Description

Nowadays, popular music makes heavy use of synthesized music and sounds. The advance in computer technology makes this easier and cheaper than ever. Rather than using a recorded sound, digital synthesizers compute the shape of sounds using various mathematical methods.

First a signal that has the desired period is generated and that signal is then multiplied by an envelope to give a finite time evolution and span (instrument notes usually have a specific frequency but only last a finite length of time).

Sounds generated like that can bear some resemblance to real musical instruments, but often sound dull. To make them more pleasing to the ear, one must process them using filters which affect their frequency content.

When we ear a real instrument, the sounds that reach our ears are a superposition of the sound coming directly from the instrument and the sounds reflected inside the instrument itself as well as the walls of the room in which we are. To generate a warm artificial sound, one must simulate that process using artificial reverberation.

The aim of the project will be to study the mathematics behind the creation of an artificial wave form as well as the design of digital filters and artificial reverberation. These mathematical tools are effectively all that is needed to build a synthesizer using a computer.

The project will involve some python programming to implement various types of filters and process some sound files.

Prerequisites

• Analysis in many variable II (MATH2031)
• Mathematical Modelling II. or some good knowledge of python

Resources

• Music: a Mathematical Offering (Chapter 7 and 8)
• Introduction to Digital Filters, (Julius O. Smith III)
• Digital Filters (3rd edition) R.W. Hamming. Dover Publication (1989)
• Digital Signal Processing (M.H. Hayes, Schaum's outlines).
• Several Web resources including wikipedia.