Basic Waveforms | Fundamental Frequencies & Harmonics
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In the previous lessons, we learned about what a frequency is, and the different properties a frequency can have. Such as Amplitude and Polarity. We also looked at Phase, and how that can affect the sounds we hear.
In this lesson we’ll look at different types of basic waveforms, as well as what Harmonics are, and how they play a role in shaping waveforms and “coloring” the sounds we hear.
- There are 5 basic waveforms.
-Sine,
-Triangle,
-Sawtooth,
-Square,
-Pulse.
But before we get too deep into waveforms let's talk about what Harmonics are.
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So, by now I'm hoping you already know what a frequency is, and that Humans can hear from 20Hz - 20kHz. We’re about to expand on this greatly.
At its definition, a Harmonic is a whole number multiple of a fundamental frequency.
Let's dive into this! So we know what a frequency is, but what is a fundamental frequency? At its core, a fundamental frequency is the lowest frequency relative to its harmonics.
A fundamental frequency can have an infinite amount of harmonics, but humans can only hear so high (20kHz). Thankfully this typically allows us to work with fairly small numbers.
When referring to harmonics we typically refer to them as their multiple, or the 2nd, 3rd, 4th, 5th, 6th, 7th… harmonic.
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For example, let's start with 60Hz:
60x2= 120Hz - this would be the 2nd harmonic.
60x3= 180Hz - this would be your 3rd Harmonic.
60x4= 240Hz - this is your 4th harmonic.
60x5= 300 - this is your 5th harmonic.
60x6= 360 - this is your 6th harmonic.
In all of these equations, 60Hz is our fundamental frequency. You see, The fundamental is the lowest, or the foundational frequency that the harmonics were built on top of. Notice the harmonics stared at the 2nd multiple. This is because the first, the origin, is the lowest frequency and therefore is the fundamental.
Without 60Hz in our example, we wouldn't have the ability to calculate a harmonic. Without the fundamental, there are no harmonics.
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Another example, let's look at the piano. I'm picking G4. This note relates to 391.99Hz. I picked this number because it's already relatively high. Lets look at it's harmonics.
So 391.99Hz is our fundamental now. To find our 3rd harmonic, we multiply 391.99Hz X 3. To find the 5th harmonic, we multiply by 5.
1st harmonic (fundamental): 391.99 Hz
2nd harmonic: 783.98 Hz
3rd harmonic: 1175.97 Hz
4th harmonic: 1567.96 Hz
5th harmonic: 1959.95 Hz
6th harmonic: 2351.94 Hz
7th harmonic: 2743.93 Hz
This essentially goes on into infinity, but again, we can only hear up to 20kHz. So if 391.99Hz is our fundamental or, the lowest frequency relative to the harmonics. The last harmonic we would be able to physically hear would be the 51st harmonic,
which is 391.99 x 51 = 19,991.45Hz.
However, if we started with a lower fundamental, the number of the highest audible harmonic would be greater. Simply because it took more Multiples for the lower number to reach past 20kHz than it did for the already high numbered frequency.
- Now that we've covered harmonics, let's take a look at some different wave shapes!
First off, and most importantly.
The Sine or Sinusoidal Wave.
This is the wave that we are very familiar with already. The Sine wave is the purest of waveforms. It is mathematically considered a pure tone because it only contains A single frequency. In other words it only contains a fundamental with no harmonics. This is the wave that we've been looking at for the past few lessons due to this fact.
The Triangle Wave.
This wave looks very close to a Sine wave. The difference between the sine wave and the Triangle wave isn't just its shape. The Triangle wave contains harmonics! Here's the catch though, it only contains the odd numbered harmonics. So 3, 5, 7, 9, 11, and so on to infinity..
The next thing to know, is that the amplitudes of the harmonics are not equal to the amplitude of the fundamental. In fact, there is an equation to find the amplitudes of each harmonic in order to produce the waves correctly.
The fundamental divided by the square of the harmonic multiple
Don't worry! We don't have to memorize equations Like this in order to work with audio. We have many tools available to us that will both generate these waves perfectly for us as well as allow us to see the harmonics too. It’s important to at least have a foundational understanding of what is truly happening under the hood.
The Saw / Sawtooth Wave.
This wave is my personal favorite. A saw wave contains All harmonics. Both the even and odd. It gets its name from its shape, with its ramped up and instantaneously dropped amplitude, just like a saw blade. This creates a very full and rich sound which we will demonstrate later.
The Square Wave.
This wave has immediate changes in amplitude. Think of turning a light switch on and off.
Much like the triangle wave, the square is made of only the odd Harmonics. The difference between the square and the triangle, are the amplitudes of the harmonics themselves. With the square wave, the amplitudes of its harmonics are much higher, and more consistent through the higher ranges. The amplitudes of the triangle wave’s harmonics decrease more rapidly the higher they go. This makes the square wave a sharper, fuller sound. Because it contains more harmonics that we can hear because they are simply louder.
The Pulse wave.
This wave is also square wave BUT, the main difference between a pulse and a square is the Duty Cycle of the positive amplitude. The duty cycle is how long the positive amplitude stays Positive. If we look at a square wave, or any of the other basic waves for that matter. They are +50% positive and -50% negative for the most part. A pulse wave is more so +20% / -80%. Or is can change to +40% / -60%, or it could even be +1% / -99%.
A Pulse wave’s positive amplitude can vary from 1% to 100%. You’ll hear this wave also called a Pulse Width, because of this particular feature of the pulse wave.
- Okay, we just learned a lot of new things. Let's sum it all up by adding the new terms along with the terms we learned from the previous lessons!
Harmonics
So what are harmonics? Harmonics are frequencies that are the whole number multiples of a fundamental frequency while reducing in amplitude the higher the multiple is.
Sine Wave
A sine wave is considered a pure wave because it only contains a fundamental frequency and has no harmonics.
Triangle Wave
A Triangle wave consists of a fundamental frequency and all of its odd numbered harmonics (3, 5, 7..). The amplitude of each harmonic is inversely proportional to the square of the harmonic. (1/n^2)
Saw Wave
A Saw wave consists of a Fundamental frequency and all of its harmonics, both the even and the odd. The amplitudes of each harmonic reduces proportionately to the number of the harmonic. (1/n)
Square Wave
Square wave consists of a fundamental frequency and only the odd numbered harmonics. The amplitude of each harmonic decreases in proportion to the number of the harmonic.
(1/n)
Pulse wave
A Pulse wave consists of the same structure as a square wave, with the major difference being the variable Duty Cycle of the positive amplitude. Instead of a 50/50 cycle like a square wave, a pulse can be anywhere from 1/99 or 49/51.