# Moog Monday - On Synthesizers: Linear And Exponential Control Modes

*(This column originally appeared in the September 1978 issue of Contemporary Keyboard magazine.)*

In my last two months' columns I discussed some basic principles of voltage control, using the voltage-controlled amplifier as an illustrative device. I defined the control mode as the mathematical relationship that tells how the voltage-controlled parameter (*amplifier gain* in the case of the VCA) changes as the control voltage increases. I talked a bit about the linear mode, the mode in which a graph of the control voltage versus the parameter change is a straight line. Fig. 1a shows what such a graph looks like.

In this column I will explain another, even more important control mode: the exponential mode. In the *exponential* mode, the controlled parameter changes by a certain ratio whenever the control voltage changes by a certain *amount*. Fig. 1b shows the control characteristic of a typical exponential mode VCA. Mathematically speaking, this graph has three features that are of interest to electronic musicians. The first is that, as the control voltage goes negative, the controlled parameter approaches but never reaches zero. Second, as the control voltage goes positive, the controlled parameter increases at an increasing rate, theoretically without limit, And third, the "middle" of the curve (the part we're usually interested in) is smooth, with no abrupt corners or jogs.

An important reason that devices with exponential control modes are musically useful is that many aspects of the way we hear can be accurately described by exponential functions. For instance, if we increase the amplitude of a sound by equal *ratios*, then the loudness seems to increase in equal steps. Fig. 2 shows this relationship. The graph has the same shape as Fig. 1b, but the axes are labeled differently. The vertical axis is a measure of how big the sound vibrations actually are, while the horizontal axes are two measures of how loud the sound seems to us. The bottom horizontal axis is the familiar set of score dynamic markings, while the top line is marked in acoustic decibels; I have discussed decibels at length in an earlier column. For the purpose of this column I will point out that decibels are defined as being logarithmically related to the actual sound amplitude. Logarithm is the inverse (exact opposite) of exponent: if decibels are logarithmically related to sound amplitude, then sound amplitude is exponentially related to decibels. An increase of six decibels corresponds to the doubling of the sound amplitude (or audio voltage applied to a speaker), and is roughly equivalent to one notch up the scale of musical dynamic markings.

Another important reason that voltage-controlled devices, especially voltage-controlled amplifiers, are musically useful is that "natural" sounds frequently build up and decay exponentially. For instance, the vibration of a guitar string, once plucked, decays exponentially. That is, the vibrational energy in the string decreases by a fairly constant ratio during each second of vibration. Furthermore, the high overtones of a guitar tone that are set into motion when the string is plucked decay exponentially at a more rapid rate than those of the fundamental and low overtones.

I'll now illustrate these basic features of the exponential control mode with a couple of simple patches using one or two exponential mode VCAs. While doing so, I'll point out some of the advantages of the exponential mode over the linear mode. Fig. 3a shows the simplest use of an exponential mode VCA. A variable voltage source under control of the musician is used to determine the VCA gain. A constant-amplitude audio tone is fed to the VCA's signal input. Fig. 4b, which comes from combining Figs. 1b and 2, shows that the *perceived* loudness of the exponential mode VCA output varies smoothly from *pp* to* ff* as the control voltage is varied from 0 to +6V. The middle value of control voltage (+3V) gives an output loudness of mf, the middle of the loudness range. On the other hand, if the VCA had a linear control mode, the loudness would vary drastically from complete silence to within one dynamic marking of its maximum output as the control voltage went through the first half of its range. This is shown in Fig. 4a, which comes from combining Figs. 1a and 2. Thus, an exponential mode VCA often makes a more satisfactory, smoother-operating electronic volume control than a linear mode VCA.

Next we'll add another control voltage (see Fig. 3b). Modular VCAs are generally equipped with two or more control inputs; the VCA gain is determined by the sum of the control voltages applied to these inputs. Our second control voltage is a negative-going ramp, a control signal whose voltage starts at some positive value and decreases at a steady rate, like a single sawtooth wave cycle. The VCA output is shaped into an exponentially decaying tone. The rate at which the ramp voltage descends determines the tone's rate of decay ("percussiveness"), while the other control voltage, which does not change rapidly, determines the overall loudness of the tone. If the ramp repeats regularly (slow sawtooth), then you have "repeat percussion." If the ramp voltage increases instead of decreasing, the VCA output increases exponentially, thus producing a "reversed tape" type of exploding sound. The important feature of this simple patch, however, is that a single VCA performs two functions: envelope shaping and overall dynamic control. Within the gain range of the VCA, there is no interaction between the overall dynamic and the envelope shaping control voltages. When linear modes VCAs are used, two separate VCAs are needed to perform the envelope shaping and the overall dynamic control functions.

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