Fundamental frequency

Vibration and standing waves in a string, The fundamental and the first six overtones

The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids (e.g. Fourier series), the fundamental frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as f₀ (or FF), indicating the lowest frequency counting from zero.^[1][2][3] In other contexts, it is more common to abbreviate it as f₁, the first harmonic.^{[4][5][6][7][8]} (The second harmonic is then f₂ = 2⋅f₁, etc. In this context, the zeroth harmonic would be 0 Hz.)

Explanation

All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. The period of a waveform is the T for which the following equation is true:

$x(t)=x(t+T){\text{ for all }}t\in {\mathbb {R}}$

Where x(t) is the value of the waveform at t. This means that this equation and a definition of the waveforms values over any interval of length T is all that is required to describe the waveform completely.

Every waveform may be described using any multiple of this period. There exists a smallest period over which the function may be described completely and this period is the fundamental period. The fundamental frequency is defined as its reciprocal:

$f_{0}={\frac {1}{T}}$

F0leftclosed

Since the period is measured in units of time, then the units for frequency are 1/time. When the time units are seconds, the frequency is in $s^{-1}$, also known as Hertz.

F0rightclosed

For a tube of length L with one end closed and the other end open the wavelength of the fundamental harmonic is 4L, as indicated by the top two animations on the right. Hence,

$\lambda _{0}=4L.$

Therefore, using the relation

$\lambda _{0}={\frac {v}{f_{0}}}$ ,

where v is the speed of the wave, we can find the fundamental frequency in terms of the speed of the wave and the length of the tube:

$f_{0}={\frac {v}{4L}}.$

F0bothclosed

F0bothopen

If the ends of the same tube are now both closed or both opened as in the bottom two animations on the right, the wavelength of the fundamental harmonic becomes 2L. By the same method as above, the fundamental frequency is found to be

$f_{0}={\frac {v}{2L}}.$

At 20 °C (68 °F) the speed of sound in air is 343 m/s (1129 ft/s). This speed is temperature dependent and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature (1.1 ft/s for every increase of 1 °F).

The velocity of a sound wave at different temperatures:-

• v = 343.2 m/s at 20 °C
• v = 331.3 m/s at 0 °C

In music

In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. The fundamental may be created by vibration over the full length of a string or air column, or a higher harmonic chosen by the player. The fundamental is one of the harmonics. A harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The reason a fundamental is also considered a harmonic is because it is 1 times itself. ^[9]

The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Together they form the harmonic series. Overtones which are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere near a harmonic, and are just called partials or inharmonic overtones.

The fundamental frequency is considered the first harmonic and the first partial. The numbering of the partials and harmonics is then usually the same; the second partial is the second harmonic, etc. But if there are inharmonic partials, the numbering no longer coincides. Overtones are numbered as they appear above the fundamental. So strictly speaking, the first overtone is the second partial (and usually the second harmonic). As this can result in confusion, only harmonics are usually referred to by their numbers, and overtones and partials are described by their relationships to those harmonics.

Mechanical systems

Consider a spring, fixed at one end and having a mass attached to the other; this would be a single degree of freedom (SDoF) oscillator. Once set into motion it will oscillate at its natural frequency. For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness; (providing the system is undamped). The radian frequency, ω_n, can be found using the following equation:

$\omega _{{\mathrm {n}}}^{2}={\frac {k}{m}}\,$

Where:
k = stiffness of the spring
m = mass
ω_n = radian frequency (radians per second)

From the radian frequency, the natural frequency, f_n, can be found by simply dividing ω_n by 2π. Without first finding the radian frequency, the natural frequency can be found directly using:

$f_{{\mathrm {n}}}={\frac {1}{2\pi }}{\sqrt {{\frac {k}{m}}}}\,$

Where:
f_n = natural frequency in hertz (cycles/second)
k = stiffness of the spring (Newtons/meter or N/m)
m = mass(kg)
while doing the modal analysis of structures and mechanical equipment, the frequency of 1st mode is called fundamental frequency.

See also

• Hertz
• Missing fundamental
• Natural frequency
• Oscillation
• Harmonic series (music)#Terminology
• Pitch detection algorithm
• Scale of harmonics

References

1. "sidfn". Phon.ucl.ac.uk. Retrieved 2012-11-27.
2. "Phonetics and Theory of Speech Production". Acoustics.hut.fi. Retrieved 2012-11-27.
3. "Fundamental Frequency of Continuous Signals" (PDF). Fourier.eng.hmc.edu. Retrieved 2012-11-27.
4. "Standing Wave in a Tube II - Finding the Fundamental Frequency" (PDF). Nchsdduncanapphysics.wikispaces.com. Retrieved 2012-11-27.
5. "Physics: Standing Waves" (PDF). Physics.kennesaw.edu. Retrieved 2012-11-27.
6. "Phys 1240: Sound and Music" (PDF). Colorado.edu. Retrieved 2012-11-27.
7. "Standing Waves on a String". Hyperphysics.phy-astr.gsu.edu. Retrieved 2012-11-27.
8. "Creating musical sounds - OpenLearn - Open University". Open University. Retrieved 2014-06-04.
9. John R. Pierce (2001). "Consonance and Scales". In Perry R. Cook. Music, Cognition, and Computerized Sound. MIT Press. ISBN 978-0-262-53190-0.