Understanding Sound Waves and Waveforms
Waveforms are the base of any sound—therefore the base of synthesis. In any synthesizer, the oscillator is the module in charge of generating repeating waves from scratch, at various amplitudes and frequencies, which will then be altered through other processes of synthesis.
There are virtually infinite types of waveforms, although here we’ll cover the most common ones. Most synths use the basic shapes like sine, square, triangle, and saw waves (which I’ll cover later) but some of them provide you with access to all kinds of waveforms (this is especially true when it comes to wavetable synths; for instance, Serum even allows you to import any image and it will create custom waveforms from it).
Each one has different characteristics in their sound, and knowing what they sound like will be extremely useful when choosing a specific waveform as a start point; selecting the correct waveshape will allow you to achieve the fundamental character of the sound you are looking for. So let’s present an explanation of the science behind each of them on this brief tutorial.
What is a Waveform?
Technically, the definition of a waveform is that it’s a graph that’s constructed by points that go up and down around an equilibrium point like this:
In this graph, the x-axis represents time and the vertical y-axis represents amplitude (which, not-technically speaking, is equal to loudness). This is called a time-domain representation of audio — it represents movement over time.
When the wave presents itself in the physical world — for example, through your speakers — it translates into sound (sound is, basically, air particles moving back and forth very fast). Without a waveform, you wouldn’t get any sound.
Note: although we represent waveforms on our screens as they’re going up and down, in the physical space, in order to generate sound, the air particles actually move back and forth (they are longitudinal waves); think of how your speakers push the air for example. If you play music loud enough, you’ll see their cones moving, especially along the bass line. Remember that sound always requires and needs a medium to travel through, and that is normally the air.
But, in music, how is sound produced?
In sound synthesis, sound waves are created by an oscillator. Note that this is the waveform generator of the synthesizer and it creates perfect waves — such as the sine wave showed before — which is impossible to find in the real analog world (for example, a guitar string will never produce the same wave every time you pluck it; there will be changes each time, extremely small changes that you won’t notice).
Now let’s go through amplitude and frequency, which are the most fundamental characteristics of a sound wave.
The amplitude determines how much the wave will move from the equilibrium point. It’s a value between 0 (silence) and 1 (maximum displacement). It’s a bipolar measurement, which means it can go +1 and -1.
As I said earlier, the amplitude is similar to loudness. The difference is that loudness is based on human perception (perceived loudness), while the amplitude is an established value.
When we transpose it to the physical space, the higher the amplitude the more air particles will be moved by, for example, your speakers.
The frequency of a wave is basically the rate at which it moves in and out; How fast will be its motion.
All waves consist of a continuous series of cycles or periods. For instance, in the previous image, the sine wave completed one cycle or period (it moved back and forth one time, ending at the equilibrium point: it’s a constant pattern). Then, the frequency is how many periods the wave completes in one second.
While the cycle or period (T) is measured in seconds (for example, it takes 0.2 seconds for a wave to complete a single cycle) the frequency () is measured in Hertz (Hz), which equals 1/s or s–1. This means that if we have a signal with a frequency of 20 Hz, that wave is moving back and forth 20 times every second. We say that the frequency of a wave it’s the same to the inverse of its cycle or period ( if T = 0.2 s then = 1/0.2 Hz = 5 Hz ).
The frequency is related to the pitch of a sound: the higher the frequency, the higher the pitch. The same as amplitude and loudness, frequency and pitch differ on the fact that pitch is related to the human perception of the tone, while the frequency is a measurable value.
It’s a general rule that humans can hear any sound between 20 Hz and 20.000 Hz. Nevertheless, you probably won’t hear a frequency of 20 Hz on almost any situation (you would feel it, for example, in a concert, where your entire body actually vibrates), so 30 Hz is probably a more realistic boundary. On the other hand, you probably won’t be able to hear a frequency of 20.000 Hz, since humans, as we get older, we lose our perception of the highest frequencies.
Each note on, for example, a piano or a guitar, has a specific frequency. For instance, A4 — whose frequency is, in fact, used as a reference for tuning all the other notes — vibrates at a frequency of 440 Hz. Additionally, when we duplicate the frequency, we also get a note that is an octave higher: in this case, we get A5, which vibrates at a frequency of 880 Hz.
Fun fact: the lowest note of some large pipe organs is a C that has a frequency of around 16 Hz, which means that we are not able to hear its fundamental tone!
Note that the waveforms produced by synthesizers are periodic, which means that their periods are repeated perfectly in order to produce a constant tone. This differs from the generation of the sound of organic instruments since, as I explained earlier, each period of these will vary, even though they’re incredibly small variations.
