Mathematical music theory

Jeffrey Giansiracusa - email: jeffrey.giansiracusa@durham.ac.uk

Description

Music is an expressive art, but it is filled with mathematical structures.  In this project we'll learn about how ideas from algebra and geometry can illuminate and describe some of these structures. 

When mathematicians say the name 'Riemann,' they almost certainly mean Bernard Riemann (1826-1866), who was the inventor of modern differential geometry.  But we're going to investigate ideas that originated with the music theorist Hugo Riemann (1849–1919).  He initiated some fascinating links between group theory in mathematics and the structure of harmony and chords in music.  H. Riemann's ideas were further developed in the 20th cetnury into what is now called neo-Riemannian theory. It gives us some exciting mathematical perspectives for appreciating music.

The notes of an octave are C, C#, D, D#, E, F, F#, G, G#, A, A#, B.  If we identify these with the numbers 1,2,...,12, then we can think about notes in terms of the group Z/12 of integers modulo 12.  The dihedral group of order 24 acts on Z/12 by transposition and inversion, and so in music this is called the T/I group.  Individual notes are a little boring, so we're going to look at chords, which are subsets of Z/12.  The place to start is with the major and minor triads, which are subsets of size 3.  Then we start thinking about how various permutations of Z/12 act on the set of subsets of size 3 and interact with the T/I group.  

Most pieces of music progress through a sequence of chords, and the group threory of the transformations that produce this sequence can tell us interesting things about the music.  But this is only the beginning.  Orbifolds (a kind of manifolds with singularities) enter the story when we think about how choirs sing, and many more surprising pieces of mathematics.  

Prerequisites

• This project has no strict module prerequisites, but Algebra II (MATH2581) would be beneficial.

• A knowledge of basic music theory at the level of roughly Grade 4 (see the syllabus here) would be helpful, but not entirely required if you are enthusiastic to learn.

• You will need to know how to read music (notes on the treble and bass clef), but you do not need to play a musical instrument.  

Resources

You can start with the wikipedia page for neo-Riemannian theory in music: https://en.wikipedia.org/wiki/Neo-Riemannian_theory

Tom Fiore, Music and Mathematics, https://www-personal.umd.umich.edu/~tmfiore/1/musictotal.pdf

John Baez, This week's finds in mathematical physics, Week 234, June 12, 2006, https://math.ucr.edu/home/baez/week234.html

Richard Cohn, Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective
Journal of Music Theory, Vol. 42, No. 2, Neo-Riemannian Theory (Autumn, 1998), pp. 167-180
https://www.jstor.org/stable/843871?seq=8

Dmitri Tymoczko, A geometry of music harmony and counterpoint in the extended common practice
Available online through the library: https://discover.durham.ac.uk/permalink/44DUR_INST/k3s6qp/alma991010371049107366

Some additional resources and lists of links
http://www.ams.org/publicoutreach/math-and-music
https://ncatlab.org/nlab/show/music+theory