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Mathematics and Music

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Mathematics and Music

BA
  • UCAS code GW13
  • Option 3 years full time
  • Year of entry 2021

The course

“There is geometry in the humming of the strings. There is music in the spacing of the spheres.” – Pythagoras

Looking to keep your love of music alive whilst exploring the true breadth of mathematical ideas and methods? Since Pythagoras developed his theory of the unity of arithmetic, geometry and music around the fundamental laws of proportion and harmony, in the 6th century BC, the two disciplines have influenced and informed each other. They are both concerned with the creation, appreciation and analysis of abstract patterns and logical ideas. Our joint honours degree allows you to keep your career options open and pursue your mathematical and musical interests in a 50/50 split, under the guidance of inspiring teachers from both departments.

Throughout your course, you will have the flexibility to tailor your studies to your own particular interests. Alongside our core Mathematics modules in year 1, which will give you a thorough grounding in all the key methods and concepts that underpin the subject, we offer a wide range of practical and academic modules from one of UK's top music departments (Complete University Guide 2015). In years 2 and 3 this flexibility increases, allowing you to specialise in the areas of mathematics and music that interest you the most. You will develop your musicianship and mathematical skills to an advanced level, and gain a host of transferable skills such as data handling and analysis, logical thinking, communication, creativity and problem-solving.

Our Department of Mathematics is internationally renowned for its work in pure mathematics, information security, statistics and theoretical physics, and our joint BA course spans pure and applied mathematics, statistics and probability. It also offers you to chance to carry out project work on chosen topics. Meanwhile, in the Department of Music, you will be able to pursue performance and composition whilst exploring the broader historical, sociological, ethnographic and philosophical elements of music. For keen singers and instrumentalists, we offer a wide range of exciting and diverse performance opportunities and you will have access to our well-equipped studios, practice rooms and recording facilities.

We offer a friendly and motivating learning environment and a strong focus on small group teaching and academic support. You will take part in group tutorials, problem-solving sessions, practical workshops and IT classes, as well as practical music lessons and lectures. You will also benefit from generous staff office hours and a dedicated personal adviser to guide you through your studies, plus a CV writing workshop and competitive work placement scheme.

Our flexible degree programmes enable you to apply to take a Placement Year, which can be spent studying abroad, working or carrying out voluntary work. You can even do all three if you want to (minimum of three months each)! To recognise the importance of this additional skills development and university experience, your Placement Year will be formally recognised on your degree certificate and will contribute to your overall result. Please note conditions may apply if your degree already includes an integrated year out, please contact the Careers & Employability Service for more information. Find out more

  • Combine your love of mathematics and music and benefit from a varied and flexible, modular curriculum.
  • We are a friendly department with a strong focus on small group teaching.

Core Modules

Year 1
  • In this module, you will develop an understanding of the key concepts in Calculus, including differentiation and integration. You will learn how to factorise polynomials and separate rational functions into partial fractions, differentiate commonly occurring functions, and find definite and indefinite integrals of a variety of functions using substitution or integration by parts. You will also examine how to recognise the standard forms of first-order differential equations, and reduce other equations to these forms and solve them.

  • In this module you will develop an understanding of the calculus functions of more than one variable and how it may be used in areas such as geometry and optimisation. You learn how to manipulate partial derivatives, construct and manipulate line integrals, represent curves and surfaces in higher dimensions, calculate areas under a curve and volumes between surfaces, and evaluate double integrals, including the use of change of order of integration and change of coordinates.

  • In this module you will develop an understanding of the fundamental algebraic structures, including familiar integers and polynomial rings. You will learn how to apply Euclid's algorithm to find the greatest common divisor of two integers, and use mathematical induction to prove simple results. You will examine the use of arithmetic operations on complex numbers, extract roots of complex numbers, prove De Morgan's laws, and determine whether a given mapping is bijective.

  • In this module you will develop an understanding of basic linear algebra, in particular the use of matrices and vectors. You will look at the basic theoretical and computational techniques of matrix theory, examining the power of vector methods and how they may be used to describe three-dimensional space. You will consider the notions of field, vector space and subspace, and learn how to calculate the determinant of an n x n matrix.

You will take four from the following:
  • Theory and Analysis
  • Practical Musicianship
  • Creative Composition Techniques
  • Practical Composition Skills
  • A Very Short History of Music
  • Introduction to Historical Musicology
  • Introduction to World Music
  • Contemporary Debates in Music
  • Solo Performance
  • Creative Ensemble Performance
Year 2
  • In this module you will develop an understanding of the basic concepts of graph theory and linear programming. You will consider how railroad networks, electrical networks, social networks, and the web can be modelled by graphs, and look at basic examples of graph classes such as paths, cycles and trees. You will examine the flows in networks and how these are related to linear programming, solving problems using the simplex algorithm and the strong duality theorem.

