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Lab #5: Synthesis of Sinusoidal Signals solution

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ECES-352

Lab    #5:        Synthesis    of    Sinusoidal    Signals    –    Music    Synthesis
(Lab    Report    Due    Beginning.of.Next.Lab)
This    is    the    official    Lab    #5    description;    it    is    similar    to    the    one    in    Appendix    C.3    of    the    text,
but    the    piece    Fugue    #5    for    the    Well-Tempered    Clavier    has    been    chosen    for    the
synthesis.        Formal    Lab    Report:        You    must    write    a    formal    lab    report    that    describes
your    approach    to    music    synthesis    (Section    4).

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ECES-352

Lab    #5:        Synthesis    of    Sinusoidal    Signals    –    Music    Synthesis
(Lab    Report    Due    Beginning.of.Next.Lab)
This    is    the    official    Lab    #5    description;    it    is    similar    to    the    one    in    Appendix    C.3    of    the    text,
but    the    piece    Fugue    #5    for    the    Well-Tempered    Clavier    has    been    chosen    for    the
synthesis.        Formal    Lab    Report:        You    must    write    a    formal    lab    report    that    describes
your    approach    to    music    synthesis    (Section    4).
Important:        When the    lab    instructs    you    to    get    an    updated    matlab    file    (like
specgram.m),    please    refer    to    http://dspfirst.gatech.edu/matlab/toolbox/
Introduction
This    lab    includes    a    project    on    music    synthesis    with    sinusoids.        The    piece,    Fugue    #5
for    the    Well-Tempered    Clavier    by    Bach    has    been    selected    for    doing    the    synthesis
program.        The    project    requires    extensive    programming    effort    and    should    be
documented    with    a    lab    report.        A    good    report    should    include    the    following    items:        a
cover    sheet,    commented    MATLAB    code,    explanation    of    your    approach,    conclusions,
and    any    additional    tweaks    that    you    implemented    for    synthesis.        Since    the    project
must    be….
evaluated by listening to the quality of the synthesized song, the criteria for judging a good song are given
at the end of this lab description. In addition, it may be convenient to place the final song on a web site so
that it can be accessed remotely by a lab instructor who can then evaluate its quality. If you would like to try
other songs, the DSP First CD-ROM includes information about alternative tunes: Minuet in G, Fur Elise ¨ ,
Beethoven’s Fifth, Jesu, Joy of Man’s Desiring and Twinkle, Twinkle, Little Star. CD-ROM
MUSIC
SYNTHESIS The music synthesis will be done with sinusoidal waveforms of the form
(1)
so it will be necessary to establish the connection between musical notes, their frequencies, and sinusoids.
A secondary objective of the lab is the challenge of trying to add other features to the synthesis in order
to improve the subjective quality for listening. Students who take this challenge will be motivated to learn
more about the spectral representation of signals—a topic that underlies this entire course.
2 Pre-Lab
In this lab, the periodic waveforms and music signals will be created with the intention of playing them out
through a speaker. Therefore, it is necessary to take into account the fact that a conversion is needed from
the digital samples, which are numbers stored in the computer memory to the actual voltage waveform that
will be amplified for the speakers.
2.1 Theory of Sampling
Chapter 4 treats sampling in detail, but this lab is usually done prior to lectures on sampling, so we provide
a quick summary of essential facts here. The idealized process of sampling a signal and the subsequent
reconstruction of the signal from its samples is depicted in Fig. 1. This figure shows a continuous-time input
C-to-D
Converter
D-to-C
Converter
Figure 1: Sampling and reconstruction of a continuous-time signal.
signal , which is sampled by the continuous-to-discrete (C-to-D) converter to produce a sequence of
samples , where is the integer sample index and is the sampling period. The sampling
rate is where the units are samples per second. As described in Chapter 4 of the text, the ideal
discrete-to-continuous (D-to-C) converter takes the input samples and interpolates a smooth curve between
them. The Sampling Theorem tells us that if the input signal is a sum of sine waves, then the output
will be equal to the input if the sampling rate is more than twice the highest frequency in the input,
i.e., . In other words, if we sample fast enough then there will be no problems synthesizing the
continuous audio signals from .
