On Oscillator Linearity and Musicality
Oscillators, like most things, are imperfect. This page looks at a
class of oscillators called VoltagetoFrequency Converters, which is a
fancy way of saying VCO.
To whet your appetite, here are a couple of excellent app notes by the legendary
Jim Williams (RIP) that provide some superb examples of highquality analogue design:
In the majority of VtoF converter designs the control response is linear,
where the output frequency is simply
$$
F_\textit{out} = k V_\textit{in} + \textit{offset}
$$
Linearity is a measure of how well the actual VtoF oscillator
conforms to the ideal. Analog Devices define linearity of a VtoF as
The preferred method of specifying nonlinearity error is in terms
of maximum deviation from the ideal relationship after calibrating
the converter at full scale. This error will vary with the full
scale frequency and the mode of operation.
(AD654 datasheet)

It is usually quoted as a percentage, and in typical
datasheet fashion will include typical and worstcase manufacturing spread as well
as over temperature.
Which is all very well, but for a musical oscillator, where we concern
ourselves with semitones and cents, how does linearity percentage equate to
oscillator linearity? Well, this page presents the tools and tables to help
you answer two questions:
 "I want a musical VCO that tracks to better than C cents, what VtoF linearity percent do I need?"
 "This VtoF has L percent linearity. How many cents tracking will it give me?"
The relationship between frequency and cents is
$$
\textit{freq} = \textit{freq}_\textit{base} \times 2^{\frac{\textit{cents}}{1200}}
$$
and where there are 100 cents per semitone, 12 semitones to an octave.
With this fundamental relationship we can now answer these two questions.
I want a musical VCO that tracks to better than C cents, what VtoF linearity percent do I need?
If we rearrange the above equation to compute the linearity percentage,
we get the following equation:
$$
\textit{linearity(%)} = ( 2^{\frac{\textit{cents}}{1200}}  1) \times 100
$$
If we apply this over a range of cents accuracy, from 0.01 cents to 10 cents,
we get the following results:
Cents  Linearity (%) 
0.01  0.00058 
0.02  0.0012 
0.05  0.0029 
0.1  0.0059 
0.2  0.012 
0.5  0.029 
1  0.058 
2  0.12 
5  0.29 
10  0.58 
This VtoF has L percent linearity. How many cents tracking will it give me?
Working out the answer to this question requires turning our frequency
equation inside out. With some rearranging we arrive at
$$
\textit{cents} = \frac{1200}{\textit{log}_{10} 2}(\textit{log}_{10}(L + 100)  2)
$$
where L is the VtoF linearity in percent. Applying the above to a
range of linearities we get the following table:
Linearity (%)  Cents 
10  165.004 
5  84.467 
2  34.283 
1  17.226 
0.5  8.635 
0.2  3.459 
0.1  1.730 
0.05  0.865 
0.02  0.346 
0.01  0.173 
0.005  0.087 
0.002  0.035 
0.001  0.017 
What does this all mean?
Firstly, from a musical perspective, it is commonly stated that a musical
oscillator only needs to track to within ±5 cents. From the above we
can see that we need a VtoF linearity better than 0.29%.
Secondly, we can use the second table to qualify how good an offtheshelf
VtoF converter is in a musical context.
Note: these figures come from datasheets, so treat with caution.
Singlechip VtoF Converters
Roughly in increasing order of linearity. Note prices are from
UK Farnell website as of June 2014 for DIP parts, and are presented as a guide to
relative cost only.
Device 
Linearity (typical %) 
Cents 
Comments 
Guide Price (GBP) 100off qty 
MC14046 (Mot) 
1.0% 
17.2 
Supposedly a "goldenoldie"... but not according to the datasheet! 
N/A 
CD4046 (TI) 
0.4% 
6.9 
Cheap as chips, but poor linearity. Thomas Henry seems to like them though. 
0.163 
XR4151 (Xicor) 
0.05% (precision mode) 
0.87 
Pretty decent little 100kHz VtoF, needs a bit of work to
get the stated linearity (external opamp, diode, etc) but for
the price it's a real bargain! 
0.68 
AD654 
0.06% <250kHz 
0.52 
Pretty good linearity over an 80dB range. Price is good for
a linear VCO that is usable over 13 octaves. 
4.23 
0.2% <500kHz 
3.5 
ADVFC32 (AD) 
0.05% < 100kHz 
0.87 
Analog Devices' clone of the VFC32. 
6.80 
0.2% <500kHz 
3.5 
LM331 
0.01% < 11kHz 
0.17 
Wellknown VtoF converter (I believe designed by Bob Pease?).
Very good linearity over the 1Hz to 11kHz range (13 octaves).
Fairchild make a lowcost clone: the KA331 
1.05 (KA331: 0.254) 
VFC32 (TI) 
0.025% <100kHz 
0.43 
The industrystandard VCF32. Excellent linearity over
a 100dB range (17 octaves), for a similar price to the
AD654 but in a larger package. 
4.14 
0.05% <500kHz 
0.87 
VFC320 (TI) 
0.025% <100kHz 
0.43 
Faster version of the VCF32, up to 120dB range (20 octaves),
for about twice the price. 
14.34 
0.05% <500kHz 
0.87 
0.1% <1MHz 
1.73 
Jim Williams App Note 14
The classic AN14
describes a number of linear VtoF, most of which are compared in the table below:
Circuit  Linearity (%)  Cents 
Figure 1 (1Hz to 100MHz)  typ. 0.06%  1.04 
Figure 5 (1Hz to 2.5MHz)  typ. 0.05%  0.87 
Figure 8 (<10kHz)  typ. 0.005%  0.087 
Figure 10 (100kHz to 1.1MHz)  typ. 0.0007%  0.012 
Figure 14 (1Hz to 100kHz)  typ. 0.1%  1.7 
All of these circuits are truly amazing at what they achieve. Jim was truly an
analogue guru of the highest order.
