of Microprocessors

82 MUSICAL ApPUCATIONS OF MICROPROCESSORS
allowed would be quite difficult to deal with. Typical sources oferror in this
range are thermoelectric voltages, voltages induced from stray magnetic
fields, and thermal noise in resistors. What is worse, we expect these voltage
levels to travel freely through patch cords without degradation. Thus, it is
apparent that directly proportional voltage control is impractical for a wide
control range and great relative accuracy.
Exponential Relation
Another relationship that makes more sense from a lot of viewpoints is
an exponential one. Stated first in musical terms, such a relationship could be
something like a 1 V/octave. In mathematical terms, this would be F = 2VPo,
where F is the output frequency, V is the control voltage in volts, and F 0 is
the basis frequency for this relative scale. For a basis frequency of 20 Hz, a
voltage range of 0 to 10 V would cover the audible range. An interesting
property of such a scale is that the 1% relative accuracy desired corresponds
to about 14.5 mV independent of the frequency range. Thus, rather than a
liberal error allowance at high frequencies and astingy one at low frequencies,
the margin for error is a constant, manageable value.
This property alone would be sufficient persuasion for adopting the
exponential relationship, but there are many more desirable characteristics.
In Chapter 1, ie was noted that the sensation of pitch was an approximately
exponential function of frequency. Using an exponential voltage-controlled
oscillator, a linear increase in control voltage of, say, 1 V/sec would result in
a reasonably steady rise of pitch. A linear VCO, on the other hand, would
very rapidly sweep through all of the low frequencies and then seem to
require a considerable amount of time to complete the upper end of the
sweep.
Even if the ear's response did not resemble an exponential curve, the
equally tempered musical scale is precisely exponential. One octave, of
course, corresponds to 1 V and a half-step corresponds to 1/12 V or about
83.3 mY. Likewise, a fifth corresponds to 7/12 V and a major third is 1/3 V.
Thus, if some arbitrary voltage level produces a particular pitch, a voltage
1/12 V higher produces a pitch one half-step higher.
One application of this property is in transposition from one key to
another. Consider a melody played in the key of C on a keyboard that outputs
control voltages. If a constant voltage of 1/6 V is added to the keyboard
output, the melody would actually sound in the key of D. The constant
voltage may be injected with a mixer between the keyboard and the oscillator,
although most VCO modules have a basis frequency knob that does the
same thing.
Injection of vibrato is also considerably simplified with an exponential
VCO. A 1% vibrato, for example, would require a 15 mV peak-to-peak
vibrato voltage independent of what note was played. With a linear veo,
the vibrato would be excessive on the low notes and nearly inaudible on the