This module is not currently running in 2024 to 2025.
This module is divided into two - one part is about the mathematics of sound, both acoustic and digital, and the other is about the structure of music as it affects musical composition.
The mathematics of sound includes the study of the linear wave equation, in particular, the mathematics of drums and Chladni patterns. We then move on the mathematics of digital sound - the discrete Fourier transform, the short time Fourier transform and the Gabor transform. Here we can answer questions like, does Louis Armstrong play the trumpet the same way he sings? And, how to slow down music without losing pitch?
The mathematics of rhythm and harmony are two very different fields of study. Many world music rhythms can be studied using the Euclidean algorithm. Finally, the harmonic progression of a musical composition can be modelled as a path in chord space. In this part of the module, we will look at how simple geometric ideas are used to model voice leading and harmony . For this last part, familiarity with the keyboard would be helpful but is not a prerequisite.
Indicative syllabus:
Part 1
a. The mathematics of the drum
i. Solutions of the linear wave equation in two dimensions in terms of Bessel functions
ii. Standing waves and Chladni patterns
b. The mathematics of digital music processing
i. Aliasing, Sampling, Filtering
ii. Discrete Fourier Transform, Convolutions
iii. Gabor transform and applications
iv. Spectrograms and applications
Part 2
c. The mathematics of rhythm: Euclidean rhythms in world music
i. The mathematics of harmony in tonal music: Introduction to a mathematical chord space, the Tonnetz.
At level 7, topics will be studied and assessed to greater depth.
42
80% Examination, 20% Coursework
D. Benson, Music: A Mathematical Offering Cambridge University Press, Cambridge, 2006.
G. Loy, Musimathics: The Mathematical Foundations of Music MIT Press, Vols 1 and 2, 2007.
N. Collins, Introduction to Computer Music, Wiley, 2010.
J.S. Walker and G.W. Don, Mathematics and Music: Composition, perception and performance, CRC Press, 2013
D. Tymoczko, A Geometry of Music, Oxford University Press, 2011.
G. Toussaint, The Geometry of Musical Rhythm, CRC Press, 2013.
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes.
On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of discrete Fourier analysis, the geometry of world rhythms and rhythmic tilings, and the geometry of harmony space;
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: Chladni patterns, digital signal processing, the mathematical construction of world rhythms;
3 apply a range of concepts and principles in discrete Fourier analysis in loosely defined contexts, showing good judgment in the selection and application of tools and techniques;
4 make effective and well-considered use of Maple and musical composition software as appropriate.
The intended generic learning outcomes.
On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make effective use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material effectively;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.
University of Kent makes every effort to ensure that module information is accurate for the relevant academic session and to provide educational services as described. However, courses, services and other matters may be subject to change. Please read our full disclaimer.
