Podcast Summary: "Why Numbers are Music to Our Ears (Update)"
People I (Mostly) Admire is a captivating podcast series where host Steve Levitt engages with high achievers to uncover the intriguing intersections of their lives and passions. In the episode titled "Why Numbers are Music to Our Ears (Update)," released on January 11, 2025, Levitt reunites with mathematician Sarah Hart to delve deeper into the harmonious relationship between mathematics and music.
Introduction to Sarah Hart and Her Mission
In this bonus episode, Steve Levitt welcomes back Sarah Hart, a distinguished mathematician who has previously appeared on the show in 2021 and 2023. Hart is renowned for her efforts to make mathematics accessible and enjoyable to the general public. She holds dual academic positions: a professor of mathematics at Birkbeck College, University of London, and the professor of geometry at Gresham College, a historic role dedicated to providing free public lectures.
Sarah Hart:
"I've been on a crusade to make math education more engaging, and you're someone who's actually figured out how." [01:38]
Mathematics and Music: A Symbiotic Relationship
Demonstrating Musical Octaves
Levitt and Hart begin their discussion with a practical demonstration of octaves on a keyboard. By playing a middle C and a high C (an octave higher), and then both simultaneously, Levitt illustrates how these notes, despite being different in pitch, sound remarkably similar to the human ear.
Sarah Hart:
"The crazy thing is it really sounds like you're playing one note." [03:31]
Steve Levitt:
"What we do when we're learning something or experiencing something is we need to actually see it happening or to hear it. We need something to grab onto." [03:36]
This experiment underscores the foundational mathematical principle that octaves are related by a frequency ratio of 2:1, a relationship that dates back to Pythagorean theories on musical harmony.
The Mathematics Behind Musical Intervals
Frequency Ratios and Pleasant Harmonies
Hart and Levitt explore why certain musical intervals, such as the octave (2:1) and the perfect fifth (3:2), are inherently pleasing to our ears. These intervals are derived from simple whole-number frequency ratios, which the brain recognizes and processes as harmonious.
Steve Levitt:
"When you multiply by three over two and you do that 12 times, you'll get like three to the power 12 divided by two to the power 12. That is not 128, it's 129.7 and a bit. Right. 12 fifths are not equal to seven octaves." [10:42]
This discussion highlights the complexity of musical tuning systems and the historical challenge of aligning mathematical precision with auditory perception.
Historical Integration of Mathematics in Music
Pythagorean Influence and Beyond
The conversation traces the integration of mathematical concepts in music back to the Pythagoreans, who believed in a universal harmony governed by numerical ratios. This belief system influenced not only music but also other disciplines like astronomy and geometry.
Steve Levitt:
"These very early experiments about the lengths of different strings making these lovely ratios, it was all bound up in this idea of how everything has a wonderful universal harmony." [12:42]
However, Levitt also acknowledges the limitations and obstacles faced by early mathematicians and musicians, such as theological constraints that hindered the acceptance of astronomical discoveries contradicting geocentric models.
Constraints as Catalysts for Creativity
Mathematical Constraints in Music Composition
Levitt and Hart delve into the role of constraints in fostering creativity, both in music and literature. They discuss how imposing mathematical rules or structures can lead to innovative and compelling works.
Sarah Hart:
"The idea that mathematics applied to astronomy probably held us back hundreds of years in actually figuring out what was going on." [12:59]
Steve Levitt:
"If you constrain everything, then you're trapped in a really rigid box. There's no room for you to be creative with absolutely no constraints at all." [27:58]
Literary Constraints and Mathematical Patterns
The episode also explores how literary forms, such as haikus and sonnets, incorporate mathematical structures. Hart explains the significance of prime numbers in haikus and the combinatorial possibilities in sonnet constructions.
Sarah Hart:
"You have 17 sounds, split into 5, 7, and 5. Those are prime numbers, which makes for a cleaner break and more interesting structure." [22:23]
Mathematics in Storytelling and Media
Choose Your Own Adventure and Graph Theory
Levitt discusses the mathematical underpinnings of interactive storytelling, such as choose-your-own-adventure books and the Netflix film Bandersnatch. He explains how graph theory, which studies networks and connections, is essential in designing these branching narratives.
Steve Levitt:
"They design it and they draw these Graphs themselves by hand... it's a mix of creativity and structure." [40:43]
This segment underscores the balance between creativity and mathematical structure necessary to create engaging and manageable interactive experiences.
Sarah Hart's Contributions and Future Projects
"Once Upon a Prime" and Beyond
Sarah Hart shares insights from her book Once Upon a Prime, which explores the delightful intersections between mathematics and literature. She emphasizes the joy of uncovering mathematical patterns in literary works, enhancing readers' appreciation of both disciplines.
Steve Levitt:
"Mathematics is really our way of understanding structure and pattern... literature has... rhythms and patterns." [21:31]
Upcoming Ventures
Looking ahead, Hart discusses her plans to further investigate the mathematical foundations of music in her next book, continuing her mission to reveal the beauty and creativity inherent in mathematical concepts.
Sarah Hart:
"I'm planning my next book, which will be all about the beautiful mathematics behind music." [48:10]
Conclusion
In this engaging episode, Steve Levitt and Sarah Hart illuminate the profound and often surprising connections between mathematics and music. Through practical demonstrations, historical context, and discussions on creativity and constraints, they reveal how numerical relationships underpin some of the most pleasing and structured aspects of both disciplines. Hart's dedication to making mathematics accessible and enjoyable shines through, offering listeners a renewed appreciation for the mathematical harmony that resonates in everyday life.
Notable Quotes:
-
Steve Levitt:
"What we do when we're learning something or experiencing something is we need to actually see it happening or to hear it. We need something to grab onto." [03:36] -
Sarah Hart:
"The crazy thing is it really sounds like you're playing one note." [03:31] -
Steve Levitt:
"Mathematics is really our way of understanding structure and pattern." [21:31] -
Sarah Hart:
"You have 17 sounds, split into 5, 7, and 5. Those are prime numbers, which makes for a cleaner break and more interesting structure." [22:23]
This summary captures the essence of the episode, highlighting the key discussions and insights shared by Steve Levitt and Sarah Hart. It provides a comprehensive overview for those who haven't listened to the podcast, offering a glimpse into the fascinating interplay between mathematics and music.