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Mathematics and Music

Mathematics and Music

BA
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If we make you an offer for this course for 2022 entry, we guarantee to confirm your place even if one of your final A-level results is one grade below those you have received in that offer. Equivalencies and exclusions apply. Full details here.

Key information

Duration: 3 years full time

UCAS code: GW13

Institution code: R72

Campus: Egham

UK fees: £9,250

International/EU fees: £21,900

The course

Mathematics and Music (BA)

“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.

  • 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.
From time to time, we make changes to our courses to improve the student and learning experience, and this is particularly the case as we continue to respond to the Covid-19 pandemic. If we make a significant change to your chosen course, we’ll let you know as soon as we can.

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:
  • This course aims: 1. to develop basic music-analytical literacy, 2. to introduce basic concepts concerning counterpoint, harmony, melody and form that underpin the analysis of music, 3. to put these concepts into practice in the analysis of pieces from a variety of repertories. The course addresses the contrapuntal, melodic, harmonic and formal elements of tonal music. Weekly lectures, in which students are introduced to analytical concepts and then practise deploying them, through listening, score study and the completion of practical exercises, are supplemented by private study based on Moodle and recommended readings, to consolidate concepts learnt in the lectures and provide further opportunities to practise new skills.

  • The aim of this course is to develop students' awareness of music theory through practical exercises and musical analysis. Through practical exercises that focus on musical literacy as well as aural awareness, students develop the ability to identify and analyse the musical parameters of metre, rhythm, pitch, harmony, counterpoint and form. In-class exercises may focus on listening, whereas exercises for self-study or small-group work may include written exercises. Tasks set for private study between classes provide a basis for students to continue their own practical training throughout their musical careers.

  • This course introduces students to some fundamental techniques of music composition. The precise topics taught may change depending on the research interests of the staff responsible for teaching the course, but typically include:

    • Soundworlds and scale formations

    • The vertical dimension: chords and simultaneities

    • The horizontal dimension: melody and voice leading

    • Developments in rhythm

    • Developments in harmonic vocabulary and tonalities

    • Form in contemporary composition

    • Acoustic timbre and texture

  • This course gives students the opportunity to practise the art of musical composition and develop skills in independents creative work. The precise topics taught may change depending on the research interests of the staff responsible for teaching the course, but typically include:

    • Writing for solo instrument or voice with accompaniment

    • Writing for small chamber ensemble

    • Introduction to media and film music

    • Studio techniques

  • This course introduces a wide range of repertories within the history of music. It stimulates students to relate features of musical compositions and performances to their wider historical contexts and gives students a fundamental knowledge of specific musical cultures. It provides students with opportunities to develop skills in research and information retrieval and in critical reading of primary and secondary literature, to receive formative feedback on those skills, and to build a foundation for higher-level study. The course will offer students a conceptual map of musical styles, composers and practices by introducing them to a wide chronological range of repertories, from early music to music of the twentieth century. It will emphasise questions of change, interaction and transmission through the study of specific forms and repertories in their historical context. Lectures will be designed around major repertorial moments (e.g. Stravinsky in 1910) or problems (e.g. the post-Beethovenian symphony), to bring together questions of form, style, performing practice and historical context.

  • This course introduces concepts underlying the historical and critical study of music. It enables students to begin thinking critically about the priorities that underlie historical texts from different intellectual traditions and stimulates them to relate features of musical compositions and performances to wider historical contexts. It provides students with opportunities to develop skills in research and information retrieval and in critical reading of primary and secondary literature, to receive formative feedback on those skills, and to build a foundation for higher-level study. This course introduces students to the different kinds of historical question that we can ask about music, and interrogates some of the terminology and categories frequently used in the secondary literature (e.g. canonisation, reception, tradition, nationalism, exoticism, the work concept). Case-studies are used to illuminate specific topics and problems in the historiography of a wide variety of musics.

