Even and Odd Harmonics (Synthesis and Sound Design Training)

An Omega student offers his perspective on learning about sound at The Omega Studios’ School

Written by Jeremy Quintin for The Omega Studios’ School

A big hello to all the readers out there! My name’s Jeremy and I’m a new student at The Omega Studios’ School. Before coming to this school, I already had a bit of background in electronic music and sound design, with my personal projects in Fruity Loop Studios (FL). I make a lot of goofy electronica basically. Having developed that experience on my own definitely helped me keep up to pace with some of the early work involved in the school’s educational program. Yet even with this pre-existing knowhow, I’ve already been given a lot of new information.  This has definitely expanded my comprehension of some of the most fundamental characteristics of sound waves.

Here’s a good example: before taking synthesis and sound design classes at Omega, I never knew what to call all those additional frequencies you see in complex waveforms. You know what I’m talking about, right? No? If you ever opened up a spectrogram before then you absolutely know, but here’s a picture of a spectrogram for those who don’t:

“Oh you mean that visualizer with the lines!” Yeah exactly! Although it’s a bit more than a visualizer. It’s a tool one can use to analyze the frequency, or frequencies, of sounds and songs. Every white line on the above diagram represents a frequency of one instrument.  As this sadly small picture points out, the bottom line is the fundamental frequency, while every line above it is an overtone (the two sections, upper and lower, represent the left and right in a stereo recording). Some of the overtones are known as harmonics, these are whole multiples of the fundamental frequency (if ƒ is the fundamental, then 2 x ƒ, 3 x ƒ, and 4 x ƒ represent the first three harmonics in the series). Harmonics are generally thought to be the most pleasant overtones to accompany the fundamental. These frequencies working together create what most people understand as the unique sound of an instrument, which is called the “timbre.” If you don’t understand anything else of what I just said, know that a violin sounds the way it does because it has a unique collection of those overtones playing together. If that unique collection is changed at all, you begin to feel that you are no longer listening to a violin.

I knew about Timbre, but that all those lines were assigned names such as “fundamental” and “harmonics” was brand new information to me!

“This guy’s an electronica composer and he didn’t know about harmonics? Wow.” I hear you say, as if I can’t sense your judgment through the internet. Well, I never knew much about the process of how harmonics were determined, but I did know that each of those lines represented a frequency, and that was all I needed to know once upon a time. You don’t expect a pianist to know what the harmonic spectrum of a piano is in order to play it, do you?

That said, now that I do know the names and the properties of these overtones, I can say that I’ve become very interested in them. I was especially interested in the idea of even harmonics (even numbered multiples), which are said to sound more pleasing than odd harmonics (odd numbered multiples), which are said to sound more dissonant. Now, if you’re like me, you can’t help but find it peculiar that a sound such as a guitar or a piano, which most folks find pleasant sounding, might be harboring some dissonant tones, which most folks understand synonymously with poor composition.

I got to talking with my teacher Neal Keller about even and odd harmonics, and he mentioned to me by way of example that square waves are composed entirely of odd harmonics. That seemed pretty crazy to me because I think square waves sound pretty good! “And you’re telling me they’re filled with dissonant tones?” Well, Neal didn’t say “dissonant”, he suggested “uneven” instead. Trying to get to the bottom of this, I decided to take a look at a square wave using FL‘s spectrogram analyzer Wave Candy. Sure enough, the harmonics series emphasizes F, F3, F5, etc. Furthermore, if you separate out those frequencies using a band-pass filter (there’s my EDM production knowledge kicking in) to listen to them separately, they are very clearly different notes as opposed to all one note at multiple octaves.

Just to make sure I wasn’t misreading anything, I decided to try and recreate the square wave using Fl‘s 3xOSC, a synthesizer which can generate single sine waves. Since any single frequency is a sine wave, and since Fourier’s Theorem states that all complex sounds are composed of combinations of simple sine waves, it should be possible for me to rebuild a square wave. Using Fruity‘s Spectrogram Wave Candy, I should be able to pick out the correct frequencies alongside 3xOSC and to write those frequencies down. I began at an A2 of 110 Hz to make sure I had a lot of room, started stacking my multiples in the piano roll, and sure enough a square wave began to form in the oscilloscope:

(Word of Warning: FL‘s piano roll does not allow you to manipulate the frequencies of notes, so notes were chosen based on being the closest frequency representations of the harmonics I had to record. This can put the frequency off ± 5 hz in early low overtones to ± 100 hz in much higher overtones)

The waveform pictured above is formed only after adding about four odd harmonics above the fundamental.  Proper square waves have dozens.

Now that’s pretty strange, but it’s important to understand a couple things about overtones and additive synthesis, which is essentially what I’m doing here. For one, overtones aren’t just different frequencies. They also have different amplitudes. If all the overtones of a sound were at the same amplitude, or the same volume, then you’d probably get some mad destructive and constructive interference all over your waveform, making it look more like an indecisive worm and sound more like a horn being jammed into your ear (as I learned the hard way). Each of the harmonics I added in have actually had their amplitudes cut by about half of the preceding harmonic, which seemed to be close to how the amplitudes measured up in the pre-constructed square wave I used as a guide. The reduction in amplitude not only reduces how much the complex wave gets shaped by those additional frequencies, but also how obvious that dissonance is.

For two, in music theory dissonant notes aren’t really described as dissonant until they are in harmonic opposition with a chord or progression. More simply put, notes don’t sound bad until other notes make them sound bad. Even knowing this, dissonant notes have their place in music theory. They serve to create more interesting progressions, as well as jarring chords, which hold the listener’s attention (if not their interest). Some people like dissonant notes and others do not, and sometimes people don’t even realize when a note is dissonant because the composition makes it sound pretty good anyway.

So what does this say about sound? Basically, these waveforms which we already describe as complex are a lot more complex than we give them credit for. Musical composition, as I’ve pointed out, already makes room for dissonant sounds, and as it turns out, so does sound itself! The very common instruments which we use to craft either harmonic or dissonant chords can in fact have harmonic or dissonant elements within them, all dedicated to creating unique voices that say “hey, this is a violin.” The implications of that are pretty cool. If you learn how to count harmonics, then you can pretty much play with the building blocks of sound. It’s the equivalent of a physicist playing with atoms to make T-Rexes –except he has to call on us sound designers to make the roar.  I don’t want anyone getting carried away here, though. Creating a dinosaur roar requires far more complex manipulations than simply adding harmonics, just as splitting atoms is not going to create life. Even so, knowing how to add and remove harmonics properly can allow you to create your own original synthesized sounds. Plenty of good use there for the gaming industry music or EDM scene!

Jeremy Quintin