1. Design of a Guitar Tab Player in MATLAB Background Lecture
Module 1: Modeling a Guitar Signal

2. Multitude of guitar tabs on the Internet Some are good some are not
The Problem
Multitude of guitar tabs on the Internet
Some are good some are not
Wasted time and frustration for an amateur guitar player trying to learn a song
Having a way to check quickly if a guitar tab is good or not

3. Typical Electric Guitar
Low E D
frets g b
High e
to amplifier
magnetic pickups
A typical electric guitar has 6 strings and 24 frets

4. e|------------------------1-0----------|
Guitar Tablature
e| |
B| p1-0--|
G| |
D| |
A| |
E| |
Guitar tablature is a form of musical notation that is
well suited for playing the guitar.
It shows the six strings of the guitar. The number and its location
corresponds to the string to pluck and the fret where to place the finger.
Fret 0 represents the case where one leaves the string open

5. Playing the Guitar Tablature
B| p1-0--|
G| |
D| |
A| |
E| |
Low E
fret 2
Low E
(open string)
Low E
(open string)
Low E
fret 3
Low E A D G B
High e

6. Guitars and Guitar Signals
What Do We Know About
Guitars and Guitar Signals

7. Guitar String Vibrations
First Harmonic (Fundamental)
Second Harmonic
Third Harmonic
The string must make an integer number of half sines between the two points it is attached too

8. Fundamental Frequency for a String
f0 is the frequency of the sound signal in Hertz (Hz)
m is the linear mass of the string given in kilograms per meter (kg/m)
L is the length of the string in meters (m)
T is the tension on the string giving in Newtons (N)
A guitar string is tuned by tightening or loosening the string (increasing or decreasing T)
The frequencies produced by a bass are lower than that of a guitar. This is why a bass has longer strings than a guitar (greater L value).

9. Fundamental Frequencies for a Guitar
String
Fundamental Frequency
Low E
82 Hz A
110 Hz D
147 Hz G
196 Hz B
247 Hz
High e
330 Hz
The guitar will be tuned when the strings produce the fundamental frequencies above.
(The fundamental frequencies for a bass are 41, 55, 73, and 98Hz)

10. Fundamental Frequencies with Frets
fret 3 fret 2 fret 1
When placing the finger on a fret, the sound has a higher pitch (higher frequency) than that of the open string. The frets are designed such that the fundamental frequencies produced are:
where f0 string = 82, 110, 147, 196, 247, or 330Hz depending on the string.
Example: Plucking the Low E string with the finger on the second fret produces a sound with fundamental frequency:

11. How Can the Guitar Signal Be Modeled?

12. Mathematical Formulation:
Pure Tone or Sinewave A T0
Fundamental Period t -A
Mathematical Formulation:

13. Single Sinewave Model
String
Guitar Recording
Low E
82 Hz A
110 Hz D
147 Hz G
196 Hz B
247 Hz
High e
330 Hz
Modeling the sound of a guitar string using a single sinewave does not sound satisfactory. It seems that the guitar string has a higher “pitch” than the single sinewave.

14. Adding Several Sinewaves (Fundamental and Harmonics)

15. Signal Obtained from Adding Sinewaves

16. Issues How many harmonics to use? Are the amplitudes all equal to 1?
Does the signal obtained look like the actual signal?
More importantly: Does the signal obtained sound like the actual signal?
Take measurements
to check/improve the model

17. Measuring Actual Guitar Signals Using a Computer Sound Card
guitar output jack plug
PC/laptop microphone
or line input jack connector

18. Sampling in an Analog Signal (MIC/Line Input):
Analog to Digital Conversion
Sampling Theorem:
One must sample at least twice as fast as the highest frequency contained in the signal

19. Producing an Analog Signal (Speaker Output):
Digital to Analog Conversion

20. Sound Card Control (Volume Control - Playback) D/A Control
Programs->Accessories->Entertainment->Volume Control
Typical sound cards can produce electrical signals up to +/- 2 volts maximum. This explains why one needs external speakers with power supply for louder sounds.

