Square One: FM Basic Training

Algorithms and operators aren’t intuitive, but they are fathomable.
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In the March 1999 issue of EM, I discussed the basics ofmodulation synthesis and explored amplitude modulation (AM) and relatedtechniques. This article examines another form of modulation synthesis,frequency modulation (FM).

FM synthesis, in a rather limited form, is possible withvoltage-controlled analog synthesizers. But the musical potential of FMdidn’t become fully apparent until John Chowning’spioneering work on the digital implementation of FM in the 1970s. Adecade later, when Yamaha introduced the DX7 synthesizer and its manyrelatives, 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 theJanuary 1999 “What’s New” column), recently predictedan FM synth revival. If he’s right, then we’ve picked agood time to reexamine the subject.

Boot Camp Revisited

For those who missed (or forgot) the earlier article, let’sstart with a rapid review of the general characteristics of modulationsynthesis. Modulation synthesis is a waveshaping technique in which anaudio-rate signal called the modulator controls some parameterof another audio signal, called the carrier. In FM, thefrequency is the modulated parameter. The modulation process generatesnew sine-wave components, called sidebands, in the spectrum ofthe output signal. The power of the sidebands is governed by themodulation index (discussed shortly). The index is defineddifferently for AM and FM, but in both cases it is related to theamplitude of the modulator.

Sideband frequencies can be calculated by taking the sums anddifferences of the frequencies of carrier and modulator components. Theresultant spectra fall into two broad classes. In a harmonicspectrum, all components are members of a harmonic series, that is,they are integer multiples of some fundamental frequency. In otherwords, their frequencies are 2, 3, 4, and so on, times the frequency ofthe fundamental. In an inharmonic spectrum, some or allcomponents 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 amodulation spectrum will be harmonic or inharmonic. Here we repeat Rule1 for fledgling modulation synthesists:

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

Now let’s relate these generalities to the specifics ofFM.

Simple FM Spectra

One reason for FM’s popularity is that interesting spectra canbe synthesized with limited resources. In FM, a sine carrier andmodulator generate a theoretically infinite number of sidebands. Byvarying one simple parameter, the modulation index, we can createcomplex variations in the spectrum. Sine-wave FM with dynamic indexcontrol (that is, an index that changes over time) is the basis of mostcommercial FM synthesizers.

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

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

Upper sideband frequencies, which are the sum of Fc and everyinteger multiple of Fm (Fc + Fm, Fc + 2Fm, Fc + 3Fm, and soon).

Lower sideband frequencies, which are the difference of Fcand 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 Hzcarrier and a 400 Hz modulator. The ratio Fc:Fm reduces to 5:4,so according to Rule 1, this will be a harmonic spectrum. Figure2a represents the first three sideband pairs around the carrier.Notice that some lower sidebands have negative frequencies. This willoccur whenever Fm or one of its multiples is greater thanFc. A signal with frequency –F is simply inverted(180 degrees out of phase) with respect to a frequency F. Inthis particular example, the negative frequencies wouldn’t affectthe sound.

In Figure 2b, negative components are represented as havingnegative amplitudes (downward lines). In fact, they are positive, but180 degrees out of phase, as noted earlier. This makes it easy to seethat the spectrum is harmonic, as predicted. It consists of a 100 Hzfundamental, with odd-numbered harmonics. Don’t confuse thecarrier component (500 Hz) with the fundamental (a 100 Hzsideband).

Figure 2c shows how negative frequencies affect the sound.This is another harmonic spectrum, because Fc and Fm areboth 100 Hz (Fc:Fm = 1:1). Although the sideband series aroundthe carrier extends infinitely, only the first three sideband pairs areshown. The first lower sideband has a frequency of 0, which is aninaudible DC component. As the brackets show, the 100 Hz carriercomponent 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 negativecomponent of unequal amplitude. The summation of the correspondingpositive/negative components produces a partial cancellation, orattenuation, of every positive component in the spectrum. Figure2d illustrates this result. The audible spectrum has a fundamentalof 100 Hz, with all harmonics present.

Here’s a quick quiz: compute the first three FM sideband pairswhere Fc = 500 Hz and Fm = 202.61. Is this spectrumharmonic or inharmonic? (Answer: The lower sidebands will appear at297.39, 94.78, and -107.83. The upper sidebands are 702.61, 905.22, and1107.83. The spectrum is inharmonic.)

The Modulation Index

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

The instantaneous frequency of the modulated carrier deviates aboveand below Fc in proportion to the amplitude of the modulator. Inlinear FM, the positive and negative frequency excursions,measured in hertz, are equal. I will assume linear FM in thisdiscussion, because digital FM implementations are normally based onlinear FM.

