Definition
While math and music each correlate to opposite sides of the brain, the two can sometimes come together to reveal interesting information. For example, listening to music while doing math has been proven to help students learn, and any musician knows that music can be stripped back to patterns, timing, and formulae. Music can be considered both an art and a science because it applies the principles and logic of math. However, rather than using numbers, it uses sounds, pitch, and tempo. [1]
Who
There are many famous musicians who could credit their ability to a background in mathematics and science. A prime example is Brian May, the lead guitarist of Queen, who studied physics and mathematics at Imperial College London and is the author of A Survey of Radial Velocities in the Zodiacal Dust Cloud, as well as being one of the best guitarists in the world. Another example is Dan Snaith, who has the stage name Caribou and who also completed a Ph.D. in mathematics. [1]
What
The mathematics of music is a fascinating intersection of two seemingly disparate fields. Here’s an overview of some of the key ways that mathematics and music intersect:
- Pitch and Frequency:
- Musical pitches correspond to specific sound wave frequencies, which can be described mathematically.
- The relationship between pitches, such as the perfect fifth interval, can be expressed using ratios of frequencies.
- Mathematical concepts like the harmonic series help explain the overtone structure of musical sounds.
- Rhythm and Meter:
- Musical rhythms and time signatures can be represented and analyzed using mathematical concepts like fractions, ratios, and modular arithmetic.
- Patterns in rhythm, such as polyrhythms, can be described and understood through mathematical modeling.
- Techniques like Fourier analysis can be used to decompose complex rhythmic structures into their fundamental components.
- Musical Form and Structure:
- Concepts from set theory, group theory, and other branches of mathematics have been used to analyze the formal and structural aspects of music.
- Symmetry, transformations, and other mathematical properties can be used to describe and understand musical forms, such as fugues, sonatas, and 12-bar blues.
- Tuning and Temperament:
- The mathematical properties of different tuning systems, such as Pythagorean tuning and equal temperament, have been studied extensively.
- Exploring the mathematical relationships between different notes and intervals has led to the development of various tuning and temperament systems.
- Generative Music and Algorithmic Composition:
- Mathematicians and computer scientists have developed algorithms and generative models to create music, often drawing on concepts from areas like formal grammars, Markov chains, and cellular automata.
- The field of algorithmic composition explores how mathematical and computational techniques can be used to generate novel musical compositions.
- Musical Acoustics and Instrument Design:
- The physical properties of sound waves and the behavior of musical instruments can be described and analyzed mathematically.
- Concepts from fields like signal processing, fluid dynamics, and materials science have been applied to the design and optimization of musical instruments.
The interplay between mathematics and music has led to important advancements in both fields, and continues to inspire new avenues of research and creative expression. By exploring the mathematical underpinnings of music, researchers and musicians can gain deeper insights into the nature of sound, the structure of music, and the cognitive processes involved in musical perception and creation. [2]
Why
To mathematicians, mathematics is an intrinsically beautiful subject. Its objects are pure and precise and seem worthy of study unto themselves. However, to many people, these objects may seem dry, remote, and irrelevant. Still, it is demonstrably clear that mathematics is extraordinarily relevant and essential to nearly every human endeavor; science and engineering, medicine, politics, business, etc.. Indeed, mathematics provides a common language to express so many important aspects of human knowledge that in order to attain a better understand of progress in those areas, one must be capable of grasping basic mathematical concepts. [3]
See Theoretical Knowledge Vs Practical Application.
How
Music is basically a series of varying sound wave frequencies that collectively produce harmonics. Musical concepts such as harmony, melody, and rhythm have mathematical underpinnings. In fact, musical notations such as fractional notes and their orderly arrangements on a music sheet are actually mathematical in nature. For example, the relational values between notes or chords can be arranged into major and minor keys, which may include semi-tonnes. This means that music has mathematical order that can be expressed in abstract notations that musicians can play in an instrument or sing. [1]
Many of the References and Additional Reading websites and Videos will assist you with understating and applying mathematics to music.
As some professors say: “It is intuitively obvious to even the most casual observer.“
References
[1] What Math Can Teach Us About Music. 2024. Arduino Education. Accessed June 15. https://www.arduino.cc/education/what-math-can-teach-us-about-music.
