In Part 2 of this series, we discussed basic audio theory and focused on the building block of digital FM synthesis: the operator. We also learned that an operator we can hear is called a carrier. But sometimes the output signal coming from an operator is not routed to your instrument’s output or headphone jacks, but is instead routed to the input of another operator, like this:
These kinds of operators are called modulators, and they cannot be heard. Instead, we hear their influence on the operator (or, in some cases, operators) they are affecting, or modulating. Other than that, there is absolutely no difference between a carrier and a modulator—they have precisely the same internal software components: an oscillator, an amplifier, and an envelope generator.
As stated in Part 2, digital FM synthesizers always offer multiple operators. In the case of modern instruments like the MONTAGE and MODX, there are eight of them, all functionally identical, labeled OP1, OP2, OP3, etc. (Even early FM instruments like the original Yamaha DX7 offered six operators.) So how do we know which operators are acting as carriers, and which as modulators? The answer lies in the algorithm being used.
“Algorithm” is a word that tends to strike fear in the hearts of newcomers to digital FM, but it’s actually just a simple computer term that means “a set of instructions designed to carry out a particular task.” For example, if you instruct a computer (in code) to add together two numbers, that’s an algorithm, albeit a very simple one. If you instruct it to put up a picture of the Mona Lisa on your screen and display it in 12,000 different shades of purple, that’s also an algorithm—albeit a much more complex (and much more useless) one.
We’ll be discussing algorithms in much greater detail in Part 5 of this series, but for now you really only need to know a couple of basic facts about them:
Up until now, we’ve only been working with Algorithm 1. That’s because this is the default algorithm when you select an initialized Performance such as “Init Normal (FM-X),” which has served as the template for all the exercises we’ve been doing thus far.
Here’s what Algorithm 1 looks like:
As you can see, all eight operators are on the bottom row, meaning that in this algorithm, they all serve as carriers. (Sharp-eyed observers will also notice a line around Operator 1; this is called a feedback loop, and is something we’ll also be discussing in detail shortly.)
Since this installment is all about modulators and modulation, this algorithm isn’t going to be of much use to us. Instead, for the purposes of this article, we’ll be working with Algorithm 67, which looks like this:
In this algorithm, the odd-numbered operators (1, 3, 5 and 7) are all sitting above the even-numbered operators (2, 4, 6 and 8), with lines indicating that Operator 1 is sending its output signal into Operator 2, Operator 3 into Operator 4, Operator 5 into Operator 6, and Operator 7 into Operator 8. (Operator 1 also has a feedback loop.) In other words, the odd-numbered operators are acting as modulators, and the even-numbered operators below them are the carriers they are modulating.
Armed with this basic information, we’re ready to take a closer look now at …
Of course, the best way to understand this is to hear it. Accordingly, in your MONTAGE or MODX, call up the “Part 2_01” Performance you created in Part 2 of this series, or click here to download it from the Soundmondo website. (This is simply “Init Normal (FM-X),” saved without reverb.)
Play a few notes. You’ll hear a sine wave. Which operator are we hearing? To find out, do the following:
We’ll spend much of the rest of this article exploring the effect that a modulator has on a carrier, but for now, set Operator 1’s level to 0 and the level of Operator 2 (a carrier in this algorithm) to 99, following the steps outlined above, then save the Performance as “Part 3_01.” (You can also find this Performance on the Soundmondo website by clicking here.)
First, though, let’s discover the answer to this basic question …
FM is an acronym for “Frequency Modulation,” and it’s something that existed long before there were computers or digital synthesizers. For more than 70 years, it’s been widely used for radio transmission when applied to waves with rates of millions of cycles per second (as opposed to the 20 to 20,000 cycle-per-second range that we can hear—see Part 2 for more information.) But for much longer than that—millennia, in fact—it’s been an integral part of music, where it’s used to create vibrato.
