Part 4: Frequency Content

Looking back at the waveform from Part 2, you may have wondered why the signal is not a pure sine wave (like an ideal tuning fork would produce). The waveform display seems to indicate that there are additional oscillations in the signal which cause secondary peaks and other shapes to appear. In this part of the lab, we will consider the frequency spectrum of the signal and how it affects the way the string sounds. In short, the fundamental frequency calculated in Part 2 is not the only frequency present in the signal, but there is energy at other frequencies as well. The amplitude versus frequency is an important function called the spectrum, which affects the timbre of the sound. For example, a ``brighter'' tone corresponds to more energy in the high-frequency overtones.

1. Close patch 1-1, and do not worry about needing to save any information stored in the patch. Download lab patch 1-2 (1-2.pd) and save it to a local directory. Then open 1-2.pd with pd.

2. Also, double check that pd is not in edit mode by clicking the Edit menu and making sure that ``edit mode'' is not checked.

3. Pluck the string. The frequency window shows a plot which indicates the spectral content of the signal as the signal is read into the sound card.

   Peaks in the plot indicate energy in the signal at that particular frequency. For example, in the plot below, peaks occur at around 165 Hz, 330 Hz, 495Hz, .... An example is shown in Figure 6. The peaks are called partial overtones. They are nearly harmonically related, as long as the string is not stiff, so they are also called harmonic overtones, or simply harmonics. If they were exactly harmonically-related, then the $n$th harmonic would be given by the following for $n=1,2,3,...$:
   

   $$f(n) = nf_0$$ (10)

   
   

   Figure 6: Pd plot showing harmonics in the magnitude spectrum of the recorded string vibration in response to a pluck.

   

4. Pluck the string once firmly and once softly. Look at the peak corresponding to the fundamental frequency calculated in Part 2. How does the amplitude of this harmonic change over time? Is the amplitude of the fundamental frequency greater in the firm or soft pluck? Can you see any change in the harmonic peak locations between just after the pluck and late in the decay?

   Record your observations below:
   

5. Pluck the string again and now consider the amplitude of the next peak in the plot with a frequency higher than the fundamental frequency. This peak is called the second harmonic, and it occurs at twice the frequency of the fundamental frequency.
   1. Carefully measure the frequency of the fundamental frequency and the second harmonic. Is the ratio exactly 2.000? If not, what do you measure for this ratio? How can this be explained?

      Explain below:
      

   2. Pluck the string softly and more firmly, and consider the strength of the second harmonic relative to the fundamental frequency. Is there a difference? Explain below:
      

6. Acquire a guitar slide or something similar (a smooth glass bottle will work) and press it against the end of the string roughly 3 inches from the bridge. Pluck the string. Now slide the object closer to the bridge, essentially changing the effective length of the string. How does the sound of the string change? Can you see a change in the spectrum?

   Record Your Observations Below:
   

7. Each of the harmonics is due to a standing wave on the string. Note that for the even harmonic standing waves, the string does not move at all at its midpoint. However, for the odd harmonic standing waves, the string always moves at its midpoint. Consider what would happen to the pitch if it were possible to mute only the odd harmonics while allowing the even harmonics to ring?

   Pluck the string once and quickly mute the string at its midpoint for a moment so that the string continues to ring but with a different pitch. Since the underside of your index finger is soft, one way to mute the string is to gently but steadily press the underside of your finger against the midpoint of the string. It may take several tries to find the exact midpoint. It will however be obvious once you achieve it.

   This muting action should allow you to mute the fundamental frequency. How does the pitch change? How would you expect this change in pitch to be reflected in the spectrum? Does the frequency plot indicate what you would expect?

   Record Your Observations Below:
   

   This technique is often used by guitarists in order to create a unique and higher pitched sound. An example plot of the spectrum with the fundamental frequency muted is shown in Figure 7.

   Figure 7: Plot of the spectrum of the second harmonic of string vibration in response to a pluck.