Biology for Photographers: Why is the Aperture Scale Logarithmic?

One of the main stumbling points for new photographers is the seemingly random series of numbers that we have come to know as the f-stop scale or aperture scale. Things start out innocently enough f/1, f/1.4 (just add 0.4 every time, right?), but things get ugly quickly — f/2, f/2.8, f/4. Why would anyone invent such an arbitrary scale?

To answer, we must go back to the second century BC. It was during this period 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 a 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 certainly exceeds any film or sensor invented so far. If the human eye could distinguish small linear increments of brightness, there would be no way to maintain the same wide dynamic range.

The logarithmic nature of human perception was known as early as the 1800s and was eventually summarized by German psychologists as the Weber-Fechner law. The law has implications that apply to many different human processes — vision, hearing, and mental processing. Modern psychologists believe that before children are taught the linear number scale (1,2,3…), their natural tendency is to think in terms of a logarithmic scale (2,4,8,16). For a particularly mind blowing description of this, just listen to this Radiolab podcast, which describes an entire tribe in the Amazon who uses a logarithmic number scale for everyday life. Ask them for a number half way between 1 and 9 and they will say 3.

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 early 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. The figure below 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. Voila!

The f-stop numbering scheme may seem clumsy and awkward, but it is a necessary consequence of our human biology. Hipparchus would be proud of us.

Image credit: 100mm Series E focus and aperture scales by realblades, stars by davedehetre, Counting Fingers by Dreckman Digital

  • Deltaflux

    To add to the confusion, the last two aperture areas in the diagram should be x/16 and x/32, not 10 and 20 as shown.

  • Craig Lester

    I thought it was because the aperture area is round and the whole thing is based on the square root of 2 (1.414). Huh.

  • sdg

    Great article, do more about photographic science?

  • Susan H

    loved this article. thanks for making it interesting, petapixel!


    “It has been estimated that the human eye can effectively process 10 f/stops of light levels — an extraordinary range which certainly exceeds any film or sensor invented so far.”

    It is true the human eye has a higher dynamic range than a CMOS sensor but the figures above are mistaken. DxO Mark counts currently 135 cameras with a range above 10 f/stops, the best being close to 14 f/stops – it allows them to handle a range of illumination of about 16,000:1. Depending on the source, the eye is credited with 10 to 14 f/stops of instantaneous dynamic range, so new sensors are clearly on par. If you take into account our eyes’ ability to adjust in very low or very bright light, you go above 20 f/stops.

  • Karmative
  • Michael Zhang

    Thanks for catching that. We’ve fixed the diagram

  • Erik Turner

    this article seems to suggest the aperture series has something to do with human perception. this isn’t true. it’s just geometry. don’t make it more complicated than it… well it isn’t. it’s middle school geometry.

  • Don Dement

    I agree with Erik. The article’s author introduces irrelevant, if interesting, history and by doing so re-elevates it all to a mystery again. Light varies based on the area of the aperture that we have chosen to measure and name by its diameter. The result – 40% increases or 1.4 x – is just simple geometry based on using a simple ruler-type linear instrument to measure the physical aperture. Areas vary as the square of the diameter.

    NEF2JPG comment misses the point by relating to camera apertures and then claiming the numbers are wrong. We’re comparing the eye’s dynamic range limits with sensor limits, not camera apertures, which could be made in any range, as he notes.

  • Kevin Martini

    Great article!

    I like how it ties together concepts of logarithmic perception, since the decibel scale is also a logarithmic representation of human perception.

  • Geoff

    Why do they complicate things by including diameter and square roots? Using simple 1,2,3.. would work just fine. These numbers could represent the log or power for the aperture sizes. It may not accurately represent the aperture sizes off the bat but not as if the current system does anyway.

  • Steve S

    It is. f-stops are the ratio of the focal length to the aperture *diameter* (hence always written f/—). But the light is proportional to the *area* of the aperture, which is proportional to the square of the diameter). To halve the light, you halve the area, which means reducing the aperture diameter by a factor of square root of 2. Since the aperture is in the denominator of the f-stop, the f-stop increases by this factor. So the f-stops go like sqrt(2)^n for n=0,1,2,3,etc

    Why are f-stops written as this ratio? Because the amount of light (per unit area) is fixed, for a given ratio. At an f-stop of 2.8, for example, a lens will produce the same amount of light on the sensor regardless of the focal length of the lens.To set your shutter speed for a correct exposure, you don’t need to know the aperture diameter, you need to know the ratio of the focal length to the aperture diameter, i.e. the “f-stop”. Of course at a given f-stop, as the focal length gets larger, so must the aperture diameter. So a 400mm f/2.8 lens needs a lens element 8 times bigger (in diameter) than a 50mm f/2.8. Which is why the 400mm 2.8 is $7k, whereas the 50mm 2.8 (if one existed) might only be $100.

  • Steve S

    Well, the perception angle does explain why the series goes like 1, 1/2, 1/4, 1/8, etc. instead of 1, 1/2, 1/3, 1/4, 1/5, etc. or any other series. If your ruler had its major divisions marked 1,2,4,8,… instead of 1,2,3,4,… you might wonder why.

  • Seven Bates

    This is superb photography blogging.

  • Matt Fisher

    Bear in mind that, within the limits of reciprocity, each f-stop lets in either twice as much or half as much light as its nearest neighbor. This is also true of the somewhat arbitrary seeming divisions in the shutter speed options. So while you’re right, it doesn’t really matter WHAT you number the f-stops as, the divisions aren’t arbitrary. And because light increases/falls off through the aperture as an inverse square, these values are necessary. As opposed to light falling off/increasing as a function of shutter speed, which is linear (for the kinds of values cameras use) resulting in unconfusing shutter speed options.

  • Cortex

    Why don’t modern cameras have linear increments in shutter speed? Let’s say that a given aperture falls just short of reaching perfect exposure at 1/60 — wouldn’t it make sense for a camera in 2011 to offer a shutter speed of 1/59? The leap from 1/40 to 1/30, for instance, is enormous.

  • Cortex

    Why don’t modern cameras have linear increments in shutter speed? Let’s say that a given aperture falls just short of reaching perfect exposure at 1/60 — wouldn’t it make sense for a camera in 2011 to offer a shutter speed of 1/59? The leap from 1/40 to 1/30, for instance, is enormous.

  • MartinF

    Its not biological, its mathematical. And its not logarithmic, its exponential… 

  • Basil Glew-Galloway

    Not in one image, though…

  • Basil Glew-Galloway

    Humans created this scale the way we did for a reason…

  • Basil Glew-Galloway

    no…  that would not make sense…

  • Thinktank

    is it just me, but hasn’t anybody reflected on the differencies between logaritmic curves and exponential curves?
    The increments mentioned (2,4,8,16) describes an exponential curve. The logaritmic curve could be described with the figures 2, 4, 5, 5.5, 5.25…
    The logaritmic curve raises fast initially and then flattens out, the exponential curve is initially flat and then raises faster and faster.

  • johnconnor

    Its logarithmic too, log to the base 2… oops i should have commented a year ago!!!