  • This course introduces students to the socio-cultural contexts, functions, philosophies, techniques, and organising principles of a variety of musics of the world; musics from at least three continents will be studied. These musical traditions will be approached from both theoretical and practical perspectives, also giving a variety of opportunities for hands-on experience. Course content will vary from year to year according to staff interests, availability of musicians to provide workshops, and to ensure freshness of approach. A typical curriculum might cover the following regions and theoretical themes:

    • World Music - Introduction (culture, contact & concepts)
    • South America: Andes to Amazon (exchange)
    • Africa: Jaliya and Mbira (the musician)
    • Indonesia: Sundanese Gamelan (temporal organisation)
    • North India: The Classical Tradition (improvisation)
    • Papua New Guinea: The Kaluli (music and ecology)
    • Iran: The Persian Classical Tradition (music & religion).
  • This course introduces students to a range of key debates and issues in contemporary musicology and to a range of key issues concerning music in the contemporary world. It encourages students to think about music’s relation to social and cultural contexts and introduces them to unfamiliar musical styles and repertoires as well as broaden understanding of those closer to home. It hones students’ skills in reading a wide variety of critical and theoretical writing about music. This course will survey some of the key contemporary issues in music that have arisen from the changes of the modern world, as well as contemporary debates in musicology. The twentieth century in particular has seen a transformation of musical cultures across the world, and this course looks at a range of the issues and controversies that have emerged as a result. The study of music has broadened to include many more social, cultural and political. This course will introduce students to truly contemporary ways of studying music, combining approaches and issues traditionally associated with musicology, ethnomusicology and popular music studies, divisions which are becoming increasingly blurred. Lecture topics may include:

    • Ideas of ‘authenticity’ in music

    • Value judgements about music

    • Protection and preservation of music

    • Heritage and revivals

    • Music and tourism

    • New forms of fusion and hybridity

    • The idea of ‘world music’.

    • Music and identity

    • Music and gender

    • Music and race

    • Music and nationalism

  • This course aims to further students’ skills as performers through regular (typically weekly) one-to-one vocal/instrument lessons with an approved visiting teacher.  Students will be offered opportunities to perform in practical seminars where matters of interpretation and stage manner will be discussed.  Constructive critical feedback given as well as developing students' skills in delivering feedback. 

    Students’ will develop the capacity to reflect on what constitutes good programming and fine performance.  Participation in College music events is fostered through ensemble and other activities. 

     The course consists of regular individual instrumental or vocal lessons with a teacher approved by the Department.  A series of practical seminars is run in which students perform and discuss suitable repertory under the supervision of the course co-ordinator, develop skills in the writing of programme notes to a high standard as well as concert reviews and  engage with  'professional preparation’ consisting of the development of stage presence and other relevant concerns.

  • The module aims to develop a broad range of innovative, practical, creative and collaborative musical skills. It promotes student initiative and creativity, while developing focused, critical, technical and context sensitive perspectives on selected musical repertoires/traditions/genres. It seeks to explore, reflect upon, extend and/or challenge specific musical performance conventions. The module will commence with at least two plenary lectures/seminars at the start of term one, when the module aims will be clarified, followed by fortnightly workshops and plenary meetings through terms 1 and 2. A list of student performance interests/skills will be circulated immediately after the first meeting. Students will then be requested to form their own groups. Flexibility in membership will be permitted until the end of term one when students must commit to a group with whom to be examined. Any student not integrated in a group will be allotted to one by the module tutor. All students will be required to regularly document their experience of group participation and creative practice in a performance diary.

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:
  • The module introduces a range of important concepts for analysing music, and of the published secondary literature in music analysis. It puts these concepts into practice in increasingly sophisticated analysis of score-based and recorded pieces from the Western musical tradition and in the reading of more complex analyses. It lays foundations for further analytical and technical work in options modules and in final-year special studies. The analytical systems and repertories to be studied will vary from year to year, but students may expect to build on theoretical and analytical foundations established in the first year, by broadening their knowledge (through scores and recordings) of a wide range of Western musical repertoire, to learn and then apply standard analytical methods in order to gain a deeper understanding of the music's construction and expressive effect, and to learn the vocabulary and technical proficiency necessary for reading and evaluating analyses of music by scholars from those traditions. The module may address pre-tonal, tonal or post-tonal music.

