An Introduction to FM Synthesis

 

Frequency Modulation synthesis (FM synthesis) is a method for sound creation that generates very complex and unique timbres containing many spectral components. As a matter of fact, the first FM algorithm was invented by John Chowing in 1967 and the technique was made popular when Yamaha released the first commercial digital synthesizer, the DX7, in the early 1980s.

 

DX7

 

All things considered, FM is perhaps the most fragile and temperamental synthesis technique. The basic idea is to periodically control one oscillator’s frequency with another oscillator at a rate that is fast enough to hear as a tone rather than a rhythmic pulse (like what LFOs do). Therefore, the oscillator that is generating the center frequency is called the “carrier” and the oscillator that is generating the control frequency is called the “modulator”.

This is what a frequency modulated waveform looks like in NI FM8. Notice the number of cycles per time period varies periodically. 

 

Frequency Modulated Waveform

 

Singer’s Analogy 

 

Hypothetically, if you were to “vibrato” faster than 20 Hz (20 times per second), the effect would become a timbral change. Likewise in FM (and AM), when you push the modulator frequency into auditory rate, the effect is a timbral change. In the example below, you can hear the effect go from rhythmic to timbral as I gradually increase the rate of modulation into auditory rate. 

 

 

This timbral change happens because modulating pitch of an oscillator at an auditory rate adds spectral components to the output called “sidebands”. In addition, the amplitudes and position of these “sidebands” can be calculated mathematically using “Bessel Functions” –  functions involving the ratio of the carrier frequency to the modulator frequency (C:M ratio), “frequency deviation”, and the modulation “index”. However, no math is necessary to create sound. This is the jist of it…..

  • Whole number C:M ratios like 1:2 produce more harmonic and consonant sidebands
  • Irregular C:M ratios like 1:7.13 produce noisier, dissonant content.
  • Slightly inharmonic C:M ratios like 1:2.5 often produce clangorous, metallic sounds
  • The deeper the modulation index, the more sidebands.

For example, if your carrier (the center frequency) is 220Hz and your modulator (the control oscillator) is 440Hz, the C:M ratio is 1:2 and the result will be harmonious. Some FM synths like NI FM8 modify the modulating frequency to match the ratio for each note played so that the timbre stays more or less consistent across the keyboard. FM8 also depolarizes sidebands to avoid confusion (it makes all the frequencies and amplitudes positive). Negative sidebands are only necessary for understanding Chowning’s mathematical models for prediction.

 

Where are these parameters in FM8?

 

FM’s “matrix” is where you decide the number of oscillators (FM8 calls them Operators). You can also change the depth and routing in the “expert” section.

 

FM 1

FM 3

 

In the picture above, the lighter colored bottom row is the output. Any operator that is routed to that (F in this case) will be “carried”. Right click on operators to enable them. Moreover, to modulate an operator, click and drag in the box that “lines up” with both operators, like the “33” box above. “33” is arbitrary. You can select any value from 1 – 100.

In addition to this, within the expert tab, you can determine the C:M ratio in the highlighted box below. Each operator can have its own ratio; however, having a ratio other than 1 on the carrier will transpose the fundamental frequency.

 

FM 4

 

In the “spectrum” tab, you can see the unique sidebands that are being produced by your FM sound in a spectrograph (amplitude over frequency) and the corresponding waveform (amplitude over time). You will notice when experimenting that the sidebands are more evenly spaced in “nice number” C:M ratios and unevenly spaced with irregular C:M ratios.

 

FM 7

 

Enveloping Operators in FM

 

FM8 allows the user to customize the temporal envelope (amplitude over time) of each oscillator. This is useful when trying to emulate bell-like and “plucked” sounds because most of the noisy, inharmonic energy in these sounds exists within the initial impulse, leaving the tone to ring and decay over a significantly longer amount of time. On the contrary, making ambient, evolving, spacey sounds merits linking envelopes together. These sounds don’t need that initial, characteristic impulse because there is no distinction between the sound and what caused it. For further temporal control, try enveloping the filter (operator Z). Here is where you’ll find the enveloping options in FM8.

 

FM 5

 

So, here are a few basic patches I made in FM8. 

“Panbell Flute”

 

“Vocal Organ”

 

“Plucked Lead”

 

You can download the patches here…..

 

 

 

Reference:

Chowning. Bristow. “FM Theory and Applications” By Musicians for Musicians. Chapter 4. Yamaha Music Foundation. 1986