Square One: FM Basic Training

Apr 1, 1999 12:00 PM, John Duesenberry

In the March 1999 issue of EM, I discussed the basics of modulation synthesis and explored amplitude modulation (AM) and related techniques. This article examines another form of modulation synthesis, frequency modulation (FM).

FM synthesis, in a rather limited form, is possible with voltage-controlled analog synthesizers. But the musical potential of FM didn’t become fully apparent until John Chowning’s pioneering work on the digital implementation of FM in the 1970s. A decade later, when Yamaha introduced the DX7 synthesizer and its many relatives, FM won mass acceptance in the music world.

The FM craze of the 1980s has abated, but one industry pundit, impressed by Yamaha’s new FS1R synth ($999.95; discussed in the January 1999 “What’s New” column), recently predicted an FM synth revival. If he’s right, then we’ve picked a good time to reexamine the subject.

Boot Camp Revisited

For those who missed (or forgot) the earlier article, let’s start with a rapid review of the general characteristics of modulation synthesis. Modulation synthesis is a waveshaping technique in which an audio-rate signal called the modulator controls some parameter of another audio signal, called the carrier. In FM, the frequency is the modulated parameter. The modulation process generates new sine-wave components, called sidebands, in the spectrum of the output signal. The power of the sidebands is governed by the modulation index (discussed shortly). The index is defined differently for AM and FM, but in both cases it is related to the amplitude of the modulator.

Sideband frequencies can be calculated by taking the sums and differences of the frequencies of carrier and modulator components. The resultant spectra fall into two broad classes. In a harmonic spectrum, all components are members of a harmonic series, that is, they are integer multiples of some fundamental frequency. In other words, their frequencies are 2, 3, 4, and so on, times the frequency of the fundamental. In an inharmonic spectrum, some or all components do not fit into a harmonic series.

The ratio of the carrier and modulator signals’ frequencies, which can be represented as Fc:Fm, determines whether a modulation spectrum will be harmonic or inharmonic. Here we repeat Rule 1 for fledgling modulation synthesists:

Rule 1. If Fc:Fm is a ratio of simple integers, the modulation spectrum will be harmonic. Otherwise, the spectrum will be inharmonic.

Now let’s relate these generalities to the specifics of FM.

Simple FM Spectra

One reason for FM’s popularity is that interesting spectra can be synthesized with limited resources. In FM, a sine carrier and modulator generate a theoretically infinite number of sidebands. By varying one simple parameter, the modulation index, we can create complex variations in the spectrum. Sine-wave FM with dynamic index control (that is, an index that changes over time) is the basis of most commercial FM synthesizers.

Figure 1 shows the output waveform that is created from a typical sine carrier and modulator. The waveshaping effect, a kind of “bending” of the sine waveform as its instantaneous frequency changes, is apparent. Though this may be visually interesting, the spectrum that results is even more so.

The resulting spectrum of an FM process is easy to predict. Given a sine carrier of frequency Fc and a sine modulator of frequency Fm, the FM spectrum will consist of the following components:

Upper sideband frequencies, which are the sum of Fc and every integer multiple of Fm (Fc + Fm, Fc + 2Fm, Fc + 3Fm, and so on).

Lower sideband frequencies, which are the difference of Fc and every integer multiple of Fm (Fc – Fm, Fc – 2Fm, Fc – 3Fm, and so on).

The original carrier frequency, Fc.

Let’s consider the spectrum resulting from FM with a 500 Hz carrier and a 400 Hz modulator. The ratio Fc:Fm reduces to 5:4, so according to Rule 1, this will be a harmonic spectrum. Figure 2a represents the first three sideband pairs around the carrier. Notice that some lower sidebands have negative frequencies. This will occur whenever Fm or one of its multiples is greater than Fc. A signal with frequency –F is simply inverted (180 degrees out of phase) with respect to a frequency F. In this particular example, the negative frequencies wouldn’t affect the sound.

In Figure 2b, negative components are represented as having negative amplitudes (downward lines). In fact, they are positive, but 180 degrees out of phase, as noted earlier. This makes it easy to see that the spectrum is harmonic, as predicted. It consists of a 100 Hz fundamental, with odd-numbered harmonics. Don’t confuse the carrier component (500 Hz) with the fundamental (a 100 Hz sideband).

Figure 2c shows how negative frequencies affect the sound. This is another harmonic spectrum, because Fc and Fm are both 100 Hz (Fc:Fm = 1:1). Although the sideband series around the carrier extends infinitely, only the first three sideband pairs are shown. The first lower sideband has a frequency of 0, which is an inaudible DC component. As the brackets show, the 100 Hz carrier component is matched with a sideband of its inverted frequency (–100 Hz), as is the 200 Hz component.

