Category Archives: Frequency Modulation – It’s Not So Bad

Frequency Modulation – It’s Not So Bad Part I: Introduction

I don’t claim to be an expert in FM; I don’t claim to be especially good at it, for that matter. I do claim, however, that FM is not as hard or scary as you might have heard and I hope to prove this over the course of three tutorials. From my tinkering with FM, I have come to the conclusion that it is difficult to learn because of the dirth of good user interfaces, rather than any property of the method, itself.

The trouble is that FM synthesizers is that they, by necessity, have more parameters that need to be set than most others; presenting them all intuitively is difficult. Conceptually, FM is actually simpler than subtractive, in my opinion. Whereas with subtractive, you build a complex sound to then take away from, with FM, you just build a complex sound.

For these tutorials, we’ll be using Thor as our FM sythesizer (it’s not ideal but it’ll do) and in this first tutorial, we’ll be getting ourselves acquainted with some basic FM concepts. Create a Thor, initialize the patch, remove the filter, set the amp sustain to full, and replace the analog oscillator with an FM pair oscillator.

Observations on Basic FM Sounds

The math averse among us need not fear; we’ll be using our ears. For the following exercises, it would be best to stick around middle c.

The first observation I’d like to make is that raising and lowering the modulation amount produces very different changes in timbre than raising or lowering a filter cutoff frequency. If we were to perform a filter sweep, only the harmonics above the cutoff frequency would be affected and they would be affected linearly, that is, their amplitudes would change at a steady rate and in just one direction (resonance notwithstanding). If we sweep the FM amount knob on the oscillator, we’ll find that this is not the case for FM. All harmonics are affected and the effect on their respective amplitudes is non-linear.

Those with an ear for this sort of thing will have noticed that the default ratio (1:1) produces the full harmonic spectrum. Now let’s observe what happens when the frequency of the modulator is raised. You should hear that gaps in the spectrum appear. With ratios above 1:10, or so, sounds can take on an enharmonic quality, though you’re actually just hearing higher harmonics than you’re used to (observation 3).

For observation 4, we’ll compare two ratios, 1:2 and 1:4. (We’ll be sticking with 1:n ratios, as they’re the most musical.) If you listen closely, you may be able to tell that they produce the same harmonics (odds) but at different levels. If you can’t hear it, try comparing 1:2 with an FM amount of ’44’ to 1:4 with an FM amount of ’19;’ the sounds should be quite similar. Different ratios can produce the same harmonics, they just won’t be at the the same amplitudes.

Next, we’ll turn the FM amount knob back to ‘0’ and create a second FM pair and connect it as a modulator in the mod matrix, with an amount of ~35. You should hear something similar to a filtered sawtooth. So far we’ve only used perfect 1:n ratios. Listen to what happens when you detune the modulator (+/-10 cents). You should hear beating (observation 5).

Now disconnect the second pair from the first and connect it to the output. Set the two pairs to different 1:n ratios and turn up the FM amount. Make careful note of the sound. Disconnect the second pair, turn the FM amount down, and set the carrier frequency to that of the first pair’s modulator. Create a third FM pair and set the carrier frequency to that of the second’s modulator. If you connect the second and third pairs as modulators to the first pair in the mod matrix, you should be able to produce the same sound as you just heard, because two modulators modulating a carrier is the same as two modulators modulating separate carriers of the same frequency (observation six).

To recap:

1. Amplitude changes of a harmonic are nonlinear.

2. 1:1 produces the full harmonic series; other 1:n ratios above have gaps.

3. High 1:n ratios can sound enharmonic, though they aren’t.

4. Different ratios can produce the same harmonics but at different amplitudes.

5. Detuning the operators results in beating but only of the overtones.

6. Two or more modulators modulating a single carrier is the same as two modulators modulating separate carriers of the same frequency.

All FM Is Not the Same

This observation has a big effect on FM in Thor, so I’ve given it its own heading: there are different kinds of FM. The FM pair oscillators and the mod matrix use different algorithms and so produce different sounds. The oscillators use the same technology as the DX synthesizers, whereas the matrix is based on analog technology. The analog based algorithm has two key deficiencies: artifacts at low FM amounts and a dulling of the sound as you play up the keyboard. There’s no avoiding the former but the latter can be compensated for, somewhat, with key-scaling.

These deficiencies will become important in tutorials II and III.

Some Math, for the Curious

This is the equation for the frequencies generated:

Equation Image 2


In English, there are sidebands (new partials) of frequencies (‘w’) equal the carrier plus and minus integer multiples (‘n‘) of the modulator. So, for 1:1, there are positive sidebands of 2c (c+1m), 3c (c+2m), and so on. Negative – yes, negative – sidebands are 0c (c-1m), -1c (c-2m), and so on. The negative sidebands are “reflected” back at the reverse polarity.

The reason different ratios can produce the same harmonics but at different amplitudes is that the harmonics will be generated as different sidebands. In the case of 1:2 and 1:4, the first positive sidebands will be the third and fifth harmonics, respectively; the amplitudes of the sidebands will be the same but they’re representing different harmonics.

A detuned modulator will create beating because they generate detuned sidebands. Let’s take a ratio of 1:99, as an example: positive sidebands will be 1.99c (c+1m), 2.98c (c+2m), 3.97c (c+3m), and so on; reflected sidebands will be .98c (c-2m), 1.97c (c-3m), 2.96c (c-4m), and so on. Obviously, this is to the detriment of intonation. Happily, though, a detuned carrier is mathematically the same and mitigates the flattening effect.

The equation below determines the amplitude of sidebands:

Equation Image 3

In English, the modulation index (‘β‘), i.e., brightness, is equal to the change in the carrier’s frequency divided by the modulator’s frequency. Using this equation to actually calculate the amplitude of a given sideband requires can’t really be done with pen and paper but, in practice, it’s not necessary. However, I would like to make note of the denomitator, the modulator’s frequency. As it increases, i.e., as you play up the keyboard), the modulation index (‘β‘), i.e., the number and aplitude of sidebands, decreases. Dedicated FM synthesizers compensate for this; analog and analog based synthesizers do not.

Part II: Synthesizing Bells

Part III: Synthesizing Brass

Equation images courtesy of Gordon Reid.