2.2 D-to-A Conversion
Most computers have a built-in analog-to-digital (A-to-D) converter and a digital-to-analog (D-to-A) converter (usually on the sound card). These hardware systems are physical realizations of the idealized concepts of C-to-D and D-to-C converters respectively, but for purposes of this lab we will assume that the
hardware A/D and D/A are perfect realizations.
2
The digital-to-analog conversion process has a number of aspects, but in its simplest form the only
thing we need to worry about at this point is that the time spacing between the signal samples must
correspond to the rate of the D-to-A hardware that is being used. From MATLAB, the sound output is done
by the soundsc(xx,fs) function which does support a variable D-to-A sampling rate if the hardware on
the machine has such capability. A convenient choice for the D-to-A conversion rate is 11025 samples per
second,2 so seconds; another common choice is 8000 samples/sec. Both of these rates satisfy
the requirement of sampling fast enough as explained in the next section. In fact, most piano notes have
relatively low frequencies, so an even lower sampling rate could be used. If you are using soundsc(), the
vector xx will be scaled automatically for the D-to-A converter, but if you are using sound.m, you must
scale the vector xx so that it lies between . Consult help sound.
(a) The ideal C-to-D converter is, in effect, being implemented whenever we take samples of a continuoustime formula, e.g., at . We do this in MATLAB by first making a vector of times, and then
evaluating the formula for the continuous-time signal at the sample times, i.e., if
. This assumes perfect knowledge of the input signal, but we have already been doing it this
way in previous labs.
To begin, create a vector x1 of samples of a sinusoidal signal with , , and
. Use a sampling rate of 11025 samples/second, and compute a total number of samples
equivalent to a time duration of 0.5 seconds. You may find it helpful to recall that a MATLAB statement
such as tt=(0:0.01:3); would create a vector of numbers from 0 through 3 with increments of
0.01. Therefore, it is only necessary to determine the time increment needed to obtain 11025 samples
in one second. You should use the syn sin() function from a previous lab for this part.
Use soundsc() to play the resulting vector through the D-to-A converter of the your computer,
assuming that the hardware can support the Hz rate. Listen to the output.
(b) Now create another vector x2 of samples of a second sinusoidal signal (0.8 secs. in duration) for
the case , , and . Listen to the signal reconstructed from these
samples. How does its sound compare to the signal in part (a)?
(c) Concatenate the two signals x1 and x2 with a short duration of 0.1 seconds of silence in between.
You should be able to use a statement something like:
xx = [ x1, zeros(1,N), x2 ];
assuming that both x1 and x2 are row vectors. Determine the correct value of N to make 0.1 seconds
of silence. Listen to this new signal to verify that it is correct.
(d) To verify that the concatenation operation was done correctly in the previous part, make the following
plot:
tt = (1/11025)*(1:length(xx)); plot( tt, xx );
This will plot a huge number of points, but it will show the “envelope” of the signal and verify that the
amplitude changes from 100 to zero and then to 80 at the correct times. Notice that the time vector
tt was created to have exactly the same length as the singal vector xx.
(e) Now send the vector xx to the D-to-A converter again, but change the sampling rate parameter in
soundsc(xx, fs) to 22050 samples/second. Do not recompute the samples in xx, just tell the
D-to-A converter that the sampling rate is 22050 samples/second. Describe how the duration and
pitch of the signal were affected. Explain.
2
This sampling rate is one quarter of the rate (44,100 Hz) used in audio CD players.