  • This module will: develop your knowledge of a range of fundamental techniques of musical composition with particular focus on structure, harmonic control and the manipulation of rhythmic and melodic material provide an opportunity to practise the art of musical composition and to develop skills in creative work. Developing on areas covered in first-year Composition modules, this module will provide a framework in which you will be introduced to a number of techniques from diverse schools of composition in order to encourage you to explore and develop your own creativity. Key works from the past few decades will be studied and used as models or springboards for your own musical invention. You will create a portfolio of technical exercises and a short composition written in response to a given brief.

  • This module furthers students’ understanding of music history, by exploring two case-studies defined chronologically or thematically. It develops students’ ability to critically think and write about music in historical contexts that are both familiar and unfamiliar and explores more advanced concepts underlying the historical and critical study of music. It encourages students to put these concepts into practice in increasingly sophisticated historical and critical writing about music and it lays the foundations for further historical and critical writing in options modules and in particular for final-year special studies. This module will probe moments of music history that expose the complex relationship between musical repertories and historical contexts, or the nuanced processes of historical continuity, change, and cause and effect. The case-studies will vary year by year, but sample topics include: The Rise of Musical Notation in Medieval Europe; Music and the Reformation; Monteverdi: Between Renaissance and Baroque; Nationalism in Late Romantic Music; Popular and Art Music of the 1960s.

  • This module expands students’ knowledge of concepts characteristic of ethnomusicology and equips them with stimulating approaches to understanding, enjoying and studying their own music as well as that of others. It broadens students’ understanding of the possibilities of music as human activity and of the wider contexts in which music exists in the world. It raises issues concerning the political and ethical challenges involved with studying and writing about music across the globe, whether historically or in the contemporary world, and develops students’ ability to critically think and write about music in contexts that are both familiar and unfamiliar.

    This module will involve a combination of the study of musical repertoires from different parts of the globe and introduction to a range of methodologies that might be applied to a broad range of musics and contexts. Particular repertoires and areas will vary, but approaches and issues may include: the idea of music as culture/society; looking at music beyond concepts of ‘art’; understanding the strengths and problems of fieldwork as a methodology; looking at musical change and hybridisation; issues relating to music and gender, sexuality, class, ethnicity or religion; studying musical instruments; mapping music geographically, socially and historically; and the colonial legacy of ethnomusicology and ethical issues of contemporary research.

  • This module explores ways that electronic media and technology have brought about change in and opened up new possibilities for musical production, consumption, sounds, practices, experience, contexts and meanings. It considers the role of electronic media and technology in preserving and documenting musical culture as well as in changing it, and it examines how developments in electronic media and technology have affected and continue to affect dynamics of power in musical production and consumption across the world. The module explores the effects of electronic media and technology on popular, traditional and classical musics and introduces concepts and techniques for the study of the interaction of music, media and technology. It also encourages a deeper and more critical understanding of music, music making and musical culture through study in both familiar and unfamiliar cultures and contexts. The module will introduce students to a range of ways in which electronic media and technology have affected and transformed musical cultures across the world through their fundamental ability to: record and store musical sound; create new sounds, new ways of combining sounds and new ways of synching sound with other media; turn musical sound into a commodity; separate musical sound from live performance context; amplify music; mass produce music; mass disseminate music; and greatly alter dynamics of power in the production and consumption of music. The module will cover a range of key phenomena and issues in contemporary musical culture that are inextricably linked to electronic media and technology. Exact topics will vary, but may include: popular and mass-mediated music; recorded music; electronic music; the impact of technology on compositional practices; music industries; piracy; film music, video and multimedia; music and the Internet; globalisation; debates on the value of mass mediated music; and questions of power and representation.

  • This course will require the student to undertake the study of an instrument or voice, with the aim of developing technical ability and musical interpretation expressed through performance.  The student will consciously and actively address concerns such as the acquisition of technical competence in performance, the development of powers of interpretation, strategies of practice and performance, effective communication and so on.  These skills are further developed through the writing of programme notes, concert reviews and developing critical feedback skills in performance seminars.

