Dominant Frequency

The dominant frequency is the frequency of the peak of greatest amplitude in the spectrum of a sound. It must not be confused with the fundamental frequency, which sets the perceived pitch: the two often coincide, but not always.

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How to measure it in class

FizziQ offers side by side a “dominant frequency” instrument, a “fundamental frequency” instrument and a sound spectrum. Comparing the three on the same sound is the most direct way to understand the difference.

Steps:

Open the Dominant frequency instrument and check its operation on a pure tone produced by FizziQ’s synthesizer, for example 440 Hz: both instruments display the same value
Switch to the sound spectrum and play a sustained note on a real instrument (flute, guitar, singing voice): identify the comb of regularly spaced peaks
Record the frequency of the tallest peak (the dominant) and that of the first peak of the comb (the fundamental), then compare them
Repeat with a low male voice, a bell sound, or a sound played through a small phone loudspeaker: the dominant is then often no longer the first peak
Record the observed difference and the dominant/fundamental ratio, which must be a whole number for a harmonic sound

Scientific activities on this topic

Calculate the frequency of a sound by studying the oscillogram of a tuning fork sound (see the activity)

Timbre and harmonics of a musical instrument - compare the spectra of several instruments playing the same note and identify where the dominant peak lies
Speed of sound measured in a test tube - identify the dominant frequency of an air column and deduce the speed of sound from it

Learn more

A real sound is a sum of frequencies

Except in textbook cases (tuning fork, synthesizer), a sound is never monochromatic: it is composed of a multitude of components. Fourier analysis consists of decomposing the signal into a sum of sinusoids and representing the amplitude of each as a function of its frequency - this is the spectrum. When the frequencies present are all whole multiples of a single frequency f₀, the sound is said to be harmonic; the spectrum then looks like a regular comb.

Dominant and fundamental: two different definitions

The fundamental is read in the spacing of the peaks: in a harmonic spectrum, the constant gap between two neighboring peaks equals f₀, even if the peak located at f₀ is weak or absent. The dominant is read in the height of the peaks: it is the position of the tallest peak. Nothing requires the tallest peak to be the first one. On a singing voice, on an oboe or on a trumpet, the dominant is frequently the harmonic of rank 2, 3 or higher.

The case of the missing fundamental

This is the most spectacular phenomenon, and the heart of the distinction. If the component at f₀ of a harmonic sound is completely removed, leaving only 2f₀, 3f₀, 4f₀…, the ear continues to perceive the note f₀. The brain reconstructs the pitch from the periodicity of the whole set of harmonics, not from the presence of a peak at f₀. This is called the missing fundamental or virtual pitch.

This is not a laboratory curiosity. A phone loudspeaker hardly goes below 300 to 500 Hz: it reproduces practically nothing of the fundamental of a male voice (approximately 110 Hz), and yet the voice remains perfectly recognizable and at the right pitch. Bells provide another example: their “hum” partial and their inharmonic spectrum mean that the perceived note - the strike note - often corresponds to no peak in the spectrum. In both cases, the dominant frequency measured by the device and the note heard by the ear are markedly different.

Inharmonic sounds: the fundamental no longer exists

For a plate, a drum, a bell or a noise, the natural frequencies are not in whole-number ratios. There is then no fundamental frequency in the strict sense, and the notion of pitch becomes blurry. The dominant frequency, however, always remains defined: there is always a maximum in the spectrum. This is precisely why it is useful in vibration analysis, environmental acoustics or mechanical fault detection, where one is not looking for a note but for a signature.

How to see a sound wave (FizziQ blog) compares oscillogram and spectrum on everyday sounds.

Formula

Decomposition of a periodic sound of frequency f₀ into a Fourier series:

s(t) = Σ Aₙ sin(2π n f₀ t + φₙ), with n = 1, 2, 3…

Dominant frequency: the frequency f_d at which the amplitude of the spectrum is maximal:

f_d = n_max × f₀ where A(n_max) = max(Aₙ) for a harmonic sound

Frequency resolution of a Fourier analysis over N samples:

Δf = fₛ / N

Rank of the dominant harmonic:

n = f_d / f₀ (a whole number for a harmonic sound)

where:

s(t): sound signal (arbitrary unit)
f₀: fundamental frequency (Hz)
Aₙ: amplitude of the harmonic of rank n
f_d: dominant frequency (Hz)
fₛ: sampling frequency (Hz), typically 44,100 Hz
N: number of samples in the analysis window
Δf: frequency resolution (Hz)

Application examples

Tuning fork at 440 Hz: the sound is nearly pure. The dominant and the fundamental both equal 440 Hz, and the spectrum shows only one peak. This is the reference case where the distinction cannot be seen.
Male voice on the phone: fundamental around 110 Hz, but the passband starts around 300 Hz. The measured dominant frequency is often located around 300 to 600 Hz, that is harmonic 3 to 5, while the perceived pitch remains 110 Hz.
Clarinet: the spectrum is rich in odd harmonics. In certain registers, harmonic 3 exceeds the fundamental and becomes the dominant.
Church bell: inharmonic spectrum. The perceived note can lie an octave below the most intense partial, and correspond to no measured peak.
Noise from an engine or a fan: the dominant frequency reveals the rotation speed. A fan at 1,500 rpm fitted with 7 blades produces a dominant peak around 1,500/60 × 7 = 175 Hz.
Helmholtz resonator: by blowing across the neck of a bottle, you obtain an almost pure sound whose dominant directly gives the resonance frequency of the cavity.

FAQ

Q: Are dominant frequency and fundamental frequency the same thing? A: No, even though they often coincide. The fundamental is the lowest frequency of the harmonic series, the one that gives the perceived pitch. The dominant is the frequency of the most intense peak in the spectrum, which can be any harmonic. On a pure tone, the two are identical; on a voice or a brass instrument, they regularly differ.

Q: Why does FizziQ sometimes display a dominant frequency that jumps around? A: Because the relative amplitudes of the harmonics evolve during the note, especially at the attack. When two harmonics have similar amplitudes, the maximum switches from one to the other and the display flips. This is not an error: the spectrum really is changing.

Q: Can we hear a note whose frequency is not present in the sound? A: Yes. This is the missing fundamental phenomenon: the ear deduces the pitch from the regular spacing of the harmonics. A sound composed of 400, 600 and 800 Hz is heard at 200 Hz, even though no energy is emitted at that frequency.

Q: How precise is the measurement? A: It is set by the resolution Δf = fₛ/N of the Fourier analysis, on the order of 10 Hz with common settings. That is excellent for a high-pitched sound and poor for a low-pitched one. To refine in the low range, the analysis window must be lengthened, at the cost of a slower response.

Q: What kind of sound should be measured to get clean results? A: A sustained, stable sound, loud enough relative to the ambient noise, recorded a few tens of centimeters from the microphone. A brief sound or a sound gliding in pitch does not produce a clean peak.

Fundamental Frequency - Harmonic Sound - Spectral Analysis - Fourier Transform (FFT) - Timbre - Pitch (Sound Height) - Pure Tone - Spectrogram