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2.1. The Overtone Series

Go ahead and open up Audacity. On the "Generate" drop-down menu, select the "Tone..." option. A dialog box like this one pops up:

Generate Wave Dialog Box

After you match these parameters, click okay. When you press play, you should hear a very pure and stable sound. When you zoom in on the waveform, a clean-looking sine wave like this one is observed:

Sine Wave in Audacity

This is in contrast to what we observe when a real-life musical instrument is recorded at the same pitch. For example, this is what the waveform looks like when I play a stable A 440 pitch on my trombone:

Trombone Sound Wave

Now some of the differences between the sine wave and the trombone waveform can be explained by the natural failings of microphones and ambient noise in the room. However, there has to be some secret that explains why the trombone wave shape is this odd. Our secret lies in something called the overtone series. To explain this, let's do an experiment.

2.1.1. Building a New Level of Sound

In the same Audacity file you have open, go ahead and click the "Add New>Audio Track" option from the "Tracks" drop-down menu several times, until you have ten or so mono tracks on your screen. If you are a Windows user and a lover of keyboard shortcuts, Ctrl+Shift+N will save you a bunch of time. On each one of these tracks, generate a sine wave that is a whole-number multiple of 440 Hz. For example, in track two generate a sine wave at 880 Hz, track three is at 1320 Hz, and so on. We call 440 Hz a fundamental, because it is what all the other pitches are based off of. Press play and see what your project sounds like after you add each new wave.

By the time you've filled your tracks with waves, the sounds your computer is spitting out will have gotten nasally and gross-sounding. Notice, however, that it sounds like only one tone is generated, not ten separate tones. Our brains are trained to combine a sinusoid series into one coherent sound, rather than perceiving each individual wave as an independent piece of information. This fact is the first piece to solving our trombone waveform riddle.

Notice on the left of the plots of our waves is a box full of information and options, and in it is a slider from "-" to "+", like this:

Plus Minus Slider in Audacity

This slider controls the volume of each of our sinusoids. Mess around with this and make some of the waves louder and softer than others. Try each of these suggestions and compare them to the sound of the original:

  1. Keep the first and lowest-pitched wave where it started, and make each succeeding wave quieter.
  2. Do as in number 1, but mute every odd-numbered track.
  3. Do as in number 1, but mute every even-numbered track.

Notice how each of these changes adds a new character to the sound produced. When you've found a wave with a sound you like, export it to your favorite audio format. Then reopen that file in Audacity and zoom in on that waveform. You should see something entirely new looking - this is what I found:

Combined Waveform in Audacity

Because waveforms add (as discussed in the very first lesson here), destructive and constructive interference has given rise to this new waveform (a waveform that happens to have the same frequency as our initial sine wave). The lesson learned here is that sine waves related to each other proportionally add to create a new tonal color, called timbre. The amplitude of each of the sine waves greatly affects the final sound produced, but leaves the original pitch unchanged. Timbre is a new quality of sound, independent of pitch - it is determined by wave shape, not frequency.

This is exactly the same thing as what is going on in our trombone waveform. In a trombone, the shape of the instrument causes vibrations at set intervals. This produces overtones, which in turn add together to produce the timbre we recognize as a trombone. Some of the intervals have a stronger tendency to vibrate, and thus produce louder overtones.

We've discovered a hidden feature of our universe - a trombone (or any musical instrument) does not just produce one sine wave, it generates a bunch of sine waves, all with proportional frequencies. When these sine waves are added together, a complex waveform, like the one we observe in Audacity, emerges. The series of waves a trombone produces are distinct from the waves a piano produces, and so trombones and pianos sound different from each other. To use our new vocabulary, trombones and pianos have different timbres. Why this happens will be discussed in later sections.

2.1.2. Harmonics vs. Overtones

So far in this article, I have made a deliberate effort to only use the word "overtone" to describe the different pitches that are present in each tone a musical instrument produces. The word overtone refers to any pitch above the fundamental that is perceived as being an integral part of that fundamental's timbre. So far, the only overtones we've studied have been related to the fundamental by the series 1, 2, 3, 4, 5, 6, etc. (\(1 \times 440\), \(12 \times 440\), \(3 \times 440\), etc.). Overtones that follow this pattern are called harmonics, because they follow the harmonic series (\(\frac{1}{1}\), \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), \(\frac{1}{5}\), etc.).

Oftentimes, when physicists talk about musical instruments they call overtones harmonics, because in almost all cases modern musical instruments produce overtones that follow the harmonic series. Instruments are designed explicitly with this in mind, because we perceive the harmonic series as producing more pleasant tone colors.

To clarify, all harmonics are overtones, but not all overtones are harmonics.

But, like everything in science, nothing is as simple as the harmonic series. Trombones, for example, do not naturally follow the harmonic series. If you were to play a metal pipe like a brass instrument, the overtones would follow a different pattern - they would not be harmonics. Idiophones (percussion) are even worse. The ratios that their overtones follow are dependent on the speed of sound in whatever material that happens to be vibrating. Instrument makers shave wood off of xylophones and specifically shape trombones to make the overtones follow the harmonic series, again because simple ratios are more pleasant to our ears.

In the next section, we will discuss how to break complicated waves up and look at all the overtones that they are composed of.

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