Sound theory: Waveforms

Pure sinusoids have been fully described in the previous paragraphs [Properties of sound ] . It is perceived as a frequency tone equal to the sinusoid's frequency. It is easily generated electronically and is often used as a testing instrument.

This waveform's rate and harmonic content are described in the following figure:

Square wave

As we can see the harmonic content of a square wave is composed purely by odd harmonics. The amplitude decreases at a rate of 1/f. Empirically speaking, this means that the third harmonic (the one that has triple the frequency of the fundamental, since the one that has double the frequency is not present) has an amplitude of 1/3 of the fundamental, the fifth harmonic has 1/5 and so on.

The following are the sounds of a square wave, one having a frequency of 440 Hz (the equivalent of an A note) and one having a 1 KHz frequency:

Square wave (f=440 Hz)  [Track 4]

Square wave (f=1 KHz)  [Track 5]

Sawtooth wave

Sawtooth waves contain all the harmonics with the amplitude decreasing at a rate of 1/f.

The following are the sounds of a sawtooth harmonic: one has a frequency of 440 Hz (the equivalent of an A note) and one having a 1 KHz frequency:

Sawtooth wave (f=440 Hz)  [Track 6]

Sawtooth wave (f=1 KHz)  [Track 7]

Triangular wave

A triangular wave's harmonic content is very similar to that of a square wave. The only difference is that its amplitude decreases with at a rate of 1/f².

The following are the sounds of a triangular wave, one has a frequency of 440 Hz (the equivalent of an A note) and one having a 1 KHz frequency:

Triangular wave (f=440 Hz)  [Track 8]

Triangular wave (f=1 KHz)  [Track 9]

The main difference between harmonics and hypertones is that the latter are not strictly related with the fundamental frequency whilst harmonics are their multiples.

Hypertones strongly depend on the musical instrument that has generated them and they contribute to characterizing sound regardless of their amplitude being inferior to that of harmonics.