  • In this module you will develop an understanding of vectors and matrices within the context of vector spaces, with a focus on deriving and using various decompositions of matrices, including eigenvalue decompositions and the so-called normal forms. You will learn how these abstract notions can be used to solve problems encountered in other fields of science and mathematics, such as optimisation theory.

You will take at least two from the following:
  • Studies in Music Analysis
  • Studies in Composition
  • Studies in Music History
  • Studies in Ethnomusicology
  • Studies in Music, Media and Technology
  • Practical Performance I
Year 3. You will take one of the following:
  • Special Study: Dissertation
  • Special Study: Theory and Analysis
  • Special Study: Performance
  • Special Study: Composition

Optional Modules

There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.

Year 1
  • All modules are core
Year 2
  • In this module you will develop an understanding of the concepts of scalar and vector fields. You examine how vector calculus is used to define general coordinate systems and in differential geometry. You will learn how to solve simple partial differential equations by separating variables, and become familiar with how these concepts can be appield in the field of dynamics of inviscid fluids.

  • In this module you will develop an understanding of statistical modelling, becoming familiar with the theory and the application of linear models. You will learn how to use the classic simple linear regression model and its generalisations for modelling dependence between variables. You will examine how to apply non-parametric methods, such as the Wilxocon and Kolmogorov-Smirnov goodness-of-fit tests, and learn to use the R open source software package.

  • In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

  • In this module you will develop an understanding of the concepts arising when the boundary conditions of a differential equation involves two points. You will look at eigenvalues and eigenfunctions in trigonometric differential equations, and determine the Fourier series for a periodic function. You will learn how to manipulate the Dirac delta-function and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

  • In this module you will develop an understanding of ring theory and how this area of algebra can be used to address the problem of factorising integers into primes. You will look at how these ideas can be extended to develop notions of 'prime factorisation' for other mathematical objects, such as polynomials. You will investigate the structure of explicit rings and learn how to recognise and construct ring homomorphisms and quotients. You will examine the Gaussian integers as an example of a Euclidean ring, Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

  • In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

  • In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

  • In this module you will develop an understanding of the convergence of series. You will look at the Weierstrass definition of a limit and use standard tests to investigate the convergence of commonly occuring series. You will consider the power series of standard functions, and analyse the Intermediate Value and Mean Value Theorems. You will also examine the properties of the Riemann integral.

  • Solo Performance
  • Ensemble Performance
  • Composition Portfolio
  • Practical and Creative Orchestration
  • Choral Conducting
  • Composing with Technology 1
  • Introduction to Jazz: Theory, Practice and Contexts
  • Popular Music and Musicians in Post-War Britain and North America
  • Korean Percussion Performance
  • Practical Ethics
  • Musical Aesthetics
  • Mozart's Operas
  • Issues in Sound, Music and the Moving Image
  • Intercultural Performance: Theory and Practice
  • Music and Society in Purcell's London
  • Contemporary Music Performance
  • Music, Power and Politics
  • Ideas of German Music from Mozart to Henze
  • Music and Gender
  • Hearing the Orient: Critical and Practical Approaches to the Middle East
Year 3
  • You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 7,000 words in length, and give a ten-minute presentation outlining your findings.

  • In this module you will develop an understanding of a range of methods for teaching children up to A-level standard. You will act act as a role model for pupils, devising appropriate ways to convey the principles and concepts of mathematics. You will spend one session a week in a local school, taking responsibility for preparing lesson plans, putting together relevant learning aids, and delivering some of the classes. You will work with a specific teacher, who will act as a trainer and mentor, gaining valuable transferable skills.

  • In this module you will develop an understanding of how prime numbers are the building blocks of the integers 0, ±1, ±2, … You will look at how simple equations using integers can be solved, and examine whether a number like 2017 should be written as a sum of two integer squares. You will also see how Number Theory can be used in other areas such as Cryptography, Computer Science and Field Theory.

  • In this module you will develop an understanding of a range methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind state-of-the art factorisation methods. You will also look at how to analyse the complexity of fundamental number-theoretic algorithms.

  • In this module you will develop an understanding the different classes of computational complexity. You will look at computational hardness, learning how to deduce cryptographic properties of related algorithms and protocols. You will examine the concept of a Turing machine, and consider the millennium problems, including P vs NP, with a $1,000,000 prize on offer from the Clay Mathematics Institute if a correct solution can be found.

