User:John R. Brews/Fourier series

Fourier series (music)

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The amplitude of a musical note varies in time according to its sound envelope.

The ideas of Fourier series have interesting application in music, for example, in illuminating the differences in pitch between musical instruments and in the construction of music synthesizers. A laboratory determination of pitch is made by a subject listening to a note and to the tone of a single sinusoidal wave of a variable frequency, and identifying at what frequency the note and the sinusoid sound alike.

The frequency spectrum of a musical instrument playing a particular note varies with the instrument and with the way that it is played. The manner of playing determines the sound envelope of a note, and therefore the amplitude of its harmonics.

Some instruments (like the flute or the violin) exhibit a fundamental frequency and its harmonics in varying amplitudes and phase, and others (like the cymbal or the drum) do not.

Helmholtz' theory

A single note is often referred to as a tone, the lowest frequency present as the fundamental or prime partial tone and harmonics of this frequency that appear when a note is played on an instrument as harmonic upper partial tones or upper tones or partial tones. Where only one frequency is present, as with a tuning fork, it is called a simple tone:

It is well known that this union of several simple tones into one compound tone, which is naturally effected in the tones produced by most musical instruments, is artificially imitated on the organ by peculiar mechanical contrivances. The tones of organ-pipes are comparatively poor in upper partials. When it is desirable to use a stop of incisive penetrating quality of tone and great power, the wide pipes (principal register and weithgedackt) are not sufficient; their tone is too soft too defective in upper partials; and the narrow pipes (geigen-register and quintaten) are also unsuitable, because, although more incisive, their tone is weak. For such occasions, then, as in accompanying congregational singing, recourse is had to the compound stops. In these stops every key is connected with a large or smaller series of pipes, which it opens simultaneously, and which give the prime tone and a certain number of the first upper partials of the compound tone of the note in question. Helmholtz, p. 91

These words identify the organ as a form of mechanical rather than electrical music synthesizer. They also lay out the characterization of a musical note by its constituent frequency components, in the same way that a Fourier series expresses a periodic waveform in terms of its harmonic components.

Fourier series

A musical note is a a periodic function in time and in space, modulated by a sound envelope. That is, the note can be expressed as:

$n(x,\ t)=s(x-vt)f(x-vt)\ ,$

where s is the sound envelope that expresses the duration of the note at a fixed location or its extent in space at a fixed time, and f is a periodic function of its argument:

$f(x+\lambda -vt)=f\left(x-v(t+T)\right)=f(x-vt)\ ,$

with λ the wavelength of f and T = λ/v the period of f and v the speed of propagation of f through space.

A periodic function of a variable ξ satisfies the relation:

$f(\xi +P)=f(\xi )\ ,$

where P is called its period. Fourier's theorem states that any such (real) periodic function can be expressed as a sum of sinusoidal functions with periods related to P in what is now called a Fourier series:

$f(\xi )=a_{0}+\sum _{1}^{\infty }a_{n}\cos \left({\frac {2\pi }{P/n}}\xi +\varphi _{n}\right)\ ,$

a series of cosines with various phases {φ_n}.

If the function is a fixed waveform propagating in time, we may take ξ as:

$\xi =x-vt\ ,$

where x is a position in space, v is the speed of propagation and t is the time. The period in space at a fixed instant in time is called the wavelength λ=P, and the period in time at a fixed position in space is called the period T=λ/v. Thus, a function periodic in time with period T can be expressed as a Fourier series:

$f(t)=a_{0}+\sum _{1}^{\infty }a_{n}\cos \left(n\omega _{0}t+\varphi _{n}\right)\ ,$

where ω₀ = 2π/T is called the fundamental frequency and its multiples 2ω₀, 3ω₀,... are called harmonic frequencies and the terms cosines are called harmonics. A function f(x) of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral sub-multiples of λ (i.e. λ, λ/2, λ/3, etc.):

$f(x)=a_{0}+\sum _{1}^{\infty }a_{n}\cos \left({\frac {2\pi }{\lambda /n}}x+\varphi _{n}\right)\ .$

The connection between Fourier series and music is that the periodic function f decides the characteristic sound of the instrument that differentiates one instrument from another. As the remarks of Helmholtz suggest, the sound of a note played upon each instrument is characterized by not only its fundamental tone or frequency, which is the same for all instruments, but by the higher partial tones or harmonics in the note, which are different in amplitude and in phase for each instrument, making them sound different.

References

• Thomas D. Rossing (2007). Springer Handbook of Acoustics. Springer, p. 541. ISBN 0387304460. 

• concert hall design
• timbre
• timbre
• first partial
• clarinet and violin
• Helmholtz