The secret behind repeating beats

Can the pitch of a percussion be altered by beating rates?

Most percussions such as bass drums are known to be indefinite pitched instruments, such that the pitch is unchangeable and irrelevant in musical applications. It sounds impossible to change the tone without actually modifying the system since the mechanical characteristic of the “instrument” was unchanged, that is, no matter the beating pattern, the tone should remain the same, right? Nevertheless, as shown in the demo video below, if you strike an “instrument” repeatedly, you may notice a tone rising with the beating rate. The question arises as to whether or not the pitch of the instrument was changing or it was just some acoustical illusions?

The secret behind repeating beats

Response of single impulse

Before stepping into the question, it is helpful to first analyze the acoustic response caused by a single strike, namely the impulse response. Depicts as the figure below, a single impulse applying to the instrument will excite a series of oscillations, interact through physical systems, and are eventually recorded as microphone signals.

Propagation path from the excitation to the receiver (microphone)

For objects such as a random piece of paper, the impulse response is usually broadband in the sense of no apparent resonance frequency exists. As an example, the impulse response of the system shown in the demonstrative video at the beginning was illustrated on the left of the figure below. In complement, the corresponding spectrum of the signal and the background noise are calculated and plotted on the right-hand side. One should notice that the response is indeed broadband and spreads over a certain range of frequency.

Impulse response and the corresponding spectrum of a random paper struck by an electric fan, the spectrum implies a broadband response covering up to a few kHz

The response caused by repeating impulses

Composed of time-invariant dynamic systems which are further assumed to be linear, the response of multiple impulses can be viewed as a linear combination of each impulse response correspondingly [1]. In the following figure, for the system shown in the video, a segment of response caused by multiple impulses (~20Hz) was depicted on the left.

While the normalized spectrum of another segment with a beating rate of around 90Hz was shown on the right, although with a similar envelope, one could easily discover peaks occurring at beating frequency and its integer multiplication. This is the tone that makes it sounds like it was rising/ falling with the beating rate!

The figure on the left shows the response caused by repeating strikes with the black arrow roughly indicating the input timing, the figure on the right shows the spectrum of the segments while the beating rate was around 90Hz, notice the peaks at the integer multiplication pointed by arrows

This phenomenon can be further emphasized by viewing the temporal spectrum (spectrogram or time-frequency plot) generated below. The pitch of the concentrated areas is clearly rising (and then falling) with the rate of the beating.

Spectrogram (Time-Frequency plot) of the full acoustic signal from the demo video

But why?

At this point, you may already correctly guess that a repeating beat didn’t actually “create” signals but rather highlights the parts that match its frequency through the interference between responses. A more rigorous discussion will be given in the next section.

Realization in the frequency domain

From a time-domain point of view, parts of the impulse response that matches the integer multiplicated frequency of beat will constructively build up and stands out among other bands.

Alternatively, from the frequency domain, the spectrum of the repeating impulses (refer to as the Dirac comb function) was also a repeating impulse spaced with the repeating frequency. With the aid of the convolution theory, the convolution results in the time domain could be acquired equivalently by a point-wise product in the frequency domain [3]. This implies that the spectrum of the effective output excited by the repeating impulse is only occupying specific areas shown in the lower-left corner of the following figure and inherits the envelope from a single impulse response with proper normalization.

Linear system excited by repeating impulses, the path above shows an interpretation in the time domain and the path below follows the frequency domain

Those peaks located at the integer multiplication of the beat rate were exactly the strip patterns one may observe from the spectrogram of the demo video.

What if the impulse response was narrowband?

So far the case with a broadband impulse response was discussed, but most instrument (including an indefinite pitched instrument) usually has a narrowband response with one or more explicit resonant frequencies. This is shown as the response signals of a typical drum in the figure below. It turns out that whether or not the impulse response was a broadband signal is a critical factor affecting the beat-rate dependent tone. This could be informally demonstrated in two different scenarios.

Impulse response and the corresponding spectrum of a drum striking with two different positions [2], there are clearly few resonant frequencies

1. At a very low repeating beating rate, the “comb” in the frequency domain will be packed tightly and recovers the original spectrum of the impulse response as the output which is not a surprise.
2. At some carefully selected frequency, the effective excitation could never meet any of those bands, causing a silent output. This limits the possible output band and breaks the demonstration. Luckily a random piece of paper was used instead of a fine-tuned drum in the demo video and avoids this issue.

Conclusion

In this article, the nature of the output of a repetitive excited system and the spirit of convolution theory was demonstrated through some simple household objects, namely an electric fan and an arbitrary piece of paper. Interestingly, in order to have a recognizable tone that follows the beating rate, an “instrument” with broadband impulse response is mandatory, which also answers the question in the subtitle. I hope you found this experiment fun!

References

[1] C. T. Chen, Linear System Theory and Design. Oxford University Press, 2014.

[2] P. Richardson & R. Toulson, ‘Fine Tuning Percussion — A New Educational Approach’, 06 2022.

[3] C. D. McGillem & G. R. Cooper, Continuous and Discrete Signal and System Analysis. Holt, Rinehart, and Winston, 1984.