Don’t be Phased by Waveform Distortions

For many years there has been an ongoing debate about the importance of phase changes in music. Some people claim that we cannot hear the effects of phase errors, others claim that we can. Who is right? The figure below shows a waveform of a perfect fifth, with components in the ratio ${3 : 2}$ for various values of the phase-shift. Despite the different appearances, all sound pretty much the same.

What is phase? A pure sine-wave is represented by the function

$\displaystyle A \sin (\omega t - \phi )$

where ${A}$ is the amplitude, ${\omega}$ is the frequency and ${\phi}$ is the phase. The phase determines the moments at which the wave form passes through zero. The two waves

$\displaystyle A \sin (\omega t) \qquad\mbox{and}\qquad A \cos (\omega t ) = A \sin (\omega t + \pi/2 )$

are ${90^\circ}$ degrees (or ${\pi/2}$ radians) out of phase.

Sound waves are comprised of a multitude of sine-waves. If all the components of a sound wave are phase-shifted by an amount proportional to the frequency, the overall sound wave is not changed in form, just delayed or advanced in time. Apart from the time-change, we hear no difference in the sound.

Sound Speed and Wavelength

Sound travels at about 340 meters per second. So, a tone of 340 Hz has a 1 metre wavelength. The 88th note on a piano, the highest note ${C_8}$ , has frequency 4186 Hz, greater by a factor of about 12.3 and its wavelength is about 8cm. Thus, a change of this distance from source to listener, perhaps a nod of the head, changes the phase by ${2\pi}$ .

Our hearing is not sensitive to phase changes of the components of a waveform. This is just as well, for if phases were perceptible, listeners sitting in different positions in an auditorium would perceive different sounds. If the human ear were sensitive to relative changes of phase, great practical difficulties would arise in tuning of phase as well as pitch. Fortunately, this is not the case.

Reconstructing a Triangle

As an example of the effects of phase changes, we can generate a triangular waveform by combining all the odd harmonics:

$\displaystyle \cos t + \frac{\cos 3 t}{3^2} + \frac{\cos 5 t}{5^2} + \cdots$

If we shift each component by ${90^\circ}$ , all the cosines are changed to sines:

$\displaystyle \sin t + \frac{\sin 3 t}{3^2} + \frac{\sin 5 t}{5^2} + \cdots$

The appearance of the waveform, as it would be displayed on an oscilloscope, changes dramatically, but the perceived sound is not much changed.

Amazingly, the sound perceptions of these two waveforms are virtually indistinguishable. You can confirm this by generating the two waveforms and playing the corresponding sounds. This can be done by visiting the following website:

https://meettechniek.info/additional/additive-synthesis.html

and using the additive synthesis waveform generator there.

Conclusion

For normal music and speech, changes of phase are generally inaudible, so that listeners have difficulty perceiving phase distortions. Thus, in most circumstances, phase distortion is not particularly damaging to the quality of music. However, there are specific conditions in which it may be intolerable. For example, components of short duration, such as those produced by percussion instruments, produce transients, and phase distortions can change the character of the resulting sounds.

ThatsMaths