Harmonics

Harmonic frequencies are an essential part of sound. If you have read some of my previous posts and experiment summaries, you will probably have noticed references to different harmonic modes, but I have not dedicated a post to the concept yet. Harmonics are effectively different modes of vibration, sounds with different frequencies, that can exist in a medium (on a string or in a tube, for example). These different harmonics are often combined to create more complicated sounds in instruments through Fourier's Theorem. In this sense, harmonics are able to change the timbre of sound and to distinguish instruments. A good example of this is a comparison between the flute and the clarinet. The flute is effectively a column that is open at both ends, while a clarinet is effectively a column that is closed at one end. This means that the clarinet creates only odd harmonics, while the flute is able to create both odd and even harmonics. What does this mean?

Let's take a look at what the different harmonic modes look like on a string:

(The Physics of Music and Color)

The first harmonic is not pictured, but would have one antinode in the middle. As you can see, the harmonic mode number corresponds to the number of loops in the pattern. An even harmonic is a harmonic with an even number of loops, and an odd harmonic is the opposite. While this should provide a basic understanding of what harmonics are, we can look at them through a mathematical lens to understand the relationship between them.

In the image above, we see a full wavelength in the second harmonic. The first harmonic is half of a wavelength. We can break down the relationship between the length of the string and the harmonic mode number to look for a pattern:

Harmonic Mode	Wavelength in Terms of Wave Velocity (V) & String Length (L)
1	V/2L
2	2V/2L
3	3V/2L
4	4V/2L
5	5V/2L

While the 2L remains constant, the V coefficient increases by an interval equal to the harmonic mode number. From this observation, we can derive the equation:

(The Physics of Music and Color)

Works Cited:
1. Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.