3
2.3 Structures in MATLAB
MATLAB can do structures. Structures are convenient for grouping information together. For example, run
the following program which plots a sinusoid:
x.Amp = 7;
x.phase = -pi/2;
x.freq = 100;
x.fs = 11025
x.timeInterval = 0:(1/x.fs):0.05;
x.values = x.Amp*cos(2*pi*(x.freq)*(x.timeInterval) + x.phase);
x.name = ’My Signal’;
x %—- echo the contents of the structure “x”
plot( x.timeInterval, x.values )
title( x.name )
Notice that the members of the structure can contain different types of variables: scalars, vectors or strings.
2.4 Debugging Skills
Testing and debugging code is a big part of any programming job, as you know if you have been staying up
late on the first few labs. Almost any modern programming environment provides a symbolic debugger so
that break-points can be set and variables examined in the middle of program execution. Of course, many
programmers insist on using the old-fashioned method of inserting print statements in the middle of their
code (or the MATLAB equivalent, leaving off a few semi-colons). This is akin to riding a tricycle to commute
around Atlanta.
In order to learn how to use the MATLAB tools for debugging, try help debug. Here is part of what
you’ll see:
dbstop – Set breakpoint.
dbclear – Remove breakpoint.
dbcont – Resume execution.
dbstack – List who called whom.
dbstatus – List all breakpoints.
dbstep – Execute one or more lines.
dbtype – List M-file with line numbers.
dbquit – Quit debug mode.
When a breakpoint is hit, MATLAB goes into debug mode. On the PC
and Macintosh the debugger window becomes active and on UNIX and VMS
the prompt changes to a K. Any MATLAB command is allowed at the
prompt. To resume M-file function execution, use DBCONT or DBSTEP.
To exit from the debugger use DBQUIT.
One of the most useful modes of the debugger causes the program to jump into “debug mode” whenever
an error occurs. This mode can be invoked by typing:
dbstop if error
With this mode active, you can snoop around inside a function and examine local variables that probably
caused the error. You can also choose this option from the debugging menu in the MATLAB editor. It’s
sort of like an automatic call to 911 when you’ve gotten into an accident. Try help dbstop for more
information.
Download the file coscos.m and use the debugger to find the error(s) in the function. Call the function
with the test case: [xn,tn] = coscos(2,3,20,1). Use the debugger to:
4
1. Set a breakpoint to stop execution when an error occurs and jump into “Keyboard” mode,
2. display the contents of important vectors while stopped,
3. determine the size of all vectors by using either the size() function or the whos command.
4. and, lastly, modify variables while in the “Keyboard” mode of the debugger.
function [xx,tt] = coscos( f1, f2, fs, dur )
% COSCOS multiply two sinusoids
%
t1 = 0:(1/fs):dur;
t2 = 0:(1/f2):dur;
cos1 = cos(2*pi*f1*t1);
cos2 = cos(2*pi*f2*t2);
xx = cos1 .* cos2;
tt = t1;
2.5 Piano Keyboard
Section 4 of this lab will consist of synthesizing the notes of a well known piece of music.3 Since these
signals require sinusoidal tones to represent piano notes, a quick introduction to the frequency layout of the
piano keyboard is needed. On a piano, the keyboard is divided into octaves—the notes in one octave being
twice the frequency of the notes in the next lower octave. For example, the reference note is the A above
Middle-C A-440
C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 G5 A5 B5
28 30 32 33 35 37 39 40 42 44 45 47 49 51 52 54 56 57 59 61 63
41 43
OCTAVE
Figure 2: Layout of a piano keyboard. Key numbers are shaded. The notation means the C-key in the
fourth octave.
middle-C which is usually called A-440 (or ) because its frequency is 440 Hz. (In this lab, we are using
the number 40 to represent middle C. This is somewhat arbitary; for instance, the Musical Instrument Digital
Interface (MIDI) standard represents middle C with the number 60). Each octave contains 12 notes (5 black
keys and 7 white) and the ratio between the frequencies of the notes is constant between successive notes.
