The Harmonic Series

A musical tone is characterized by pitch, loudness, timbre and duration.  These are psychoacoustic descriptions.
Loudness is a very complex psychoacoustic variable since it depends on pitch and timbre also !! I will avoid discussing loudness …. but conentrate on pitch and timbre.
A musical tone is usually very complex when we analyze its frequency content.
We recognize a pitch in it. It is a surprizing fact that the pitch is the s c fundamental frequency (or “fundamental”) in a “harmonic series”.
Thus: when we say that a musical tone has a certain pitch, we mean that this is a tone composed of harmonics and that we identify the fundamental as its “pitch” !
Two tones may sound very different yet have the same pitch, meaning the same fundamental.
Another curiosity:  the loudness of the fundamental is not important ! the ear (or the brain) will conclude which is the fundamental even if it is not present!
F ex if we listen to tone compsed of frequencies 200, 300, 400  then we identify the pitch to 100 !! which is the fundamental !!

Chapter 8, pg 132
The math behind harmonic series is Fourier analysis but for those without mathematical training , the idea can be understood from the following:

The Fourier Theorem: Any periodic waveform may be built of a harmonic series i e sine waves with multiple integer frequencies of the fundamental.

Acoustically it sounds like a steady sound f ex the vocals, the whistling, because the periodic wave does exactly the same thing over and over again.
Is this the typical sound of musical instruments ?
Musical instruments produce transients too but they always have a steady region of the wave.
Percussion instruments produce none-periodic sounds.
With a synthesizer you can shape a sound as you want.

The individual sine components may have any amplitude and any relative phase.
The repetitive frequency of the wave itself is the lowest frequency of the harmonics, i e the fundamental.

Any set of sine waves whose frequencies do not belong to a harmonic series will combine to nonperiodic waves. The sound sounds impure and unsteady, it is not musical.

White noise: this is defined as a wave that spectrum having equal amplitudes at all frequencies, it has a continuous spectrum (within a bandwidth)

A vibrating string exhibits many frequencies, all of them are standing waves, are harmonic series and perfect sine functions!
It does not matter how the vibrations was started: these frequences are always the same each time you excite the string and let it vibrate by itself.
All this is quite remarkable.
Mathematically we can describe a standing wave as a result of two identical travelling waves that move in opposite direction and superpose.

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