An amateur's investigations into a few aspects of mathematics -- group theory, integrals of logarithms, continued fractions, matrices and their eigenvalues, analysis, undergraduate math problems.

Also music: the acoustics of the violin family of instruments, music created by computer, plus some paintings, drawings and local history ..... sharing my personal interests!


Welcome to John Coffey's web site of interesting finds.

This website is for maths students and amateurs who enjoy exploring maths, music and acoustics for interest.  I am neither a professional mathematician nor musician, so there are no high-powered research papers here -- just the work of an enthusiast.

The latest additions are

1) the local history of Cross Town, Knutsford, Cheshire focusing on six of the grand mansions built by Victorian merchants and industrialists,

2) some more paintings,

3) an investigation of the paradoxical rapid oscillations of a mandarin orange and other flat-bottomed objects when tilted and released,

4 ) an article on sums of powers of integers and the Euler-Maclaurin summation formula, and

5) is a revision article on Fourier series and transforms. In it I give an illustrated account of the representation of periodic functions by Fourier series, and extend this to the Fourier transform of a non-periodic continuous function. Then I illustrate sampling at discrete data points and the discrete Fourier transform. There is a brief discussion of convolution and deconvolution with relevance to recovering true signals from signals mixede with noise. An appendix describes in some detail the fast Fourier Transform algorithm. The article is here.

"What makes a good tune?" That is a question I have recently asked myself. I selected nearly 60 tunes from across the spectrum of Western music and dissected them to see what features they have in common. The article is here. There is also a subsidiary article about the shape of musical phrases and how mathematical curves can be fitted to these shapes.

A related and ongoing project is to write a suite of computer programs which carry out some of the basic functions in creating 'classical' music. This is still work in progress, but I have placed here an account of progress so far. It uses none of the sophisticated methods of artificial intelligence, but instead takes a step by step approach to creating a sequence of major and minor triads, converting them to 4-part harmony (like a hymn tune), then decorating the chords with rhythm and melody.

Not long ago I added a section on matrices and their eigenvalues. That page hosts two articles, one on numerical algorithms for calculating eigenvalues and eigenvectors, and the other, recently completed, is on some elementary aspects of random matrices. Over the last 60 years random matricies have found application in many branches of number theory, nuclear physics, biological systems, phase transformations, and finance.

There is a section of figure drawings which I had made several years ago. I also uploaded a slightly revised version of the article on Kepier water mill -- it's about the long-gone water corn mill at Kepier, Durham City, England, on the new Local History page. There is another article about the CrossTown area of Knutsford, Cheshire.

Acoustics of the violin family: Here is a long article describing how sound is radiated from vibrating objects. You can download it here.   I have also painted a few more pictures.

Some years ago I investigated the acoustics of the violin and viola. See the section. I made an experimental plywood violin and a viola to novel designs and they sound quite good. I have used experiment and finite element analysis (FEA) with the LISA, Mecway and Strand7 programs to model the vibrational behaviour on the computer. There is an article on the Helmholtz A0 air resonance in the f-holes.   If you are learning finite element analysis, you may find these articles interesting.

There is also my account of how we hear the sound of a violin. It describes how our ears detect sounds and how we can recognise pitch and loudness. There are some sound clips of how the pitch and tone of low notes on a viola are perceived by our ears, explaining the Paradox of The Missing Fundamental !

Other sections are:

  • Sequences and Series is an introduction to mathematical analysis.   Analysis is that vast area of maths concerned with rigorous proof of the existence of limits of sequences, tests for convergence, continuity of functions, differentiability, and integration.   Here are the full texts of two books on analysis by my colleague Dr John Reade of the University of Manchester.  They are an undergraduate introduction to analysis, and a new, previously unpublished book on uniform convergence. My articles on the Euler-Maclaurin sum formuila and on Fourier series and the Fast Fourier Transform is here too.

    Continued Fractions were much studied in the 19th century, but now are largely forgotten.  They have many intriguing properties and applications in number theory, including giving the most economical approximation by rational numbers. Back in 2013 I added sections on Thiele's method for interpolating a given data set by fitting a rational function, and Gauss's hypergeometric function, plus other thoughts on continued fractions of functions f(x).

    Group Theory has interactive programs to 'teach-yourself' the basics of mathematical group theory.  The programs, called PermGroups and Word Groups, come with detailed instructions and fully worked examples.  Great for first year undergraduates who want to 'get a feel' for finite groups and representation theory.
  • Integrals of log(x) is a study of definite integrals of rational functions involving the logarithm, log(x), and polynomials, using complex variables.   Many fully worked examples of complex contour integration.  Useful for undergraduates who want to practice complex contour integration.
  • Puzzles and Problems are maths challenge problems with my solutions.
  • Paintings: I also paint pictures when I find time.
  • Music: I enjoy European classical music in an amateur way.  Here are three short pieces I have written
        i)  this is me live, playing a Baroque-style two part invention (mp3, 1.3 MB).
       ii)  a fugue played on flute, oboe and vibes and marimba.
      iii)  a more modern piece for oboe and piano -- "Chase".

    Here also are three examples of multi-track recording, with me playing all the instruments. A Trio Sonata movement by Handel, the Prelude to a concerto grosso by Corelli, and a minuet and trio from Haydn's quartet Op 20 No 4. I made these during the lock-down for Covid19 between Spring 2020 and Spring 2021.

John Coffey, Cheshire, England, January 2022