FM Synthesis - An explanation without the maths.

Decitronic

Not sure how many of you'll be interested but I thought I'd post it here if it was any use, it's my latest blog post. I tried to explain FM Synthesis without having to resort to diagrams and mathematical equations.

FM Synthesis - An explantion without the maths.
Well, an attempt. This is still going to be cryptic and not strictly true at parts [for the sake of explanation] but if you don't want to delve into the maths of it all just to grasp a better understanding then this might be for you.

First of all my reason for jumping head first into FM Synthesis was that I intended on making a program that could be run with extremely limited processing resources [I.e an old net book, or modded psp] yet still attain interesting and complex results. This would allow for me to have many instruments on stage for live performances without the need for expensive computers or large and heavy devices. For this purpose FM Synthesis proved to be the best choice.


A bit of history first. John Chowning discovered the technique while at Stanford University in 1967-1968. Yamaha bought the rights to use it after it was patented by Chowning in 1975. Due to the nature of Analog oscillators and the fact that they drift it is not feasible to use them for FM Synthesis (although not impossible by any means), hence it is an almost entirely Digitally practiced form of synthesis. A famous example of a vintage FM synth is the legendary DX7 from Yamaha. In 95 the patent expired and left it open to all developers.

To understand Frequency Modulation synthesis it may be easier to explain it in comparison to other forms of synthesis. I will use the examples of Subtractive , Additive and later AM Synthesis. Subtractive, as the name suggests, alters sounds by taking away harmonics from a source. It does this mainly through filters and volume envelopes. For example, an oscillator producing a sine wave could be run through a low pass filter, at lower frequencies this would give for a much deeper sound if the cut off is set low. Resonance of the filters then stimulate frequencies around the cut off and variations upon these practices give many different sounds. Additive synthesis, again as the name suggests, alters sounds through means of addition. It is significantly more complex than subtractive means. It adds sinusoidal overtones together, this can be used to create quite realistic sounds modelling real life examples of overtone patterns or it can be used to create sounds that would never naturally occur. The overtones dynamics can also be controlled.

FM Synthesis doesn't work by directly changing a sound like Subtractive or Additive, it works by using the source along with a modulation source in the audio range and using the difference in the signals to modulate the frequencies and create a much more complex waveform. Changing a sound with a modulation source is present in other forms of synthesis, such as using an LFO in Subtractive synth to control parameters, but it is the significance of the modulator's frequency being in the audible range that is important. To work with harmonic sounds the modulating signal must be able to react with the original signal by having a harmonic relationship in some way, I.e being similar in pitch. The more modulation placed on the sound, the most complex the outcome is. Atonal and inharmonic sounds are often worked with to create bell like tones and percussive sounds. FM Synthesis is particularly popular for this.

Basically in frequency modulation the user modulates the frequency. Vibrato and Glissando do this already on many non-FM synths but it's taken to a much further and more indepth level with FM synthesis itself. John Chowning said that the idea for FM Synthesis came to him while fiddling with vibrato. He had sped it up so fast that it began to create audible sidebands which altered the sounds timbre as opposed to the increased speed of vibrato as you would expect, this is perceived as a timbral change. Now the vibrato at this speed still was just a change, a modulation, of the original frequency but due to the speed and depth of it the listener hears it as a timbral change, and there in lies one of the simplest examples of FM synthesis.