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

As I increases, the number of significant sidebandsincreases, while the carrier component is weakened. The sound of FMwith a slowly increasing index is distinctive, resembling an elaboratecrossfade between the sine-wave carrier and a number of partials aboveand below it. As I decreases, of course, the spectrum evolves inthe opposite direction, and the sidebands disappear.

This overall change in sideband power doesn’t mean that theamplitudes of the individual sidebands all change by the same amount.In fact, as I changes, the amplitude of each sideband pairevolves in a different pattern. As some sidebands gain amplitude,others lose amplitude and disappear. In addition, there may becancellation effects caused by phase-inverted sidebands. This accountsfor the complex “churning” quality of a dynamicallychanging FM spectrum.

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

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

Operators and Algorithms

To obtain a dynamic index, you need only control the modulatoramplitude 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. Figure3a illustrates the simplest configuration of two operators: theoutput of operator 2 is routed to the frequency-control input ofoperator 1. The envelope generator within operator 1 controls the finaloutput amplitude of the patch.

Yamaha FM synthesizers feature various fixed configurations ofoperators. These configurations are called “algorithms”(which sounds more impressive than “patches”). Figure3b shows our two operators as part of a Yamaha-style algorithm. Inthis algorithm, operator 2 modulates operator 1; operator 4 modulatesoperator 3; and operator 6 modulates itself (via feedback) and operator5. 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 ifwe are to understand more complex FM synthesizers, which usually employat least four operators patched in many different ways. The DX7 had sixoperators and 32 algorithms; the new Yamaha FS1R has eight operatorsand 88 algorithms. Not wishing to seem too Yamaha-centric, I shouldpoint out that there are other approaches. The original Synclavier, forexample, used a sine modulator and a complex carrier that was createdusing additive synthesis. Software synthesis systems place fewrestrictions on the number of FM oscillators, their waveforms, or theirinterconnections. Let’s consider some of the possibilities.

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

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

The use of complex modulators, such as sampled or additivelysynthesized signals, can also be regarded as examples of parallel,multiple-modulator FM. In such cases, just think of the complexmodulator 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 inturn modulates operator 1. In practice, serial-modulator FM spectra arevery similar to parallel-modulator spectra.

Other FM Options

Modulating the frequency of a periodic carrier with a noise signalgenerates random sidebands above and below the carrier. This is anexcellent way to obtain a “pitched noise” effect that issimilar to noise filtered through a narrow bandpass filter.

Feedback FM, where the output of an oscillator is fed backinto its own frequency-control input, is a technique patented byYamaha. In feedback FM, the number and amplitude of the sidebands tendto increase in a more linear relationship to the modulation index. Thisspectral evolution is closer than simple FM to the natural evolution ofacoustic instruments’ spectra. Feedback FM with a very highmodulation index can yield extremely rich spectra, sometimes resemblinghigh-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 impossibleto obtain on most analog synthesizers. Analog oscillators are typicallydesigned to have a 1-volt-per-octave response to keyboard controllers.This is usually accomplished by putting an exponentialconverter on the frequency-control input. Because of theconverter, an incoming sine modulator will drive the carrier frequencyasymmetrically. The positive frequency deviation will be greater thanthe negative deviation. This raises the perceived pitch of themodulated signal. Consequently, changing the modulation index has apitch-bending effect, and it is difficult to get an exponential FMpatch to sound in tune across a wide pitch range. Exponentialconversion also distorts the modulator signal, turning a sinusoidalmodulator 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, butthey’ve never been plentiful. The Moog modular 901B VCO had alinear input, as did VCOs made by Serge Modular and Gentle Electric.With other gear, it may be possible to modify the hardware and bypassthe exponential converter.

Valediction

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

John Duesenberry’selectronic music isavailable through the Electronic Music Foundation. Check the EMFcatalog at www.emf.org.

FM Synthesis Tools

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

You can find older Yamaha FM synthesizers second-hand at bargainprices; the models are too numerous to list here. I personally preferthe 6-operator models—I’ll never give up my ten-year-oldTX816—but 4-operator units are cheaper. The Yamaha SY-seriesproducts allow modulation of FM operators by AWM2 samples.

If you want to hook up your own operators and algorithms, softwaresynthesis is the way to go. You can find links to MIT’sCsound (cross-platform) and James McCartney’sSuperCollider (Mac) at TomErbe’s Mac software Web site. GUI-based synthesis/DSPprograms include Jim Bumgardner’s Syd (Macintosh; alsoavailable from Erbe’s site), Seer Systems’ Reality(Windows), Synoptic’s VirtualWaves,Digidesign’s TurbosynthSC (Macintosh), Cycling74’s MSP (Mac), and Symbolic Sound’s Kyma System(which uses dedicated hardware and Mac or Windows software). All ofthese support FM synthesis in some form.