As a result, this ratio must be . Since middle C is 9 keys below A-440, its frequency is approximately
261 Hz. Consult chapter 9 for even more details.
Musical notation shows which notes are to be played and their relative timing (half, quarter, or eighth).
Figure 3 shows how the keys on the piano correspond to notes drawn in musical notation. The white keys
are all labeled as , , , , , , and ; but the black keys are denoted with “sharps” or “flats.” A sharp
such as is one key number larger than ; a flat is one key lower, e.g., is key number 48.
3
If you have little or no experience reading music, don’t be intimidated. Only a little music knowledge is needed to carry out this
lab. On the other hand, the experience of working in an application area where you must quickly acquire knowledge is a valuable
one. Many real-world engineering problems have this flavor, especially in signal processing which has such a broad applicability
in diverse areas such as geophysics, medicine, radar, speech, etc.
5
A4 = (A-440)
TREBLE
BASS
F-SHARP
HALF NOTE
QUARTER NOTE
EIGHTH NOTE
D5 A4
C4 (middle-C)
D4
E4 F#4
G4
42 44 46 47 49
39 37 35 34
30 28
32
40
51 52 54
C5
B3 A3 G3 F#3 E3 D3 C3 B2
27
B4
Figure 3: Musical notation is a time-frequency diagram where vertical position indicates which note is to be
played. Notice that the shape of the note defines it as a half, quarter or eighth note, which in turn defines the
duration of the sound.
Another interesting relationship is the ratio of fifths and fourths as used in a chord. Strictly speaking the
fifth note should be 1.5 times the frequency of the base note. For middle-C the fifth is G, but the frequency
of G is about 392 Hz which is not exactly 1.5 times 261.6. It is very close, but the slight detuning introduced
by the ratio gives a better sound to the piano overall. This innovation in tuning is called “equallytempered” or “well-tempered” and was introduced in Germany in the 1760’s and made famous by J. S. Bach
in the “Well Tempered Clavier,” which is the source of the song for this lab.
You can use the ratio to calculate the frequency of notes anywhere on the piano keyboard. For
example, the E-flat above middle-C (black key number 43) is 6 keys below A-440, so its frequency should
be Hertz.
3 Warm-up
3.1 Note Frequency Function
Now write an M-file to produce a desired note for a given duration. Your M-file should be in the form of a
function called key2note.m. Your function should have the following form:
function xx = key2note(X, keynum, dur)
% KEY2NOTE Produce a sinusoidal waveform corresponding to a
% given piano key number
%
% usage: xx = key2note (X, keynum, dur)
%
% xx = the output sinusoidal waveform
% X = complex amplitude for the sinusoid, X = A*exp(j*phi).
% keynum = the piano keyboard number of the desired note
% dur = the duration (in seconds) of the output note
%
fs = 11025; %– or use 8000 Hz
tt = 0:(1/fs):dur;
freq = %<=============== fill in this line
xx = real( X*exp(j*2*pi*freq*tt) );
For the freq = line, use the formulas given above to determine the frequency for a sinusoid in terms
6
of its key number. You should start from a reference note (middle-C or A-440 is recommended) and solve for
the frequency based on this reference. Notice that the xx = real( ) line generates the actual sinusoid
as the real part of a complex exponential at the proper frequency.
Instructor Verification (separate page)
3.2 Synthesize a Scale
In a previous section you completed the key2note.m function which synthesizes the correct sinusoidal
signal for a particular key number. Now, use that function to finish the following incomplete M-file that will
play scales:
%— play_scale.m
%—
scale.keys = [ 40 42 44 45 47 49 51 52 ];
%— NOTES: C D E F G A B C
% key #40 is middle-C
%
scale.durations = 0.25 * ones(1,length(scale.keys));
fs = 11025; %– or 8000 Hz
xx = zeros(1, sum(scale.durations)*fs+length(scale.keys) );
n1 = 1;
for kk = 1:length(scale.keys)
keynum = scale.keys(kk);
tone = %<============= FILL IN THIS LINE
n2 = n1 + length(tone) – 1;
xx(n1:n2) = xx(n1:n2) + tone; %<=== Insert the note
n1 = n2 + 1;
end
soundsc( xx, fs )
For the tone = line, generate the actual sinusoid for keynum by making a call to the function
key2note() written previously. It is important to point out that the code in play scale.m allocates a
vector of zeros large enough to hold the entire scale then inserts each note into its proper place in the vector
xx.
