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A harmonic spectrum is a
A harmonic spectrum is a spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
containing only frequency components whose frequencies
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 ...
are whole number multiples of the fundamental frequency
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 ...
; such frequencies are known as 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. "The individual partials are not heard separately but are blended together by the ear into a single tone."Benward, Bruce and Saker, Marilyn (1997/2003). ''Music: In Theory and Practice'', Vol. I, p.xiii. Seventh edition. McGraw-Hill. .
In other words, if is the fundamental frequency, then a harmonic spectrum has the form
:
A standard result of Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
is that a function has a harmonic spectrum if and only if it is periodic.
See also
*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 ...
* Harmonic series (music)
A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''.
Pitched musical instruments are often based on an acoustic resonator su ...
* Periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
* 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. ''Z ...
* Undertone series
In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones must ...
References
{{Signal-processing-stub Functional analysis Acoustics Sound