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Harmonics and Resonance

Harmonics​

When an object vibrates, it often creates not just a single sound wave, but several waves at different frequencies. These additional waves are known as harmonics. The primary wave, or the lowest frequency, is called the fundamental frequency. Harmonics are integer multiples of this fundamental frequency. Because all other harmonics other than the fundamental frequency are higher-pitched than the fundamental frequency, they are also referred to as overtones. Module 1 shows a visualization of the harmonics for a string.

Module 1
An illustration of harmonics for a string.

Fundamental vs. Dominant Frequencies​

Module 2 shows the C3 note recorded from a digital piano. The fundamental frequency as measured through software is 131.2 Hz, which is very close to the theoretical 130.8 Hz. You can see peaks at 261.2 Hz (roughly two times the fundamental frequency), 392.4 Hz (roughly three times the fundamental frequency), and so on.

Module 2
The frequency components of a C3 note played on a digital piano.

Note that in Module 2 the fundamental frequency also happens to be the loudest frequency, which is known as the dominant frequency. Despite its name, the dominant frequency is not the frequency you perceive the sound to be. The fundamental frequency is usually the one that will stand out. Figure Y shows the same audio recording but with a non-fundamental frequency boosted in order to become the dominant frequency. The audios in Figure X and Figure Y sound extremely similar despite having different dominant frequencies.

Module 3
The frequency components of a C3 note with the 392.4 Hz component boosted.

How loud does the fundamental frequency need to be to be audible over the dominant frequency? Of course, at some point the fundamental frequency would be so quiet that it gets overpowered. We can test this by generating audio with both 200 Hz and 400 Hz frequency components and varying the volume level until the 200 Hz frequency "disappears". Module 1 shows the result of this experiment.

Module 4
200 Hz and 400 Hz frequencies played together, with the 200 Hz frequency gradually decreasing in volume. At around 8 seconds, the fundamental frequency will feel like it disappears even though it still contributes to 20% of the volume of the sound.

Odd Harmonics​

When you press a key on a piano, a hammer inside the piano strikes a string. The vibration of this string is what produces sound. However, where should the hammer strike the string? The position of the strike greatly impacts the sound that is produced.

As it turns out, piano hammers are intentionally aligned off-center so that all harmonics can sound when a note is played. If the hammer strikes directly in the middle of the string, then none of the even harmonics would play. This is because the 2nd, 4th, 6th, etc. harmonics will not occur at the center of the string since there is a node at that position. Nodes are places on the string where there is no vibration, i.e. the string is "standing still" at these spots. Antinodes are places on the string where the vibration is the most intense.

Module 2 shows the nodes and antinodes for the 3rd harmonic of a string.

Module 5
The nodes and antinodes for the 3rd harmonic of a vibrating string.

Something interesting happens if you do strike the middle of a string. In this case, the resulting sound wave will be a weighted sum of the odd harmonics of a sine wave. This results in a waveform that is approximately a square wave, as shown in Module 2.

Module 6
The weighted sum up to the Nth harmonic, including only odd harmonics. The strange effect as the corners of the rectangles is known as the Gibbs phenomenon.

Sympathetic Resonance​

Sometimes vibrations from one source can induce vibrations in a receiver. This is known as sympathetic resonance. A fun example of this (and also a magic trick for children) is to "transfer" the sound from one tuning fork to another, as seen in this YouTube video.

Sympathetic resonance can occur in certain instruments, where plucking/hammering one string causes another string to start vibrating. Sometimes this is unintentional and can cause unwanted sounds to appear in the recording. If you know what you are doing, you can use this effect to your advantage to produce interesting music. An example of sympathetic resonance on the piano is showin in this YouTube video.

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