Introduction
Frequency modulation (FM) is elegant in its simplicity, yet impressive in the complexity that lies within. It is less intuitive for sound design and therefore intimidating to many who may have unexplored FM instruments in their collection. The Yamaha DX7 (see Figure 5.1)—the runaway success of the 1980s that introduced FM synthesis to the masses—came with a great collection of presets, but few players dug into the editing capabilities that formed those patches because it was so different than the subtractive synthesis they were familiar with. Our goal here is to help you get over the initial learning curve so that you will have confidence to explore and design great FM sounds of your own.
How FM Works
For starters, FM synthesis works in the same manner as FM radio transmissions. The frequency you select on the radio dial is called the carrier frequency, and it is being modulated by the appropriately named modulator signal, which is the audio program being broadcast. Conventionally in FM synthesis, the carrier is a sine wave and the modulator is one or several sine waves (see Figure 5.2).Figure 5.1
The Yamaha DX7 (photo courtesy of Yamaha Corporation of America).
At its most basic, FM synthesis requires only two sine waves. But rather than adding them together, as would be the result when two VCOs are summed in a mixer, the carrier is being frequency modulated in a fashion similar to how the LFO is used to make vibrato. However, unlike the LFO that generates infrasonic waves (those below the audible frequency range), the modulator for FM is coming from a VCO producing sine waves within the audible spectrum. A
curious thing happens when the modulator shifts from the infrasonic territory into the audible range: Rather than continuing to hear a vibrato effect, instead we hear added frequencies called
sidebands that mix with the carrier. With the modulator set to a low level, the result will
comprise the carrier frequency plus the modulator’s frequency for one sideband and the carrier frequency minus the modulator’s frequency for the other sideband—this is referred to as the sum and difference. Figure 5.2 AM vs. FM. Left: a sine-wave carrier has been amplitude modulated by a sine-wave modulator; right: a sine-wave carrier has been frequency modulated by a sine-wave modulator (x axis, time; y axis, amplitude/intensity). Figure 5.3 Spectral analysis comparison. Left: the carrier of pitch C5 (522 Hz) with a modulator introduced at a low level set to C4 (261 Hz) producing sum and difference sidebands. Right: the same carrier modulated at a higher level resulting in multiple sidebands across a wider frequency range. As the intensity of the modulation signal is increased, more sidebands are generated at the sum and difference of the carrier and a potentially infinite number of integer multiples (2x, 3x, 4x, etc.) of the modulator. If a negative number results from a “difference” calculation it will still appear in the spectrum with its phase inverted at the equivalent positive value (yikes!).
Figure 5.3 shows a sine wave carrier at 500 Hz being modulated by a sine-wave modulator at 250 Hz. With a relatively low modulator level, only the sum and difference values of 750 Hz (500 + 250) and 250 Hz (500 – 250) are present. As the level of the modulator increases, more sidebands appear. Table 5.1 shows the predictable pattern of sidebands as the modulation level is increased.
Table 5.1 Calculating FM Sidebands
Carrie r Fre que ncy (C) Modulator Fre que ncy (M)
C + M C – M C + 2M C – 2M C + 3M C – 3M C + 4M C – 4M* *etc., as modulation intensity increases. Does your hair hurt yet? The good news here is that you don’t need to go through the math to design sounds with FM, but the basics help you gain an understanding of the underlying principles at play.
To generalize a bit: First, the more of the modulator added, the more complex the resulting spectrum becomes. Second, when there is an integer–ratio relationship between the carrier and the modulator, the resulting timbre will be more harmonic in nature; conversely, noninteger ratios result in timbres that contain more inharmonic frequencies that produce noise. The harmonic timbres in FM tend to be bell-like and have a purity that was not available in the subtractive instruments that preceded, thus the enormous popularity of FM when this new, vast palette of sounds became available. The FM electric piano with its tine-like quality became all the rage in the 1980s and was used extensively (overused?) on pop recordings of the era.
Given the mathematical predictability of FM it is understandable that it was not utilized much prior to the advent of digital oscillators. FM was possible with analog instruments—and it’s very likely that, either intentionally or by accident, many patch connections were made on modular synths from the output of one VCO to the modulation input of another, with the resulting sound considered either an “oops” or something interesting but nonsensical. Analog oscillators were inherently unstable; their pitch was imprecise to begin with and often drifted as the instrument heated up. As a result, recreating a FM patch or having it remain stable while performing was nearly impossible. Digital oscillators provided the pitch stability necessary to create sounds with predictable and repeatable results.
Algorithms
The timbral complexity offered by a single carrier and modulator is remarkable but merely the starting point for FM. The range of sounds offered by the DX7, for example, and instruments that would follow benefit from chaining a sequence of modifiers in series and/or parallel to two or more carriers so that the same kind of layering that we used in subtractive synthesis would be possible here as well. For the DX7, Yamaha incorporated 32 algorithms that contain the essential carrier–modulator combinations for producing a wide range of sounds (see Figure 5.4)
Figure 5.4
Figure 5.5
Figure 5.6
As a case study, let’s look at Algorithm 1 (see Figure 5.5), which shows a total of six numbered blocks: two stacked on the left and four stacked on the right. The bottom blocks in all DX7 algorithms are carriers, those above are the modulators.
The left-hand column of blocks in Algorithm 1 is a straightforward modulator-to-carrier configuration. The right-hand column has three modulators: one fed by another modulator that is fed by yet another modulator. It is helpful to start at the bottom of the algorithm diagram with the carrier and work up when designing a sound. In fact, a great way to learn FM synthesis is to deconstruct preset patches you like by turning off all but one carrier, and step-by-step introducing each modifier and (if present) additional carrier(s) until you understand the purpose of each component. Back to Algorithm 1 (Figure 5.4): Notice that the patch out from block 6 is not only feeding into block 5, it’s also feeding back into itself! This is how noise is generated in FM. Noise, of course, is useful in many sound designs of percussive instruments or used as a layer, perhaps, with a short attack–decay envelope to act as the attack portion of a pitched sound.
Operators
Let’s retire the term “blocks” at this point and replace it with Yamaha’s term: operator. Each algorithm of the DX7 consists of six operators in thirty-two different configurations. Figure 5.6 shows the constituent parts of an operator: a sine-wave oscillator fed with pitch information from the MIDI controller (commonly your keyboard), a modulation input (used for some operators in a given algorithm), and an EG that is multiplied with the oscillator in a VCA.
Figure 5.7
Why is there an EG in every operator? If the operator is used as a carrier, the EG is simply shaping its volume over time. If the EG is part of a modulator, it will shape the intensity of the modulation so that the complexity of the overtones will vary over time. Using EGs as much as possible in your operators will add dynamic interest to the patches you design.
It is worth noting here that a discussion of filters may seem conspicuously absent in this chapter about FM. The harmonic complexity and control of the spectrum coming from the algorithms alone makes a filter somewhat redundant and unnecessary. Learning to craft a timbre by choosing the best algorithm option and shaping the operators accordingly will yield better results than applying a filter. That being said, some newer software instruments like NI’s FM8 do offer LPFs with resonance control as an additional creative element.