Why is the standard f-number system not consistently rounded?

Question

The standard f-numbering system can be calculated by the square root of powers of two. Commonly used today is the 1/3 system, as it can be found here: https://en.wikipedia.org/wiki/F-number#Typical_one-third-stop_f-number_scale

An interesting observation is however, that the numbers do not follow a consistent schema, how they are rounded from the calculated value.

Of course, there are the trivial f-numbers, which are powers of two, such as 1, 2, 4, 8, 16. Then, I presume for reasons that you only want two digit numbers, all numbers above 9 are simply rounded to an integer number. But here starts the inconsistencies: f/22 should actually be rounded up, to be f/23, as its exact value is 22.627. And also the numbers below 10 are not consistent: f/1.2 should be f/1.3, f/3.5 should be f/3.6 and f/5.6 actually f/5.7 - if the numbers were correctly rounded. As the other numbers are rounded correctly, the question is: what makes these numbers special, that they are not rounded correctly?

According to the history section on Wikipedia 1:, the numbering system used today stems from the Royal Photographic Society. In the linked book on page 104 2, a table is shown which gives already the 5.6 number, but interestingly also f/12.5 (which is presumably today's f/13). The value of f/11 was however described as f/11.3 on page 100. Thus, also back then rounding was not consistent... To my surprise it also shows an aperture f/7.7 - possibly to provide a convenient conversion value from the Goerz, Voigtlander, and Dallmeyer systems.

What are the reasons for rounding the numbers in such this special, unpredictable way? Was it just convenient to use 5.6 instead of 5.7 for example? Funny enough, the table shown on the Wikipedia article notes this as f/5.66.

Answer 1

Probably for the same reason exposure times are inconsistently rounded. The difference between the "actual" target f-numbers as powers of √2 and the rounded numbers we use is so trivial as to be beyond the limits of the vast majority of cameras in existence to accurately differentiate.

Why would anyone care if the rounded numbers used to describe the powers of the square root of two were imprecise to a lesser degree than cameras could actually achieve? Between variations in purity and age of chemicals, temperature control when developing and printing, as well as the limitations of manufacturing tolerances at the time, even half-stop accuracy was a pipe dream when several different systems for describing entrance pupil size, including using f-numbers, were being created. The rounded numbers were still more accurate than the capabilities of cameras to actually execute them, much less the more mathematically precise target values. Eventually the f-number system we all use today was standardized as alternate systems fell by the wayside. But even today the actual apertures achieved by even high end cameras and lenses are less precise than the inconsistently rounded numbers we use.

In other words, when we tell the camera to use f/22, it targets f/(√2)^9 (or approximately f/22.627), but where it actually lands could be anywhere from, say, f/21.5 to f/23.7. This is still within the range of 1/6-stop accuracy (roughly f/21.3574 - f/23.9729 for "f/22") that has been the general expectation for even high-end interchangeable lens cameras used to take artistic, documentary, historical, etc. photos since the latter part of the 20th century.^{1, 2}

5.6 is twice 2.8, just as (√2)^5 is twice (√2)^3.

22 is twice 11, just as (√2)^9 is twice (√2)^7.

90 is twice 45, just as (√2)^13 is twice (√2)^11.

The numbers are just approximations of the actual targeted entrance pupil openings, where the camera/lens will attempt to make every full stop exactly half as large in area as the previous one.

When we tell the camera to stop down to f/5.6, it aims for f/(√2)^5.
When we tell the camera to stop down to f/11, it aims for f/(√2)^7.
When we tell the camera to stop down to f/22, it aims for f/(√2)^9.

Just as:

When we tell the camera to expose for 1/30 it aims for 1/32 (or 1/2^5 which is also 2^-5).
When we tell the camera to expose for 1/125 it aims for 1/128 (or 1/2^7 also 2^-7).
When we tell the camera to expose for 15" it aims for 16 seconds (or 2^4).
When we tell the camera to expose for 30" it aims for 32 seconds (or 2^5).

(You can easily test those last two with a stopwatch. Some cameras actually target the rounded value of 30 seconds, but many cameras target the actual power of 2 of 32 seconds.)

There are some f-numbers that are rounded to the same integer for some 1/3-stop scales and some 1/2-stop scales, but they're not the same target apertures with both the 1/3-stop scale and the 1/2-stop scale. For example, f/13 on the one-half stop scale (f/13.454) is slightly narrower than f/13 on the one-third stop scale (f/12.699). This is because the 1/2-stop scale is based on the fourth root of two, while the 1/3-stop scale is based on the sixth root of two. The 1/4-stop scale is based on the 8th root of two. 1/6-stop graduations are based on the 12th root of two.

_{¹ If you want more precise than that, you'll need to go with lab grade instruments that typically have fixed aperture openings. If you want to change the f-number, you must swap out the aperture diaphragm and then precisely align it in the center of the lens' optical path, rather than opening or closing an iris type diaphragm with multiple blades. For precision of exposure times, you'll control it using the duration of very precisely controlled lighting in an otherwise darkened enclosure, rather than by using a focal plane shutter.}

_{² There are reports scattered about that most lenses with electronically controlled apertures are calibrated in 1/8-stop steps. Whichever 1/8-stop value is closest to the 1/3-stop value selected by the photographer is the one that is targeted when 1/3-stop intervals are selected by the camera's user. When 1/2-stop intervals are selected, the 1/8-stop calibrations line up, with every fourth step being either a full or half stop value.}