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# Signal Processing Toolbox

## Generating Guitar Chords using the Karplus-Strong Algorithm

This example shows how to generate realistic guitar chords using the Karplus-Strong Algorithm and discrete-time filters.

### Setup

Begin by defining variables that we will be using later, e.g. the sampling frequency, the first harmonic frequency of the A string, the offset of each string relative to the A string.

```Fs = 44100; A = 110; % The A string of a guitar is normally tuned to 110 Hz Eoffset = -5; Doffset = 5; Goffset = 10; Boffset = 14; E2offset = 19; ```

Generate the frequency vector that we will use for analysis.

```F = linspace(1/Fs, 1000, 2^12); ```

Generate 4 seconds of zeros to be used to generate the guitar notes.

```x = zeros(Fs*4, 1); ```

### Playing a Note on an Open String

```When a guitar string is plucked or strummed, it produces a sound wave with peaks in the frequency domain that are equally spaced. These are called the harmonics and they give each note a full sound. We can generate sound waves with these harmonics with discrete-time filter objects.```

Determine the feedback delay based on the first harmonic frequency.

```delay = round(Fs/A); ```

Generate an IIR filter whose poles approximate the harmonics of the A string. The zeros are added for subtle frequency domain shaping.

```b = firls(42, [0 1/delay 2/delay 1], [0 0 1 1]); a = [1 zeros(1, delay) -0.5 -0.5]; ```

Show the magnitude response of the filter.

```[H,W] = freqz(b, a, F, Fs); plot(W, 20*log10(abs(H))); title('Harmonics of an open A string'); xlabel('Frequency (Hz)'); ylabel('Magnitude (dB)'); ```

To generate a 4 second synthetic note first we create a vector of states with random numbers. Then we filter zeros using these initial states. This forces the random states to exit the filter shaped into the harmonics.

```zi = rand(max(length(b),length(a))-1,1); note = filter(b, a, x, zi); ```

Normalize the sound for the audioplayer.

```note = note-mean(note); note = note/max(abs(note)); hplayer = audioplayer(note, Fs); play(hplayer); ```

### Playing a Note on a Fretted String

```Each fret along a guitar's neck allows the player to play a half tone higher, or a note whose first harmonic is 2^(1/12) higher.```
```fret = 4; delay = round(Fs/(A*2^(fret/12))); b = firls(42, [0 1/delay 2/delay 1], [0 0 1 1]); a = [1 zeros(1, delay) -0.5 -0.5]; [H,W] = freqz(b, a, F, Fs); hold on plot(W, 20*log10(abs(H)), 'r'); title('Harmonics of the A string'); legend('Open A string', 'A string on the 4th fret'); ```

Populate the states with random numbers.

```zi = rand(max(length(b),length(a))-1,1); ```

Create a 4 second note.

```note = filter(b, a, x, zi); ```

Normalize the sound for the audioplayer.

```note = note-mean(note); note = note/max(note); hplayer = audioplayer(note, Fs); play(hplayer); ```

### Playing a Chord

```A chord is a group of notes played together whose harmonics enforce each other. This happens when there is a small integer ratio between the two notes, e.g. a ratio of 2/3 would mean that the first notes third harmonic would align with the second notes second harmonic.```

Define the frets for the C chord.

```fret = [3 3 2 0 1 3]; ```

Get the delays for each note based on the frets and the string offsets.

```delay = [round(Fs/(A*2^((fret(1)+Eoffset)/12))), ... round(Fs/(A*2^(fret(2)/12))), ... round(Fs/(A*2^((fret(3)+Doffset)/12))), ... round(Fs/(A*2^((fret(4)+Goffset)/12))), ... round(Fs/(A*2^((fret(5)+Boffset)/12))), ... round(Fs/(A*2^((fret(6)+E2offset)/12)))]; b = cell(length(delay),1); a = cell(length(delay),1); H = zeros(length(delay),4096); note = zeros(length(x),length(delay)); for indx = 1:length(delay) % Build a cell array of numerator and denominator coefficients. b{indx} = firls(42, [0 1/delay(indx) 2/delay(indx) 1], [0 0 1 1]).'; a{indx} = [1 zeros(1, delay(indx)) -0.5 -0.5].'; % Populate the states with random numbers and filter the input zeros. zi = rand(max(length(b{indx}),length(a{indx}))-1,1); note(:, indx) = filter(b{indx}, a{indx}, x, zi); % Make sure that each note is centered on zero. note(:, indx) = note(:, indx)-mean(note(:, indx)); [H(indx,:),W] = freqz(b{indx}, a{indx}, F, Fs); end ```

Display the magnitude for all the notes in the chord.

```hline = plot(W, 20*log10(abs(H.'))); title('Harmonics of the C chord'); xlabel('Frequency (Hz)'); ylabel('Magnitude (dB)'); legend(hline, 'E','A','D','G','B','E2'); ```

Combine the notes and normalize them.

```combinedNote = sum(note,2); combinedNote = combinedNote/max(abs(combinedNote)); hplayer = audioplayer(combinedNote, Fs); play(hplayer); ```

`To add a strumming effect we simply offset each previously created note.`

Define the offset between strings as 50 milliseconds.

```offset = 50; offset = ceil(offset*Fs/1000); ```

Add 50 milliseconds between each note by prepending zeros.

```for indx = 1:size(note, 2) note(:, indx) = [zeros(offset*(indx-1),1); ... note((1:end-offset*(indx-1)), indx)]; end combinedNote = sum(note,2); combinedNote = combinedNote/max(abs(combinedNote)); hplayer = audioplayer(combinedNote, Fs); play(hplayer); ```