Instructor Verification (separate page)
3.3 Spectrogram: Two M-files
In this part, you must display the spectrogram of the scale synthesized in the previous section. Remember
that the spectrogram displays an image that shows the frequency content of the synthesized time signal. Its
horizontal axis is time and its vertical axis is frequency.
(a) Generate the signal for the scale with play scale.m.
(b) Use the function specgram(xx,512,fs). Zoom in to see the progression of three consecutive
notes in the scale (help zoom), and identify the note A-440 in your spectrogram. The second
argument4 is the window length which could be varied to get different looking spectrograms. The
4
If the second argument is made equal to the “empty matrix” then its default value of 256 is used.
7
spectrogram is able to “see” the separate spectrum lines with a longer window length, e.g., 1024 or
2048.5
Instructor Verification (separate page)
(c) If you are working at home, you might not have the specgram() function because it is part of
the “Signal Processing Toolbox.” In that case, use the function plotspec(xx,fs) which can be
downloaded from Web-CT (you also need to download spectgr.m).6 Show that you get the same
result as in part (b). Explain why the result is correct. If necessary, add a grid so that frequencies can
be measured accurately.
Note: The argument list for plotspec() has a different order from specgram, because plotspec() uses an optional third argument for the window length (default value is 256).
4 Lab: Synthesis of Musical Notes
The audible range of musical notes consists of well-defined frequencies assigned to each note in a musical
score. Five different pieces are given in the book, but we have chosen a different one for the synthesis
program in this lab. Before starting the project, make sure that you have a working knowledge of the
relationship between a musical score, key number and frequency. In the process of actually synthesizing the
music, follow these steps:
(a) Determine a sampling frequency that will be used to play out the sound through the D-to-A system of
the computer. This will dictate the time between samples of the sinusoids.
(b) Determine the total time duration needed for each note, and also determine the frequency (in hertz) for
each note (see Fig. 2 and the discussion of the well-tempered scale in the warm-up.) A data file called
fall02 fugue.mat will be provided with this information stored in MATLAB structures; this contains the portion of the piece needed for this lab. A second file called fall02 fugue short.mat
has the same information for the first few measures of the piece; you may find this useful for initial
debugging. Both of these files are contained in a ZIP archive called fall02 fugue.zip which is
linked from the lab page.
(c) Synthesize the waveform as a combination of sinusoids, and play it out through the computer’s built-in
speaker or headphones using soundsc().
(d) Make a plot of a few periods of two or three of the sinusoids to illustrate that you have the correct
frequency (or period) for each note.
(e) Include a spectrogram image of a portion of your synthesized music—probably about 1 or 2 secs—so
that you can illustrate the fact that you have all the different notes. This piece has many sixteenth
notes, so a window length of 512 might be the best choice for specgram(). In addition, the
spectrogram M-files will scale the frequency axis to run from zero to half the sampling frequency, so
it might be useful to “zoom in” on the region where the notes are. Consult help zoom, or use
the zoom tool in MATLAB-v5.3 figure windows.
5
Usually the window length is chosen to be a power of two, because a special algorithm called the FFT is used in the computation. The fastest FFT programs are those where the signal length is a power of 2. 6
Actually, you should download the ZIP file with all the new/updated M-files for ECE-2025.
8
4.1 Spectrogram of the Music
Musical notation describes how a song is composed of different frequencies and when they should be played.
