This is an important video where we put the math aside and talk more generally about sinusoids.
And we’ll do that by addressing these fundamental questions, questions you probably have had
lingering at the back of your mind?
Why did we talk about mathematical periodic functions in the context of audio?
What does a sine wave actually represent in the real world?
And why does sinusoidal motion create a pure tone?
Let’s address the first question and I’m sorry to say this, but we’ll have to throw
away a lot of what we learnt over the last few videos.
The reason we talked about triangles and circles previously was to derive the wave equation,
an equation that we can plugin into an audio system and generate any type of sinusoidal
waveform.
So we’ll keep the wave equation because it’s relevant for audio purposes.
But what does a rotating circle have anything to do with audio and how it’s generated?
Well it doesn’t really.
The possible conclusion that you may have arrived at, after having watched the last
couple of videos is that sin waves are intricately linked with circles and originate from circles.
That’s not strictly true.
In fact the opposite is true, where circles are the consequence of sinusoids.
Let's look at sinusoidal motion along a single dimension, like you see here, where the dimension
is along the y axis.
Here’s the same body with the same motion along the x-axis.
Now, let's combine the 2 dimensions of motion together.
We get the body following the path of a circle.
This is exactly what we described previously when we said that every point on a circle
can be equated to a coordinate of the cos and sin functions of a radian angle.
But now, we’re implying that the circle is only in existence due to the combined 2
dimensional harmonic motion of a body.
It’s a chicken and egg problem essentially, it’s all a matter of perspective.
Motion described by a sine function, or simple harmonic motion happens everyday in real life,
in the example of a weight pulling down a spring, or a pendulum swinging or a string
vibrating, we can see that the motion is naturally fast at the resting point if the body were
not moving, and naturally slow when grinding down to a halt at the extreme ends of it’s
range.
The sin function merely describes and quantifies this shape or motion.
If we were to make the motion linear, for example, you can see how robotic, awkward
and unnatural the motion looks.
The next question, which is quite a profound one, is what does the wave equation actually
represent.
We know that y(t) = Asin(2pift + phi) describes a simple sine wave and we know that this periodic
wave produces a pure tonal sound.
But what is y(t) here, what physical attribute in the real world does this wavey motion represent?
It’s not the amplitude or volume, since we have a stablly loud signal.The answer will
most definitely spill over into several other modules since it is fundamental to different
areas of study, but I’ll try to race through it here anyway, just to cover all the basics.
The answer to the question is, that it depends.
It depends on the domain and the context.
In the domain of the physical world, where we live and breathe, the wave motion y(t)
represents the rise and fall of pressure.
In the analog and electrical world of cables and wires and speakers, it represents voltage.
In the digital world of computer and audio interfaces, it represents arbitrary bits of
0’s and 1’s.
We haven’t talked about pressure or voltage or data bits before, and we will in more detail.
But just glossing through, when a guitar string is plucked or when a vocal chord is strained,
it starts vibrating, which inturn starts resonating and vibrating the air around it as well.
The molecules of air around the source are imparted some amount of energy and are pushed
away from the source and they eventually collide and transfer some of their own energy to other
molecules they meet kickstarting a chain reaction.
This causes the molecules to first crowd together and then move apart.
The alternate crowding together and moving apart of these molecules in turn affects the
surrounding air pressure.
This fluctuation of air pressure is propagated outwards till this resonance is transferred
to the air next to our ear canal and it’s interpreted as sound.
So the wave equation, in the physical domain, represents the sinusoidal motion of the change
in pressure over time, or more simply it describes a pressure wave.
Now let's address the last question of why sinusoidal waves create a pure tone.
The sinusoidal function describes the most basic form of periodic motion, motion that
repeats itself periodically at regular intervals.
And I can’t stress this enough, that periodic motion is what gives rise to a perception
of pitch.
It doesn’t have to be just a simple sine wave though.
A complex waveform, which is still repeating itself at regular intervals produces the perception
of pitch.
We’ve already seen and heard the sine wave before, countless times.
So let’s take the example of a harp playing a note, and here is it’s waveform zoomed
in.
We can see that this general shape seems to be repeating itself over and over again.
Here’s another example of a clarinet playing the same note.
The waveform here is sufficiently more complex, but you can just about make out that it’s
a repeating waveform.
Now here’s an example of a crash cymbal, which sounds inharmonic to say the least.
You can’t really associate this crash sound having any sort of pitch.
And the waveform proves it right.
There is no discernible pattern which repeats itself.
This awkward notion that repeating patterns produce the perception of pitch may sit uneasy
with you.
But there is a good mathematical backing for this behavior and we’ll find out more when
we talk about the harmonic series when we discuss other waveform shapes.
But instead of trying to identify patterns by the tedious task of peering hard into complex
waveforms, we can subject audio signals to a complicated mathematical transformation
called the Fourier transforms which can decompose a complex signal into many simple signals
of distinct values.
These values can be plotted on a frequency response graph to see what a snapshot of the
waveform looks like.
And for the harp and the clarient, we can clearly see distinct frequencies that stand
out which we perceive and recognize as pitch.
But the frequency response of a cymbal is all over the place, proving that there is
no specific and discernible pitch associated with it.
What about a sine wave then ? If we decompose the sine wave with the help of the Fourier
Transform, we get 1 value, and no more.
Just one distinct value in the frequency response graph to showcase that the sine wave was decomposed
into 1 simple signal, which is itself.
This shows us that mathematically and physically, the sine wave represents the simplest possible
periodic wave.
And our ear is miraculously just a beautifully designed frequency analyzer, a mathematical
transformation machine, which lets us experience an auditory sensation of what these graphs
visually represent.
Hello there folks.
We’ve come to the end of the module on the origins of the sine wave, and I’d like to
thank you for sticking with it!
I’ve got a whole series of technical topics in the pipeline, all geared towards audio
engineering, programming and signal processing.
So if you fancied the topics, the aesthetics and approach of the previous video, please
do consider subscribing cos there are going to be a lot more.
And if you are feeling really generous, I’ve got a Patreon link down below.
Until next time then.