Last month, I talked about understanding FM intuitively, exploring sounds by identifying the different audio chunks. This is a practical and useful approach - but it leaves some questions unanswered. What is actually going on between the operators? How does one operator modify the sound of another?

I believe that practice is the most solid way of building an understanding of FM voicing, but a little knowledge of the theory goes a long way in preparing you for what to expect when you start tweaking or creating new sounds. The theory of FM is not that difficult to understand, especially if you can think in terms of harmonics or overtones when you describe sounds, so in this article let's take a look at what makes FM tick...

FM Synthesis has the reputation of being rather complicated, but there are just a couple of basic parameters that govern the sound - they are Modulation Index and Frequency Ratio.

1. Modulation Index - this is the amount by which a modulator modifies a carrier. It is directly related to the Output Level of the modulators, represented as TL in the MA Voice Editor, which stands for Total Level. The carriers don't modulate anything of course, so their output level is just that... the audio output level.
2. Frequency Ratio - this is the ratio between the frequencies of the modulator and carrier, usually represented as an integer. The frequency in Hertz of an operator is its frequency ratio number times the frequency of the note (or midi note number) being played. In the MA Voice Editor this is represented by MT, which stands for "multiplier".

These are the main fundamental variables in FM. The envelopes, LFO's and so on are just different ways of controlling these fundamental variables. I should mention that the MA FM synth does actually offer a third variable and that is operator waveform, but to keep things simple while we're looking at the theory, we'll just deal with the case where operators are set to produce sine waves.

Lets begin our look inside the algorithm with modulation index. The actual math is simplified, but it's accurate enough for an ear test and some basic understanding...

Consider a two-operator FM system, with each operator producing a sine wave at the same frequency. They will be each be denoted as having a frequency ratio of 1; that means each operator is producing a sound (sine wave) at 1 x the frequency of the pitch of the note played on the keyboard. The amount by which the Modulator affects the Carrier, which is directly related to its output level, is called the MODULATION INDEX.

So what actually happens to the output of an FM pair, frequency ratio 1:1, when you play middle C (261.5Hz) on your keyboard and the Modulation Index is gradually increased from zero? Well, at zero modulation index, the output will be just the sin wave from the carrier at 261.5Hz (middle C) - but as the Modulation Index is increased, side bands, or harmonics, begin to appear, (just like FM radio) at integer ratios of the carrier frequency. The more the index is increased, the more harmonics are generated. They appear to ripple out from the center (carrier) frequency in waves as modulation index goes up. Take a look at diagram 1...

The harmonic sidebands on the right are recognizable as part of the familiar frequency spectrum - probably the most common technical method we use to describe a static sound. But what about the sidebands on the left? What are these harmonics with "negative" frequency? What does a harmonic at 0Hz mean? Well, the 0Hz harmonic disappears - you don't hear much at a frequency of nothing, and the negative harmonics just represent sin waves that are out of phase. They can be thought to reflect around the 0Hz point to combine with their positive counterparts, as shown in diagram 2.

The actual mixing in of the reflected harmonics is not simply additive; some sidebands add while others subtract their energy, and the actual math is a little more complex than suggested here. The result, though is still a general non-linear expansion of bandwidth as the modulation index increases. The aural effect is that the sound gets brighter, but in a complex, non-filter like way. This is one of the features of FM that make it sonically so interesting. Diagram 3 gives an impression of how the extra harmonics get added as the modulation index goes up. Notice how the harmonics close to the fundamental go up and down in energy, as modulation index increases.

In short, the modulation index governs the number of harmonics, or the frequency bandwidth of the spectrum that is produced by an FM pair of operators.

What does the frequency ratio do? This is the other fundamental variable in FM so let's take a look at what happens when the frequency ratio between the operators is not 1:1, but for example, 2:1, as shown in Diagram 4. The modulation index is set at a level such that sidebands duly appear, but this time, the interval between the sidebands is 2 x the carrier frequency. Again, the negative bands reflect around the 0Hz line and combine with their positive counterparts to build a normal frequency spectrum, but in this case you can see that every other harmonic is missing. This gives a characteristic "clarinet-like" or 'hollow" sound. The number of harmonics, or bandwidth, is still governed by the modulation index (the output level of the modulator).

Diagram 5 shows another example, with a 3:1 ratio. Note how the negative sidebands reflect around the 0Hz line. They do not fall "on top of" their positive counterparts but in this case, between the gaps adding new harmonic components to the spectrum and giving yet a different flavor to the tone produced. You can also notice that for the same modulation index, the harmonics cover a wider spread, meaning that there will be higher frequency, (brighter) components to the sound.