Vibrato is a subtle, pulsating change of pitch—a form of musical expression that every trained singer and most instrumentalists regularly apply, typically to held notes, by moving their larynx, finger or hand. (Pianists, alas, and some percussionists don’t have any means of adding vibrato due to the construction of their instruments, but most every other musician can do so.) And it was a fascination with vibrato and how it affected spatialization (the ability to move a sound source in a three-dimensional field, and the way the human ear distinguishes those movements) that led to Stanford professor Dr. John Chowning’s discovery of digital FM, as described in Part 1 of this series.
“I was searching for sounds that had some internal dynamism,” he explains, “because for localization one has to have sounds that are dynamic in order to perceive their distance. The direct signal and the reverberant signal have to have some phase differences in order for us to perceive that there are in fact two different signals. Vibrato is one of the ways that one can do that.”
As related in Part 1, Chowning’s “ear discovery,” as he called it, occurred one evening in 1967 as he was using Stanford’s mainframe computer to digitally model the sound of two sine wave oscillators in a simple modulation configuration, one slowly altering the pitch of the other to produce vibrato. Curious as to what would occur if he increased the rate and/or depth beyond what was possible with the human touch, he issued instructions to the computer to double and triple some of the numbers. And that’s when a curious thing happened: At the point when the rate of the vibrato increased to where it could no longer be perceived as a cyclical change, the sound changed from simple pitch fluctuation into a timbral change—a change in tonality. What’s more, as the rate and depth increased further, he heard more and more timbral complexity.
Ready to replicate what happened that fateful evening? Fire up your MONTAGE or MODX, call up the “Part 3_01” Performance you created earlier (or download it from Soundmondo) and let’s make it happen!
Vibrato is defined by two factors: its speed and its intensity. In the case of digital FM, we can see (and hear) clearly that its speed is determined by the frequency of the modulator. But what determines its intensity? The answer is simple: the output level of the modulator.
This is a key concept to understanding how digital FM works: Carrier output level determines how loud the sound will be, while modulator output level determines how timbrally complex the sound will be. In other words, the higher the modulator output level, the more overtones are generated. Let’s run a simple exercise to demonstrate this:
At first, there isn’t much change, but when the level of Operator reaches a value of 65 and higher, the sound becomes buzzier and buzzier. That’s because more and more overtones are being generated as the amount of modulation increases.
So far, all of our experiments in this article have been conducted with the modulator in Fixed Frequency mode—in other words, without it receiving pitch input from the keyboard (or a connected MIDI source). But, as we saw in Part 2, operators that are set to Ratio mode “track” the keyboard—in fact, that’s the way they work in most digital FM sounds. And, as we learned, the Ratio value you enter is used as a multiplier, so that, for example, if you set it to 1.00 and you play A above middle C, the operator multiples 440 (that note’s frequency) by 1.00, resulting in an output of 440 Hz (440 times 1). If, on the other hand, you set the Ratio to 2.00 and play A above middle C, the frequency of the operator’s output will be 880 Hz. We also discussed the fact that a Fine control allows you to also enter fractional Ratio values, so that if an operator’s Ratio is set to 1.5 and you play A above middle C, for example, its output will have a frequency of 660 Hz (440 times 1.5).
In a modulator/carrier configuration, this means that both the source of the modulation (the modulator) and the source of the audio signal (the carrier) are changing frequency as you play different notes on the keyboard. This enables you to craft complex timbres that do not change regardless of the note you play (very important!) and also allows you to predict, with complete mathematical certainty, the kind of timbre you create (even more important!).
As with other aspects of digital FM, this concept is easier to hear than explain, so call up the “Part 3_01” Performance you created earlier (or download it from Soundmondo). This Performance was created before we began experimenting with Fixed Frequency mode, so both Operator 2 (the carrier) and Operator 1 (the modulator) are still in Ratio mode, with identical values of 1.00. (In other words, the ratio—small “r”—between them is 1 to 1, usually indicated with a colon between the two numbers, with the modulator value first; i.e., 1:1, 2:1, 3:1, etc.) Since in this case the ratio between the two operators is 1:1, if we play A above middle C on the keyboard, the frequency of both operator’s outputs will be 440 Hz.
