The file fugue.mat is available on Canvas. It contains numerical data that encode about $30-$

seconds worth of Johann Sebastian Bach's Fugue #$2$ for the Well$-$Tempered Clavier. You are to

write MATLAB code that loads this file, synthesizes the notes with correct timing, and then

plays the notes.

You will also prepare a document discussing possibilities for the computer$-$generated synthesis

of music.

In order to complete this project, you will need a computer with both MATLAB and sound. If

you use a computer in the ECE department, you may need a set of headphones.

This project is a rewritten version of a project given on the CD accompanying the book:

J$.$ H$.$ McClellan, R$.$ W$.$ Schafer, and M$.$ A$.$ Yoder, Signal Processing First, Prentice Hall, $2003.$

Each student will submit his$/$her own work. You may discuss this project with other students.

But your submissions must be the fruit of your own labor.

Description of Project

The file fugue.mat contains a structure array called voice with three elements. $($In musical

terms, this fugue contains three voices.$)$ These elements are:

voice$($m$),$ where m $=1,2,$ or $3$

Each of these elements is a structure. Each of the elements represents a sequence of notes that is

to be played. Each structure contains three fields:

voice$($m$).$pitch

voice$($m$).$start

voice$($m$).$duration

Each of these fields is a vector. For a given structure $$ for example, voice$(2)$ the three

vectors $$ pitch, start, duration $$ are of equal length. These three vectors together

define a sequence of notes. The vector start defines the start time of each note. The vector

duration defines the duration of each note. The vector pitch defines the pitch of each

note.

$2$

For example, the first number in the vector voice$(2).$start

voice$(2).$start$(1)$

is the start time in seconds of the first note in the sequence of notes defined by voice$(2).$ The

first number in the vector voice$(2).$duration

voice$(2).$duration$(1)$

is the duration in seconds of this first note. The first number in the vector voice$(2).$pitch

voice$(2).$pitch$(1)$

is the pitch number of this first note.

Using the symbol p to represent the pitch number voice$($m$).$pitch$($n$),$ the pitch number

for the n$-$th note of the m$-$th voice, the frequency of this note is given by

$440\backslash$times $2(49)12$ Hz

A couple of examples might be helpful. When $=49,$ the frequency is $440$ Hz$.$ When $=37,$

the frequency is $220$ Hz$.$

The vector start tells you when each note should start, and the vector duration tells you

how long each note should last. Both of these vectors measure time in seconds.

In musical terms, each of the pitch numbers $($numbers in the vector pitch$)$ represents a key on

the piano keyboard. This scheme accounts for both black and white keys. The pitch number

increases as your finger moves from left to right on the keyboard. Middle C has been assigned

the pitch number $40.$ When you move your finger on the keyboard to the right by $12$ keys, the

frequency doubles. For example, your finger moves $12$ keys to the right in shifting from the key

associated with pitch number $37$ to the key associated with pitch number $49,$ and the

corresponding frequencies are $220$ Hz and $440$ Hz$,$ respectively. A set of $12$ consecutive keys

$($including both black and white keys$)$ is called an octave, corresponding to a doubling of the

frequency.

Based on the above comments, the following idea should be forming in your mind. The file

fugue.mat contains the same information normally found on sheet music. The program that

you write will essentially replace three instruments. Your program will take the information

from fugue.mat and use this information and the sound capability of the computer to play this

fugue. The actual playing of the fugue is accomplished by passing a signal vector, which your

program constructs, to the function sound. The signal vector will also be recorded.

$3$

Specific Instructions

You will first copy the file fugue.mat from Canvas into the folder that you will use as the

current folder while working on this project.

You will write a script m$-$file and a function m$-$file. The function will be called from the script.

Within the script, you will load the file fugue.mat into the MATLAB workspace using load.

The script will also define a variable representing the sampling frequency and assign this

variable the value $11025($hertz or$,$ equivalently, samples per second$).$ This sampling frequency

is only one$-$fourth the sampling frequency $(44\mathrm{.}1$ kHz$)$ used on a CD$,$ but $11025$ Hz is adequate

for the purpose of this project.

Your function will produce a signal vector representing one note. There will be three input

arguments to this function. The first of these is the pitch number. The next argument is the

duration in seconds. The final argument is the sampling frequency in hertz. This function will

not need the start time because the signal vector produced by this function will later be shifted to

start at the correct time. $($That shifting will occur in the script