    The course consists of the study of appropriate repertory with an individual instrumental or vocal teacher approved by the Department, the writing of programme notes and concert reviews, the development of ensemble musicianship and/or music-administrative skills through membership of college ensembles and/or the in-house concert administration team.

Year 3. You will take one of the following:
  • This module lets students write a detailed essay of 10,000–11,000 words (including footnotes or endnotes and excluding appendices and bibliography) on a topic of a critical, aesthetic, technical, historical, cultural, analytical or theoretical nature relating to music; or on a topic of an ethnomusicological nature; or to make a transcription of one or more historical sources relating to music and to write a detailed accompanying commentary of 5000-5500 words (including footnotes or endnotes and excluding appendices and bibliography). The content of the module is the process, in all its detail, of the preparation and production of an extended piece of academic work.

    You will undertake an extended piece of academic work at the level appropriate to the final year of an undergraduate degree programme, carried out independently under the guidance of a supervisor, and laying the foundations for possible further work in the field at postgraduate level.

  • The learning objective is to write a detailed essay on a topic of a technical, analytical or theoretical nature relating to music.

    You will undertake an extended piece of academic work at the level appropriate to the final year of an undergraduate degree programme, carried out independently under the guidance of a supervisor, and laying the foundations for possible further work in the field at postgraduate level.

  • This course will develop and refine students’ abilities as solo performers at an advanced level through weekly seminars in which performances will be subjected to critical scrutiny by the course tutor and members of the class.  Students will develop the ability to manage the occasion of performance at a professional level and will engage in the study and performance of music by twentieth-century or contemporary composers writing in particularly challenging or complex musical styles.

    The course consists of regular instrumental or vocal lessons with a teacher approved by the Department, regular two-hour practical seminars in which students perform suitable repertory according to a rota that requires appearance in front of their peers at least twice a term, thereby gaining platform experience in preparation for the final recital.  ‘Professional preparation’, consists of the development of stage presence and other relevant concerns, such as preparation for an audition, performance practice, interpretation and communication.  Students share participation in a public lunchtime recital.  The dates of the recitals are arranged by the Concert Office in the preceding summer vacation.

    The writing of programme notes and concert reviews to professional standard as well as the development of ensemble musicianship and/or music administration and concert management skills are key requirements.

  • This module develops students' knowledge of advanced compositional techniques with particular focus on structure, harmonic control and the manipulation of rhythmic and melodic material. It provides opportunities to practise the art of musical composition and to develop skills in independent creative work and increases students' awareness, knowledge and understanding of issues related to contemporary composition practice in a variety of contexts. Developing on areas covered in MU2213 Composition Portfolio, the module will provide a framework for you to further explore the possibilities in your own compositional method. You will complete a structured portfolio that will properly demonstrate your increased awareness, knowledge and understanding of contemporary art music and related compositional issues. You will be assisted in acquiring a deeper confidence in experimenting with a range of compositional methods and techniques whilst being encouraged to explore the possibilities of your own compositional voice in the hope that this trend will continue into your professional life. During workshops you will be given the opportunity to have your work rehearsed and recorded by professional musicians. It is hoped that through these workshops you will discover more about the possibilities of instrumentation and the many practical compositional issues facing composers today. You should also seek to develop your own opportunities for the performance of your music in order to develop your confidence and professional activity.

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.

A Levels: AAB-ABB

Required subjects:

  • A-level Mathematics at grade A
  • A-level grade Music or pass in Grade 7 Music Theory
  • Grade 7 in performance (where applicants are unable to take Grade 7, video evidence may be acceptable)
  • Applicants without A-level in Music or pass in Grade 7 Music Theory may be eligible for the Intensive Theory entry. This requires Music GCSE grade A/7 or equivalent, plus performance at Grade 7 level. In term 1 you will be required to take Fundamentals of Music Theory, an intensive music literacy course.
  • Students wishing to take Solo Performance options will need to be of Grade 8 level at point of entry
  • At least five GCSEs at grade A*-C or 9-4 including English and Mathematics

Where an applicant is taking the EPQ alongside A-levels, the EPQ will be taken into consideration and result in lower A-level grades being required. For students who are from backgrounds or personal circumstances that mean they are generally less likely to go to university you may be eligible for an alternative lower offer. Follow the link to learn more about our contextual offers.

