Vibration Tonal Quality, or Why Are the Drums Not Rectangular?

Science has a huge, infinite number of advantages, and one of them is that it is she who is able to briefly, accurately and succinctly answer the most unexpected in its simplicity, children’s questions. For example: “Why is the drum round?” If you have children or you yourself asked this question in childhood, then in this translation, which we decided to share before the start of the flagship Data Science course, you will find two answers: the first is a detailed one, in formulas, the second is a short answer that the child can understand.

I’ve always been fascinated by how the properties of any waveform relate to how we perceive them. Especially the shape of the waves and the shapes from which the waves emanate. We know that sound is a mechanical wave and a medium is needed for its propagation. The vibration source moves air molecules by vibrations that cause compression and depression (high and low pressure areas). These pressure changes interact with our eardrum and we perceive sound. So, if all kinds of vibrations only vibrate air molecules, then there must be something special in the properties of the waveform that helps us distinguish between the two sounds. What makes the guitar sound different from the piano, or why my voice is different from yours.

I will look at this problem from two perspectives. In the first approach, I compare two real samples from different instruments and analyze their properties. In the second, I work from scratch and show how solutions to wave equations in various scenarios manifest themselves in these properties that we perceive.

The piano is a stringed instrument. Vibrations begin with a hammer hitting its strings. The same behavior is observed in the waveform graph. The hammer creates a high impulse force on the strings, and the result is a sudden increase in amplitude. Over time, the amplitude and therefore the loudness of the signal decays.

The picture shows the sounds of violin notes [включая полутон в середине, который по историческим причинам не считается нотой с формальной точки зрения, поэтому в названии вы видите диапазон A4-B4, все звуки в нём не обозначены] with increasing frequency. The frequency response is not captured in the waveform graph, but the waveforms are different from those of a piano. Unlike a piano, the amplitude of all notes gradually increases, as the sound is emitted by the bow, which gently rubs against the string. And the decrease in amplitude is also sharp, as the violinist mutes one string while playing the other. And so we do not hear any lingering sound or any thinningas we call it in music.

To find more differences in sound, we have to move into the frequency domain. The best way to see these differences is with a spectrogram, which is a visual representation of the frequency spectrum as the signal changes over time. To create a spectrogram, the audio signal is split into smaller segments, and FFT can be applied to each of these segments. Let’s start by looking at a simple 440 Hz sine wave.

Sine wave with a frequency of 440 Hz

The sine wave tone and the piano tone are at exactly the same pitch, at 440 Hz. However, the sine wave listen not so interesting. Sounds like just a continuous tone of sound without any dynamics or saturation. Although many interesting sounds can be obtained by interference and applying transformations on sinusoids. A synthesizer is a device that allows you to do this.

import numpy as np 
import matplotlib.pyplot as plt
import librosa, librosa.display
sr = 22050 # sample rate
T = 2.0    # seconds
t = np.linspace(0, T, int(T*sr)) # time variable
x = 0.5*np.sin(2*np.pi*440*t)
def plot_spectrogram(signal, name):

    spectrogram =     librosa.amplitude_to_db(np.abs(librosa.stft(signal)))
    plt.figure(figsize=(10, 5))
    librosa.display.specshow(spectrogram, y_axis="log")
    plt.colorbar(format="%+2.0f dB")
    plt.title(f"Log-frequency power spectrogram for {name}")
    plt.xlabel("Time")
    plt.show()
plot_spectrogram(x, 'sine wave')

By observing the spectrogram of the sine wave, one can find that the only frequency present is 440 Hz.

piano, srp = librosa.load('Piano.mf.A4_trimmed.wav')
violin, srv = librosa.load('Violin.arco.mf.sulA.A4B4.wav')
plot_spectrogram(piano, 'piano')
plot_spectrogram(violin, 'violin')

By plotting the spectrogram of the piano, we can see that the most dominant frequency is 440 Hz, but at the same time many other frequencies are present. These are known as overtones, and for a single vibrating string, it can be shown that these overtones are actually integer multiples of the fundamental frequency, in this case 440 Hz.

In the case of the violin, the spectrogram is the same story. There is a fundamental frequency, and then there are overtones in all three notes played. The number of overtones is much higher than on a piano spectrogram. And this is one of the reasons why the piano sounds different from the violin, even though they are both stringed instruments. The fundamental frequency increases as it transitions from the first note to the third played note. Also here overtones are integers that are multiples of the fundamental frequency.

Thus, it can be said that the frequency of the overtones has something to do with the tonal quality of the sound. Playing one note on a string instrument, we see that there are integers multiples of the fundamental frequency in the sound. If these overtones are not in whole multiples or in specific frequency ratios (such as when chords are played on multiple keys), it is difficult for the human ear to discern whether the sound has any tonal quality or not.

Let’s go back to the original question. Why are most drums round? From tabla, congo, bongo before the usual drum kit – everything is round. What’s so special about a round or rectangular shape? To figure this out, we’ll have to look at the vibrations coming from the rectangular membrane and see if we can generalize something.

Suppose that the membrane is installed in the plane xy, but u (x, y, t) – vertical deflection of the membrane relative to the equilibrium position (x, y) in time t… With constant t surface z = u (x, y, t) sets the shape of the membrane at the moment t…

Consider a stretched membrane mounted on a rectangular frame of length a and width b… To study its fluctuations, you need to solve the problem of the initial value.

Here equations (2) and (3) are boundary values, and equations (4) and (5) are initial values. To find simple solutions, we can use separation of variables satisfying homogeneous boundary conditions, and then apply the principle of superposition to construct a series of solutions that also satisfy the initial conditions. We can look for solutions of the form u (x, y, t) = X (x) Y (y) T