Something similar came up in one of my classes today: why is it that there are so many books that gloss over the mathematical concepts of page layout, but few or none that actually explain it in detail? Do people who write layout books not realize that some design schools have no math department and offer no geometry classes?

Came by chance across this little book today:
http://www.youworkforthem.com/product.php?sku=P0194

It might be of interest to you as it seemed to go a little further than Bringhurst.

The first and second are based on the Fibonacci scale and rounded up, not actually doubling.
http://www.typophile.com/node/15148

Tim

The series of numbers used in the Half Pica modular scale are shown on page 158 at letter (e). It is a double-stranded Fibonacci series. The snapshot of the series here -- it starts at 4 and 5 -- does not show how the two lines originate, which might in fact be the key to the puzzle.

When I examine the second Modular stick, the first value starts as 0.5. If I let this go as its own golden series:

0 0.5 0.5 1 1.5 2.5 4 6.5 10.5 17

This seems to work since these numbers all appear on the left side of the stick as shown on page 168.

Likewise on the right side where the first number shown is 1:

0 1 1 2 3 5 8 13 21 34 51 85

This also holds true. But we must be careful: the number '2' doesn't show up on the right side of this Half Pica stick in print. The line is there marking its place but no number is shown. So we have to consider that Bringhurst sometimes omits numbers on the scales where it might bump up against other numbers in print. Numbers may be a part of the series but do not show up on this page.

Recall that this Half Pica Modular scale, along with the Whole Pica, are based on the ratios 1:PHI and 1:2. What this appears to mean is that each line by itself is really just a 1:PHI extrapolation. But because the starting-place of the left-side is 1/2 of the starting-value on the right -- the left starting at 0.5 and the right at 1 -- each value on the left is 1/2 a value on the right.

So much for the Half Pica Modular Scale. Tim's point is well-taken -- these are values rounded up since PHI is an irrational number. But for our purposes in trying to figure out how these modular sticks work we can take the values as-is bearing this in mind for the first two sticks at least.

I think I've got it. After a day and a half the verdict seems to be: Bringhurst rounds the values off and, in many cases, arbitrarily.

I'll show you what I have so far.

Starting off with the assumption that each side of the stick is distinct and occupied by a single series of numbers, and also assuming that the 1:2 ratio is not responsible for a number line of its own but is rather a function of the starting-point, as in the Half Pica Modular Scale, we start with this series from '1':

1 1.5 2.25 3.375 5.0625 7.59375 11.39 17.08

On the right side of the stick what we see though is:

1 (2) 3 4.5 7 10 15

There's a line marker where (2) should be on the right side. It may be safe to assume that (2) is part of the series.

Well the ratio is 2:3, or re-written, 1:1.5.

1 X 1.5 = 1.5 1.5 X 1.5 = 2.25 2.25 X 1.5 = 3.375 etc.

What Bringhurst seems to do is ignore the decimals unless it is an even .5:

1 1.5 2 3 4.5 6 9

This breaks down at 13.5 which is not on the stick. '14' however is. Also, 1.5 is on the left side here, something we didn't want.

If you look at Bringhurst's values on the right side by themselves:

1 2 3 4.5 7 10 15 22.5

Each is more or less separated by a multiplier of 1.5:

1 X 1.5 = 1.5 (2). 2 X 1.5 = 3. 3 X 1.5 = 4.5 4.5 X 1.555 = 7 7 X 1.4285 = 10

On the left side:

1.5 X 1.5 = 2.25 (above the 2 on the stick) 2.25 X 1.5 = 3.75 (4); 4 X 1.5 = 6 6 X 1.5 = 9 9 X 1.5 = 13.5 (14) 13.5 X 1.5 = 20.25 (20) 20 X 1.5 = 30

What all these numbers designate is a rounding off of the values whenever it seems fitting to balance the 2:3 and 1:2 ratios. In the line just above where 9 X 1.5 = 13.5 is shown, notice how 14 is used on the stick since it's a double of 7, but 13.5 is used in the next calculation, since 20 is a double of 10.

It looks like these modular sticks are all likely approximations only as good as the starting ratios and the finesse with which you preserve those ratios in rounding off.