There are two main ways to visualize waveforms: the spectrum analyzer and the oscilloscope. There are some pieces of free software you can download for each of them.
The spectrum analyzer describes the frequencies in the x-axis (there can be multiple sound waves with different frequencies playing at the same time) and the amplitude in the y-axis.
The spectrum analyzer that I recommend is SPAN by Voxengo which is a free plugin that, apart from having an excellent spectrum analyzer (which you can find in pretty much any EQ plugin), it provides more information such as phase correlation and RMS metering, and it also allows you to isolate certain frequencies when hearing in the context of an entire song or instrument.
This is what a sine wave with a frequency of 440 Hz looks like in SPAN:
And this is what a sine of 440 Hz and another one of 880 Hz look like:
Secondly, we have the oscilloscope. This tool displays the displacement of the waveform (the movement) over time. In this case, the x-axis is time, while the y-axis is the signal itself.
You can get the oscilloscope s(M)exoscope by Smartelectronix, which is also free yet extremely useful for visualizing waveforms and also detecting phase issues.
This is what a sine wave looks like on s(M)exoscope:
The Sine Wave
The sine wave is the simplest of all waves. As we’ll see later, it’s the “mother” of all of them, since every other wave is made up of a combination of sine waves of different frequencies (harmonics). It is even described by a trigonometric equation called the sine function. This formula describes it’s movement through time.
It looks and sounds simple since it only has one frequency. You can see how it looks like on the images I showed earlier. Then, let’s listen to it live:
Sine wave played at a frequency of 440 Hz; that is, A4
In sound design, sine waves are often used to create sub-bass sounds. This is because there’s not much detail on the lower frequencies, therefore a simple wave like a sine wave fits this role perfectly. This is how a sub-bass sounds like — you will have to listen to this on a pair of headphones or with a high-end pair of speakers since some cheap sound systems don’t reproduce these lower frequencies at all.
Sine wave played at a frequency of 49 Hz; that is, G1
Before moving on to the more complex waveforms, we have to stop to explain harmonics.
As I said earlier, all waveforms are made up of adding sine waves of higher frequencies. These higher sine waves are the harmonics or overtones.
Every waveform has a fundamental frequency (which is, taking for example the voice of a singer, the frequency with which we “sing” along: in other words, it conforms the pitch), or first harmonic, followed by a bunch of higher frequencies or harmonics that are multiples of the fundamental one. These upper harmonics don’t determine the pitch, but they determine the timbre of the sound.
For example, if we play an A4 at a frequency of 440 Hz, the second harmonic would be 880 Hz, the third one would be 1320 Hz, and so on.
Note: the difference between a harmonic and an overtone is that the fundamental frequency actually is a harmonic, while it’s not an overtone. Therefore, the second harmonic mentioned before can be called first overtone, the third harmonic would be the second overtone, and so on.
Visualized in SPAN, harmonics look like this:
This is a Triangle Wave visualized through SPAN. I’ll explain this waveform later.
Now we can move on to other types of basic waveforms, where the difference between each of them is the combination of these upper harmonics.
The Square Wave
The square wave, like all waveforms, moves back and forth. What makes it special is that it jumps between the highest and lowest values, no medium point. Note: actually, there are medium points in between, since instantly jumping between two values is physically impossible — air molecules cannot teleport. That’s why it is not possible to create or even listen to a perfect square wave, but what you hear are almost exact approximations.
This shape produces only odd harmonics. This means that, for example, if we play an A3 which has a frequency of 220 Hz, the harmonics would be: 660 Hz (third harmonic), 1100 Hz (fifth harmonic), 1520 Hz (seventh harmonic) and so on. Additionally, the amplitude of a given harmonic is equal to the inverse of its harmonic number. Then, the third harmonic would have an amplitude of 1/3, the fifth would have 1/5, and so on:
In this example, we are taking a square wave oscillating at a frequency of 1 Hz. You can use a calculator for getting the harmonics.
It is a very “bright” sound. It is the typical 8-bit sound (old Nintendo soundtracks were made almost entirely by square waves). In modern music, they are also used for creating distorted basses (it the end, a Square Wave is a distorted version of the Sine).
This is what it looks like on SPAN:
And this is what a square played at 220 Hz (A3) sounds like:
There also are variations of the square wave that can be made through pulse width. These control symmetry or spacing between the two “squares” of a wave. This effect produces a pulse wave, that looks like this:
This little variation to the wave make a huge difference on the sound — it makes it feel more Nintendo-ish:
The Triangle Wave
The triangle wave looks like a mix between the square and the sine wave; and it also sounds that way.
Such as the square wave, the triangle has only odd harmonics. The difference is that these fade out quicker than the Square. It sounds like if we applied a low pass filter to a square wave.
