Mathematics & Music

"There is geometry in the humming of the strings, there is music in the spacing of the spheres."
–Pythagoras

$Mathematics and Music poster$ 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 below.

Videos

Majesty of Music and Math. "Explore the interconnectedness of music and mathematics. The Majesty of Music and Math features remarks by Santa Fe Institute mathematician and computer scientist Cris Moore and musical selections by The Santa Fe Symphony Orchestra with Principal Conductor Guillermo Figueroa."

Music + Math: Symmetry. "From Pythagoras' observations of the fundamental mathematical relationship between vibrating strings and harmony to the digitized musical world we enjoy today, The Majesty of Music and Mathematics with the Santa Fe Symphony and the Santa Fe Institute will explore the remarkable interweaving of the languages of music and mathematics." View the entire Santa Fe Institute series, covering the tritone, harmonics, ratios and more.

Geometry in Music; Dmitri Tymoczko. "What could be a better medium to communicate math to the public than the universal language of music? Ever since Pythagoras used numerical terms to express intervals between notes and derived musical tones from geometrical patterns, mathematicians have linked music to numbers." At the SIAM Annual Meeting in 2010, Tymoczko used graphics and sound to connect math to the music of Chopin, Mozart, and Schubert. Video by Adam Bauser, Bauser Media Group.

Combining Math and Music. Eugenia Cheng, a mathematician who also is a concert pianist, describes how a mathematical breakthrough enabled Johann Sebastian Bach to write "The Well-Tempered Clavier" (1722). At the time that the video was recorded, Cheng was a visiting senior lecturer in mathematics at the University of Chicago. Cheng also has an 11-part video series, Math in Music, hosted by WFMT, 2020, with topics such as "Feeling the Commutativity of Multiplication," "Math to Build New Ideas," "Symmetry in Music," "Fractions Give Us Feelings!," "Math Can Also Sound Bad," and "Harmonics As Special Effects."

David Kung on "Symphonic Equations: Waves and Tubes." David Kung (St Mary's College of Maryland) presents "Symphonic Equations: Waves and Tubes" -- a miniexcursion into math and music, presented as a MAA Distinguished Lecture at the Carnegie Institution for Science.

The Science Behind the Arts: The Maths Behind Music. University of Surrey, England.

More Videos

Turning math into music. Sean Hardesty (Rice University) plays the opening of the Sibelius Violin Concerto and discusses the relationship between mathematics and music.

The world's ugliest music: Scott Rickard at TEDxMIA. Scott Rickard has degrees in mathematics, computer science, and electrical engineering from M.I.T. and MA and PhD degrees in applied and computational mathematics from Princeton University. Rickard says that he is "passionate about mathematics, music and educating the next generation of scientists and mathematicians."

Robert Schneider - Reverie in Prime Time Signatures (August 2013, Banff Centre). Robert Schneider (Apples in Stereo/Elephant 6) composed this theme for "MSI (Math Sciences Investigation): Anatomy of Integers and Permutations," a play by mathematician Andrew Granville and screenwriter Jennifer Granville. "As the title indicates, the piece is written in prime-numbered time signatures; which is to say, there is a prime number of beats in each measure. The main theme plays in the time signature 7/4, which indicates 7 beats per measure, with an interlude that passes through the signatures 2/4, 3/4 and 5/4 as well. From the constraints imposed by these rhythmic patterns, melodies emerged naturally as I composed, special to each prime..."

Music and math: The genius of Beethoven – Natalya St. Clair TED Ed. Natalya St. Clair uses the Moonlight Sonata "to illustrate the way Beethoven was able to convey emotion and creativity using the certainty of mathematics."

The Math Behind Music. Ethan Thompson and David Hamilton explain the math behind music in a fun, concise way in this finalist entry in the 2015 Math-O-Vision contest.

Listen Up (Podcasts)

What Do Equations Sound Like on the Piano? Pius Wong (Austin Community College) discusses Chordinates, a web tool he created that generates melodies from the graphs of math equations.
Making Beautiful Mathematics. In this podcast interview, Rob Schneiderman (Lehman College, CUNY) discusses the metaphorical connections between math and music.
Marcus du Sautoy: music and symmetry. Marcus du Sautoy (Simonyi Professor for the Public Understanding of Science and a professor of mathematics at the University of Oxford) talks to a concert audience about the connections between music and mathematics.
Putting Music on the Map. Listen to Dmitri Tymoczko (Princeton University) speak about using topology to represent musical chords as points in a space.

A Medley of More Mathematics and Music

"The Smile of Math: When Categories Inspire," by Maria Mannone
"Phase transitions: The math behind the music," Phys.org, May 23, 2019
"No Static at All: Frequency modulation and music synthesis," by David Austin
"Time-frequency Analysis of Musical Rhythm," by Xiaowen Cheng, Jarod V. Hart, and James S. Walker
"Musical Actions of Dihedral Groups," by Alissa S. Crans, Thomas M. Fiore, and Ramon Satyendra
"Music: Broken Symmetry, Geometry, and Complexity" by Gary W. Don, Karyn K. Muir, Gordon B. Volk, James S. Walker
"Was Something Wrong with Beethoven’s Metronome?" by Sture Forsén, Harry B. Gray, L. K. Olof Lindgren, and Shirley B. Gray
Emily Howard, articles and compositions posted as Artist in Residence, Department of Mathematical Sciences, University of Liverpool
"The mathematician’s patterns, like those of the composer, must be beautiful," by Peter Lynch, The Irish Times
"Math and the Musical Offering," by Tony Phillips
"The mathematics of piano tuning," by Tony Phillips
"Surface Topology in Bach Canons: Part I and Part II," by Tony Phillips
PRiSM: The Centre for Practice & Research in Science & Music at the Royal Northern College of Music, directed by composer Emily Howard (Professor of Composition, RNCM) and co-directed by mathematician Marcus du Sautoy (Simonyi Professor for the Public Understanding of Science and Professor of Mathematics, University of Oxford)
"The magical mathematics of music," by Jeffrey Rosenthal
"Can One Hear the Sound of a Theorem?," by Rob Schneiderman
"This Bird’s Songs Share Mathematical Hallmarks With Human Music," by Helen Thompson
"The Generalized Tonnetz," by Dmitri Tymoczko
"Neo-Riemannian theory," Wikipedia

If you would like to recommend more websites on mathematics and music, please email AMS Outreach.

Resources available by subscription or purchase

Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance
Mathematics and Music, by David Wright
Many more books that can be found via a search on the web or your preferred online bookstore!

Also see: Society for Mathematics and Computation in Music. The Society was founded in 2006 as an International Forum for researchers and musicians working in the trans-disciplinary field at the intersection of music, mathematics and computation. The website includes information about joining and attending meetings, and online newsletters.