One of the most disturbing matter for new photographers is the seemingly random series of numbers that we have come to know as the f-stop scale or aperture scale. Why would anyone invent such an arbitrary suite ?
Let’s go back to the second century BC, when a greek astronomer named Hipparchus developed the first system for organizing stars by their apparent brightness. He ranked stars on a scale from 1 to 6 based on the brightness he observed. Centuries later, when astronomers developed methods to quantify the actual brightness of each star, they noticed something strange. A category one star was not six times brighter than a category six star: it was 100 times brighter. Every step on the apparent brightness scale yielded an actual brightness increase of 2.5x. It turns out that the human eye is not very good at picking out small differences in brightness. In order to see any difference, we must change the brightness a lot, like two and a half times its original value.
What Hipparchus discovered by accident, was the logarithmic nature of human perception. Somewhere within us, we are hardwired to perceive level changes only when they are many times less than or greater than the next level. The visual advantage we gain from this is dynamic range. It has been estimated that the human eye can effectively process 10 f/stops of light levels. An extraordinary range which exceeds from far any film or sensor.
The logarithmic nature of human perception was known in the nineteenth century, and has been expressed by German psychologists as the Weber-Fechner law. The law has implications that apply to many different human processes: vision, hearing, and mental processing. Much more, the logarithm scale can be observed in a lot of domains, like sound (dB scale), chemistry (pH scale), relevance of changes in biological processes, etc.
Which brings us back to f/stops. At the same time psychologists were musing about the logarithmic human perceptions, early photographers were quantifying the optical principles of their cameras. Fairly early on, it was determined that the area of the aperture hole needed to vary by a factor of 2x in order to yield perceptibly brighter or darker photographs from one f/stop to the next.
Here a figure which shows the progression of aperture areas going from largest to smallest. For each progression, the area is divided in half until we get to the smallest aperture which is 1/32nd the size of the original one. The diameter of each of these apertures is proportional to the square root of the aperture area. Thus, by taking the square root of the aperture areas, we see some familiar numbers: 1, 1.4, 2, etc.
√2 = 1,4142 is thus the multiplying factor between two adjacent apertures.
The f-stop numbering scheme may seem awkward, but it is a necessary consequence of our human biology.
Hipparchus would certainly agree.