21. Sound Card Control (Volume Control - Recording) A/D Control
Signals at the MIC or line input should not exceed +/- 150 mV

22. Microsoft Windows Sound Recorder
Programs->Accessories->Entertainment->Sound Recorder
Sound signals are most commonly stored into .wav files.
The samples are obtained using Pulse Code Modulation (PCM).
One has the flexibility to specify the sampling frequency and the number of bits

23. Measurement of the Open Low E String

24. Three Sinewaves Model
actual recording
model

25. Issues
It sounds better than the single sinewave model but it is difficult to tell from the time waveform:
How many harmonics to use
The amplitudes of the sinewaves to use
If the frequencies of the sinewaves are as expected
The Fourier Transform can be used to show the frequencies and amplitudes contained in the signal.
The results are shown in a what is called the Frequency Spectrum graph

26. Frequency Spectrum of Harmonic Signals
Single sinewave (single tone)
Frequency Spectrum
Math Formula
Several sinewaves (fundamental and harmonics)
Frequency Spectrum
Math Formula

27. Measured Frequency Spectrum of the Open Low E String
f = 82 Hz, amplitude =
f = 164 Hz, amplitude =
f = 246 Hz, amplitude =
f = 328 Hz, amplitude =
fundamental
(82 Hertz)
second
harmonic
(164 Hertz)
third
harmonic
(246 Hertz)
forth
harmonic
(328 Hertz)
fifth
harmonic
(410 Hertz)
becoming negligible
The measured frequency spectrum is obtained by running a Fast Fourier Transform (FFT) program on the time samples of the guitar string.
One can clearly see the fundamental and harmonics, their frequencies and amplitudes.

28. Other Measured Frequency Spectrums
Frequency spectrum of a human voice (somewhat harmonic)
Frequency spectrum of a drum type sound (not harmonic)

29. Measurement Based Model

30. Mathematical Model of an Open String
The guitar signal is modeled as a sum of several sinewaves (fundamental and harmonics):
The fundamental frequency f0 depends on the string and is given for a tuned guitar by:
String f0
Low E
82 Hz A
110 Hz D
147 Hz G
196 Hz B
247 Hz
High E
330 Hz
The amplitudes A1, A2, A3, etc of the sinewaves in the model are obtained by running the FFT program on the measured time samples of the open string as shown in the next slide.

31. Getting the Amplitudes A1, A2, A3, etc for the Model
Run the FFT program on the measurement .wav file of the open string. Then locate the peaks and report the fundamental/harmonics amplitudes A1, A2, A3, A4, etc using the Frequency Spectrum graph produced.
Check that depending on the string, f0 = 82, 110, 147, 196, 247 or 330Hz.

32. Modeling any String/Fret Combination
The model can be extended to any of the string/fret combinations. The main difference comes from the fundamental frequency to use. For example calling fx the fundamental frequency for a string with a given fret gives the general model:
Note that the general model also works for open strings since
String f0
Low E
82 Hz A
110 Hz D
147 Hz G
196 Hz B
247 Hz
High E
330 Hz

33. Design Choices
To complete the general model, one needs to define the sinewave amplitudes A1, A2, A3, etc. There are two straightforward methods:
Take samples of all 144 string/fret combinations and extract the sinewave amplitudes from the measured frequency spectrums obtained running the FFT program.
Reuse the same sinewave amplitudes as those obtained for the open strings.
Pros/Cons: Solution 1 is more accurate than solution 2 but will require a lot more work and a lengthier program. Solution 2 should be sufficient for the application: playing quickly guitar tabs to check if the tab is valid or not.

34. Effects of Decay
The general model does not go down after some time as does the actual guitar string signal.
We need to add something in the model that can make the signal go down over time.

