fundamental frequency

All sinusoidal and many non-sinusoidal waveforms repeat exactly over time – they are periodic. The period of a waveform is the smallest positive value $T$ for which the following is true:

{{block indent|1=$x(t)\; =\; x(t\; +\; T)\backslash text\{\; for\; all\; \}t\; \backslash in\; \backslash mathbb\{R\}$}}

Where $x(t)$ is the value of the waveform $t$. This means that the waveform's values over any interval of length $T$ is all that is required to describe the waveform completely (for example, by the associated Fourier series). Since any multiple of period $T$ 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:

{{block indent|1=$f\_0\; =\; \backslash frac\{1\}\{T\}$}}

When the units of time are seconds, the frequency is in $s^\{-1\}$, also known as Hertz.

For a pipe of length $L$ with one end closed and the other end open the wavelength of the fundamental harmonic is $4L$, as indicated by the first two animations. Hence,

{{block indent|1=$\backslash lambda\_0\; =\; 4L$}}

Therefore, using the relation

{{block indent|1=$\backslash lambda\_0\; =\; \backslash frac\{v\}\{f\_0\}$}}

where $v$ 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:

{{block indent|1=$f\_0\; =\; \backslash frac\{v\}\{4L\}$}}

If the ends of the same pipe are now both closed or both opened, the wavelength of the fundamental harmonic becomes $2L$ . By the same method as above, the fundamental frequency is found to be

{{block indent|1=$f\_0\; =\; \backslash frac\{v\}\{2L\}$}}

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.{{cite book |title = Music, Cognition, and Computerized Sound |author-link=John R. Pierce |last=Pierce |first=John R. |chapter=Consonance and Scales |editor-first=Perry R. |editor-last=Cook |publisher=MIT Press |date=2001 |isbn=978-0-262-53190-0 |chapter-url= https://books.google.com/books?id=L04W8ADtpQ4C&q=musical+tone+harmonic+partial+fundamental+integer&pg=PA169}}

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 {{em|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.

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, {{var|ω}}₀, can be found using the following equation:

{{block indent|1=$\backslash omega\_\backslash mathrm\{0\}\; =\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}\; \backslash ,$}}

where:

• {{var|k}} = stiffness of the spring
• {{var|m}} = mass
• {{var|ω}}₀ = natural frequency in radians per second.

To determine the natural frequency in Hz, the omega value is divided by 2{{var|π}}.

Or:

{{block indent|1=$f\_\backslash mathrm\{0\}\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}\; \backslash ,$}}

where:

• {{var|f}}₀ = natural frequency (SI unit: hertz)
• {{var|k}} = stiffness of the spring (SI unit: newtons/metre or N/m)
• {{var|m}} = mass (SI unit: kg).

While doing a modal analysis, the frequency of the 1st mode is the fundamental frequency.

This is also expressed as:

{{block indent|1=$f\_\backslash mathrm\{0\}\; =\; \backslash frac\{1\}\{2l\}\; \backslash sqrt\{\backslash frac\{T\}\{\backslash mu\}\}\; \backslash ,$}}

where:

• {{var|f}}₀ = natural frequency (SI unit: hertz)
• {{var|l}} = length of the string (SI unit: metre)
• {{var|μ}} = mass per unit length of the string (SI unit: kg/m)
• {{var|T}} = tension on the string (SI unit: newton){{Cite web|url=https://www.wirestrungharp.com/material/strings/about_string_calculator/|title=About the String Calculator|website=www.wirestrungharp.com}}

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

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Category:Musical tuning

Category:Acoustics

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