10. Line the Fractions up (In Order)

I remember once in a computer class, a fellow student asked the lecturer about “vulgar fractions”. It immediately became obvious that the lecturer didn’t know what a vulgar fraction was. I would not expect that a reader of this post didn’t know what a vulgar fraction is, but the lesson of that class was that one has to allow for the possibility. A vulgar fraction is one that has a horizontal line with a numerator on top and a denominator below. When I was little, this was just called a “fraction”, but in that class all those years ago, the default format for a fraction was a decimal fraction.

One thing about decimal fractions: you know what order they go in. They really do serve us best for many practical purposes for which vulgar fractions used to be more common. I have to admit that I cannot immediately imagine seventeen sixty fourths of an inch. (To save you scratching your head, it is 6.746875 mm) On the other hand, sometimes the vulgar fraction is more natural. My first career was as an acoustician. There used to be a rule of thumb for the design of a rectangular room for listening to music in. The adage used to be “avoid whole number ratios of room dimensions”. There were two things wrong with this. The first, that as a rule of thumb, it really wasn’t good enough, and the second was that it wasn’t really obvious how to do that.

The size and shape of a cuboidal room can be fully specified by three numbers: height h, width w, and length l. The shape (without the size) can be specified by any two of three ratios: w/h, l/w, and l/h. The relationships between these ratios are determined by a single multiplication:
l/h = w/h  x  l/w
The task of exploring this relationship with a single multiplication suggested a slide rule to me. I converted to decimal all vulgar fractions with integer numerators and denominators in the range 1 to 16. There were some simplifications. I chose to look only at fractions that were greater then one. (I could have looked at fractions that were all less than one, and then hung it upside down.) I then sorted the fraction values into numerical order. I placed these on two strips of paper with a log scale. These worked like the “C” and “D” scales on an ordinary slide rule. If you chose a value for (say) the ratio of width to height for a room (w/h), and placed the “1” of the top scale on that value on the lower scale, then complying with the adage amounted to finding a spot where both scales were free of vulgar fractions. This point would correspond to l/h on the top scale, and l/w on the bottom scale.

It was interesting that the vulgar fractions were not evenly spaced, and there were gaps: that is there were opportunities to “avoid whole number ratios”

I have recently recalculated the value of vulgar fractions with numerators and denominators in the range 1 to 16, and I have tabulated them. Here is just a tiny bit of the table I generated:

You can see the whole table here .

In the application, it seemed that the lower the whole numbers were, the more significant the whole number ratio was. I devised the “weight” to give each ratio, evaluated as follows:
Weight = 1/(Numerator * Denominator)
A strange and interesting pattern emerged on my slide rule. I might figure out how to reproduce this pattern to show it to you. I will come back to this.

Some years later, another application for the slide rule emerged. The task was to design a frequency shift keyed (FSK) modulator. The usual scheme at that time was to use a VCO from a phase locked loop chip, and feed the data stream into the voltage control pin. This was easy to set up to nominally give the correct frequencies for a Mark and for a Space, but in the days when the usual tolerances (without paying lots extra) for resistors was plus or minus 5%, and for capacitors was plus or minus 10%, the error budget for the modulator didn’t look too good.

An alternative scheme, was to divide a crystal derived frequency, and to change the divide ratio according to whether a Mark or a Space is required. In principle, this is easy. If the two specified frequencies are:
Mark Frequency = fM
Space frequency = fS,
Then we find the smallest frequency fC for which we can write:
fM =  fC / kM                     (1)
and
fS =  fC / kS                       (2)
where kM and kS are integers.
We can then start with a clock at frequency fC and divide it by kM to get a Mark, and divide it by kS to get a space.
This is all very well in principle. In practice, we might find that it is not convenient to obtain a clock at exactly fC, but it is convenient to get a clock at some frequency that is close to it. Another problem is that if the integer constants (divide ratios) kM and kS are large numbers, then the hardware required to implement the frequency divisions become excessive, and the frequency fC is inconveniently high. Our error budgeting, will easily show that we can allow ourselves to get pretty sloppy and yet still produce a much more accurate result than with the VCO. How would we choose constants so that instead of the equations (1) and (2), we establish close approximations that give us a very much simpler design. The result might be simpler, but how do we arrive at it? With the reverberant room slide rule! I dug that out, and chose very close approximations for kM and kS, which yielded a design with frequency errors that were a decade lower then with the VCO design, and at lower cost, and no value-critical passive components.

These days we could do all this in software. If we have a spare pin on a processor that is already on the board, we might build the modulator at zero cost. The method for obtaining kM and k might still apply.

Recently I was reminded of all this when another completely different application arose for a clear picture of how the vulgar fractions fall when placed in order. I wonder how many different applications there might be.