There is still one more typical type of ratio to look at, and that is when the CARRIER is higher than the MODULATOR. Lets take a look at a ration of 1:4. This time, the carrier will play a sin wave at 4 times the frequency of the note you are playing, the modulator, 1 times the frequency. Starting with a MODULATION INDEX of zero, you will hear a sin wave at 4 x 261.5Hz (if you are playing middle C).

As you can see from Diagram 6, the result is a very different shape, with the sidebands bunched more evenly around the carrier frequency. An important point however, is that the fundamental frequency is not the carrier frequency in this case. The interval between the harmonics is still 1 times 261.5Hz, (the ratio of the modulator times the frequency of the note you are playing). It is this interval which describes a particular harmonic series and defines what note your actually hear or perceive. When the ratio of the frequencies of the carrier and modulator are an integer ratio, the interval between harmonics in the series will be the lowest common denominator of the two numbers. This is usually 1; therefore the fundamental note you will hear will be (1 x) the frequency of the note you are playing. You can test this by constructing a spectrum for say a ratio of 5:3. You will find, that after reflections have been taken into account, the harmonics still fall on the series given by 1 x the frequency of the note you are playing.

Summing up, the frequency ratio between carrier and modulator determines what harmonics will appear in the spectrum and the modulation index (output level of the modulator) decides how many harmonics there will be and what their energies, or levels, are. Controlling these two parameters, you can build many different types of sounds with a simple FM pair. Dynamic changes to the sound are brought about by applying an envelope to each of the operator outputs, causing the modulation index to vary with time and I've added some appropriate exercises to demonstrate this. Here is what you need to do to hear the theory at work...

When you are set up, try the following exercises. They will really help you to get familiar with underlying parameters of FM synthesis work to make different sounds.

• FM Aural Exercise 1. Modulation Index:
  Select Operator 1. Activate TL by clicking on it - now the Up/Down or Right/Left arrow keys on your PC keyboard can be used to adjust this value while you click repeatedly on the soft keyboard note C5 to monitor the sound. Now you are listening to the effect of changing the modulation index between an FM pair with frequency ratio of 1:1. Familiarize yourself with this timbral change.
• FM Aural Exercise 2. Ratio of Modulator:
  Set the TL value of operator 1 (the modulator) to about 30. Click on the MULT slide bar to activate it and use the arrow keys on you PC keyboard to adjust the frequency ratio up from 1 to 15 and back. Familiarize yourself with this timbral change.
• FM Aural Exercise 3. Ratio of Carrier:
  Reset the MULT value of Operator 1 to 1 and click on the Operator-2 tab to make it active. Click on its MULT slide bar and move the values up to 15 and back down again. Familiarize yourself with this timbral change. Notice how it is different to changing the ration of the Modulator.
• FM Aural Exercise 4. Feedback on the Modulator:
  Reset the MULT value of Operator 2 to 1 and click on the Operator-1 tab to make it active. Leave the TL setting at around 30 and select FB on the right-hand side of the operator controls panel. Click on its slide bar to make it active and use the arrow keys to adjust its level up to 7 and back down again. Familiarize yourself with this timbral change. Notice the similarities and differences to changing the modulation index.
• FM Aural Exercise 5. Envelope Generator on the Modulator:
  Reset the FB value of Operator-1 to 0. Set the TL (output) value of Operator 1 to about 20 and click on the Attack Rate (AR) slide bar to make it active. Now adjust the AR from 15 down to 1 and back. Notice that the effect is a change in timbre through the attack portion of the sound. [Come back to this one again later and set the feedback set to 7 and TL at 30. Can you hear the "brass" when the AR gets down to around 8 or 9?]
• FM Aural Exercise 6. Envelope Generator on the Carrier:
  Reset the AR value of Operator-1 to 15 and click on the Operator-2 tab to make it active. Now select AR of operator-2 to make it active and use the arrow keys to adjust it down to 1 and back up again. Notice that the effect is a change in volume through the attack portion of the sound.

If you are patient enough to spend an hour or so on these exercises, really observing how the timbres change, then you are well on the way to mastering FM programming. The rest is familiarity with how the other parameters link to these sonic basics. However, if you don't have time for the exercises, take a quick listen to the SMAF example (EARTRAINING_mmf.zip).