Exponential Converters

All classic analogue synthesizers have exponential converters built into their oscillators to convert the linear voltage scale we use to control them into the exponential scale we hear. The 1 volt per octave standard has this transformation built into it. This is a critical part of the synthesizer system because, without it, pitches would sound flat for a given voltage interval or fall entirely outside the control voltage range. This has to do with the fact that we hear pitches in an exponential scale. This can be heard using a slew limiter switched to linear on a classic synth. Exponential control of filter frequencies is just as important for the same reasons. So controlling the pitch in a synthesizer with an exponential function is important. Now, to date, the common way of achieving such a conversion uses some of the most uncommon parts and very sophisticated circuit designs that utilize matched components and little heater ovens and very fine adjustments with precision manufactured resistors and precision voltage and current references. This is all for the sake of accuracy, and elimination of drift caused by changes in temperature. We’re going to blow that all away and start with the simplest most accessible approach using the central mechanism at the heart of these systems.

At the heart of an exponential converter is a simple lowly diode. That’s it. A lesser known fact about diodes is that they have a nonlinear current to voltage relationship that pretty much matches an exponential function. See the exponential curve in the top right quadrant of the graph when V approaches Vd.

Diode-IV-Curve

So we have at our disposal a linear voltage to exponential current converter. This turns out to be an advantage because the typical sawtooth oscillator is actually current controlled rather than voltage controlled, because the frequency depends on how fast the timing capacitor charges, and that rate of charge is dependent on the rate of electron accumulation at the plate, i.e. current. More theory on relaxation oscillators can be found on my blog post.

The problem with the diode approach for classic synthesizer designs is that the V to I function is heavily temperature dependent. According to Hal Chamberlin (my elder in these matters) for a typical VCO design a 1 degree Celsius change in temperature would result in nearly a half step change in pitch. Pretty significant if you consider that the ambient temperature in your average terrestrial environment can vary between 4 and 40 degrees Celcius, which would result in a possible pitch drift of roughly 1 1/2 octaves. Is this too much drift for a chaotic noise box, or is it the range of instability that we find actually advantageous? If the pitches drifted into an inaudible range it would render the device useless at certain temperatures.  If you wanted the device to function in outer space it could be a problem, but it looks like here on earth it would produce meaningful values in any condition short of a sub zero meat locker. Good to know.

If it turns out that the drift is too much there is a pretty simple approach to eliminating the bulk of the drift using a second diode which cancels out the changes. There will still be nonlinearities and since we won’t bother to source matched components it won’t be perfect, but the bulk of the drift will have been eliminated.

If you’re interested in this topic I recommend checking out Hal Chamberlin’s book “Musical Applications of Microprocessors.” It’s out of print so you’ll have to search the library system, but it will be well worth the effort as it has an extensive section on classic analogue synthesizer circuit design. If you are thinking about building a synthesizer I also recommend checking out the ASM-1 site as they have a whole modular synthesizer design that is open source, developed and well tested.

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One Response to “Exponential Converters”

  1. The Steiner Resonant Filter & A New Adaptation « Tristan Dabbles Says:

    [...] next project will be to build an exponential converter which will make playing the light controlled synth a little [...]

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