Now hold down A above middle C on the keyboard and slowly raise the level of Operator 1 (using the INC/YES button, data dial or control slider, as described previously) to its maximum value of 99, listening carefully as you do so. This is the “sweep” you’ll hear as the level (that is, the degree of modulation) increases:
Obviously, the timbre at the end is quite different from the one at the beginning. As we learned in Part 2, the starting signal (that of the carrier alone, before it is modulated) is a sine wave, which looks like this:
The wave at the end, however, looks quite different:
In-between, too, the waveshape has undergone several transformations, at one point looking very much like the traditional sawtooth wave provided by the oscillators in most analog synths:
What happens if we alter the Ratio value of one of the operators? Let’s try it:
This time, the waveform at the end is considerably more complex than in our last experiment, indicating the presence of more overtones:
Perhaps even more intriguing is the shape of the waveform when the Level is set to a value of 66:
This looks very much like the square wave most analog synth oscillators offer. It’s a waveform that’s typically used to fashion woodwind-like sounds (just as sawtooth waves are used to fashion brass sounds), and it sounds like this:
If you’d like to try it out for yourself, you can download the Performance named “Part 3_05” from Soundmondo. Played in context, it sounds somewhat clarinetish … although, as we’ll learn in the next installment of this series, we can get it a lot closer to clarinet when we start tweaking envelopes.
Curious as to what happens when you raise the Ratio of the modulator (Operator 1, in this case) even further? Try changing it to values of 3.00, 4.00, and higher, each time slowly raising Operator 1’s level so that you can hear the sonic differences. Here’s what the sweep at a ratio of 3:1 sounds like:
Compare it to what you hear with a ratio of 6:1:
Of course, you can stop the sweep at any time when you discover a useful timbre, at which point you should save it as a Performance for future use in your musical creations. For example, here’s the tonality you get (this is downloadable from Soundmondo as the Performance named “Part 3_06”) when you set the ratio to 6:1 and set Operator 1’s Level to a value of 77:
As you might expect (and as you can clearly hear), the higher the ratio (that is, the greater the difference between the frequency of the modulator and that of the carrier), the higher the overtones generated. But keen-eared listeners will notice a kind of sonic weirdness when increasing modulator levels to near maximum when using high ratios. This a form of digital distortion called aliasing, the result of overloading, a byproduct of which is the production of inharmonic overtones (as discussed in Part 2, these are overtones that are not exact multiples of the fundamental frequency, but instead have only a random relationship).
Inharmonic overtones are the essence of clangorous sounds like bells, chimes, cymbals, etc.—all sounds that digital FM synths are especially good at creating. More importantly, the digital FM process gives you a great deal of control over which inharmonics are generated, as well as their strength, allowing you to precisely craft the sounds you want to hear. Let’s find out how to pull off this feat of sonic wizardry.
The key to creating sounds with inharmonics is to use non-integer (non-whole number) ratios between modulator and carrier—that is, fractional ratios such as 1.01:1, or 3.87:1, or 6.92:1. As occurs when using integer ratios, the higher the ratio, the higher the frequency of the overtones generated (only in this case, the overtones are inharmonics), and the greater the level of the modulator, the greater the strength of the overtones (again, in this case, inharmonics) generated.
Let’s see how this works. Call up the “Part 3_01” Performance you created earlier (or download it from SoundMondo) and do the following:
Clearly, as this (non-integer) ratio increases, so too does the frequency of the inharmonic overtones being generated; in other words, their type changes. (At a certain point, you even begin hearing undertones—inharmonics below the fundamental frequency; the reason for this is mathematically complicated and beyond the scope of this article, but it’s worth noting.) At the highest Fine value of 127 (which results in a ratio of 1.99: 1), you hear beating once again because the ratio is close to being an integer value (in this case, 2:1).
