In my discussion on square waves, when I mentioned the properties of
its Fourier components, I glibly suggested “We’ll ignore phase”. In
truth, we need to look a little more closely at it. I’ll start by
defining phase.

Imagine in your mind’s eye a graph of a sine
wave. If this were to represent a sound wave, the vertical axis would
represent the air pressure oscillating up and
down, and the horizontal axis would represent the passage of time. If
you were to capture that sound wave with a microphone, the voltage
waveform that the microphone would create would look just the same.
Imagine now that you could actually “see” the sound waves, as though
they were waves on the surface of a pond. Imagine yourself standing
next to the microphone, observing the sound waves as they travel toward
you and the mic. Imagine that you can make out the individual peaks and
troughs. You can watch an individual peak approach and impinge on the
microphone. At the instant it does so, you observe the output voltage
of the microphone. You will see that the output voltage also goes
through a peak. You watch as the peak passes by and is replaced by its
trailing trough, and you observe the voltage from the microphone
simultaneously decay from a peak to a trough. I hope this is all pretty
obvious.

Now we are going to add a second microphone. Except
that we are going to align this one a short distance behind the first
one. Now, by the time the peak of the sound wave impinges on the second
microphone, it has already passed the first one. The output voltage
from the first microphone has already passed its peak and is on its way
down to the trough. But the output voltage of the second microphone is
only just reaching its peak. Over a period of time, both microphones
capture exactly the same oscillating sine wave, but the output of the
second is always delayed slightly compared to the output of the first.
This is the Phase of the sine wave in action. The phase represents the
time alignment. And, as you can work out for yourself, it is not the
absolute phase which is important, but the relative differences between
phases.

Interesting things happen when we add two nominally
identical sine waves together. Let’s specify two sine waves which have
the exact same magnitude and frequency, but differ only in phase. I
won’t go into the mathematics of this, but what you get depends strongly
on the phase difference. If both phases are identical the two sine
waves add up, and the result is a sine wave of exactly twice the
magnitude of the original. However, if the two sine waves are time
aligned such that the peak of one coincides with the trough of the
other, then the end result is that both sine waves cancel each other out
and you end up with nothing. These two extreme conditions are often
referred to as “in-phase” and “out-of-phase”. However, there are a
whole spectrum of phase relationships between these two extremes. As
the phase difference gradually varies from “in-phase” to “out-of-phase”,
so the magnitude of the resultant signal gradually falls from twice
that of the original, down to zero.

Lets now go back to the
square-wave example of yesterday’s post. Yesterday’s example was
concerned with the summing of individual sine waves (a fundamental and
its odd harmonics), and how the more of these odd harmonics you added
in, the closer the result approached to a square wave. We also saw that
by limiting the number of harmonics, what we got was a close
approximation to a square wave, but with some leading and trailing edge
overshoot, and a bit of ripple. However, in all of that discussion we
took no account of phase. Or, more specifically, we allowed for no
phase difference between the different harmonics. What happens if we
start to introduce phase differences?

The result is that we no
longer get a nice clean square wave. As we mess with the phase
relationships we get all sorts of odd-looking waveforms, some of them
bearing precious little visual relationship to the original square wave.
Yet all of these different waveforms comprise the exact same mixture
of frequencies. The obvious question arises - do they sound at all
different? In other words, can we hear phase relationships? This is a
tricky question.

My own experiments have shown me that I have
problems detecting any differences between test tones that I have
created artificially, comprising identical assemblies of single
frequency components added together with different - and sometime
arbitrary - phases. But from a psychoacoustical perspective this is
perhaps not surprising. Our brains are not wired to recognize test
tones, and cannot therefore create a sonic reference from memory with
which to compare them, so it is not surprising that I would find it very
difficult to hear differences between them.

The conventional
wisdom is that relative phase is inaudible, but this is hard to test
with anything other than synthesized test tones. With real music there
is no possibility to independently adjust the phase of individual
frequency components (for reasons I won’t get into here) so the question
of the audibility of relative phase is one which remains either
unproven, or proven as to its inaudibility, depending on your
perspective.