A harmonic spectrum is a
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
containing only frequency components whose
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
are
whole number
An integer is the number zero ( 0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number ( −1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative ...
multiples of the
fundamental frequency
The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
; such frequencies are known as
harmonic
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
s. "The individual partials are not heard separately but are blended together by the ear into a single tone."
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 Joseph Fo ...
is that a function has a harmonic spectrum if and only if it is
periodic.
See also
*
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
*
Harmonic series (music)
The 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''.
Definite pitch, Pitched musical instruments are often based on an Acoust ...
*
Periodic function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
*
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. ''Zho ...
*
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 mus ...
References
{{Signal-processing-stub
Functional analysis
Acoustics
Sound