In this case, the amplitude of a given harmonic is equal to the square of its harmonic number. Taking again the example of a triangle wave played at a frequency of 220 Hz, the third harmonic (660 Hz) will have an amplitude of around 0.1 (1/3^2), the fifth harmonic will have an amplitude of 0.04 (1/5^2), and so on.
In this example, we are taking a triangle wave oscillating at a frequency of 1 Hz
This waveform sounds really good on lower frequencies therefore it’s commonly used to create bass lines, but it also sounds pretty good on the lead melody line of a song. They are also often used for frequency modulation, due to its simple shape.
This is what a triangle wave looks like on SPAN:
And this is what a triangle wave at a frequency of 220 Hz (A4) sounds like:
The Saw Wave
The Sawtooth (sometimes called “ramp wave”) is the “fuller” waveform since it contains all (even and odd) harmonics, which means it’s the most complex of these four basic waveforms. It sounds really harsh and bright, even more than the square.
Just like the square wave, the amplitude of a given harmonic is equal to the inverse of its harmonic number. Therefore, the second harmonic will have an amplitude of 1/2, the third one will have an amplitude of 1/3, and so on.
This wave is pretty often the one you will choose when doing subtractive synthesis because, due to its richness in terms of harmonics, you have a lot to subtract and shape.
The saw wave is used in almost every kind of electronic instrument, although it’s not very common on sub-bass sounds. Its texture is somehow similar to a trumpet. One of its most famous uses is to create a supersaw lead, which consists of layering several saw waves (around 16), each of them slightly detuned (this process is made with the unison tool, which you can find in most synthesizers).
This is what a saw wave looks like on SPAN:
And this is what a saw wave played at 220 Hz sounds like:
Noise is not a waveform, but an option for sound generation that most synthesizers have. It is an oscillator that produces random waves.
Since it’s completely randomly generated, it has no consistency. Therefore, it has no harmonics nor frequency/pitch. Additionally, this sound will cover the entire frequency spectrum, unless you cut it out through filters.
There are various types of noise. The one showed below is called white noise. There’s also, for example, what’s called pink noise, which is the same as white noise, with the difference that the amplitude will be the same for every frequency along the spectrum.
This is what white noise looks like on SPAN:
And this is how it sounds like:
In music production, noise is commonly high-passed using audio equalization techniques, in order to fill up the higher frequencies of the song, which creates a lot more energy. You usually won’t use it on the low-end because, as I said earlier, there’s not much definition down there, so you don’t want such a chaotic mess on the lower frequencies (although it’s sometimes used there to create some effects like wind sounds or explosions).
So these are the most basic waveforms that you will find in almost any synthesizer and are the base of almost every sound. Now that you know the basics of waveforms, and to end this article, let’s go through two concepts that I find essential to understand: phase and voicing.
The phase of a wave is basically, musically speaking, at which point its period starts when you trigger it with your keyboard. In other words, it’s the amount of offset applied to a wave (how much we “push it” back or forward), and it’s measured in degrees.
Most synthesizers will let you choose what this point will be. Some of them, like Serum, let you randomize this point, in order to avoid problems.
Knowing this concept is key because, if we play two equal waveforms at an exact opposite phase, they will cancel each other when they add on. This effect is known as phase canceling, and it’s based on the fact that 1 + (-1) equals 0.
Voicing comes in when you hit more than one note on your keyboard at the same time. This means that you are adding waves on top of each other. For example, take a look at these two sine waves of different frequencies, that when combined, create a much more complex waveform.
This may seem pretty simple, but remember that we are using sine waves (the most basic waveform), and only two of them.
Imagine what you can create if you combine more than two waves (in fully produced songs there are tons of them sounding at the same time) and using more complex waveforms such as the square or saw. That way you have endless possibilities on what you can create combining waves.
Waveforms are signals that, when translated to the real physical world, move the air particles in order to produce audible sound.
The sine wave is the most basic yet the most important waveform. Every musical sound (and, basically, every sound in the universe) is made up of tons of sine waves of different frequencies added on top of each other. Even other basic waveforms consist of a lot of harmonics which at the same time are sine waves.
If you are a sound designer, it’s important to understand how each wave sounds like in order to choose the right one when doing synthesis.
There are much more complex waves (such as the bell waveform or the acid ones) which we’ll cover on other articles. But, as always, those are also created by the addition of sine waves.
If you want to learn more about sound design, you can check our article on the basics of synthesis and sound design, where we go through the most important concepts of this world (and we even have a dedicated article for each of them!).
Finally, I hope this article gave you a better idea about the physics and mathematics of sound waves, especially for sound designing.