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The Facts on Linear vs. Exponential

As rambled by David Hillel Wilson, Curator, New England Synthesizer Museum

There are a lot of different perspectives that this problem can be viewed from; I have tried to cover them all. If you don't see what you're looking for, keep reading.

What is Linear?

Linear means "a straight line", "a proportion", "a common sense relationship between two variables". Linear is intuitive, it's what you expect. For example, If you're driving 50 MPH for 1 hour, you will travel 50 miles. If you drive 50 MPH for 2 hours, you will go 100 miles. 3 hours, 150 miles. If you're plugging Christmas tree lights into an outlet, then if the first set of lights uses 1 Amp and the second set uses 2 amps, then the total current draw is 1+2 = 3 amps. Linear is the relationship that we, as educated humans, take for granted. Linear is exactly what you would expect things to be without thinking.

Why, therefore, do we even care about exponential responses? Nature would seem to be linear. By the way, what _is_ an exponential response, anyways??

What is Exponential?

O.K. here it is. The Human ear is not linear. It just ain't. What do I mean? Here are some examples:

  • If a note is 100 Hz then the note 1 octave up will be 200 Hz. If the ear were linear, then the next few octaves would be 300 Hz, 400 Hz, etc. However, as many of you already know, the frequency _doubles_ with each octave. Octaves are 100 Hz, 200 Hz, 400 Hz, 800 Hz, etc. This is not linear.
  • If your amplifier makes a sound from a 1 Volt pp signal, then the same sound, made loud enough to "sound" twice as loud to the ear, requires you give the amplifier _not_ a nice linear 2 Volts pp, but rather 10 Volts pp.

So the reason we even bother with exponentiation is that we want our synthesizers to look (to us) like they have the linear relationships that we accept, are used to, and understand intuitively. If we made an oscillator that produced 100 Hz for 1 volt input and gave it a linear response, we would get the following:

  • 1 Volt 100 Hz (original pitch)
  • 2 Volts 200 Hz (one octave above)
  • 3 Volts 300 Hz
  • 4 Volts 400 Hz (two octaves above)

So if I want to raise the pitch of this VCO by one octave, I can't just add "one octave's worth" of volts; If it is running at 100 Hz I need to add 1 volt to get it to go up one octave; but if it is already making 200 Hz I must give it 2 volts to make it hit the next octave. So a nice, cheap linear VCO appears to behave very strangely to our ears because our ears are not linear. We build exponential VCO's because they 'seem' linear to us, and we like linear. This example linear VCO is said to have a 100 Hz/Volt sensitivity.

Exponential VCO's

To give a VCO a response that sounds natural (linear) to us, we must make the VCO jump through hoops so that when our ear interprets the pitch of the VCO, it _seems_ linear to us, and is very intuitive to use. This fancy VCO response is called exponential, because the formula for it involves an exponent. An exponent is simply the power that you use when you "raise a number to a power". In "Two to the third power", the exponent is 3. The equation for the frequencies of the notes on a piano keyboard is

frequency = 440.0 * (2 ** (NoteNumber/12)) Hz

where NoteNumber is 0 for concert A, 1 for A#, -1 for G#, etc. You don't need to know or understand this equation; All I want you to get is that the NoteNumber appears to the right of the "raise to a power" symbol, "**". Thus, exponential VCO's sound more natural when programming a synthesizer.

Note that I said _Programming_; If you are _listening_ to a synth produce a 440 Hz concert A tone, it doesn't matter how that frequency was arrived at; All 440 Hz sine waves sound the same; so do all 440 Hz Sawtooth waves, no matter how the pitch of the oscillator is determined. Thus, the linear vs. exponential debate has nothing to offer listeners; it is strictly for composers, programmers, and synth designers to bat around, curse at, etc. (There are a few exceptions; Glide, vibrato, and FM sound different on a linear synth than on an expo synth).

