How to Sculpt Sounds With FM Synthesis

If you’ve already read my other blog on FM synthesis, then you’re ready for part two: an extensive tutorial on how to design sounds using FM synthesis. To give ambitious beginners a helping hand, I’ll be covering terms like operator, carrier, modulator, envelope, scaling and feedback, giving you all the basic knowledge you need to use either FM-based hardware such as real synthesizers, or FM-based software such as the Native Instruments FM8 virtual synth (included with Komplete) to sculpt your own sounds.

Operators: Carriers and Modulators

In part one, I explained basic waveforms (audible sound) and fast vibrato — two concepts that have a different name in the context of FM synthesis. Here, a carrier represents an audible sound, while fast vibrato is referred to as a modulator. In other words, if an FM synth produces a sound, that’s always the consequence of a carrier, while a modulator is primarily used for fast vibrato, so it’s not something you can actually hear directly, but you can hear the effect of it. Technically speaking, a modulator can be used as a carrier if your synth or plug-in allows for it, but this isn’t something you’ll actually run into much. More on that in a bit. Schematically, modulators and carriers are represented by squares and arrows. Here’s a basic example:

Modulators Modulate Carriers

The rightmost icon above represents a speaker. Formally, this is the most complete way to notate FM. In reality, most FM synthesizers and plug-ins don’t include the speaker icon since it’s little more than a formality, which is also why I won’t be bringing it up again in the rest of this article. As you can see, the example above includes one carrier, which is more or less comparable to the standard oscillator of an analogue, virtual-analogue or sample-based synthesizer. The modulator in the scheme represents a fast vibrato. In any other synthesizer, this would probably be called an LFO, but that’s actually not really a good comparison since in FM synthesis, the modulator usually isn’t audible. The modulator simply serves to modulate the frequency of the carrier. Also, schemes like the one above can also come in vertical form:

Operators and Algorithms

In FM synthesis schemes, every square is an operator, no matter the function. The way the operators are linked is what we refer to as the algorithm. The scheme, or algorithm seen above includes four operators. The position of each operator determines whether it’s a carrier or a modulator. In this case, 1 and 3 are the carriers since they’re the final stop. As mentioned earlier, the carriers represent audible sounds which means they could easily represent two oscillators. Operators 2 and 4 are the modulators here. Now let’s look at something completely different:

Single Carrier with Multiple Modulators

The algorithm above also has four operators, but this time, they’re linked differently. This results in a different sound. Here, there’s only one carrier (operator 1) and three modulators (operators 2, 3 and 4), where modulator 4 has a direct effect on modulator 3 and an indirect effect on modulator 2 since it modulates the modulator that comes before it. The carrier is indirectly influenced in an even more complex way, since modulator 4 modulates two other modulators on its way down to the carrier. Compared to a more traditional synthesizer, there’s only one audible oscillator here while the timbre is modified in three different ways. Also, bear in mind that we’re not factoring in the output level yet. If the output setting for modulator 2 is set to a low value, the modulators that come before it obviously won’t impact the carrier as much as when the output is set to a higher value.

Two Equally Effective Modulators

The algorithm above puts a spin on the single-oscillator algorithm we’ve just described. Here, operators 3 and 4 hold the same amount of power over operator 2, but the difference is that operator 4 doesn’t have an effect on operator 3. As such, the resulting sound will be different from the sound produced by the previous algorithm.

A Mass of Options

Take what you’ve just learned and imagine how many different ways you can put four operators together. That’s FM synthesis in a nutshell. It’s easy to grasp but the options are near endless. The challenge lies in recognising and managing all of that potential. Remember the Yamaha DX7 I mentioned in part one? That’s actually a 6-operator, 32-algorithm synthesizer, and it doesn’t even come close to what Yamaha did with the 8-operator, 88-algorithm FS1r back in the late nineties. If you consider each algorithm to be a different planet with its own lifeforms, any multi-algorithm FM synth is basically a massive solar system brimming with sonic life. At the same time, this is also why fixed FM algorithms were a bit of a problem back in the day. If you didn’t know which algorithm you needed before starting a project and hit a brick wall somewhere down the road, you were basically doomed. You could still switch algorithms of course, but chances were that any operators you had already set wouldn’t work as intended. This could even end up turning certain modulators into carriers. These days, it’s hard to imagine that the designers of the DX7 never saw these programming issues coming. Even the now ultra-rare-and-ultra-expensive Jellinghaus DX Programmer – a custom DX7 controller of which only 25 were ever made – only made things just a little bit easier. Go figure.

Tweaking Operator Properties

So far, we’ve only looked at FM synthesis in its most basic form. Since using modulators and carriers alone won’t get you more than static waveforms, next up, we’re going to look at ways you can augment your operators. The list of parameters below roughly matches what most hardware and software-based FM synthesizers offer up and I’ll go over them one by one.

