Fundamental Frequency

The fundamental frequency, written f₀, is the lowest frequency of the harmonic series of a periodic sound; all the other components are integer multiples of it. It is what determines the perceived pitch of the sound. It is expressed in hertz (Hz) and equals f₀ = 1/T.

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How to measure it in class

FizziQ has a “fundamental frequency” instrument that detects the periodicity of the signal, a “dominant frequency” instrument, and a sound spectrum. Running them together on the same sound is the key experiment.

Steps:

Calibrate the measurement chain on a pure tone produced by FizziQ’s synthesizer, at 440 Hz for example: both instruments should display 440 Hz
Sing a sustained vowel, or play a clear note on an instrument, and record the fundamental frequency displayed
Open the sound spectrum and check that the spectral lines are evenly spaced, the gap between two neighboring lines being equal to f₀
Compare f₀ with the dominant frequency: note the cases where they differ, and calculate the rank n = f_dominant / f₀
Vary a physical parameter (string length, air column length, tension) and plot f₀ as a function of this parameter
Check that an octave indeed corresponds to a doubling of f₀ and a semitone to a factor 2^(1/12) ≈ 1.059

Scientific activities on this topic

The FizziQ app allows you to perform a large number of experiments with a smartphone on the concept of fundamental frequency. You will use the different measurement instruments (fundamental frequency, dominant frequency, oscillogram and sound spectrum), and the sound generation tools (synthesizer and sound library).

Here are 4 activities that can be done in class or at home on the concept of fundamental frequency:

Other activities allow you to go further:

Air columns and pitch - relate the length of a tube to the fundamental frequency it produces
Helmholtz resonance: the sound of a bottle - vary the volume of air and follow the evolution of the note
Chromatic scale and frequencies: measure the frequencies of the twelve semitones and recover the geometric progression
Acoustic beats between 440 and 441 Hz - hear and measure the gap between two very close fundamentals
Spectrogram of bird song - follow the evolution of the fundamental over time
The Shepard auditory illusion - build a sound whose pitch seems to rise indefinitely

Learn more

The fundamental frequency is the lowest frequency produced by the vibration of a sound-producing object, such as a guitar string, an organ pipe, or human vocal cords. This frequency determines the base pitch of a sound that we hear.

When an object vibrates, it does not only produce a sound at its fundamental frequency, but also a series of other higher frequencies called harmonics or partials. These harmonics are integer multiples of the fundamental frequency and contribute to the richness and color of the sound, what we call timbre. For example, although two different instruments may play the same note (the same fundamental frequency), they sound different because of the variety and intensity of their harmonics.

The fundamental frequency differs from the dominant frequency, which is the strongest or most pronounced frequency in a sound spectrum, the one that stands out for its intensity compared to the others. While the fundamental frequency defines the basic musical note, the dominant frequency influences the color or timbre of the sound, as it may be one of the harmonics or another spectral component that stands out the most. This distinction is essential for understanding how sounds are produced and perceived, providing a basis for analyzing music, speech and other complex sounds in an acoustic or technological context.

In music, the theory around the fundamental frequency helps musicians understand how to tune their instruments or harmonize their voices to produce sounds pleasing to the ear. The ability to identify and manipulate the fundamental frequency and its harmonics allows the creation of complex and emotionally interesting compositions, what we call the timbre of an instrument.

The concept of fundamental frequency is also important for audio technology and architectural acoustics. In the design of concert halls or recording studios, knowledge of how sounds interact with the environment can help optimize the clarity and quality of sound. Engineers use this understanding to create spaces where the fundamental frequency and its harmonics can coexist without destructive interference, thus ensuring that music or speech is transmitted with fidelity.

The missing fundamental: pitch without the spectral line

This is the decisive experiment for understanding that fundamental and dominant are two distinct notions. Take a sound composed of 400, 600, 800 and 1,000 Hz. No energy is emitted at 200 Hz - and yet the ear hears a sound with a pitch of 200 Hz. The auditory system does not look for a line at f₀: it detects the periodicity of the whole, and 200 Hz is the greatest common divisor of the frequencies present.

Practical consequence: a smartphone’s loudspeaker reproduces nothing below 300 to 500 Hz. The fundamental of a male voice (about 110 Hz) never comes out of it, and yet the voice keeps its pitch on the phone. Likewise, for a bell, the perceived note - the strike note - often corresponds to no partial actually present in the spectrum. In these situations, a device measuring the dominant frequency will display a value clearly higher than the pitch heard: the machine reads the strongest peak, the ear reconstructs the periodicity.

Where the harmonic series comes from

A string of length L fixed at both ends can only vibrate with standing waves for which an integer number of half-wavelengths fits into L: L = n λ/2. Since v = λf, the natural frequencies are fₙ = n v/(2L). The integer ratio between the modes is therefore not a coincidence: it follows directly from the boundary conditions. The same holds for a pipe open at both ends. A pipe closed at one end, on the other hand, admits only the odd harmonics, with f₀ = v/(4L) - hence its characteristic sound and its “missing” octave. These vibrating systems are part of the science curriculum and of the physics and chemistry specialty in the first years of high school.