English language requirements

All teaching at Royal Holloway is in English. You will therefore need to have good enough written and spoken English to cope with your studies right from the start.

The scores we require
  • IELTS: 6.5 overall. Writing 7.0. No other subscore lower than 5.5.
  • Pearson Test of English: 61 overall. Writing 69. No other subscore lower than 51.
  • Trinity College London Integrated Skills in English (ISE): ISE III.
  • TOEFL iBT: 88 overall, with Reading 18 Listening 17 Speaking 20 Writing 26.

Country-specific requirements

For more information about country-specific entry requirements for your country please visit here.

Undergraduate Pathways

For international students who do not meet the direct entry requirements, the International Study Centre offers the following pathway programmes:

International Foundation Year - for progression to the first year of an undergraduate degree.

International Year One - for progression to the second year of an undergraduate degree.

With a joint Mathematics and Music degree from Royal Holloway, University of London you will be in demand for your wide range of transferable skills, including: numerical skills, data handling powers, logical thinking and creative problem solving abilities, as well as communication, teamwork, technical, time management, commercial awareness and critical thinking and analysis skills. We have a strong track record of preparing our students for the world of work and research. By combining mathematics with music you will also keep your career options open, with the opportunity to pursue your musical talents in performance, composition, production or musicology after you graduate.

Our recent graduates have gone on to enjoy successful careers in business management, IT consultancy, computer analysis and programming, teaching, accountancy, law, the civil service, actuarial science, finance, risk analysis, research and engineering, as well as in composition, music technology, publishing and the performing arts. They are working for employers as diverse as KPMG, Ernst & Young, the Ministry of Defence, Barclays Bank, English National Opera, Surrey County Arts Service, EMI and Slaughter & May. Others go on to establish themselves as successful independent musicians and/or teachers. Our Department of Mathematics is part of the School of Engineering, Physical and Mathematical  Sciences and we enjoy close ties with both the information security sector and industry at large.

We offer a competitive work experience scheme at the end of year 2, with short-term placements available during the summer holidays. You will also attend a CV writing workshop as part of your core modules in year 2, and your personal adviser and the campus Careers team will be on hand to offer career advice and guidance. The University of London Careers Advisory Service offers tailored sessions on finding relevant summer internships or holiday jobs and securing employment after graduation.

  • Keep your options open by studying both mathematics and music.
  • Mathematicians are in demand from employers, and with your additional musical skills and experience, you will have an attractive range of transferable skills to offer.
  • Our strong ties with industry and the arts sector mean we understand the needs of employers.
  • Take advantage of our summer work placement scheme and fine-tune your CV before you enter your final year.
  • Benefit from a personal adviser to guide you through your options.

Home (UK) students tuition fee per year*: £9,250

EU and International students tuition fee per year**: £21,900

Other essential costs***: £50

How do I pay for it? Find out more about funding options, including loansscholarships and bursaries. UK students who have already taken out a tuition fee loan for undergraduate study should check their eligibility for additional funding directly with the relevant awards body.

*The tuition fee for UK undergraduates is controlled by Government regulations. For students starting a degree in the academic year 2021/22, the fee will be £9,250 for that year. The fee for UK undergraduates starting in 2022/23 has not yet been confirmed.

**The UK Government has confirmed that EU nationals are no longer eligible to pay the same fees as UK students, nor be eligible for funding from the Student Loans Company. This means you will be classified as an international student. At Royal Holloway, we wish to support those students affected by this change in status through this transition. For eligible EU students starting their course with us in September 2022, we will award a fee reduction scholarship equivalent to 60% of the difference between the UK and international fee for your course. This will apply for the duration of your course. Find out more

Fees for international students may increase year-on-year in line with the rate of inflation. The policy at Royal Holloway is that any increases in fees will not exceed 5% for continuing students. For further information see fees and funding and our terms and conditions.

***These estimated costs relate to studying this particular degree at Royal Holloway during the 2021/22 academic year, and are included as a guide. Costs, such as accommodation, food, books and other learning materials and printing etc., have not been included.

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