Waveform
Pitch
Envelope
Scaling
Velocity
Output Level

Waveform

With most FM synthesizers, the waveform of your operator is a sine wave, which makes sense because sine waves are the most basic sounds you can generate. As such, working with a sine wave means you can keep things simple, but you’re also free to go over the top or go with a different waveform altogether. There are even hardware-based FM synths that offer up multiple waveforms, including the Yamaha TX81z rack module, the aforementioned FS1r, and various Yamaha SY/TG Series models. The thousands of options that a simple sine wave gives you are already pretty impressive, so you might expect that the options are doubled with every extra waveform. Sadly, that’s not exactly the case. While it’s true that every waveform comes with its own palette of sounds, the quality of those sounds differs greatly from one waveform to another. Square waves and sawtooth waves are technically too complex to turn into stunning instruments, as evidenced by Yamaha’s SY/TG Series models. These synths combine traditional FM synthesis with samples, where the samplers can be used as operators. In other words, instead of a sine wave, you have the option to load a piano sample. On paper, this sounds great. In reality, the sounds were practically unusable, and arguably even downright dreadful. Nevertheless, in the hands of experienced sound designers, even complex waveforms can yield interesting results. The best alternative operator waveforms are the waveforms that sort of keep things simple and resemble the good old sine wave, like triangle waves.

Frequency (and Ratio)

The pitch of your FM operators plays a very important role. If your carrier has an 80 Hz frequency and your modular comes in at 133 Hz, you’re never going to get a beautiful sound, but as John Chowning discovered, clean ratios can help solve this. Take a carrier at 80 Hz combined with a modulator at 160 Hz. This equals a ratio of 2:1, which along with 1:1, 3:1 and 4:1 are numbers you’ll want to remember because FM thrives on ratios like these. Just to be clear, a 2:1 ratio is essentially the same as a 4:2 and 8:4 ratio. The only difference between these ratios is that the produced pitch octaves upwards. So what about a 3.5:1 ratio, you might ask. This still works, despite the fact it includes an uneven and unrounded number, which is simply because 3.5:1 is the same as 7:2. A 263:137 ratio on the other hand isn’t going to get you any satisfying results because it can’t be reduced to something useful. For solid FM results, it’s essential that the ratio isn’t too extreme. A 14:1 ratio works well for e-pianos, while a 15:4 ratio is great if you’re creating bell-like sounds. So does the number on the left (modulator) need to be bigger than the number on the right (carrier)? Technically no, but this is usually the case since, if the typical order is flipped, the audible sound quickly moves into the referee whistle range which, more often than note, isn’t what you’re looking for — not even if you’re trying to get as close as possible to the sound of a violin or clarinet.

Envelope

Since it determines the initial attack and what happens from there, the envelope is just as vital for the sound as the frequency. Instruments that are triggered by a strum or strike (e.g. guitars and xylophones), have an envelope that pops off instantly and then continues to drop in intensity. Instruments such as violins, trumpets and flutes have an envelope that doesn’t drop in intensity, but does require an extremely fast attack. With FM synthesizers, you get an envelope for each operator. In the case of the carrier, the envelope determines the loudness, while with modulators, the envelope changes the intensity of the FM timbre, allowing you to vary the FM effect. So, if you’re running a custom, software-created 6-operator algorithm, every operator can be given its own envelope, which gives you lots of room for experimentation and shaping beautiful, comprehensive sounds.

Scaling

You might have noticed that the high frequencies produced by acoustic instruments aren’t packed with a lot of timbre. In fact, you can play the same high note on an oboe and a clarinet and it’ll be really hard to hear any difference. That’s why, for convincingly mimicked sound, it’s essential that synthesizers are able to follow changing timbres across the higher registers. To do this, analogue synthesizers use filter-scaling, where a low-pass filter makes the higher notes sound slightly duller. FM synths do something similar but also allow you to set a unique scaling value per operator. Here, scaling works just like envelopes, meaning the carrier is linked to the loudness, while modulators are linked to the timbre. Modern software usually lets you set the scaling in the form of a curve in a graph, where the output level of a given operator gradually changes as the pitch goes up. Needless to say, more primitive synths are usually limited to more basic scaling settings.

Velocity

The velocity sensitivity refers to how harshly or gently a synthesis parameter responds to keystrokes. This is one of the most powerful and expression-impacting things you can do with FM. The output of your operators can be varied depending on the power behind your keystrokes, and you can change this setting per operator too. Compared to analogue and sample synthesizers, which only allow you to work with loudness and filter depth when it comes to velocity sensitivity, the difference is huge. That’s because FM actually lets you tweak the core of the sound by coupling the velocity to the output of your modulator. This gives you the ability to go from complex (fast attack) to pure sine wave (soft attack).