Linear VCO's

One interesting side effect of a linear response VCO is that vibrato will become twice as deep if the pitch drops an octave; On an expo machine, the amount of vibrato does not change with the key being played. When an exponential VCO has a 1 Volt/octave response (the "standard" used by ARP and Moog), to transpose the pitch one octave up, you simply add in another volt. You can take two 1 V/Oct keyboards and use one to transpose the other; You can use a keyboard to transpose a sequencer, and all the time everything will stay in tune. With a linear VCO, addition does not work; you must multiply the voltage to transpose the keyboard. Thus, using two keyboards or a keyboard and a sequencer is almost impossible. Further, if you twist the pitch knob on a linear VCO such as the PAiA 4720, unless you have a voltage going into the VCO, nothing will happen. In order to track properly, the VCO _must_ produce exactly 0 Hz for a 0 Volt input, and 0 Hz is no sound. Korg faked this out on the MS series synths by not being truly modular; the keyboard is _always_ connected to the VCO, so it is always making a pitch.

Cheating with PAiA

About 10 years ago, at the height of my PAiA (linear) system, I got adventuresome and cut the long wire the goes to the top of the voltage divider board on the 2720-8 keyboard and stuck it right into the output of the 4780 sequencer. Much to my shock and amazement, I was now able to transpose the sequencer from the keyboard on an only slightly modified PAiA system! I was using the keyboard as a voltage divider, which is, after all, how all synth keyboards (linear _and_ exponential) work. Later Analog/Digital hybrid units from PAiA, specifically the P4700/J polyphonic modular synthesizer, had a DAC called the 8780. It had an input for a signal that would be multiplied by the digital number coming from the computer; The result was that vibrato on an otherwise linear PAiA machine would now be equally deep all the way across the keyboard. Unfortunately, this breakthrough came just before PAiA stopped selling all this neat stuff. (It's true that the 8780 DAC wouldn't play in tune for s__t, but that was a limitation of the device itself and not the technology or mathematics behind it).

Exponential AND Linear?

Which is better? It all depends on how you use your synthesizer. By the way, for the record, the first commercially available 1 V/oct (expo) synths were the modular Moog systems, while the first commercially available Hz/V (linear) were the modular Moog systems. What? Both? Yes. The 901 A "oscillator driver" was an expo converter, and then the multiple (usually 3) 901 B VCO's were linear, and they tracked each other in harmony - Before Korg or Yamaha, even before PAiA! Meanwhile, a larger Moog had several sets of 901 A/901 B VCO's, and they could all answer up from the same keyboard. On the kick-ass chord blast on ELP's "Tank", from their first album, you are hearing linear AND expo oscillators all tracking!! But understand the principle: Between the keyboard and each oscillator is one and only one expo converter; any other arrangement wouldn't work.

Keyboards for Linear Synths

A keyboard to drive a linear VCO must be an exponential keyboard; these are tricky to build and somewhat rare; examples include the pedals of a Moog Taurus I (but not the Taurus II), the PAiA 2720-8 and 4760 keyboards, and a few pieces by Yamaha and Korg

Cents and Decibels

By the way, when talking about linear and exponential and numbers and formulas, there are units that have the exponentiation built in to them so you can talk about exponential things like pitch as though the ear were linear. You are already familiar with one such unit, the octave. You can count the number of octaves a given keyboard has, breaking the final octave down to individual notes if the keyboard does not have an integral number of octaves (An ARP Odyssey has a 3 octave (37 note) keyboard; a MiniMoog's is 3-1/2 octaves (44 notes)). Just as octaves are divided into notes, or semitones, there is another unit called cents; There are 100 cents to the semitone, and 1200 cents to the octave. Cents are used a great deal by people who use just intonation.

Another such unit is the decibel, named for Alexander Graham Bell, the inventor of the telephone, the busy signal, the wrong number at 3:00 AM, etc. Towards the beginning of this massive text I spoke of a 1 Volt signal going into an amplifier. A gain of 20 decibels (correctly abbreviated dB) means that the voltage has been multiplied by 10. 40 dB is times 100, and 60 dB is times 1000. Thus, to make a sound twice as loud, you add 20 decibels to it instead of having to multiply. That's what decibels are, if you didn't know but were curious.

