The Harmonic Series and Timbre

In the previous article in this series, we covered some simple basics about sound: frequencies, cycles, and Hertz. And, we began speaking of overtones.

What Are Overtones?

It's important to understand you rarely hear pure single frequencies (called sine waves). Instead, there is a group of other freqeuncies, called overtones, which ring above a single low frequency called the fundamental.

Without overtones, instruments would sound pretty boring. Listen to this C major scale with sine waves and no overtones:

 

That's pretty uninteresting and one-dimensional, isn't it?

Musical instruments create sounds which are a complex mixture of many frequencies which are a result of something called the harmonic series.

What is the Harmonic Series?

The harmonic series is a simple mathematical sequence of multiples:
n x 1, n x 2, n x 3, and so on.

What does that mean for notes we play in music? Well, let's return to our example of open E on the bass guitar.

Open E rings at 41Hz (41.2034 more precisely). The note is named E after the fundamental E (41Hz) which we hear as the lowest frequency.

When you pluck the open E string, the string vibrates not only along its whole length from one end to the other, but also at many fractions of its length. These fractions are the harmonic series.

Let's do a little math. Don't get scared...it's easy enough that even I can do it!

When you play a single open E note you hear these frequencies all at once:

HarmonicFrequencyPitchInterval

41Hz x 1

41Hz

E

-

41Hz x 2

82Hz

E

Octave

41Hz x 3

123Hz

B

Fifth (+ 1 Octave)

41Hz x 4

164Hz

E

Two Octaves

41Hz x 5

205Hz

G# (slightly flat)

~ Third (+ 2 Octaves)

41Hz x 6

246Hz

B

Fifth (+2 Octaves)

41Hz x 7

287Hz

D (quite flat)

~ Flat 7th (+ 2 Octaves)

The first harmonic, called the fundamental, is the lowest, and it's why we name the note E to begin with. You certainly wouldn't want to name a single note EEBEG#BD, would you?

All of the harmonics above the fundamental are called overtones.

The lowest overtones (2, 3 and 4) are octaves of E and the fifth B. Roots and fifths are all very consonant sounding overtones. Even mother nature plays roots and fifths in the low end!

As you continue up the series, you get a major third (G#) which is slightly flat. Then another fifth (B), then a flat 7th (D). The 7th harmonic is quite out-of-tune to our ears and tuning system.

[Note: If you don't know what intervals are, see my lesson on basic intervals on bass.]

Something I've always found interesting is the notes of the harmonic series add up to a dominant 7th chord—Root, 3rd, 5th, and flat 7th. The dominant 7th is the chord I'm always telling students is the king of all chords, and this is one of many reasons why.

There are many more harmonics above these seven. The concept is the same. You could keep heading up the series infinitely, but the higher you go it becomes less and less audible.

Visual Analysis of Open E

If you look at an analysis of an open E note on bass guitar, you can see the peaks of each harmonic and how they continue and fade as they ascend:

Harmonics of the open E

The Missing Fundamental Effect

There's an interesting phenomenon in the world of psychoacoustics—especially for bass players—called the missing fundamental effect.

When you hear a tone, if the fundamental is missing, you can still "hear" the fundamental. That is, your brain somehow fills in the missing low fundamental frequency. Scientists still don't know why this happens. Just hearing the overtones gives your brain enough information to imply and imagine the absent fundamental.

Why is this interesting for bass players? Some speakers or amps may not be capable of producing really low frequencies. As a result, the low fundamentals of bass notes may vanish. But, because you can hear the overtones, you still perceive the right low note. Even though you're hearing harmonics one and two octaves above, you still imagine the low fundamental. You don't hear the note an octave higher.

Bizarre, isn't it? This is why you might still hear a low B on a 5-string bass through a tiny practice amp which can't produce frequencies that low. It just won't sound that great with the missing frequencies.

Musical Timbre

Now that you have an idea of the complexity that makes up a single note due to the harmonic series, we can understand tone better.

You probably never thought about why you can tell different instruments apart from one another. Why, for example, can you tell a piano apart from a bass guitar even when they play the exact same note?

Listen to these two notes:

 

Which instrument did you hear first? Piano or bass? You can tell the difference, right?

The reason you can distinguish them is each instrument has a unique tone or sound quality. If they didn't, the world would only need one instrument.

This unique tone is called timbre. It has the unexpected English pronunciation TAM-burr. Timbre is the unique color or character of a tone.

Instrument Timbre

An instrument's timbre comes from the unique mix of overtones it produces. Some overtones are more pronounced and others less. Some overtones fade fast and others linger. Some are louder than others. This is why one note on one instrument sounds different than that exact same note on another instrument.

The design and construction of an instrument, as well as how it is played, influences its mix of overtones. For example, when we compare grand piano to electric bass, the piano has longer strings struck by a felt hammer while the bass has shorter strings plucked with a finger or pick. Those are just two of the differences which help create their different timbres. Even different basses will have different timbres due to their different materials and design.

You can start to get a sense of how complex an instrument's tone can get. Every material, every link in the chain, has an impact on the tone produced.

In Summary

In these first two articles of the series, we've discussed that sound is all about vibrations. Things vibrate at different frequencies, or speeds. Faster frequencies produce higher pitches while slower frequencies produce lower pitches.

It's rare to hear pure single frequencies. Instead we hear a complex mix of the fundamental tone and its overtones. This complex mix of frequencies comes from the harmonic series.

The proportion of an instrument's overtones is what gives it its color, or timbre.

Now that you understand some basics of sound, frequencies, the harmonic series, and the origin of musical tone, let's look at where the bass fits in the human hearing frequency range.

 

In the next upcoming article, we'll discuss the frequency range of the bass and other instruments.