  • In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worst-case analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

  • In this module you will develop an understanding of quantum theory, and the development of the field to explain the behaviour of particles at the atomic level. You will look at the mathematical foundations of the theory, including the Schrodinger equation. You will examine how the theory is applied to one and three dimensional systems, including the hydrogen atom, and see how a probabilistic theory is required to interpret what is measured.

  • In this module you will develop an understanding of how the Rayleigh-Ritz variational principle and perturbation theory can be used to obtain approximate solutions of the Schrödinger equation. You will look at the mathematical basis of the Period Table of Elements, considering spin and the Pauli exclusion principle. You will also examine the quantum theory of the interaction of electromagnetic radiation with matter.

  • In this module you will develop an understanding of how the theory of ideal fluids can be used to explain everyday phenomena in the world around us, such as how sound travels, how waves travel over the surface of a lake, and why golden syrup (or volcanic lava) flows differently from water. You will look at the essential features of compressible flow and consider basic vector analysis techniques.

  • In this module, you will develop an understanding of non-linear dynamical systems. You will investigate whether the behaviour of a non-linear system can be predicted from the corresponding linear system, and see how dynamical systems can be used to analyse mechanisms such as the spread of disease, the stability of the universe, and the evolution of economic systems. You will gain an insight into the 'secrets' of the non-linear world and the appearance of chaos, examining the significant developments achieved in this field during the final quarter of the 20th Century.

  • In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the Neyman-Pearson framework, likelihood ratio tests, and decision theory.

  • In this module you will develop an understanding of some of the descriptive methods and theoretical techniques that are used to analyse time series. You will look at the standard theory around several prototype classes of time series models and learn how to apply appropriate methods of times series analysis and forecasting to a given set of data using Minitab, a statistical computing package. You will examine inferential and associated algorithmic aspects of time-series modelling and simulate time series based on several prototype classes.

  • In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

  • In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the no-cloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

  • In this module you will develop an understanding of how financial markets operate, with a focus on the ideas of risk and return and how they can be measured. You will look at the random behaviour of the stock market, Markowitz portfolio optimisation theory, the Capital Asset Pricing Model, the Binomial model, and the Black-Scholes formula for the pricing of options.

  • In this module you will develop an understanding of the role of mathematics and statistics in securities markets. You will investigate the validity of various linear and non-linear time series occurring in finance, and apply stochastic calculus, including partial differential equations, for interest rate and credit analysis. You will also consider how spot rates and prices for Asian and barrier exotic options are modelled.

  • In this module you will develop an understanding of some of the standard techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

  • In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

  • In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

  • In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cryptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

  • In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X2017=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

  • In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

  • In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

  • In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.

  • Musical Aesthetics
  • Mozart's Operas
  • Issues in Sound, Music and the Moving Image
  • Intercultural Performance: Theory and Practice
  • Music and Society in Purcell's London
  • Contemporary Music Performance
  • Music, Power and Politics
  • Ideas of German Music from Mozart to Henze
  • Music and Gender
  • Hearing the Orient: Critical and Practical Approaches to the Middle East
  • Practical Performance II
  • Composing with Technology 2

The course has a flexible, modular structure and you will take a total of 12 course units at a rate of four, 30-credit modules per year. In addition to our compulsory core modules you will be free to choose between a number of optional courses. Some contribute 15 credits to your overall award while others contribute the full 30.

We use a variety of teaching methods and there is a strong focus on small group teaching throughout the course. You will attend 12 to 15 hours of formal teaching in a typical week, including lectures, seminars, group tutorials, statistics and IT classes, problem-solving workshops in mathematics, and instrumental, vocal and compositional classes in music. You will also be expected to work on mathematical worksheets, musical practice and composition, revision and project work outside of these times.

Assessment is through a mixture of coursework, end-of-year examination and a portfolio of practical work, in varying proportions depending on the course units you choose to take. Statistics and computational course units may include project work and tests, and music modules may include performance or coursework components. All students will work in small groups to prepare a report and an oral presentation on a mathematical topic of their choice, which contributes towards one of the core subject marks in year 2, and two of the optional mathematics units in year 3 are examined solely by a project and presentation.

Private study and preparation are essential aspects of every course, and you will have access to many online resources and the College’s Moodle e-learning facility. You will also have a dedicated personal adviser to guide you and help you with any personal or academic issues that arise in the course of your studies.

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