#3: Harmonics: Where do they come from?

In Harmonics: the Building Blocks of Pitch, we saw how tones with a pitch are comprised of harmonics which follow a particular pattern called the Harmonic Series.

But where does the Harmonic Series come from? What causes it?

The Ideal String

To get a feel for this, let’s consider a string* (such as on a guitar, viola, harpsichord, etc.). This string is stretched taut and its ends fixed. For example:

The center portion of the string can move, but the tension on the string pulls it back toward its middle resting position.

This string, however, can move in other ways, even with both ends fixed. For example, the left and right halves of the strings can vibrate independently:

In fact, the string can vibrate with any integer division of the string:

thus, giving rise to the harmonics we hear. An analogous phenomenon happens in pipes (like a flute, organ, your vocal tract) where the air itself compresses and thins in patterns of vibrations that also divide the length of the tube by integers.

This phenomenon is an easy way to conceptualize the origin of the Harmonic Series!

Aside: A History Lesson

There is some pretty deep math that connects these harmonics to the repeating patters of vibrations we encountered last post.

The mathematician Jean-Baptiste Joseph Fourier (1768-1830) spearheaded the analysis of repeating patterns in mathematical functions, and this was later used to show that the repeating patterns of vibrations of a pitch can be viewed as a collection of sine waves that form the Harmonic Series. This provides the mathematical basis and rich theory for harmonics.

Later, Hermann Ludwig Ferdinand von Helmholtz (1821-1894) was the first to isolate harmonics acoustically and study them in detail, publishing his observations and conclusions in 1862 in the seminal work, On the Sensations of Tone.

Finally, these theories have been developed into modern techniques, such as the Fast Fourier Transform (usually abbreviated FFT), that underpin much of modern day audio software. In other words, this stuff is really useful!

Organizing Harmonics

We number harmonics starting from 1, which we assign to the lowest harmonic—the harmonic that matches the frequency of the note. We call this first harmonic the fundamental. Then, we simply number each successive harmonic. Note that not all harmonics need to be present in order for us to perceive pitch (we will discuss that in the next post), but we number them whether they are present or not.

So, taking our diagram of the vibrating string, let’s number the harmonics:

Notice that the number of the harmonic is also the number of times the string gets divided. For example, the second harmonic vibrates each half of the string separately—like we had two strings of half the length. The third harmonic vibrates the string in thirds—like three strings of one-third the length. Etc.

We can calculate the frequency of each harmonic quite simply, if we know the frequency of the first harmonic. Let’s use the pitch C2 in our example—this pitch is approximately 66 Hz. (Hz is the abbreviation for hertz, which means “cycles per second”—i.e. how many times something repeats in one second. In this example, it would be how many times in one second the pattern of vibrations forming the note C2 repeats in one second.) The frequency of a harmonic is the frequency of the fundamental multiplied by the number assigned to the harmonic. Thus, the fourth harmonic of C2 is 66 x 4 = 264 Hz. Here’s the diagram with the frequency of each harmonic labeled:

Notice how each of the harmonics is spaced an equal number of Hz apart (the Hz of the fundamental)—in this case 66 Hz apart. As we will see in detail later, this simple arithmetic relationship gives rise to the entire complexity of Harmony!

Next Steps

We have investigated the origins of harmonics, and how this gives rise to the Harmonic Series.

One of the things that makes harmonics and the Harmonic Series so fascinating is that while we can easily hear the individual harmonics when isolated, when they are all present at once, they fuse into a single tone. We will investigate this phenomenon next.

< PREVIOUS: #2: Harmonics: the Building Blocks of Pitch

(A footnote on the “ideal” string.)

*Here we are considering an “ideal” string. What is explained on this page is responsible for most of the harmonics created when a string vibrates, but it turns out that strings can vibrate in other ways, too. One example is the string can twist. Like vibrating up-and-down, this twisting also has a rest state that string naturally wants to return to, and will thus set up a different kind of vibration.

Another factor with strings is their stiffness and thickness. These factors can cause the harmonics to become progressively sharp when compared to our “ideal” string. This is a real-world issue: piano strings are stretched with high enough tension, and the lower strings are often made very thick (so that the piano can be a reasonable size), that they no longer vibrate like an idea string. On smaller pianos, the lowest notes have rather indefinite pitch because the string no longer behaves like an ideal string, and the “harmonics” produced no longer accurately mimic the Harmonic Series.

So although strings (and pipes) don’t behave perfectly “ideal” in the real world, they are close enough. The fact that we can readily tell sounds that “have a pitch” from those whose pitch is far from ideal testifies to the fact that the concept of pitch is real. There is robust math, physics, physiology, and psychology behind pitch and harmonics, and is not something “made up.”

Finally, I’d like to point out that although instruments with definite pitch are close to an “ideal” string or pipe, that doesn’t make those instruments inherently better (or worse)! Some of the beauty of a piano comes from the fact that its strings don’t behave ideally. Forcing instruments to deviate from ideal behavior is often used for expressiveness: the over-blowing of a flute, or the forceful initial attack of a bow on a violin causes an expressive “growl” or “bite”. Also, some instruments (for other reasons not yet explained), such as bells, have less definite pitches, but are considered by many to be quite beautiful. So while pitch is directly associated with the ideal harmonic series, aesthetic and artistic desirability isn’t necessarily.

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