The brackets indicate a pattern that holds throughout this spectrum: for each positive, nonzero component, there is a corresponding negative component of unequal amplitude. The summation of the corresponding positive/negative components produces a partial cancellation, or attenuation, of every positive component in the spectrum. Figure 2d illustrates this result. The audible spectrum has a fundamental of 100 Hz, with all harmonics present.

Here’s a quick quiz: compute the first three FM sideband pairs where Fc = 500 Hz and Fm = 202.61. Is this spectrum harmonic or inharmonic? (Answer: The lower sidebands will appear at 297.39, 94.78, and -107.83. The upper sidebands are 702.61, 905.22, and 1107.83. The spectrum is inharmonic.)

The Modulation Index

Although the number of FM sidebands is infinite, there is a finite number of significant sidebands. A significant sideband pair is one that has more than 1/100 of the amplitude of the carrier. The FM modulation index, or I, governs both the number of significant sidebands and their relative amplitudes. Before I explain the effect of the index, I need to define a couple of additional terms.

The instantaneous frequency of the modulated carrier deviates above and below Fc in proportion to the amplitude of the modulator. In linear FM, the positive and negative frequency excursions, measured in hertz, are equal. I will assume linear FM in this discussion, because digital FM implementations are normally based on linear FM.

The maximum change from Fc is the maximum frequency deviation, or D. You can think of D as the “depth” or “amount” of modulation. The FM modulation index I is defined as the ratio of frequency deviation to modulator frequency: I = D/Fm.

As I increases, the number of significant sidebands increases, while the carrier component is weakened. The sound of FM with a slowly increasing index is distinctive, resembling an elaborate crossfade between the sine-wave carrier and a number of partials above and below it. As I decreases, of course, the spectrum evolves in the opposite direction, and the sidebands disappear.

This overall change in sideband power doesn’t mean that the amplitudes of the individual sidebands all change by the same amount. In fact, as I changes, the amplitude of each sideband pair evolves in a different pattern. As some sidebands gain amplitude, others lose amplitude and disappear. In addition, there may be cancellation effects caused by phase-inverted sidebands. This accounts for the complex “churning” quality of a dynamically changing FM spectrum.

The amplitude of a particular FM sideband for a known value of I is given by a mathematical formula called a Bessel function. FM synthesists don’t spend their lives computing Bessel functions, of course. But if possible, take a glance at the Bessel function plots in Computer Music, 2nd edition (Dodge and Jerse; Schirmer, 1997). They will help you picture how the modulation index affects the sidebands.

What does all this mean to the musician? A dynamically changing index generates a dynamic spectrum that holds the interest of the ear.

Operators and Algorithms

To obtain a dynamic index, you need only control the modulator amplitude with an envelope that is scaled to the modulator frequency. Package this function into a little bundle called an operator, and you have the basic building block of Yamaha-style FM synthesis.

An operator can function as either carrier or modulator. Figure 3a illustrates the simplest configuration of two operators: the output of operator 2 is routed to the frequency-control input of operator 1. The envelope generator within operator 1 controls the final output amplitude of the patch.

Yamaha FM synthesizers feature various fixed configurations of operators. These configurations are called “algorithms” (which sounds more impressive than “patches”). Figure 3b shows our two operators as part of a Yamaha-style algorithm. In this algorithm, operator 2 modulates operator 1; operator 4 modulates operator 3; and operator 6 modulates itself (via feedback) and operator 5. The outputs of the carrier operators 1, 3, and 5 are summed.

Portrait of a Serial Modulator

An understanding of simple 2-operator sine-wave FM is essential if we are to understand more complex FM synthesizers, which usually employ at least four operators patched in many different ways. The DX7 had six operators and 32 algorithms; the new Yamaha FS1R has eight operators and 88 algorithms. Not wishing to seem too Yamaha-centric, I should point out that there are other approaches. The original Synclavier, for example, used a sine modulator and a complex carrier that was created using additive synthesis. Software synthesis systems place few restrictions on the number of FM oscillators, their waveforms, or their interconnections. Let’s consider some of the possibilities.

In multiple-carrier FM, a single sine modulator controls more than one carrier. The result is the sum of the three modulated carriers; their spectra are superimposed. With harmonically tuned carriers, peaks, or formants, in the spectrum can be produced. Composers such as John Chowning and Dexter Morrill have simulated vocal and brass timbres in this way. Note that modulating a complex carrier, such as a sawtooth wave, is an instance of multiple-carrier FM. In this case, the carrier can be analyzed as some number of sine components, all modulated by the same signal.

