The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest _{0}, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as _{1}, the first _{2} = 2⋅_{1}, etc. In this context, the zeroth harmonic would be 0 .)
According to Benward's and Saker's ''Music: In Theory and Practice'':

^{−1}, also known as

_{0}, can be found using the following equation:
where:
* = _{0} = natural frequency in radians per second.
To determine the natural frequency, the omega value is divided by 2. Or:
where:
* _{0} = natural frequency (SI unit: Hertz (cycles/second))
* = stiffness of the spring (SI unit: Newtons/metre or N/m)
* = mass (SI unit: kg).
While doing a

frequency
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

of a periodic waveform
In electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons
The electron is a subatomic particle
In physical sciences, subatomic part ...

. In music, the fundamental is the musical pitch
Pitch may refer to:
Acoustic frequency
* Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch"
** Absolute pitch or "perfect pitch"
** Pitch class, a set of all pitches that are a whole number of octaves ...

of a note that is perceived as the lowest partial
Partial may refer to:
Mathematics
*Partial derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

present. In terms of a superposition of sinusoid
A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathemat ...

s, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as harmonic
A harmonic is any member of the harmonic series
Harmonic series may refer to either of two related concepts:
*Harmonic series (mathematics)
*Harmonic series (music)
{{Disambig .... The term is employed in various disciplines, including music ...

. (The second harmonic is then Explanation

All sinusoidal and many non-sinusoidal waveforms repeat exactly over time – they are periodic. The period of a waveform is the smallest value of for which the following equation is true: Where () is the value of the waveform at . This means that this equation and a definition of the waveform's values over any interval of length is all that is required to describe the waveform completely. Waveforms can be represented byFourier series
In mathematics, a Fourier series () is a periodic function composed of harmonically related Sine wave, sinusoids combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an ...

.
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:
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 Hertz
The hertz (symbol: Hz) is the unit
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action ...

.
For a tube of length with one end closed and the other end open the wavelength of the fundamental harmonic is 4, as indicated by the first two animations. Hence,
Therefore, using the relation
where is the speed of the wave, the fundamental frequency can be found in terms of the speed of the wave and the length of the tube:
If the ends of the same tube are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2. By the same method as above, the fundamental frequency is found to be
At 20 °C (68 °F) the speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends s ...

in air is 343 m/s (1129 ft/s). This speed is temperature dependent and increases 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:
* = 343.2 m/s at 20 °C
* = 331.3 m/s at 0 °C
In music

In music, the fundamental is the musicalpitch
Pitch may refer to:
Acoustic frequency
* Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch"
** Absolute pitch or "perfect pitch"
** Pitch class, a set of all pitches that are a whole number of octaves ...

of a note that is perceived as the lowest partial
Partial may refer to:
Mathematics
*Partial derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

present. The fundamental may be created by vibration
Vibration is a mechanical phenomenon whereby oscillation
Oscillation is the repetitive variation, typically in time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparentl ...

over the full length of a string or air column, or a higher harmonic chosen by the player. The fundamental is one of the harmonic
A harmonic is any member of the harmonic series
Harmonic series may refer to either of two related concepts:
*Harmonic series (mathematics)
*Harmonic series (music)
{{Disambig .... The term is employed in various disciplines, including music ...

s. 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.
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 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 natural frequency, or fundamental frequency,stiffness
Stiffness is the extent to which an object resists deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (mechanics), such changes co ...

of the spring
* = mass
* modal analysis
Modal analysis is the study of the dynamic properties of systems in the frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (inform ...

, the frequency of the 1st mode is the fundamental frequency.
See also

*Greatest common divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

*Hertz
The hertz (symbol: Hz) is the unit
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action ...

*Missing fundamental
A harmonic
of a vibrating string are harmonics.
A harmonic is any member of the harmonic series (music), harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio tech ...

*Natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to Oscillation, oscillate in the absence of any driving or Damping ratio, damping force.
The motion pattern of a system oscillating at its natural frequency ...

*Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. The term ''vibration'' is precisely used to describ ...

*
*Pitch detection algorithm
A pitch detection algorithm (PDA) is an algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtract ...

*Scale of harmonics
The scale of harmonics is a musical scale
In music theory
Music theory is the study of the practices and possibilities of music
Music is the art of arranging sounds in time through the elements of melody, harmony, rhythm, and timbre ...

References

{{DEFAULTSORT:Fundamental Frequency Musical tuning Acoustics Fourier analysis