Since I write about simulating classic analog synthesizers (a process often called “virtual analog”), mostly notably in my wavetable oscillator series so far, I wanted to touch on the topic of parameter control signals. Classic synthesizers have knobs to set parameters, of course, but key parameters are controlled by voltage signals as well—primarily for oscillator and filter frequency, note volume, and perhaps oscillator pulse width.

Some more sophisticated and rare synthesizers—most often modular synthesizers—allow voltage control over such parameters as filter resonance, lag processing, and even envelope times. But the must-haves for synthesizers, certainly, are frequency and volume (pulse width is a distant third, but very useful and easy to implement, and important enough to make it requirement for most analog synthesizer designs).

### Frequency and one volt per octave

Frequency perception by the ear is logarithmic—that is, note pitch is related to the log of frequency. As frequency increases in steps by a fixed amount, the difference in pitch becomes less with each step. To look at it another way, frequency must double for each increase of an octave in pitch. To make the relationship more manageable for synthesizers, an important de facto standard was settled early on—one volt per octave control. If an oscillator is set to 100 Hz, increasing the control input by one volt takes it to 200 Hz. Increasing another volt yields 400 Hz. Each of those steps is one octave in pitch—a doubling of frequency. If control of frequency were linear, the amount of additional voltage needed to move up an octave would depend on the current frequency. And the depth of low frequency modulation—vibrato, for instance—would depend on what note was being played. This arrangement would make the system much more difficult to manage. So, pitch-based synthesizer modules—voltage-controlled oscillators and filters—contain a linear-to-exponential converters.

Note that not all classic synthesizers used exponential control of frequency. But most did, including the most recognizable, and the ones we’ll use as a model. Further, the linear controlled (Hertz per volt) systems were most often performance synthesizers, where shortcomings could be more easily worked around in a limited architecture. But we’re taking a modular approach, where exponential control is the clear winner—the main difference is that we’ll use values instead of volts for control.

For us, exponential conversion is just a bit of math, so it can be a virtual module that we can attach to any linear input. But our basic target will be to use it where the classic designs use it.

### Volume

Our volume perception is also logarithmic—loudness is related to the log of amplitude. We need to double the amplitude of a signal each time we want an increase of 6 dB in volume. So, you might think that your typical synthesizer would have developed with voltage-controller amplifiers—VCAs—that have exponential converters, and typical control sources would be linear. But while modular synthesizers often have the option of linear or exponential control, the most common synthesizer VCAs are linear. And the most common control source of an output VCA is an envelope generator with exponentially-shaped segments. An exponentially decaying voltage controlling a linear VCA gives a smooth, linear-sounding, fade out. One reason analog synthesizers evolved this way is that the shape of discharging capacitors is exponential—it’s fairly each to make exponential-shaped envelope generators from capacitors and resistors for the curves (our one-pole filters!), and transistors to switch between states.

Note that the ear is not as sensitive to volume changes as it is to frequency changes. It’s much harder to hear the asymmetry resulting from linear control with a low frequency modulation for volume that it is for pitch—just as it’s much easier to hear a singer who’s a little bit off-key in a mix than it is to to notice a vocal mixed a little too soft or loud.

Also note that even in modular synthesizers, where a VCA might have exponential control (my old Aries modular has a switch on the VCAs, to select linear or exponential control), the envelope generators typically remained exponential-only, so the VCA would be set to linear for the most common use of controlling the overall note envelope. Exponential control was especially useful for other VCA uses, such as processing control voltages.

### Mixing linear and exponential control

We’ve established that, while both frequency and volume have similar needs for exponential control, typical analog synthesizers have exponential control inputs for oscillator and filter frequency, but not for VCA amplitude. VCA are most often driven by envelope generators, which have exponential segments in classic designs. We may also use low frequency modulation for the VCA (tremolo), but our ears aren’t terribly sensitive to the resulting asymmetry. But what about the exponential segments of the envelope generators? Won’t they cause problems when we often use the same envelope generators on the exponential inputs of the oscillators?

It’s true that we’ll get a double-exponential response. Instead of the pitch sliding down at a steady rate on the decay portion of the envelope, the move starts quickly but decelerates, sliding into the target pitch relatively slowly. Fortunately, this sounds good, and mimics certain aspects of physics that we’re used to—the landing of birds and airplanes, the slowing of an elevator reaching its target floor.

### Next up

Why bring this up now? Because our next topic is envelope generators, and I wanted to establish why we’ll be looking at exponential segments. If you do a web search for *envelope generator* (and especially *ADSR*) images, you’ll see a large number of drawings with linear segments. But make no mistake—if you want to sound like a classic synthesizer, then you want exponential envelope segments.

Great article! Looking forward to your foray into envelopes. Your articles are the best I have ever read on the subject of DSP, and they’ve been an enormous help to me. Many thanks and Godspeed.