Orders of magnitude to know

Male voice: 85 to 180 Hz. Female voice: 165 to 255 Hz. Child’s voice: 250 to 400 Hz. The lowest string of a guitar (E) is at 82 Hz, the highest (E) at 330 Hz. The orchestra’s reference A is at 440 Hz. A piano ranges from 27.5 Hz (A₀) to 4,186 Hz (C₈). The human ear perceives a definite pitch from about 20 Hz to 5 kHz; beyond that, it still hears the sound but can no longer assign a reliable note to it.

Common mistake: confusing fundamental and amplitude

Many students think that a low-pitched sound is “louder” or that the fundamental is necessarily the most intense harmonic. Both ideas are wrong. Pitch (f₀) and intensity are two independent quantities: you can play a very soft low note and a very loud high note. And on a trumpet, an oboe or a loudly sung voice, the most intense harmonic is regularly harmonic 2, 3 or 4, not the fundamental.

Six science experiments with the tone generator (FizziQ blog) shows how to build custom harmonic sounds.

Formula

Relation between fundamental frequency and period of the signal:

f₀ = 1 / T

Frequencies of the harmonic series:

fₙ = n × f₀, with n = 1, 2, 3…

Stretched string of length L fixed at both ends, or pipe open at both ends:

f₀ = v / (2L)

Pipe closed at one end (only odd harmonics exist):

f₀ = v / (4L)

Mersenne’s law, for a string of linear mass density µ stretched by a force T:

f₀ = (1 / 2L) × √(T / µ)

Frequency ratio in equal temperament:

f = f₀ × 2^(k/12), k semitones above f₀

where:

f₀: fundamental frequency (Hz)
T: period of the signal (s) - not to be confused with the tension of the string
fₙ: frequency of the harmonic of rank n (Hz)
L: length of the vibrating system (m)
v: speed of the wave in the medium (m·s⁻¹) - about 340 m·s⁻¹ in air, much more in a stretched string
µ: linear mass density of the string (kg·m⁻¹)

Application examples

Guitar string: a low E string, 65 cm long, sounds at 82 Hz; the speed of the wave in the string is therefore v = 2Lf₀ = 2 × 0.65 × 82 ≈ 107 m·s⁻¹, very different from the 340 m·s⁻¹ of air. Pressing the string at the 12th fret divides L by two and doubles f₀: that is the octave.
Open organ pipe of 1.00 m: f₀ = 340 / (2 × 1.00) = 170 Hz. The same pipe stopped at one end gives f₀ = 340 / 4 = 85 Hz, one octave lower - hence the usefulness of stopped pipes to save space in the bass range.
Tuning an instrument: the tension is adjusted until the beats with the tuning fork disappear. Two sounds at 440 and 441 Hz produce a beat at 1 Hz, perfectly audible.
Speech recognition: analysis of f₀ gives the intonation and the sex of the speaker, while the resonances of the vocal tract (the formants) identify the vowel. These are two independent pieces of information carried by the same signal.
Bird song: a blackbird modulates its fundamental between 1 and 4 kHz in a few tenths of a second; only a spectrogram makes it possible to follow this variation.
Doppler effect: a passing vehicle makes the perceived fundamental drop at the moment of crossing. For a car at 90 km/h, the relative jump reaches about 15%, nearly two semitones.

FAQ

Q: What is the difference between fundamental frequency and dominant frequency? A: The fundamental is the lowest frequency of the harmonic series, the one that sets the note heard. The dominant is the frequency of the most intense peak of the spectrum, which may be a harmonic of higher rank. For a pure tone, the two are equal; for a voice or a brass instrument, they often differ.

Q: Can you hear a note whose frequency is absent from the sound? A: Yes. This is the missing fundamental: the ear deduces the pitch from the regular spacing between the harmonics, not from the presence of a line at f₀. This is what makes a voice recognizable on the phone even though the loudspeaker cannot go low enough to reproduce it.

Q: Do all sounds have a fundamental frequency? A: No. The sound must be periodic and its components must be in integer ratios. White noise, a drum, a bell or an impact have no fundamental; their pitch is undefined or ambiguous. Their dominant frequency, on the other hand, can always be measured.

Q: Why does FizziQ sometimes display double or half the note I am playing? A: This is the classic octave error of pitch detectors. If harmonic 2 is very strong, the algorithm may take it for the fundamental; conversely, a noisy signal can make it find a periodicity twice as long. A more sustained sound, closer to the microphone and in a room with little reverberation, generally corrects the problem.

Q: Do two instruments playing the same note have the same fundamental frequency? A: Yes, exactly the same - that is the definition of a note. What distinguishes them is the distribution of energy among the harmonics, that is, the timbre. A flute is poor in harmonics, an oboe is rich in them, and yet both are indeed playing 440 Hz.

Dominant Frequency - Harmonic Sound - Pitch (Sound Height) - Timbre - Standing Wave - Spectral Analysis - Fourier Transform (FFT) - Resonance Frequency - Acoustic Beat - Doppler Effect