Complex Sound

A complex sound combines several distinct frequencies, unlike a pure tone which contains only one. Almost all everyday sounds (voices, instruments, engines) are complex, and the distribution of their components determines the timbre.

Discover FizziQ

How to measure it in class

FizziQ’s frequency spectrum decomposes a recorded sound into its components and shows directly that an ordinary sound contains several of them.

Steps:

Open FizziQ’s frequency spectrum and emit a pure tone at 440 Hz with the synthesizer of a second device: note the single peak, which serves as a reference.
Sing the vowel “a” at the same pitch in front of the microphone and compare: several peaks appear even though the perceived pitch is unchanged.
Record the frequency of each peak and check whether they are integer multiples of the lowest one.
Repeat with the sound of a bell, a drum or a struck glass: the peaks are no longer in integer ratios.
Compare the oscillograms: the pure tone gives a sinusoid, the complex sound a periodic pattern of arbitrary shape.

Scientific activities on this topic

Smartphones or tablets are the ideal instrument for studying complex sounds and understanding this concept. For example, you can do the following activities:

Learn more

A complex sound is a sound that is not simply a pure sinusoid (a single frequency), but rather a combination of several different frequencies. Complex sounds are generally produced by natural or artificial sound sources that generate a spectrum of varied frequencies. They differ from pure tones, which are composed of a single frequency.

The spectrum of a complex sound is the graphical representation of the frequency components of a sound signal in the frequency domain. It shows how the energy or amplitude of the signal is distributed according to the different frequencies present in the sound. In other words, the spectrum of a complex sound reveals which frequencies are present and to what extent they contribute to the composition of the sound.

The spectrum of a complex sound can be obtained by applying a Fourier transform, generally the discrete Fourier transform (DFT) or the fast Fourier transform (FFT), to the audio recording of the sound. Once the spectrum is obtained, it is possible to analyze the different frequency components and characterize the timbre and sound properties of the sound, as well as use it for tasks such as noise suppression, audio compression, equalization, and many others.

Among complex sounds, we can distinguish:

Harmonic sounds, which are composed of a set of harmonics that are integer multiples of a fundamental frequency. In other words, the harmonics of a sound are frequencies that are integer multiples of the lowest frequency of the sound, called the fundamental frequency. Harmonic sounds are commonly associated with musical instruments and sound sources that produce a rich and tonal sound.

Noises, which are non-musical sounds, generally characterized by their unpleasant, irregular, and random quality. Unlike musical sounds, which are generally organized and have a harmonious frequency structure, noises do not follow a specific musical pattern and can be perceived as disturbing, annoying, or undesirable. For example, white or pink noise.

Formula

A periodic complex sound decomposes into a sum of sinusoids, according to Fourier’s theorem:

s(t) = A₁ sin(2π f₀ t + φ₁) + A₂ sin(2π × 2f₀ t + φ₂) + A₃ sin(2π × 3f₀ t + φ₃) + …

where:

s(t): sound signal at time t (Pa)
f₀: fundamental frequency, equal to the inverse of the period of the signal (Hz)
Aₙ: amplitude of the component of rank n (Pa)
φₙ: phase of the component of rank n (rad)

The fundamental frequency is related to the period of the signal:

f₀ = 1 / T

where:

T: period of the complex sound, readable on the oscillogram (s)
f₀: fundamental frequency, which sets the perceived pitch (Hz)

Application examples

The vowel “a” sung at 200 Hz contains components at 200, 400, 600, 800 Hz… reinforced by the resonances of the mouth around 800 Hz and 1200 Hz.
A plucked guitar string emits a complex sound whose components are practically integer multiples of the fundamental.
The sound of a bell is complex but non-harmonic: its components are not in integer ratios, hence its metallic tone.
The noise of a car engine contains a component linked to the rotation speed, for example 50 Hz at 3000 rpm, plus many additional components.
White noise is an extreme case of complex sound: its energy is spread over all frequencies, with no perceptible pitch.
Sonar and ultrasound imaging exploit the spectral analysis of reflected complex sounds to identify the media they pass through.

FAQ

Q: Are almost all sounds complex? A: Yes. The pure tone is an ideal case, produced essentially by electronic generators and approximated by the tuning fork. Voices, instruments, engines and street noises are all complex sounds.

Q: How is pitch perceived in a complex sound? A: It corresponds to the fundamental frequency, that is, to the period of the complete signal. The ear identifies it even if the fundamental is weak, or even absent from the spectrum: this is the missing fundamental phenomenon, exploited by small loudspeakers.

Q: Are all complex sounds harmonic? A: No. Strings and air columns give quasi-harmonic spectra. Bells, drums and plates give inharmonic spectra: we hear a rich tone, but the pitch is blurred.

Q: How do we obtain the spectrum of a sound? A: With a Fourier transform, computed in practice by the FFT algorithm. The device cuts the signal into slices of a few tens of milliseconds and calculates for each one the amplitude of each frequency.

Q: Are a complex sound and a noise the same thing? A: A noise is a complex sound whose spectrum is continuous and without periodic structure. A musical sound is a complex sound whose spectrum consists of well-separated peaks.

Pure Tone - Harmonic Sound - Timbre - Fundamental Frequency - Fourier Transform (FFT) - White Noise - Resonance Frequency - Pitch (Sound Height)