Overtones and harmonics are generally discussed in relationship to sound sources. These two concepts are often confused with each other and sometimes used interchangeably.
This is no surprise since in certain situations, they end up referring to the same set of frequencies. However, while it is possible for harmonics to be overtones and for overtones to be harmonics, it is also possible to have harmonics that are not overtones, and overtones that are not harmonics.
Wave Speed, Wavelength and Frequency
Before discussing harmonics and overtones, it is important to understand the fundamentals of a wave.
Waves are a disturbance in a medium, which propagates from one place to another via oscillations of points in the medium. Sound is just one example of this but so are ocean waves, waves on a string, etc.
The wave length is the distance between successive wave peaks. The wave frequency is the number of cycles per second of the wave. And the wave speed is the product of the wavelength and frequency.
If a propagating disturbance is confined within a medium, it may reflect back and interfere with itself. At certain frequencies, this creates a sustained standing wave. This happens when you pluck a guitar string, blow into a whistle or even drop a wrench on the floor – the impact of the drop causes the wrench to “ding” at a certain frequency as it vibrates briefly upon impact.
The frequencies at which such standing waves can occur are called resonant frequencies, and the values of these frequencies for a given medium depend on properties of that medium. For example, the frequency at which a standing wave on a string is created depends on the mass density of the string, the tension of the string and the length of the string.
As you’ll see in the next section, most objects have several different frequencies at which they might vibrate naturally, and those different frequencies are often related to each other and to the geometry of the object itself.
What Is an Overtone?
A resonant frequency is a natural frequency of vibration of an object. It is the frequency at which something vibrates creating a standing wave pattern. For any given object, there are usually several frequencies at which this occurs. The lowest such frequency is called the fundamental frequency and is often denoted as f1.
An overtone is the name given to any resonant frequency above the fundamental frequency or fundamental tone.
The list of successive overtones for an object is called the overtone series. The first overtone as well as all subsequent overtones in the series may or may not be an integer multiple of the fundamental. Sometimes the relationship is that simple, and other times it is more complex, depending on the properties and geometry of the vibrating object.
For example, on a circular membrane such as a drum head, there are overtones at 1.59f1, 2.14f1, 2.30f1, 2.65f1, 2.92f1 and many other values. These overtones occur at frequencies for which a two-dimensional standing wave can occur on the membrane. As you might suspect, the mathematics for deriving these values is a lot less straightforward than for determining standing wave modes on a string!
What Are Harmonics?
Harmonic frequencies are whole number multiples of the fundamental frequency, or the lowest frequency of vibration.
Consider a vibrating string. The modes of vibration are all multiples of the fundamental and are related to the string length and wave velocity. Higher frequencies are found via the relationship
where L is the string length.
From this you get the harmonic series. The second harmonic f2 = 2f1 and the third harmonic f3 = 3f1 and so on. Also note that the wave speed – the product of the wavelength and frequency – is the same for all values of n.
In this particular example with the string, all overtones are harmonics, and all harmonics are overtones. However, this is not always the case, as seen in the drum head example, and as you’ll see in the next section as well.
Difference Between Overtones and Harmonics
As discussed previously, harmonics are integer multiples of the fundamental frequency. At these frequencies, the object may or may not experience resonance. In contrast, overtones are any frequency at which resonance occurs above the fundamental. These may happen at harmonics only, or at specific harmonics only or at other values entirely.
Consider the example of standing sound waves in an open pipe (or the vibrating string): In this case, harmonics and overtones are the same. With a closed pipe, however, overtones only occur at odd harmonics.
On a rectangular or circular membrane such as a drum head, you get a little bit of everything. On a rectangular membrane, some of the overtones are also harmonics, but some are not.
For example, on a rectangular membrane with a length 1.41 times its width, the overtones occur at 1.41f1, 1.73f1, 2.00f1, 2.38f1, 2.71f1, 3.00f1, 3.37f1 and so on. On a circular membrane, most or all of the harmonics do not end up being overtones.
Vibrational modes of a drum head are examples of non-harmonic or inharmonic overtones. These also occur in cymbals and other percussion instruments.
Musical instruments including wind instruments, brass instruments, string instruments and others. They provide examples of applications of resonance and the distinction between overtones and harmonics.
Certain instruments tend to make notes at harmonics, others at odd harmonics, and others have inharmonic overtones. By using different keys on a piano, different strings on a guitar or changing the fingering on a flute, the possible overtones and harmonics change as well.
This is also why it is important to tune certain instruments periodically. The note a plucked guitar string plays depends on the mass density of the string but also the tension. After playing for a while, the string may become slightly stretched, and the tension may be changed. By readjusting the tension, the correct fundamental vibration frequency can be restored.
Timbre and Sound Quality
Timbre is the perceived sound quality of a note in music. While you might play the same note on a guitar as on a piano, your ear can tell the difference. Why is that the case even though the frequency is the same? The answer has to do with overtones.
When the guitar string is plucked, producing a given note by vibrating at its fundamental frequency, it is simultaneously vibrating at the overtone values as well but with much smaller amplitude (lower volume). Imagine a sign wave that when you zoom in on it appears “squiggly” or lined with a much smaller sign curve of its own.
The same happens when the piano key is played, and the differences in physical properties of these instruments lend to different combinations and relative strengths of overtones, creating the different timbre or sound quality that allows you to distinguish between the two instruments.
Other factors which can also affect note quality are attack, decay, sustain and release time. As a note is played the amplitude jumps up to a peak, lowers to a constant level for a while, then drops to zero when the note ends.
Attack is the time between when the note has begun to be played to the peak amplitude. Decay is the time between peak amplitude and the sustained amplitude the note is played at. Sustain is the time during which the note is played at a constant amplitude. Release is the time taken to go from the sustained amplitude to zero when the note ends.