Synthesizer Secrets: Cyclic Modulation

Figure 1: A simple synthesizer signal flow

I have summarized the core audio components in the first to ninth articles of this series into Figure 1, a simple but usable synthesizer signal flow. As you can see, the source of the audio signal is a tone generator (oscillator). The output audio signal first passes through the filter, then the amplifier, and finally is output to the outside world. The envelope generator "shapes" the brightness and loudness of the tone through the filter and amplifier in the second and third stages respectively (for ease of understanding, I use thick arrows to represent audio signals and thin arrows to represent control signals in the figure, and I omit the pitch CV, trigger and threshold signals in the figure).

Now let's imagine a musician playing a violin or cello. As you probably know, the pitch of a violin depends on how far the player presses the string against the neck of the instrument. The shorter the effective length of the string, the higher the pitch of the sound. If you've ever watched a violinist play a violin, you'll notice that the violinist often moves his fingers back and forth on the neck of the instrument as he plays. Each time the violinist moves his fingers, the length of the string will be slightly shortened or lengthened (this action also causes a slight change in the tension of the string, but we won't worry about that here). The change in string length also causes the pitch to rise or fall slightly, creating a vibrato effect. Figure 2 shows the pitch of a steady note being modulated by vibrato. Similarly, Figure 3 shows the (amplified) effect of vibrato on a waveform over time.

Figure 2: Applying this modulation waveform to the oscillator's frequency control can create a short-lived vibrato.

Figure 3: Triangle waveform with vibrato added

This effect is very simple to reproduce on an analog synthesizer. Most of you probably already know how to do it: you just apply a control voltage to the oscillator to slightly raise or lower its pitch.

Figure 4: Adding vibrato to our simple synthesizer

Looking at Figure 4, you'll notice that we're using a low frequency oscillator (LFO) to create the CV (modulation waveform). You can also see that there's a VCA that controls the amplitude of the modulation voltage, which is itself controlled by the modulation wheel. The presence of the VCA and modulation wheel in this configuration is important because without them, we wouldn't be able to easily control the amplitude of the vibrato, which would result in a very dissonant effect.

Next, let's talk about tremolo. Many people often confuse vibrato with tremolo, but the difference between the two is actually very simple. Vibrato refers to the periodic change in the pitch of the sound, while tremolo refers to the periodic change in the loudness of the sound (see Figure 5). Figure 6 has the same scale as Figure 3, but the shape of the graph is significantly different from Figure 3. You can see that the frequency of this note remains constant, but its volume changes over time.

Figure 5: Applying this control waveform to a VCA can produce a short tremolo sound.

Figure 6: Triangle waveform with tremolo added

Now that we understand the difference between vibrato and tremolo, it is not difficult to imagine how a synthesizer would differentiate between them. Vibrato requires modulation of the pitch of the oscillator, while tremolo requires modulation of the gain of the audio amplifier at the end of the signal chain, see Figure 7.

Figure 7: Adding tremolo to our simple synthesizer

A third common modulation effect is the modulation of a synthesizer's voltage-controlled filter (Figure 8). This effect is perhaps not as easy to understand as vibrato or tremolo. When using an LFO to modulate the filter's cutoff frequency, you can create three very different effects by adjusting the LFO frequency. At the slowest LFO frequency (say 0.1 Hz), you can get a slow filter sweep, which is useful for ambient music. If you increase the LFO frequency to 1 to 2 Hz, the sweep will become a wah-wah effect. If you increase the LFO frequency even further, to the upper limit of infrasonic frequencies (around 10 to 20 Hz), you can get a howling effect, which is great for simulating wind instruments.

Figure 8: Applying this control waveform to the filter cutoff frequency can create a short howling effect.

The effect of filter modulation is difficult to visualize with a waveform because the changes in the waveform due to filter modulation are so subtle (see Figures 9 and 10). You can hear the effect easily, however, and the audio effect of filter modulation is quite noticeable, but you need an oscilloscope to see that as the cutoff frequency is lowered, the edges of the waveform become slightly rounded, and the attenuation of the high frequency harmonics also results in a certain reduction in amplitude.

