The Mathematics of Musical Composition - Sarah Hart

00:00 // Introduction 03:41 // Is, What It's Pitch Should Be 07:10 // You're Writing a Piece of Music Is Repetition 10:46 // Same Gap Between Two Notes 14:10 // You Will Know That I Am a Greek Theorist 18:22 // Them Could Be Made by Just Our Favorite Translation 21:39 // Inversion Can Be Made with Your One Favorite Inversion 25:08 // There Are Four Rules That Tell You Whether a Set Is 28:22 // G and That Relationship of Having No Effect Comes 32:11 // Is Doing What We're Supposed to Do Every Few Months Is 35:37 // Notes of the Retrograde Theme Here 38:55 // Permuting All These Tone Rows 44:18 // It Says, Given Any Set of Tones 47:37 // This Associate That You See Just Write the Numbers 50:56 // Recursive Relationship 54:18 // Models Rare Events 57:55 // Group Theory, but He Is Doing Mathematics 61:38 // Of Successive Terms Pattern and structure are essential to music, from the permutations in a Bach fugue, to the structure of a round. This lecture will explore the mathematics of musical symmetries, such as the “translational symmetry” of the transposition of keys, and the “rotational symmetry” of the duet “Der Spiegel”, attributed to Mozart. It will go on to explore the last century’s experimentation with the use of mathematical forms to produce intriguing effects, including in the works of Arnold Schoenberg and Per Nørgård. This lecture was recorded by Sarah Hart on 13 October 2020. Professor of Geometry (2020 - 2024) Acting Provost of Gresham College (2025) Sarah Hart was the first woman Professor of Geometry at Gresham College, and Acting Provost between March and October 2025. She is also Professor Emerita of Mathematics at Birkbeck, University of London where she served as the Head of Mathematics and Statistics. Professor Hart studied at Oxford and Manchester, gaining her PhD in 2000. Postdoctoral research and teaching followed, including a prestigious Engineering and Physical Sciences Research Council Fellowship, before she was appointed to a lectureship at Birkbeck in 2004. She became Professor of Mathematics there in 2013, and served in various management roles including as Head of Mathematics and Statistics, Assistant Dean, and Programme Director for the MSc Mathematics. Her academic publications have been mainly in the area of pure mathematics known as group theory, which has many applications both inside and outside of mathematics, for example in coding theory and cryptography. She is actively involved in the British Society for the History of Mathematics, and has served a three-year term as President of the Society from 2021-2023. The transcript and downloadable versions of the lecture are available from the Gresham College website: https://gres.hm/maths-music Gresham College has offered free public lectures for over 400 years, thanks to the generosity of our supporters. There are currently over 2,500 lectures free to access. We believe that everyone should have the opportunity to learn from some of the greatest minds. To support Gresham's mission, please consider making a donation: https://gresham.ac.uk/support/ Website: https://gresham.ac.uk Twitter:   / greshamcollege   Facebook:   / greshamcollege   Instagram:   / greshamcollege