[2] “Fast, Helpful AI Chat.” 2024. Poe. Assistant. Accessed June 15. https://poe.com/.
[3] Naiman, Daniel. 2024. Mathematics of Music. Accessed June 15. https://www.ams.jhu.edu/dan-mathofmusic/.
Additional Reading
“7.3: Musical Scales”. 2020. Mathematics LibreTexts. Libretexts. September 12. https://math.libretexts.org/Courses/College_of_the_Canyons/Math_100%3A_Liberal_Arts_Mathematics_(Saburo_Matsumoto)/07%3A_Mathematics_and_the_Arts/7.03%3A_Musical_Scales.
The first thing musicians must do before they can play together is “tune.” For musicians in the standard Western music tradition, this means agreeing on exactly what pitch (what frequency) is an “A,” what is a “B flat,” and so on. Other cultures not only have different note names and different scales, they may even have different notes – different pitches – based on a different tuning system. In fact, the modern Western tuning system, which is called equal temperament, replaced (relatively recently) other tuning systems that were once popular in Europe. All tuning systems are based on the physics of sound. But they all are also affected by the history of their music traditions, as well as by the tuning peculiarities of the instruments used in those traditions.
“Connections: Math and Music.” 2024. The Kennedy Center. Accessed June 15. https://www.kennedy-center.org/education/resources-for-educators/classroom-resources/media-and-interactives/media/music/connections/connections/math–music/.
Math and music might seem like an odd couple. But when we take a closer look, they have more in common than you might think. You see, beneath the sound of the notes are math concepts and rules. These math ideas help organize everything from the beat (rhythm) to the tune (melody) of the music. It’s as easy as 1, 2, 3, 4!
Hoang, Lê Nguyên, and Inteano says: 2015. “The Harmonious Mathematics of Music.” Science4All. December 19. https://www.science4all.org/article/math-music/.
According to legend, the Ancient Greek Pythagoras was once walking on the streets of Samos, when the sounds of blacksmiths’ hammering suddenly gave him an epiphany. Pythagoras rushed into the shop and, as he analyzed mathematically the shapes of the blacksmiths’ hammers, he laid the foundations of music that others are building upon.
“Mathematics & Music.” 2024. American Mathematical Society. Accessed June 15. https://www.ams.org/publicoutreach/math-and-music.
Counting, rhythm, scales, intervals, patterns, symbols, harmonies, time signatures, overtones, tone, pitch. The notations of composers and sounds made by musicians are connected to mathematics. The next time you hear or play classical, rock, folk, religious, ceremonial, jazz, opera, pop, or contemporary types of music, think of what mathematics and music have in common and how mathematics is used to create the music you enjoy. Explore the connections between mathematics and music in the videos, podcasts, and articles on this web page.
“Mathematics and Music”. 2024. Simplifying Theory. Accessed June 15. https://www.simplifyingtheory.com/math-in-music/.
We decided to build this topic to show you how mathematics is related to music. Maybe you don’t like math, but don’t worry, we will try to explain each concept in a simple way, so that you realize that our sensitivity to sound is linked to the logic of our brains. This is very interesting, so let go of your prejudices. All knowledge is nice when well taught.
“Mathematics of Music: Brilliant Math & Science Wiki.” 2024. Brilliant. Accessed June 15. https://brilliant.org/wiki/mathematics-of-music/.
Sound consists of a physical wave– a sound wave. A musical note corresponds to a periodic sound wave with a specific frequency. And, as we discuss here, musical relations between notes correspond to mathematical relations between their frequencies.
“Music and Mathematics.” 2024. Wikipedia. Wikimedia Foundation. https://en.wikipedia.org/wiki/Music_and_mathematics.
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. While music theory has no axiomatic foundation in modern mathematics, the basis of musical sound can be described mathematically (using acoustics) and exhibits “a remarkable array of number properties”.
“Music Theory and Math – Understanding Connections.” 2024. Global Music Theory. Accessed June 15. https://globalmusictheory.com/music-theory-and-math-understanding-connections/.