35. Exponential Decay
The previous model is multiplied by an exponential function so that its amplitude will decrease over time:
The exponential function starts at 1 when t = 0 and drops to 0.37 when t = T. Therefore to estimate T, one checks to see when the time waveform has dropped from 1 to 0.37.
T about 1.8 seconds
0.37

36. Some Useful MATLAB Functions

37. Reading a .WAV File and Plotting the Waveform
clc;
clear;
close all;
filename = input('wav file name => ','s');
% Enter file name from keyboard
[vt, Fs, nbits] = wavread(filename);
% vt has the amplitude samples
% Fs has the sampling rate
Ts = 1/Fs; % Compute sampling time Ts
N = length(vt); % N is the number of samples
t = Ts*[0:1:N-1]; % Form times values using Ts
figure(1) % Plot vt versus t
plot(t, vt)
grid on
title('v(t)')

38. Generating a Sinewave and Saving It in a .WAV File
clc;
clear;
close all;
Fs = 8000; % Set the sampling rate Fs
Ts = 1/Fs; % Compute sampling time Ts
nbits = 16; % Set the number of bits nbits
N = 800; % Set the number of points N
t = Ts*[0:1:N-1]; % Form the time values
f0 = 10; % Set the sinewave frequency f0
vt = 2*sin(2*pi*f0*t); % Generate the sinewave
filename = input('wav file name => ','s');
% Get the filename from keyboard
wavwrite(vt, Fs, nbits, filename);
% Save the sinewave vt in the .wav file

39. Clipping Issues Clipping
One should scale the amplitude of the waveform to fit between +/- 1 before saving it to the .wav file. If not the waveform will be “clipped” as shown below.
Clipping

40. Fixing the Clipping Issue
clc;
clear;
close all;
Fs = 8000;
Ts = 1/Fs;
nbits = 16;
N = 800;
t = Ts*[0:1:N-1];
f0 = 10;
vt = 2*sin(2*pi*f0*t);
vt = vt/max(vt); % Divide vt by
% the maximum value in vt
filename = input('wav file name => ','s');
wavwrite(vt, Fs, nbits, filename);

41. Fast Fourier Transform (FFT)
The MATLAB function called fft.m can be used to compute the frequency spectrum of a signal when the time waveform samples are provided.
filename = input('wav file name => ','s');
[vt, Fs, nbits] = wavread(filename);
Ts = 1/Fs;
N = length(vt);
df = Fs/N;
t = Ts*[0:1:N-1];
f = df*[0:1:N-1]; % Create appropriate frequency values
Vf = fft(vt); % Computes the FT values (complex)
Vfnorm = Vf/max(abs(Vf)); % Normalize to +/- 1
figure(1)
plot(f, abs(Vfnorm), 'k') % Plot the magnitude versus frequency
axis([0 Fs/2 0 1]) % Limit plot from 0 to Fs/2 (x-axis)
grid on % and from 0 to 1 (y-axis)
legend('|Vnorm(f)|')
xlabel('f (Hz)')

42. Playing Sounds
To play sounds directly in MATLAB, one can use the functions sound.m, soundsc.m, or wavplay.m
SOUND(Y,FS) sends the signal in vector Y (with sample frequency FS) out to the speaker. Values in Y are assumed to be in the range -1.0 <= y <= 1.0. Values outside that range are clipped. Stereo sounds are played, on platforms that support it, when Y is an N-by-2 matrix.
SOUNDSC(Y,...) is the same as SOUND(Y,...) except the data is scaled so that the sound is played as loud as possible without clipping.
WAVPLAY(Y,FS) sends the signal in vector Y with sample frequency of FS Hertz to the Windows WAVE audio device. Standard audio rates are 8000, 11025, 22050, and Hz.
For stereo playback, Y should be an N-by-2 matrix. Values in Y are assumed to be in the range -1.0 <= y <= 1.0. Values outside that range are clipped.

43. Example of Playing Sounds
clc
clear
close all
Fs = 8000; % Sampling rate Fs
Ts = 1/Fs; % Compute sampling time Ts
N = 8000; % 8000 pts last 1 second
t = Ts*[0:1:N-1]; % Form t vector with values
f0 = 100; % separated by Ts seconds
vt = 1*sin(2*pi*f0*t);% Compute the samples of
% the sinewave
sound(vt, Fs) % Send it to the sound card

44. Example of Playing Stereo Sounds
Fs = 8000;
Ts = 1/Fs;
N = 8000;
t = Ts*[0:1:N-1];
f1 = 100;
f2 = 500;
vleft = 1*sin(2*pi*f1*t);
vright = 1*sin(2*pi*f2*t);
vt = [vleft' vright']; % vt has two columns
% vleft and vright
sound(vt, Fs)