As we’ve mentioned before, you can stop the sweep at any time when you discover a useful timbre. For example, here’s the tonality you get when you set Operator 1’s Fine value to 64 (for a ratio to 1.64:1) and set its Level value to 70:
Clearly, this is a very percussive sound (although, again, we can get it a lot closer when we start tweaking envelopes, as we’ll be doing in the next installment in this series), due in large part to the presence of a lot of inharmonic overtones. Before moving on, take a moment to save this Performance as “Part 3_07” (downloadable from Soundmondo) and then continue experimenting with different non-integer ratios between Operator 1 (the modulator) and Operator 2 (the carrier), tweaking the Level of Operator 1 as you do so in order to discover tonalities that might be useful to you in your musical endeavors. Again, you’ll find confirmation of these two guiding principles of frequency modulation: the higher the modulator/carrier ratio, the higher the frequency of the overtones, and the greater the modulator level, the more overtones.
Before we leave the subject of modulation, however, we have one more important parameter to discuss. It’s called …
As we mentioned earlier in this article, every algorithm has a line around one of the operators called a feedback loop. This indicates that the output of the operator in question (which is almost always a modulator, the exceptions being Algorithm 1 and Algorithm 55) can be routed back to its own input. (The DX7 offered a couple of algorithms whereby the output of an operator could be fed back into the input of a different operator, but since that feature had little practical value, none of the algorithms in modern digital FM instruments like MONTAGE or MODX offer this capability.)
The purpose of feedback in the digital FM process is to allow a single operator to output a more complex wave than a simple sine wave. Since by definition the ratio between an operator and itself can only ever be 1:1 (think about it), the wave being generated is exactly the same as using a 1:1 modulator/carrier ratio … but with the use of only one operator instead of two. Unfortunately, the Feedback parameter only has a range of 0 – 7 (as opposed to the Level parameter, which has a range of 0 – 127), so there are only 6 different degrees of feedback to choose from. Nonetheless, it is, as John Chowning has said, “a simple but very effective way to get an ‘edge’ to cut through.”
Let’s see how it works:
Not very exciting, is it? All you’re really hearing is a simple sine wave slowly getting buzzier, similar to what happens when you raise the output of a modulator to its maximum value in a 1:1 ratio with a carrier. But it’s what happens at the endpoint—when the Feedback value reaches its maximum of 7—that’s of interest. Here’s what that waveform looks like:
This is a nearly perfect digital replica of the traditional sawtooth wave offered by the oscillators in most analog synths—even better, in fact, than the one we achieved earlier, when we set up a 1:1 ratio between a modulator and carrier and raised the modulator level to around its midpoint.
So now we know what happens when the Feedback value is set to maximum: the operator outputs a sawtooth wave instead of a sine wave. If the feedback loop is on a modulator (as it is in most algorithms), this means that you’ll be modulating with a sawtooth wave instead of a sine wave. If your guess is that the end result will be even more complexity in the timbre created, you’re absolutely right. Let’s try it out:
What a difference! This time you hear a complex wave (the result of Operator 1 modulating Operator 2 at maximum level) getting ever more complex, until it generates into a hash of noise, the result of the generation of a ton of overtones completely obliterating the source sound. Now you know how to create noise on a digital FM synth! (Ever need noise in your musical explorations? Store this Performance as “Part 3_08”—also downloadable from SoundMondo—for future use.)
As you might expect, feedback has an ever greater effect as you increase the modulator/carrier ratio, and also can cause the production of interesting complex inharmonics when used with non-integer ratios. Try it! You may just come up with some great starter sounds … sounds that can only get better and more interesting as you apply envelopes to them. Join us here for Part 4 as we do just that.
Have questions/comments? Join the conversation on the Forum here.
Missed any part of the series? Click on the links below:
FM 101, Part One: Discovering Digital FM…John Chowning Remembers
FM 101, Part Two: The Basics
Ready to move on? Check out the next article in the series FM 101, Part 4: Going from Static to Dynamic
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