![Harmonic partials on strings](https://upload.wikimedia.org/wikipedia/commons/c/c5/Harmonic_partials_on_strings.svg)
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of a
periodic waveform
In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electronic ...
. In music, the fundamental is the musical
pitch of a note that is perceived as the lowest
partial
Partial may refer to:
Mathematics
* Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
present. In terms of a superposition of
sinusoid
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
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
0, indicating the lowest frequency
counting from zero. In other contexts, it is more common to abbreviate it as
1, the first
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
. (The second harmonic is then
2 = 2⋅
1, etc. In this context, the zeroth harmonic would be 0
Hz.)
According to Benward's and Saker's ''Music: In Theory and Practice'':
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 is true:
Where
is the value of the waveform at
. This means that the waveform's values over any interval of length
is all that is required to describe the waveform completely (for example, by the associated
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
). Since any multiple of period
also satisfies this definition, the fundamental period is defined as the smallest period over which the function may be described completely. The fundamental frequency is defined as its reciprocal:
When the units of time are seconds, the frequency is in
, also known as
Hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
.
Fundamental frequency of a pipe
For a pipe of length
with one end closed and the other end open the wavelength of the fundamental harmonic is
, 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 pipe:
If the ends of the same pipe are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes
. By the same method as above, the fundamental frequency is found to be
In music
In music, the fundamental is the musical
pitch of a note that is perceived as the lowest
partial
Partial may refer to:
Mathematics
* Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...
present. The fundamental may be created by
vibration
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic function, periodic, such as the motion of a pendulum ...
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 a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
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,
0, can be found using the following equation:
where:
* =
stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
of the spring
* = mass
*
0 = natural frequency in radians per second.
To determine the natural frequency in Hz, 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
modal analysis
Modal analysis is the study of the dynamic properties of systems in the frequency domain. Examples would include measuring the vibration of a car's body when it is attached to a shaker, or the noise pattern in a room when excited by a loudspeak ...
, the frequency of the 1st mode is the fundamental frequency.
See also
*
Greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
*
Hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
*
Missing fundamental
A harmonic sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself.
The brain perceives the ...
*
Natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all par ...
*
Oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
*
Harmonic series (music)#Terminology
*
Pitch detection algorithm
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 octave ...
*
Scale of harmonics
The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale.Yin, Wei. ''Zh ...
References
{{DEFAULTSORT:Fundamental Frequency
Musical tuning
Acoustics
Fourier analysis