Harmonic Sound

A harmonic sound is a sound that has a frequency that is an integer multiple of the fundamental frequency of a sound. For example, if the fundamental frequency of a sound is 100 Hz, then the frequencies of 200 Hz, 300 Hz, 400 Hz, etc. would be harmonics of that sound.

Discover FizziQ

How to measure it in class

FizziQ’s frequency spectrum displays the position of each component of a sound, which makes it possible to check whether they are in integer ratios.

Steps:

Open FizziQ’s frequency spectrum and have a sustained note played by a string instrument or by the voice.
Record the frequency of the lowest peak: this is the fundamental frequency f₀, the harmonic of rank 1.
Record the frequencies of the following peaks and calculate their ratio to f₀. If they equal 2, 3, 4… the spectrum is harmonic.
Also note the relative amplitude of each peak: this is what varies from one instrument to another.
Repeat with a bell, a struck glass or a drum: the ratios are no longer integers, the spectrum is inharmonic.
Compare the two sounds by ear: the first has a clear pitch, the second a pitch that is difficult to name.

Scientific activities on this topic

Several activities allow you to identify the harmonics of a sound and to distinguish harmonic from inharmonic spectra.

Learn more

Harmonic sounds are often produced by sound sources that vibrate regularly, such as violin strings or a singer’s vocal cords. They are also produced by synthetic sound sources, such as electronic keyboards and sound synthesis software.

A sound with a harmonic spectrum is recognized above all by its clear pitch: the ear assigns it a precise note without hesitation. An inharmonic sound remains perfectly audible and can be musical, but its pitch is more difficult to identify.

A harmonic sound consists of a certain number of harmonics that will determine the specific timbre of the sound. They can be identified through the frequency spectrum.

A harmonic is a frequency component that is an integer multiple of the fundamental frequency of a sound. The harmonic of rank n has frequency n·f₀. Beware of the numbering: by convention, the fundamental itself is the harmonic of rank 1. If f₀ = 100 Hz, the harmonic of rank 2 is at 200 Hz, the one of rank 3 at 300 Hz, and so on. The expression “first harmonic” is ambiguous and a frequent source of errors: it is better to say “harmonic of rank 2” than “first harmonic” to designate the component at 200 Hz. The harmonics and their relative amplitudes contribute to the timbre of the sound.

Harmonic and partial are not synonyms. A partial is any component of the spectrum, with no condition on its frequency. A harmonic is a partial whose frequency is an integer multiple of f₀. Strings and sound pipes give quasi-harmonic spectra: all their partials are, more or less, harmonics. Bells, drums and vibrating plates, on the contrary, give inharmonic spectra whose partials are not in integer ratios; this is what explains their impression of blurred pitch, difficult to name.

Among the specific frequencies of these harmonics, two particular frequencies can be cited:

The fundamental frequency is the lowest frequency of a complex sound. It determines the perceived pitch of the sound, that is, whether the sound is low or high. The fundamental frequency is often the strongest frequency component in a sound, but it is usually accompanied by harmonics that give the sound its texture.
The dominant frequency is the frequency that predominates in a complex sound in terms of intensity or amplitude. It is the frequency component that is most pronounced and contributes most to the perception of pitch and timbre of the sound. The dominant frequency can be the fundamental frequency of the sound, but in some cases, it may correspond to one of the stronger harmonics.

Formula

The frequency of the harmonic of rank n is deduced from the fundamental frequency:

fₙ = n × f₀

where:

fₙ: frequency of the harmonic of rank n (Hz)
n: rank of the harmonic, a non-zero natural number, n = 1 corresponding to the fundamental
f₀: fundamental frequency of the sound (Hz)

For a string of length L fixed at both ends, the frequencies of the normal modes are:

fₙ = n × v / (2L)

where:

L: length of the vibrating string (m)
v: speed of the wave along the string (m/s)
n: rank of the mode

Application examples

A guitar string tuned to the A at 110 Hz emits harmonics at 220, 330, 440, 550 Hz…
A male voice whose fundamental is at 120 Hz produces harmonics up to several kilohertz; they are what carry the information of the vowels.
A clarinet, a pipe closed at one end, emits essentially the odd-rank harmonics: f₀, 3f₀, 5f₀, which gives it its characteristic hollow tone.
A guitarist who lightly touches the string at its midpoint suppresses the fundamental and makes the harmonic of rank 2 sound: the resulting note sounds one octave higher.
The sound of a bell contains partials in non-integer ratios, often close to 0.5 - 1 - 1.2 - 1.5 - 2 times the nominal note: it is not harmonic.
A vibrating drumhead has normal modes in non-integer ratios: no precise pitch emerges from it.

FAQ

Q: Is the fundamental a harmonic? A: Yes, it is the harmonic of rank 1. That is why the expression “first harmonic” should be avoided to designate the component at 2f₀: it maintains a very widespread confusion. Say “harmonic of rank 2” instead.

Q: What is the difference between a harmonic and a partial? A: A partial is any component of the spectrum. A harmonic is a partial whose frequency is an exact integer multiple of the fundamental. All harmonics are partials; the reverse is false.

Q: Why don’t bells give a clear note? A: Their partials are not in integer ratios. The ear cannot group them around a single fundamental, so the perceived pitch remains blurred, but the sound remains rich and resonant.

Q: Do harmonics change the pitch of the note? A: No. The pitch is set by the fundamental frequency. Harmonics modify the timbre, not the perceived note.

Q: Up to what rank can harmonics be observed with a smartphone? A: The microphone and spectral analysis of a smartphone commonly allow you to identify about ten harmonics on a sustained note, as long as they remain within the audible band and above the background noise.

Pure Tone - Complex Sound - Timbre - Fundamental Frequency - Fourier Transform (FFT) - Pitch (Sound Height)