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Chapter 4: The Synthesis of Sound by Computer
Section 4.7: FM Synthesis
One goal of synthesis design is to find efficient algorithms, which dont take a lot of computation to generate rich sound palettes. Frequency modulation synthesis (FM) has traditionally been one of the most popular techniques in this regard, and it provides a good way to communicate some basic concepts about sound synthesis. Youve probably heard of frequency modulationit is the technique used in FM radio broadcasting. We took a look at amplitude modulation (AM) in Section 4.5.
History of FM Synthesis
FM techniques have been around since the early 20th century, and by the 1930s FM theory for radio broadcasting was welldocumented and understood. It was not until the 1970s, though, that a certain type of FM was thoroughly researched as a musical synthesis tool. In the early 1970s, John Chowning, a composer and researcher at Stanford University, developed some important new techniques for music synthesis using FM.
Chownings research paid off. In the early 1980s, the Yamaha Corporation introduced their extremely popular DX line of FM synthesizers, based on Chownings work. The DX-7 keyboard synthesizer was the top of their line, and it quickly became the digital synthesizer for the 1980s, making its mark on both computer music and synthesizer-based pop and rock. Its the most popular synthesizer in history.
FM turned out to be good for creating a wide variety of sounds, although it is not as flexible as some other types of synthesis. Why has FM been so useful? Well, its simple, easy to understand, and allows users to tweak just a few "knobs" to get a wide range of sonic variation. Lets take a look at how FM works and listen to some examples.
In its simplest form, FM involves two sine waves. One is called the modulating wave, the other the carrier wave. The modulating wave changes the frequency of the carrier wave. It can be easiest to visualize, understand, and hear when the modulator is low frequency.
FM can create vibrato when the modulating frequency is less than 30 Hz. Okay, so its still not that excitingthats just because everything is moving slowly. Weve created a very slow, weird vibrato! Thats because we were doing low-frequency modulation. In Soundfile 4.23, the frequency (fc) of the carrier wave is 500 Hz and the modulating frequency (fm) is 1 Hz. 1 Hz means one complete cycle each second, so you should hear the frequency of the carrier rise, fall, and return to its original pitch once each second.
fc = carrier frequency, m(t) = modulating signal, and Ac= carrier amplitude.
Note that the frequency of the modulating wave is the rate of change in the carriers frequency. Although you cant tell from the above equation, it also turns out that the amplitude of the modulator is the degree of change of the carriers frequency, and the waveform of the modulator is the shape of change of the carriers frequency.
In Figure 4.17 showing the unit generator diagram for frequency modulation (remember, we showed you one of these in Section 4.5), note that each of the sine wave oscillators has two inputs: one for frequency and one for amplitude. For our modulating oscillator we are using 1 Hz as the frequency, which becomes fm to the carrier (that is, the frequency of the carrier is changed 1 time per second). The modulators amplitude is 100, which will determine how much the frequency of the carrier gets changed (at a rate of 1 time per second).
The amplitude of the modulator is often called the modulation depth,
since this value determines how high and low the frequency of the carrier
wave will go. In the sound example, the fc ranges from
400 Hz to 600 Hz (500 Hz – 100 Hz to 500 Hz + 100 Hz). If we change
the depth to 500 Hz, then our fc would range from 0
Hz to 1,000 Hz. Humans can only hear sounds down to about 30 Hz, so there
should be a moment of "silence" each time the frequency dips
below that point.
Generating Spectra with FM
If we raise the frequency of the modulating oscillator above 30 Hz, we can start to hear more complex sounds. We can make an analogy to being able to see the spokes of a bike wheel if it rotates slowly, but once the wheel starts to rotate faster a visual blur starts to occur.
So it is with FM: when the modulating frequency starts to speed up, the sound becomes more complex. The tones you heard in Soundfile 4.24 sliding around are called sidebands and are extra frequencies located on either side of the carrier frequency. Sidebands are the secret to FM synthesis. The frequencies of the sidebands (called, as a group, the spectra) depend on the ratio of fc to fm. John Chowning, in a famous article, showed how to predict where those sidebands would be using a simple mathematical idea called Bessel functions. By controlling that ratio (called the FM index) and using Bessel functions to determine the spectra, you can create a wide variety of sounds, from noisy jet engines to a sweet-sounding Fender Rhodes.
Soundfiles 4.25 through 4.28 show some simple two-carrier FM sounds with modulating frequencies above 30 Hz.
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