This representation can be considered to be a time-frequency representation of the signal that synthesizes
the song. In MATLAB we can can compute a time-frequency representation from the signal itself. This is
called the spectrogram, and its implementation with the MATLAB function specgram( ) or plotspec().
To aid your understanding of music and its connection to frequency content, a MATLAB GUI is available
so that you can visualize the spectrogram along with musical notation. This GUI also has the capability CD-ROM
MUSIC to synthesize music from a list of notes, but these notes are given in “standard” musical notation, not key GUI
number. For more information, consult the help on musicgui.m which only runs in MATLAB version 5.
4.2 Fugue #5 for the Well-Tempered Clavier
Bach wrote a zillion pieces of music; this is one of the lesser known ones, but the overall sound is much like
any other Bach piece. The first few measures are shown in Fig. 4. You must synthesize the entire portion of
Figure 4: First few measures of the piece Fugue #5 for the Well-Tempered Clavier.
the Fugue #5 for the Well-Tempered Clavier given in fall02 fugue.mat by using sinusoids.7
4.3 Data File for Notes
Fortunately, a data file called fall02 fugue.mat has been provided with a transcription of the notes and
information related to their durations. The data files fall02 fugue.matand fall02 fugue short.mat
are contained in a ZIP archive called fall02 fugue.zip which is linked from the lab page. The format
of a MAT file is not text; instead, it contains binary information that must be loaded into MATLAB. This is
done with the load command, e.g.,
load fall02 fugue.mat
After the load command is executed a new variable will be present in the workspace, called ourVoices.
Do whos to see that you have this new variable.
7
Use sinusoids sampled at 11025 samples/sec (a lower sampling rate could be used if you have a computer with limited memory).
9
The variable ourVoices is a vector whose elements are structures. Each structure gives information
about a single melody in the song; in fugues, such melodies are often called “voices.” You can determine
the number of melodies in the song by calculating the length of the vector ourVoices with the command length(ourVoices). This number will also equal the maximum number of notes that are ever
simultaneously played in the song.
Measures and beats are the basic time intervals in a musical score. A measure is denoted in the score
by a vertical line that cuts from the top to the bottom of one line in the score. For example, in the top line
of Fig. 4 there are four such vertical lines dividing that part of the musical score into four measures. Each
measure contains a fixed number of beats which, in this case, equals four — the number of quarter notes in
a measure. The label “c” at the left of Fig. 4 describes this relationship and is called the time signature of
the song. By convention, “c” denotes “common time,” in which there are four beats per measure and that a
single beat is the length of one quarter note. In our representation, there are four pulses per quarter note, so
there are a total of 16 pulses per measure. (One sixteenth note corresponds to one pulse).
Each structure ourVoices(i) has three fields: notes, startingTimes, and durations.
A typical structure ourVoices(i) looks like
ourVoices(i).notes = [ # # # # … ] % Key number
ourVoices(i).startingTimes = [ # # # # … ] % Starting pulse
ourVoices(i).durations = [ # # # # … ] % Duration in pulses
The value of voices(i).notes(j) is a single note’s key number. The note’s starting pulse (where
there are four pulses per quarter note, or 16 pulses per measure) and duration in pulses is given by the
corresponding elements in the other two fields.
For example, typing ourVoices(1).notes(4) at the MATLAB command prompt returns the number 47, which describes the G in the first measure. Because the note is an eighth note and a eigth note is two
pulses long, ourVoices(1).durations(4)equals two. The value of ourVoices(1).startingTimes(4)
is 9 because this note begins eleven pulses from the beginning of the song.
