Fourier Synthesis

There is a theorem of mathematics that any periodic function (such as the pressure as a function of time of a note with a distinct frequency) can always ve written as a sum of harmonics of that frequency with each harmoic having aan integer multiple of that original frequency, each with their definite amplitude and phase. Those harmonics are sinusoidal functions. The ear (or rather the brain part of the hearing) melds together those different harmnics into one pithch experience even though the cochlea presents them to the brain as distinct vibrations coming from the basilar membraine.(except for the higher harmoics higher than about the 10th whose vibrations on the basilar membrane begin to overlap).

To show this I will take a number of oscillations whose whole "waveform" (the picture of what say the pressure looks like as a function of time"). I start with the square wave (a pressure say which has sudden (instantaneous) changes of level between which changes the function is a constant. Then a sawtooth wave, which has sudden changes in level between which the function drops down as a straight line to the next suddent change up. Because of the relatively hich amplitude of the high harmonics which make up such a square wave, or sawtooth, these sounds tend to sound very raucous. I then take an oboe like sound and a human voice recording and do the same with them. ( on the "Guitar pick" web item I do this for a recording of a guitar pluck).

Square Wave

Here is the sequence of adding together sinusoidal waves to make a ake a square wave. square wave.

We start with the fundamental with unit amplitude.

We add to this the third harmonic ( taking the definition of the first harmonic to be the same as the fundamental-- a practice endorsed by the text but at variance with most music terminology) with 1/3 amplitude and with a phase shift of 180 degrees.

to get this. Note that this addition has flattened out the tops and bottoms and increased the slope as the wave passes through zero.

We now add some of the fifth harmonic with amplitude 1/5

to get this. A closer approximation to the square wave. The top is flatter, and the sides are more nearly vertical.

Adding the seventh harmonic, again with an amplitude of 1/7 and a phase of 180degrees.

We get an even better approximation to a square wave. We can continue this process with higher and higher harmonics coming closer and closer to a square wave. Adding the first thirty harmonics ( the odd ones each with amplitude of one over the harmonic number and each second odd harmonic with a phase shift of 180 degrees) we get

a much closer approximation to the square wave. The "ringing" ( the wiggles near the vertical parts of the square wave) are a consequence of trying to approximate a vertical line ( a "discontinuity") with a sum of sinusoidal waves.

Sawtooth

WE can do the same thing with the sawtooth wave. In this case all of the harmonics are non-zero, the amplitudes are one over the harmonic number and the phases are all 90 degrees (ie all start at zero going down>

Adding the first 5 components gives us

which is a reasonable approximation to a sawtooth wave. Adding even more harmonics, again up to the thirtieth gives us

Again, because of the vertical line, we get ringing in the wave near that vertical line.

"Oboe"

Here is the waveform of a sound which sounds something like an oboe. Below the waveform, on the sliders are the amplitudes of the various harmonics which make up this waveform. Note that there is only a small amount of the fundamental, a lot of the second and third harmonics, less of the fourth and fifth and then nothing up to the 11,12,and 13th.

Voice

Here is an example of the synthesis of a voiced vowel sound -- something like me saying "oooh".

Note the strong third harmonic, and the secondary peak around the seventh harmonic in the amplitude of the various components. These "peaks" in the amplitude are called formants in the voice and arise due to resonances of the air tract between the larynx and the mouth.

Note that in each case one can estimate the highest harmonics by looking at the wiggles of the function. The length of the shortest wiggle gives a crude estimate of the highest harmonic needed to produce that sound. We note in the square and sawtooth that shortest wiggle occurs nearest the sharp rise or fall of the function. These images were produced using the program FourierSynth, written by W. Unruh.