
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 highpowered 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 flatbottomed objects when tilted and released, 4 ) an article on sums of powers of integers and the EulerMaclaurin 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 nonperiodic 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 4part 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 longgone 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 fholes. 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:
John Coffey, Cheshire, England, January 2022 