Figure 9: Howling effect on a triangle wave shape

Figure 10: Adding filter sweeps, wahs, and howls to our simple synthesizer

Of course, you can use vibrato, tremolo, and filter modulation at the same time. Many powerful synthesizers will allow you to modulate any of the three using an LFO. Some higher-end synths also provide multiple LFOs with modulation depth control, allowing you to modulate multiple parameters at the same time. Modulation depth controls can include modulation wheel, keyboard aftertouch, and pedal control. Figure 11 may look complicated, but it actually shows a synthesizer with three LFOs and multiple modulation controls in an easy-to-understand flowchart.

Figure 11: A simple synthesizer with three LFOs and multiple modulation controllers

However, in most cases we don’t need three separate controllers to control the modulation amplitude of three modulation VCAs simultaneously. Therefore, most synthesizers will include switches to change the source of modulation and the object of modulation. The sound produced by the synthesizer and the way the synthesizer plays will be affected by how the parameters are modulated and what controller is used to control the modulation amplitude.

Most powerful analog synthesizers also have another common modulation method. To understand this modulation method, let's first review a waveform I introduced in the fourth article of this series, the low-pass filter, the square wave. We call it a square wave not because its shape is a perfect square, but because it is a pulse wave with equal duration at the peak and the minimum of the waveform, as shown in Figure 12.

Figure 12: A square wave waveform

Obviously, the square wave is a special case of a type of waveform called a pulse wave. All pulse waves have the same "rectangular" shape, but different pulse waves have different durations at the "top" and "bottom". The ratio of the time a pulse wave stays at the top of the waveform to the total duration of a cycle is called the "duty cycle" of the waveform. Since the square wave stays at the top of the waveform for exactly half of the waveform's cycle, the duty cycle of the square wave is 1:2, or 50 percent. A pulse wave with a top time of one-third of the cycle has a duty cycle of 1:3, or 33.3 percent, a pulse wave with a top time of one-quarter of the cycle has a duty cycle of 1:4, or 25 percent, and so on (see Figures 13 and 14).

Figure 13: A pulse wave with a duty cycle of 33.3 percent

Figure 14: A pulse wave with a duty cycle of 25 percent

Note: You may see waveforms with a duty cycle greater than 50% in some places. In fact, for any number X between 0 and 50, the waveforms with a duty cycle of 50 + X and a duty cycle of 50 - X are exactly the same, but the phases of the two are in opposite positions. For the time being, we will not discuss the phase of the waveform.

Sawtooth waveforms with different duty cycles have distinctly different sound characteristics. Pulse waves with short duty cycles (about 5% to 10%) have a thin sound and are often used to create the timbre of instruments such as oboes. The closer the duty cycle is to 50%, the thicker the sound becomes. A square wave with a 50% duty cycle has a typical hollow characteristic and is suitable for simulating the timbre of clarinets and other "wooden" instruments.

These timbre changes are caused by the harmonic components of each waveform and the amplitude differences of each harmonic. We can simply summarize the duty cycle of the pulse wave and the harmonic distribution of the corresponding waveform with the following rule:

The pulse wave and the sawtooth wave have the same harmonic distribution. However, for a pulse wave with a duty cycle of 1 to n (n is an integer), the amplitude of every n harmonics in its spectrum is 0.

(If you don’t know what the harmonic distribution of a sawtooth wave is, review the first article in this series.)

After understanding this rule, let's go back and consider square waves. The duty cycle of a square wave is 1:2, so according to the above rule we can infer that the amplitudes of all even harmonics of a square wave are 0. In other words, a square wave does not have even harmonics, and its spectrum is entirely composed of odd harmonics such as the 1st, 3rd, 5th, 7th, etc.

Next, let's consider the two pulse waves in Figure 13 and Figure 14. Their duty ratios are 1:3 and 1:4 respectively. Therefore, the spectrum of the former is composed of the 1st, 2nd, 4th, 5th, 7th, 8th, etc. harmonics, and harmonics that are multiples of 3 do not exist. The spectrum of Figure 14 is composed of the 1st, 2nd, 3rd, 5th, 6th, 7th, 9th, 10th, etc. harmonics, and harmonics that are multiples of 4 do not exist.