#2: Harmonics: the Building Blocks of Pitch

In the previous post, A Fresh Start: Experience Harmony, we focused on the sensations of combining pitches together. Before we delve deeper into understanding these sensations, we need to gather some experience and knowledge of pitch. Harmony, after all, is about combining two or more pitches simultaneously.

What is pitch?

I find it rather difficult to define what pitch “is” without getting rather technical. Yet we all know it when we hear it. For example, here are a few common musical sounds. Which ones are pitched?

Not too hard, right? The piano (the first one) and the cello (the last one) had pitches, while the snare drum (the middle one) did not. “Why”, however, is the critical question!

It turns out that pitched sounds are highly repetitious, while unpitched sounds are not. That is, when the violin is playing a note, the wood of the instrument vibrates up and down in a particular pattern that imparts the sound of violin. How fast this pattern repeats corresponds with the pitch being played.

Here’s a violin playing a low note (G3*, the lowest note on the violin), and a higher note (D5).

Playing a note on a violin causes the string to vibrate, which in turn causes the wood of its body to vibrate, which in turn causes the air to vibrate, which our ears then hear. As a way to visualize this, let’s diagram how the air moves—for now, you can think of this as how the wood of the violin moves up and down. It might look something like this:

Diagram of sound waveform, 1/100th of a second of G3 and D5 as played on violin.
The purple segments show where the waveform repeats. (Note that the repetitions continue beyond the left and right edges of this diagram.)

The diagram shows the vibrations over the course of one hundredth (!) of a second. Notice how the vibrations repeat only twice in this time period for the G3, but six times for the D5—the pattern formed by the higher pitch repeats faster than the lower pitch. Also see how organized the motion is—practically the same motion repeats over and over again.

Let’s compare that to the unpitched snare sound:

Diagram of the sound wave made by a snare drum. Notice this waveform does not repeat, unlike the violin waveform above.

The motion (this time of the drum head) here is very chaotic—there’s very little apparent organization to this sound, which is why we don’t hear the snare drum as playing a pitch.

Harmonics

When sounds have a definite pitch, it turns out that the vibration is very highly organized. We can break the pitched sound into individual components called harmonics. Let’s take a listen.

Here’s a bassoon and a tuba playing C2 (the C two octaves below Middle C):

A bassoon (top) and a tuba (bottom) both playing C2 (the C two octaves below Middle C).

Notice that while the bassoon and tuba sound similar, you can still tell them apart. Now let’s take the above sounds and isolate the harmonics that make up the sound. We’ll listen to each of the first 16 harmonics of the same bassoon note and tuba note:

The first 16 harmonics extracted from the above recordings: first the bassoon (top) then the tuba (bottom).

Notice how the harmonics sound simple—like an organ or a flute without any color. They certainly don’t sound anything like a bassoon or tuba! Yet, if we play these harmonics simultaneously, you hear a bassoon in one case, and a tuba in the other.

Harmonics are like Legos in one sense: each harmonic is basically the same (a simple tone that has a definite pitch, but doesn’t have a particular color or timbre), yet groups of them can be assembled to sound like a violin, or a bassoon, or a tuba, or a tin whistle. When isolated, an individual harmonic of any instrument playing the same pitch will sound the same—the volumes of each harmonic might differ, but otherwise, they’ll be identical.

Let’s listen to the bassoon and tuba harmonics again.

First 16 harmonics of the bassoon (top) and the tuba (bottom) playing C2.

Also notice that harmonics that make up the pitch C2 for the bassoon and tuba both follow the same melodic pattern! This pattern is known as the harmonic series. We can notate the harmonic series like so:

The first 16 harmonics of C2. Notice that harmonics 11 and 13 can’t be accurately represented in music notation, nor do they correspond to notes on the piano. Harmonic 11 is between F and F#, while harmonic 13 is between G and A-flat. The B-flats are also noticeably flatter than what the piano plays.
A synthesizer playing C2, followed by the first 16 harmonics of the pitch.

Technically, the harmonic series keeps going forever, but for our purposes, we only need to worry about the first 16 harmonics or so. Spend some time listening to the harmonic series, and get used to its unique sound.

Next Steps

Harmony is all about understanding how the harmonics of one note interact with the harmonics of another. Being familiar with the harmonic series will help immensely. In fact, all of the sensations you experienced while singing against the drone from the A Fresh Start: Experience Harmony post are explained by how harmonics interact. That is what we’ll study next.

In the meantime, spend some time committing this “melody” of the harmonic series to memory—it will be worth it!

> NEXT: #3: Harmonics: Where Do They Come From?
< PREVIOUS: #1: A Fresh Start: Experience Harmony


For Further Study

For reference and study, here is the harmonic series for the pitches C2, D2, E2, F2, G2, A2, B2, and C3. First, you will hear a synthesizer play the note, then you will hear each of the first 16 harmonics of that note.

A synthesizer playing C2, followed by the first 16 harmonics of C2.
A synthesizer playing D2, followed by the first 16 harmonics of D2.
A synthesizer playing E2, followed by the first 16 harmonics of E2.
A synthesizer playing F2, followed by the first 16 harmonics of F2.
A synthesizer playing G2, followed by the first 16 harmonics of G2.
A synthesizer playing A2, followed by the first 16 harmonics of A2.
A synthesizer playing B2, followed by the first 16 harmonics of B2.
A synthesizer playing C3, followed by the first 16 harmonics of C3.

(A footnote on how pitches are named)

*Throughout this site, I’ll use the convention that Middle C is called “C4”. (The “4” in “C4” is pretty arbitrary—it probably got this particular number because this C starts the fourth octave on a standard piano keyboard.) Octaves are numbered with respect to C, so C4, D4, E4, F4, G4, A4, B4, C5 is the C Major scale starting on Middle C. The lowest note on the violin is G3 (the first G below Middle C). The lowest note on the cello is C2. The lowest note on the piano is A0. The A that we typically tune to (“A440”) is A4, the first A above Middle C, and nowadays we tune A4 to 440 Hz.

Note that classical musicians and scientists use the standard that Middle C is called “C4”. But MIDI most often calls Middle C “C3”, which is very confusing! So be aware that there are multiple standards. Be aware that you need to find out which standard is being used when you read something that refers to pitches using the “Letter-Name-plus-Octave” numbering system. If they don’t say, often it’s sources that don’t know better (i.e. don’t know that there is more than one standard way to number octaves), and if they are non-classical or non-scientific, they probably use the MIDI “standard” where C3 is Middle C. To make things even more confusing, MIDI does not actually define Middle C to be C3—MIDI defines note number 60 to be Middle C—the octave numbering is not part of the MIDI standard, it’s just common for MIDI software and MIDI users to prefer Middle C being called “C3”, but even that is not universal!

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