Frequency Modulation is usually done in conjunction with Amplitude modulation, the amplitude may also appear to change due to modulating the frequencies but this is more often how the ear perceives the sound as opposed to actually changing the amplitude. Along with this you have Phase modulation which is a separate but very closely related process. In fact, many synths used Phase modulation back before the patent was lifted to achieve similar results, such as Casio with their CZ series which used phase distortion synthesis. Amplitude modulation on it's own can create many new timbres for an artist to work with. We are used to having LFO's and other low frequency sources for modulation, with some LFO's even being able to be cranked up into audible ranges, but not all modulation sources need to low frequency oscillators or envelope generators. Generally only more powerful synths allow you to modulate filters, vcas, oscillators etc with higher-frequency sources but with the advent of the VST and such it is accessible to everyone now. It's important to note that Amplitude Modulation is a separate form of Synthesis to FM but is not uncommon for the two to be combined or overlap in many places hence it is a useful aspect to know in understanding Frequency modulation synthesis. An example of AM synthesis would be an audio-frequency oscillator controlling a VCA's gain. The results of AM can often be quite unexpected which makes it a powerful and interesting tool for any sound designer. As with FM synthesis, AM synthesis does not just change the sound like creating a tremolo or fade effect, it actually creates a new timbre. One example of why this is a useful addition is in a fixed frequency modulation. Assuming the Carrier and modulator have a frequency of 100Hz then the three frequencies from the AM lie at 0Hz, 100Hz, and 200Hz. These make up the Difference, the Carrier, and the Sum signals. All of these signals affect the outcome, even at 0Hz. It still has an amplitude (which is the important part) and creates an offset in the signal. This is a Direct Current offset due to having no oscillation frequency. In practice these DC offsets are quite significant.

From this you can take that AM synthesis is a tremolo in the audible range, while FM synthesis is vibrato in the audible range. We now see that the two are quite similar, just based on different approaches and as such work well together and leave for more options. FM synthesis is considerable more complex than AM synthesis in that it produces more than just 2 side bands [which is what AM produces] for this reason understanding AM leaves for way to understanding FM better. At a low frequency the modulation (the vibrato if you prefer) simply affects the source in the way you hear, a bending up and down of the frequency. It's when you move the modulators frequency closer to, the same, and further than the frequency of the source then you get much more interesting results. At higher frequencies the modulator begins to distort and affect individual cycles of the source. This is simply due to the fact that the modulator is affecting the source at a much more rapid pace with closer to, the same or even more cycles itself. As such it is able to change then entire timbre by affecting the sound as a whole [affecting nearly all the cycle], or near to it, as opposed to just changing it in parts [such as changing a large number of number of cycles at a time]. If you think of a sine wave going from 0Hz to 22Hz you hear that speeding up of the cycle's repetition as the frequency raises until you cannot discern the individual cycles and the timbre changes. It's the same process just applied differently and to another cycle, it's in this difference of the 2, or more, cycles that FM Synthesis comes about. These differences created by the side bands as a result of FM synthesis are not always harmonic, hence the often heard percussive and bell tones from FM synthesizers. The amplitude of the carrier and modulator obviously have effects also leaving AM synthesis as a component within FM synthesis to an extent in that regard. The modulator determines the timbral and spectral components of the sound, along with the amplitudes there in, this is directly proportional to the modulator's amplitude but also inversely proportional to the Modulator's frequency. As far as the carrier frequency is concerned though, the placement of the spectral components [side bands etc] is determined by the frequency of the modulator alone.

As a result of this FM Synthesis is capable of unique sounds that no other form of synthesis can obtain. When you also take into account that it is extremely suited to the modern digital world of sound production and it's processing requirements are minimal then it becomes a very attractive form of synthesis to pursue, understand and implement.

For any of you who program in Csound then here is a basic model, this might also prove as useful for understanding the process better.

instr 1
 kindx chnget "index"
 kmf chnget "mf"
 kamp expsegr 0.01, 0.1, 10000, 10, 0.01, 1, 1 
 amod oscili kindx*kmf, kmf, 1 
 acar oscili kamp, p4+amod, 1 
 outs acar, acar
 endin 

Hopefully that is useful to someone out there, for those of you more knowledgeable on FM Synthesis; feel free to correct me on any mistakes.

Cheers!

http://decitronic.blogspot.com/2011/08/fm-synthesis-explantion-without-mates.html

rcaz

Cool post!

Here's an illustration I did in Pure Data for anyone using that;



It's not a very useful patch, but it could make the starting point of the sound-making portion of an FM synth.

studiorat

Nice.

Quick question out of interest. Where's the two indexes getting their data from? I mean, where's the two changet messages coming from? Keyboard and just a knob?

kamp is the envelope, yeah? Any chance you could show us with variables, still trying to work out what the p fields in expsegr mean. Why are the last two ending in 1?