*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.