the interval between F# and C, which was considered highly dissonant and the most unrelated to the C major scale.

Why was the interval between F# and C considered dissonant?

The interval between F# and C was considered dissonant because the harmonics of the two notes clash with each other, creating a harsh and unsettling sound.

What is the harmonic series in music?

The harmonic series refers to the set of frequencies that are integer multiples of a fundamental frequency. These harmonics are the overtones that occur when a note is played on a string or another resonating body.

How are the intervals in the harmonic series related to each other?

In the harmonic series, the first harmonic is the fundamental frequency, and subsequent harmonics are integer multiples of the first frequency, such as doubling for the octave, tripling for the perfect fifth, and so on.

What mathematical challenge was faced in dividing the octave into 12 equal intervals?

The mathematical challenge was that the octave couldn't be divided into 12 equal lengths of sound waves. Instead, it required splitting the octave into 12 equal ratios, which meant calculating the 12th root of 2, a complex mathematical problem.

What significance did the 12th root of 2 have in music theory?

The 12th root of 2 was crucial because it allowed the octave to be divided into 12 equal intervals, making it possible to write music in all keys, enabling the development of modern Western music.

How did Bach take advantage of the ability to write in all keys?

Bach took full advantage of the ability to write in all keys by composing his '48 Preludes and Fugues', a collection of works in every possible key, demonstrating the new musical freedom made possible by equal temperament tuning.

What is the field of research mentioned in the script, and how is it related to mathematics?

The research field mentioned is higher-dimensional category theory, which is a branch of abstract mathematics. It deals with structures in mathematics and can be described as the 'mathematics of mathematics'.

How does the speaker feel about the connection between mathematics and music?

The speaker finds the connection between mathematics and music inspiring, appreciating how a mathematical breakthrough in tuning allowed for advances in music, particularly inspiring Bach to write his remarkable compositions.