Building Expo Converters

The way that an exponential VCO is made is as follows: You take a linear VCO (strictly speaking, it's usually a CCO, Current Controlled Oscillator, but you can ignore that distinction for this discussion) and you slap an exponential converter onto the input. And expo converters are almost all built the same way; a transistor junction is used (abused?) to produce an exponential response - one of nature's few, excluding the senses of animals and Man. Unfortunately, this response is sensitive to temperature, so early Expo synths (like the big old Moogs) tended to wander a in pitch little as they warmed up. There are two ways to stabilize these things with regard to temperature: You can A: Try to keep the transistor at a constant temperature regardless of the surrounding temperatures, or B: use a thermistor (temperature sensitive resistor) to balance out the changes in the transistor.

In machines that hold the transistor junction at a constant temperature (Newer model MiniMoogs, and EML synthesizers, for example), you usually have to turn on the machine and let it warm up for 20 minutes before playing it, so that the temperature can stabilize.

In machines that use a thermistor (All ARP, early Oberheim, early MiniMoogs), the things are usually rock solid from when the power is switched on - This is particularly true of the ARPs. This type of VCO was invented by Alan R. Pearlman, who started the ARP Instruments company. The only disadvantage here is that you can't get thermistors in small quantities. I bought a small supply from Tel Labs of Londonderry, New Hampshire, but they have since gone out of business, and the Museum is down to 1 ARP part and no Moog or Octave Cat parts. KRL still makes them, but you have a minimum order quantity of $100. If you need some, check with the Museum to see if we've bought anymore.

Now about that glide thing

Most glide circuits are just an R-C circuit, which produces an exponential response with respect to time. Thus, on an expo synthesizer, you hear the glide rush away from the previous note and then slow down as it approaches the final note. Some systems, however, produce linear glide that is always the same speed from start to finish. This can sound strange to serious Moog people. In addition, you can have either linear or expo glide on a linear synthesizer, yielding two more different-sounding combinations. A linear synth with expo glide (PAiA, for example) sounds expo when you glide up, but sounds more linear when you glide down, since as the glide slows down, the notes are crammed in closer together, and the perceived sound is of the glide spending the same amount of time at each note even though the voltage is slowing down! If you can't quite match the analog glide sound on that record with your digital synthesizer, maybe its VCO's or glide circuit are not exponential. Get a _real_ synthesizer!

Linear vs. Expo ADSR's

Now I'll ramble about linear vs. expo in ADSR's and VCF/VCAs. Those of you who have been fortunate enough to use a VCA that has both linear and exponential control modes have probably noticed that the linear control mode sounds more natural, even though the ear likes to hear exponentiated things. How can this be? The answer is in the ADSR. Most envelope generators use the same R-C used in glide circuits to make a curve that is already an exponential. This is because you can make a curve that is exponential with respect to time with just a resistor and a capacitor. Note that exponential with respect to time is easy to do; it's exponential with respect to voltage, as in an Expo VCO, that's hard to do). Thus, almost all analog ADSR's and all _good_ digital ADSR's produce this exponential curve for decay and release. If you run these exponential curves into a linear VCA, the ear gets what it wants; an exponential decay, very natural. If you run them into an exponential VCA, the ear hears a _double_ exponential, which is much less natural (although it still can be musically valid, of course).

The poor old VCF is caught in the middle! If the envelope is sweeping it, linear response is most natural. But to track the VCO's, it must be exponential. Of course, the double exponential filter sweep that results from using an expo VCF is what most of us are used to, and it therefore "sounds like a synthesizer".

One final word on ADSR's, and then I promise to shut up. As is pointed out in the Musical Engineer's Handbook from Electronotes, the attack curve on an envelope generator is usually linear, not exponential. The reason is that it becomes too difficult to determine the exact point in time at which to start the decay. For a rather drastic example of this, find an Oberheim OBx, OBxa, or OB-8. Set the VCA sustain to maximum, and the VCF decay and sustain to minimum. Finally, set the VCF attack to maximum, and press and hold down eight keys all at once. All the notes will gradually fade in as the VCF's open; They will, however, drop out one at a time at fairly random intervals with respect to each other. And these are the well regarded CEM 3310 V.C.ADSR chips; cheaper ADSR designs are even worse.

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