If that’s not complicated enough, consider multiple-modulator FM, in which a single sine carrier is controlled by several modulators. The modulators may be connected in parallel or in series. In Figure 3c, operators 2 and 3 are patched in parallel to modulate operator 1. With two modulators, each sideband generated by one acts as a carrier that is in turn modulated by the other. This can generate a huge number of sidebands, but you can keep them under control by simply using low modulation indices.

The use of complex modulators, such as sampled or additively synthesized signals, can also be regarded as examples of parallel, multiple-modulator FM. In such cases, just think of the complex modulator as the sum of a large number of sine modulators.

Modulators can also be connected in series, as in Figure 3d. In this “cascaded” or “chained” configuration, operator 3 modulates operator 2, producing a complex signal that in turn modulates operator 1. In practice, serial-modulator FM spectra are very similar to parallel-modulator spectra.

Other FM Options

Modulating the frequency of a periodic carrier with a noise signal generates random sidebands above and below the carrier. This is an excellent way to obtain a “pitched noise” effect that is similar to noise filtered through a narrow bandpass filter.

Feedback FM, where the output of an oscillator is fed back into its own frequency-control input, is a technique patented by Yamaha. In feedback FM, the number and amplitude of the sidebands tend to increase in a more linear relationship to the modulation index. This spectral evolution is closer than simple FM to the natural evolution of acoustic instruments’ spectra. Feedback FM with a very high modulation index can yield extremely rich spectra, sometimes resembling high-frequency noise.

Finally, a note to all you analog fans. The classic linear FM sound, as implemented in most digital FM synthesizers, is all but impossible to obtain on most analog synthesizers. Analog oscillators are typically designed to have a 1-volt-per-octave response to keyboard controllers. This is usually accomplished by putting an exponential converter on the frequency-control input. Because of the converter, an incoming sine modulator will drive the carrier frequency asymmetrically. The positive frequency deviation will be greater than the negative deviation. This raises the perceived pitch of the modulated signal. Consequently, changing the modulation index has a pitch-bending effect, and it is difficult to get an exponential FM patch to sound in tune across a wide pitch range. Exponential conversion also distorts the modulator signal, turning a sinusoidal modulator into a complex waveform.

This doesn’t mean that exponential analog FM sounds rotten, just that it sounds different from the FM effects we’re used to. Analog oscillators with a linear FM response have been built, but they’ve never been plentiful. The Moog modular 901B VCO had a linear input, as did VCOs made by Serge Modular and Gentle Electric. With other gear, it may be possible to modify the hardware and bypass the exponential converter.

Valediction

As I wrote this article, I enjoyed revisiting my Yamaha TX816 and taking another look inside its algorithms. I hope you’ll want to get some hands-on experience with this versatile technique, too (see the sidebar “FM Synthesis Tools”). Whether FM synthesis is in fashion or not, it has great potential for producing interesting sounds, and every synthesist should be acquainted with it.

John Duesenberry’s electronic music is available through the Electronic Music Foundation. Check the EMF catalog at www.emf.org.

FM Synthesis Tools

Mention FM synthesis and Yamaha comes to mind. If you are going to get involved with FM, you’ll certainly want to consider Yamaha products. Yamaha claims that their latest FM box, the FS1R, combines FM with “formant-shaping synthesis.” Yamaha’s current lineup also includes the EX5 (reviewed in the March 1999 issue of EM) and EX7, which feature a form of analog modeling that supports FM.

You can find older Yamaha FM synthesizers second-hand at bargain prices; the models are too numerous to list here. I personally prefer the 6-operator models—I’ll never give up my ten-year-old TX816—but 4-operator units are cheaper. The Yamaha SY-series products allow modulation of FM operators by AWM2 samples.

If you want to hook up your own operators and algorithms, software synthesis is the way to go. You can find links to MIT’s Csound (cross-platform) and James McCartney’s SuperCollider (Mac) at Tom Erbe’s Mac software Web site. GUI-based synthesis/DSP programs include Jim Bumgardner’s Syd (Macintosh; also available from Erbe’s site), Seer Systems’ Reality (Windows), Synoptic’s Virtual Waves, Digidesign’s Turbosynth SC (Macintosh), Cycling 74’s MSP (Mac), and Symbolic Sound’s Kyma System (which uses dedicated hardware and Mac or Windows software). All of these support FM synthesis in some form.