How to find Q-factor of Guitar

 Q-factor of any oscillation is the measure of how good the quality of oscillation is. A higher Q means that the oscillation sustains for longer time before the oscillation stops. A pull of string of a guitar produces a damped oscillation, that starts out with initial high amplitude and reduces gradually before finally stopping. The graph below shows waveform of damped oscillation.

amplitude decay of dampled oscillation
We can find out the quality of oscillation(Q) of guitar string can be found out using the following equation.

\(Q=\frac{w_0}{\gamma}\)   ------------>(1)

where \(w_0\) is the natural frequency of oscillation and \(\gamma\) is the damping ratio.

\(ln(\frac{A_k}{A_{k+n}}) = \frac{n \pi}{Q}\)    ------------>(2)

where \(A_k\) is the kth amplitude and k+n is the k+n th amplitude.

Consider that a string pull of a guitar generates the following signal,

\( A(t) = A_0 exp(\frac{t}{\tau})\)      ------------>(3)

where A(t) is the amplitude at time t and \(A_0\) is the initial amplitude. Suppose that the frequency produced is f = 440Hz and that the amplitude reduces from initial \(A_0=8\) and to \(A(t)=4\) in 4 seconds.

From (3) we have,

\(\tau = \frac{t}{ln(\frac{A_0}{A(t)})}\)

that is,

\(\tau = \frac{t}{ln(\frac{A_0}{A(t)})} = \frac{4}{ln(\frac{8}{4})}=\frac{4}{ln(2)}=5.77s\)

Thus in this case the decaying time \(\tau\) is 5.77 seconds.

Next we will determine the Q-factor of the guitar,

\(Q=\frac{w_0 \tau}{\gamma}=\frac{2\pi 440 \times 5.77}{2}=6 \times 10^{3} \)

Thus the quality of the guitar is high which means that clean tone(single frequency) is produced by this guitar, that is narrowband tone.

The quality of the guitar can be measured using PC with program to detect the peaks initially and certain duration.


[1] Electret Microphone Preamplifier with OPA365

[2] acquire audio signal with Arduino 

[3] Online Real-Time Signal Web Oscilloscope 

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