This idea that music is math or at least mathematical, is one that I encountered often at the university. To be fair, virtually anything can or does involve numbers and math. But music shares a famous truism with math, they’re both “universal languages”. Whether both deserve that “universal” status could be debated but there must be something to this idea that music and math are bonded in some way, right?
In general, music theory maps well to mathematical concepts. Notable sub-areas of music theory that work particularly well are: set theory, twelve-tone theory, scales, and tunings. Music did not come from math, but numeric notation and mathematic operations work well to describe music.
Pike, Jay. 2022. “The Hidden Math in Music.” Wyzant Blog. December 21. https://blog.wyzant.com/the-hidden-math-in-music/.
Music and math are not just complimentary, but as the early Greek philosophers uncovered, music is math! Aristotle famously was fascinated by music not just for its aesthetic qualities but especially for its mathematical components as well. As history progressed, lots of brilliant minds have been inspired by the enjoyment of music to better understand the nature of musical sounds and their fundamental properties in the physical world.
In this article, we will discuss four different elements of music and the math behind them: pitch, timbre, harmony, and rhythm. Each of these four elements is directly shaped by the mathematical characteristics of sound. Mathematical concepts appear everywhere in the elements of music, both in the sounds of music and the composition of music, so this description is by no means a complete list of the mathematical characteristics.
Read on and learn some amazing facts about how music and mathematics relate.
Reitman, Boris. 2021. “The Mathematical Nature of Musical Scales.” Medium. Cantor’s Paradise. May 16. https://www.cantorsparadise.com/the-mathematical-nature-of-musical-scales-f0a6536bca5d.
In this article I will explain the math for the Just intonation and Pythagorean tunings for the diatonic major scale (all the white keys on the piano).
Reitman, Boris. 2021. “The Physical Nature of Musical Sound.” Medium. Medium. May 21. https://borisreitman.medium.com/the-nature-of-musical-sound-39bf974e63b1.
This is an educational post that will be useful to anyone who wants to understand the keys on a piano keyboard from a mathematical perspective. It is a first article in three-part series, in which I focus on the physics of sound. Although I use the piano as the case-study, this explanation also applies to all musical instruments.
Videos
“How Music and Mathematics Relate.” [11 Videos] 2024. YouTube. YouTube. Accessed June 15. https://www.youtube.com/playlist?list=PLOxODW9vlVLRjofkycX3OUTivnX0DpaS_.
Gain new perspective on two of the greatest achievements of human culture—music and math—and the fascinating connections that will help you more fully appreciate the intricacies of both.
Great minds have long sought to understand the relationship between music and mathematics. On the surface, they seem very different. Music delights the senses and can express the most profound emotions, while mathematics appeals to the intellect and is the model of pure reasoning.
Yet music and mathematics are connected in fundamental ways. Both involve patterns, structures, and relationships. Both generate ideas of great beauty and elegance. Music is a fertile testing ground for mathematical principles, while mathematics explains the sounds instruments make and how composers put those sounds together. Moreover, the practitioners of both share many qualities, including abstract thinking, creativity, and intense focus.
Understanding the connections between music and mathematics helps you appreciate both, even if you have no special ability in either field—from knowing the mathematics behind tuning an instrument to understanding the features that define your favorite pieces. By exploring the mathematics of music, you also learn why non-Western music sounds so different, gain insight into the technology of modern sound reproduction, and start to hear the world around you in exciting new ways.
There is a serious mathematical problem with the tuning of musical instruments. A problem that even Galileo, Newton, and Euler tried to solve. This video is about this problem and about some of the ways to tackle it. It starts from the basic physics of sound, proves mathematically why some musical instruments can never be perfectly in tune, and then introduces the main solutions that were proposed to solve this problem, along with their upsides and downsides: Pythagorean tuning, Just intonation, the Meantone temperament, and finally – the equal temperament, which is the tuning system almost everybody uses today in the West.
A 15 minute explanation of how music is derived from maths. Discover why an octave has 12 steps & what the music of Metallica & Miley Cyrus owes to JS Bach & Pythagoras.
Music theory is supposed to be hard – it’s not. It’s actually very simple. I teach you how to build the major and minor scales with very simple algebra.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
The featured image on this page is from the Dalhousie University website.
Mathematical Mysteries 