4.3.1 Timing
Musicians often think of the tempo, or speed of a song, in terms of “beats per minute” or BPM, where the
beats are usually quarter notes. You should write the code so that the BPM is a global parameter that can be
changed easily. For example, you might let the BPM be defined with the statement:
bpm = 120;
Computer programs which lets musicians record, modify, an play back notes played on a keyboard or other
electronic instrument are called “sequencer.”8 The timing resolution of a sequencer is usually measured in
“pulses per quarter note,” or PPQ. In this lab, we will employ four pulses per quarter note. A real commercial
sequencer would have a much higher PPQ to encapsulate the subtle timing nuances of a real human playing
a real instrument. The starting times and durations of notes in the music file provided to you are specified in
terms of “pulses,” so it will be helpful helpful to compute the number of “seconds per pulse,” for instance
via:
beats_per_second = bpm/60;
seconds_per_beat = 1/beats_per_second;
seconds_per_pulse = seconds_per_beat / 4;
8
Popular commercial sequencers include Mark of the Unicorn’s Digital Performer, Emagic’s Logic Audio, Steinberg’s Cubase
and Opcode’s Studio Vision.
10
If the tempo is defined only once, then it could be changed: for example, setting bpm = 240 would
make the whole piece play twice as fast.
Another timing issue is related to the fact that when a musical instrument is played, the notes are not
continuous. Therefore, inserting very short pauses between notes usually improves the musical sound because it imitates the natural transition that a musician must make from one note to the next. An envelope
(discussed below) can accomplish the same thing.
4.4 Musical Tweaks
The musical passage is likely to sound very artificial, because it is created from pure sinusoids. Therefore,
you should try to improve the quality of the sound by incorporating some modifications. (Note: in order
to achieve the highest number of points on this lab you should implement at least one of the tweaks
described in this section, or a similarly cool tweak of your own devising.)
For example, one improvement comes from using an “envelope,” where you multiply each pure tone
signal by an envelope so that it fades in and out.
(2)
If an envelope is used it, should “fade in” quickly and fade out more slowly. An envelope such as a halfcycle of a sine wave is simple to program, but it sounds poor because it does not turn on quickly
enough, so simultaneous notes of different durations no longer appear to begin at the same time. A standard
way to define the envelope function is to divide into four sections: attack (A), delay (D), sustain (S),
and release (R). Together these are called ADSR. The attack is a quickly rising front edge, the delay is a
small short-duration drop, the sustain is more or less constant and the release drops quickly back to zero.
Figure 5 shows a linear approximation to the ADSR profile. Consult help on linspace() or interp1()
E(t)
t
A
D
R
S
Figure 5: ADSR profile for an envelope function .
for functions that create linearly increasing and decreasing vectors.
Some other issues that affect the quality of your synthesis include relative timing of the notes, correct
durations for tempo, rests (pauses) in the appropriate places, relative amplitudes to emphasize certain notes
and make others soft, and harmonics. Since true piano sounds have a second and third harmonic content,
and we have been studying harmonics, this modification is relatively simple, but be careful to make the
amplitudes of the harmonics smaller than the fundamental frequency component.9
Furthermore, if you include too many higher harmonics, you might violate the sampling theorem and
cause aliasing. You should experiment to see what sounds best.
9
In the early 80’s, a company called Digital Keyboards produced a commercial synthesizer called the Synergy in which
the user created sounds via “additive synthesis” by specifying the envelops of individual frequency components. This is
an quite powerful, albeit tedious and challenging way to create realistic sounds. American composer Wendy Carlos (best
known for Switched-On Bach and her score for A Clockwork Orange) used it extensively in her score for Tron. See
http://www.synthmuseum.com/synergy/synergy01.html
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4.5 Programming Tips
You may want to modify your key2note() function to accept additional parameters describing amplitude,
duration, etc. In addition, you might choose to add an envelope and/or harmonics. Chords are created on a
computer by simply adding the signal vectors of several notes. Although we have provided a MATLAB file
containing the note values and durations for Fugue #5 for the Well-Tempered Clavier, you are free to modify
the duration values or add notes if you think it will improve the quality of the synthesized sound.
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Lab    #5    Verification    Sheet