Output Level

I already touched on output a handful of times. This parameter literally determines the output per operator, where again, your carrier is linked to the loudness, while your modulators are linked to the timbre, or FM intensity. While output pretty much speaks for itself, its importance shouldn’t be overlooked because most of the time, you’ll end up fine-tuning your sound by fiddling with the output levels of your operators.

Lively Sound

The envelope, scaling, velocity and output level are your main determiners for injecting life into FM-created sounds. Apart from maybe physical modelling, this is also where FM shakes off other synthesis models. The musicality and tonal expression of FM is simply unprecedented when it comes to creating synthetic sounds with acoustic qualities.

Feedback

Something we’ve yet to look at is feedback, which is the hardest to visualise with an oscilloscope. In the scheme below, you can see that the output of the operator, which could be a carrier or a modulator, moves down. The red arrow indicates that the output is also fed back to the operator. In other words, the operator output modulates its own frequency. Next up, I’ll show you why this is a good thing.

Turning Sine Waves into Sawtooth and Square Waves

Back in the early days of FM, it didn’t take long before someone figured out that while modulating operators could result in thousands of unique sounds, the truly clear and pure sawtooth and square wave-based sounds that analogue synthesizers offered up weren’t on the table — unless you were lucky enough to own an FM synth that gave you those waveforms. And then feedback changed everything. The classic FM algorithm seen below – which is also known as ‘simple FM’ – incorporates feedback and is all you need to generate a solid sawtooth or square wave. The frequency ratio for a sawtooth is 1:1. The frequency ratio for a square wave is 2:1. Take your pick and after that it’s just a matter of dialling in the output of operator 2 and figuring out the exact degree of feedback. You can even go for something other than a square or sawtooth wave here. That’s the beauty of FM. In general, you can say that feedback makes an operator more complex (as well as grittier in the case of more extreme values) and generates quite a bit of noise.

Dialling in the Timbre and Loudness Separately

It’s also worth noting that a single sine wave carrier and some feedback is enough to generate a pretty good sawtooth wave. The only downside is that the loudness of the carrier, so the output level, affects the signal that’s fed back to the frequency of the carrier. This means that the degree of feedback increases along with the loudness. The same thing happens with acoustic instruments, which produce more overtones when you’re pulling out louder sounds. With synthesizers, however, you’ll usually prefer a system where you can control timbre and loudness separately, so you have more control over the sound. As such, for sawtooths, you’re better off with an algorithm like the one seen above.

Multiple Feedback Loops in a Single Algorithm

The original Yamaha DX Series synthesizers served up only one feedback loop, and it came in a fixed position within the algorithm. While that was enough to create sawtooth waves with a single carrier and a single modulator, running two carriers along with a little de-tuning for a nice chorus effect was out of the question. Thankfully, modern software like Native Instruments FM8 enables you to assign a feedback loop to each operator.

Feedback to Two Carriers?

If you’re wondering if you can use a single feedbacking modulator to control two carriers, the answer is yes. In this case, two arrows will pop out of a single modulator, each of which is linked to a different carrier. That said, if both carriers are de-tuned to some degree and the frequency of your modulator sits right in the middle, it’s actually no longer possible. After all, you’d end up with two paths where, for instance, the ratio of one is 1:0.999 while the ratio of the other is 1:1.001. That’s no bueno, because you’ll just be treated to a kind of swell that affects the core of the sound, resulting in everything but the sweet chorus-style swell you were probably looking for.

Round Robins

In part one, I mentioned that round robins are one of the perks that come with FM. Let me explain that a little further: a good FM synthesizer (e.g. the FM8 as well as any Yamaha SY/TG/DX Series model) enables you to set whether the waveform of an operator should be reset for every new note. When you opt for a fixed reset, you get the exact same sound for every note you play. If you don’t allow the reset, the sound generation picks up where the waveform left off. Put simply, without resets, the waveform starts in a different position every time. Normally, this kind of effect is negligible. After all, the waveform itself doesn’t change, it simply has a different starting point. But with FM, there’s basically no such thing as ‘normal’ since changing the phase of an operator will always subtly change the sound. While it’s nothing radical, it’s definitely noticeable. This makes round robins one of the strong points of FM synthesis, especially since it significantly contributes to the acoustic properties of FM-shaped sounds where no two performances are 100% identical.

Fixed phase (due to reset):

Shifted phase (no reset applied):

Getting Started

If you made it this far, you now have all the basic tools you need to get into FM synthesis. Pro tip: keep it simple at first by using no more than two operators. You’ll be surprised how many different sounds you can create by simply playing with the link-ups between both operators and the various parameters.

Which discoveries have you made during your FM experiments so far? Share your experiences below!