Physics says that the maximum aperture on a lens (lowest number) is equal to the focal length divided by the diameter of the iris of the lens. The iris of the lens is the opening at the front of the lens. The lens is designed to have a ring of light at the focal plane (where the sensor is located) that will cover all the corners of the lens. When that lens is put on a crop sensor camera the sensor sees an iris that is the crop factor difference in size. That is why you have to apply the crop factor to the aperture. Take it or leave it. You probably can do it your way and physics be damned.
I guess this is why the thread goes on as long as it does -- the statement above arrives at an incorrect conclusion. The f stop is calculated based on the ratio between the aperture (which is not the front opening at the front of the lens) and the focal length. An f stop is calculated on measured values, and is unaware of how the final image will be cropped, either in camera (i.e. a crop camera) or in post (Photoshop, or historically during printing). A 60mm f2. lens is always that, whether mounted on a Hasselblad or a Nikon or a Minox -- that is, the physical properties of the lens are the same regardless of the format it is used on.
Where the discussion arises is on the issue of how different sensors, especially smaller sensors, may add noise at a given exposure, and so how to 'calculate' an equivalent amount of noise in an image using different sensors. In fact, such a calculation requires some serious assumptions, assumptions that are in my opinion almost impossible to fulfill ...
Historically, ASA or ISO described a sensitivity but also described a defacto 'sensor' too -- if I shot with Kodachrome II you knew not only that my 'speed' rating was 25, but also knew what the 'grain' size (really dye cluster, but that's another topic) was. The grain size was identical no matter what format I shot, i.e. laid side by side, my 4x5 sheet film, my 120 roll film and my 35mm transparency all represented an identical 'sensor' technology -- they were different only in size. Moreover, there was no brand differentiation or advancement in any major way -- to say 'I shoot 25 ASA' was also to say I shoot with a fine grain sensor, regardless of the format. Today, such comparisons are almost impossible. I shoot currently with a 25MP crop sensor -- someone shooting with an A7 may have twice the area of sensor, but half as many pixels ... and still offer the ability to dial in 100 ISO as the sensitivity. This is different than the past, where 25 ASA film offered exactly the same pixel density (please allow the use of the word pixel here) no matter what the format -- to describe an exposure as 25 ASA f8 1/250 was also to describe the pixel density. Today, if I don't also describe the camera in my exposure reveal, the reader has no idea what pixel density I am working with. An A7S has a quarter the pixel density of the NEX7 -- the final image will have a radically different noise profile at low light levels as a consequence.
In another ten years this may not be true -- sensor technology may stabilize, as film technology did, so that to describe an ISO is also to describe a sensor density. Right now, the range is far too great to be ignored. And that brings us to the 'problem' of aperture equivalence. It is a viable factor when comparing sensors of identical density (and technology) so one could reasonably say that comparing a 25Mp crop sensor with a 50Mp FF sensor, you would/should see equivalent noise levels at apertures predictably different aperture sizes. But in practice, finding imaging chips across formats of identical density is difficult, and is further confused by advancing techonologies.
So we are left, in my opinion, with a better practice of ignoring aperture 'equivalency'. Exposure is calculated without any reference to equivalent values, and the final image's noise profile is a function of the particular model and sensor design. The Sony A7S should offer a much lower noise profile than the A7R, despite both being FF, and at identical exposure settings. Using film, such a comparison could easily be achieved by changing stock, and such a change was easily described by describing the ASA/ISO. Today, there is no short and simple way to describe that variable, and IMHO aperture equivalency most definitely not one of them!
Cheers,
GB
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